# Financial Math FM/Print version

The current, editable version of this book is available in Wikibooks, the open-content textbooks collection, at
https://en.wikibooks.org/wiki/Financial_Math_FM

# About the FM Exam
## Calculators

This is a comprehensive list of the calculators allowed during the exam. Do not assume that any non-graphing scientific calculator is allowed.
* BA-35
* TI-30Xa
* BA II Plus
* TI-30X II (IIS solar or IIB battery)
* BA II Plus Professional
* TI-30XS MultiView (or XB battery)

## More Details

For more details, see | SOA Exam FM home page.

# Time Value of Money
## Learning objectives

The Candidate will understand and be able to perform calculations relating to present value, current value, and accumulated value.

## Learning outcomes

The Candidate will be able to:
* Define and recognize the definitions of the following terms: interest rate (rate of interest), simple interest, compound interest, accumulation function, future value, current value, present value, net present value, discount factor, discount rate (rate of discount), convertible m-thly, nominal rate, effective rate, inflation and real rate of interest, force of interest, equation of value.
* Given any three of interest rate, period of time, present value, current value, and future value, calculate the remaining item using simple or compound interest. Solve time value of money equations involving variable force of interest.
* Given any one of the effective interest rate, the nominal interest rate convertible m-thly, the effective discount rate, the nominal discount rate convertible m-thly, or the force of interest, calculate any of the other items.
* Write the equation of value given a set of cash flows and an interest rate.

## Interest

Definition. (Interest) Interest is the compensation that a borrower of an asset (or capital) pays a lender of capital for its use.
Remark.
* That is, interest is the extra thing (compensation) paid to lender, in addition to the capital.
Example. Suppose a bank lends Amy $100, and one month later Amy needs to pay back $110 to the bank. Then,
* $\$100$ is the capital;
* $\$110-\$100=\$10$ is the interest;
* Amy is a borrower;
* the bank is a lender.
Exercise. Fisher A lends fisher B a fishing rod for a week, for fisher B to catch fishes. In this week, 10% of fishes caught by fisher B (decimal numbers obtained are rounded up) need to be given to fisher A in return. For simplicity, suppose all fishes are identical.

## Measurement of interest
### Terminologies

In the following, we will introduce some terminologies used in the measurement of interest.
Definition. (Principal) The principal is the initial amount of money invested.
Definition. (Accumulated value) The accumulated value (or future value) at time $t$ is the total amount received at time $t$.
Definition. (Interest (alternative definition)) The interest earned during a period of investment is the difference between the accumulated value and the principal.
Remark.
* This alternative definition is equivalent to the above definition of interest.
Example. Amy invests $100 into a fund that pays $200 one year later. Then,
* the principal is $\$100$;
* the accumulated value (after one year) is $\$200$;
* the interest (earned during this year) is $\$200-\$100=\$100$.
Exercise.
Definition. (Measurement period) The measurement period is the unit in which time is measured.
Remark.
* The measurement period is often years.
Definition. (Accumulation function) The accumulation function, denoted by $a(t)$, is the function that gives the ratio of the accumulated value at time $t$ to the principal.
Remark.
* $a(0)=1$, since the accumulated value at time 0 equals the principal, and thus the ratio is 1.
Definition. (Amount function) The amount function denoted by $A(t)$, is the function that gives the accumulated value at time $t$ of principal of nonnegative number $k$.
Remark.
* $A(0)=k$, since the accumulated value at time 0 equals the principal, which is $k$ by definition.
* Also, $A(t)=ka(t)$, since $a(t){\overset {\text{ def }}{=}}{\frac {A(t)}{k}}$.
We denote the interest earned during the $n$th period ($n$ is a positive integer) from the date of investment (i.e. from the beginning of $n$th period to the end of $n$th period by $I_{n}$.
By definition, $I_{n}=A(n)-A(n-1)$, in which
* $A(n)$ is the accumulated value at the end of $n$th period, and
* $A(n-1)$ is the accumulated value at the start of $n$th period.
Example. Define $a(t)=2^{t}$ for each time $t$.
* This is a valid accumulation function, since $a(0)=2^{0}=1$.
* Also, $a(1.5)=2^{1.5}\approx 2.83$, and
* the interest earned with principal of one from $t=0$ to $t=1$ is $a(1)-a(0)=2^{1}-2^{0}=1$, in which $a(1)$ is the accumulated value at time 1, and $a(0)$ is the principal.
Exercise.

### Effective interest rate

Definition. (Effective interest rate) The effective interest rate, denoted by $i$, is the ratio of the amount of interest earned during the period to the principal.
Remark.
* It follows that the effective interest rate during the $n$th period from the investment date, denoted by $i_{n}$, is
$i_{n}={\frac {A(n)-A(n-1)}{A(n-1)}}$
    in which $A(n)-A(n-1)$ is sometimes denoted by $I_{n}$, meaning the amount of interest.
* Unless otherwise stated, rates are expressed as annual rates..
Example. $i_{n}={\frac {a(n)-a(n-1)}{a(n-1)}}$.
Proof. Suppose the principal is $k$. Then, $i_{n}{\overset {\text{ def }}{=}}{\frac {A(n)-A(n-1)}{A(n-1)}}={\frac {ka(n)-ka(n-1)}{ka(n-1)}}={\frac {a(n)-a(n-1)}{a(n-1)}},$ as desired.
$\Box$
Example. $A(n)=A(0)(1+i_{1})\dotsb (1+i_{n})$ for each nonnegative integer $n$.
Proof. For each nonnegative integer $n$, $A(0)(1+i_{1})\dotsb (1+i_{n})=A(0)\cdot {\frac {A(1)}{A(0)}}\cdot \dotsb \cdot {\frac {A(n)}{A(n-1)}}=A(n),$ as desired.
$\Box$
Exercise. Define $i_{n}=2i_{n-1}$ for each positive integer $n$, and $i_{1}=1\%$. Suppose $A(0)=100$.

### Simple interest

For simple interest, under the simple interest rate $i$, the interest earned during each period is calculated according to the principal (and so is constant), i.e. the interest earned is $iA(0)$, i.e. $A(n)-A(n-1)=iA(0)$ for each positive integer $n$. So, $A(n)=A(0)+{\bigA(1)-A(0){\big )}+\dotsb +{\bigA(n)-A(n-1){\big )}=A(0)+inA(0)=A(0)(1+in).$ Since $A(n)=A(0)a(n)$, $a(n)=1+in$ for each nonnegative integer $n$. Intuitively, we may expect that the same form of accumulation function also holds for other nonnegative numbers.
Definition. (Simple interest) The accumulation of interest according to the accumulation function $a(t)=1+it,\quad t\geq 0$ is the simple interest, in which $i$ is the simple interest rate.
Remark.
* Since $i_{n}={\frac {a(n)-a(n-1)}{a(n-1)}}={\frac {i}{1+(n-1)i}}$ for simple interest, $i_{n}$ does not equal the simple interest rate in general.
Example. Ivan invests $10000 into a bank account which pays simple interest with an annual rate of 5%. The balance in Ivan's account after two years is $\$1000(1+2(5\%))=\$11000$.
Exercise.
Example. For simple interest, $a(s+t)=a(s)+a(t)-1,\quad s{\text{ and }}t\geq 0.$
Proof. $a(s+t)=1+(s+t)i=\underbrace {1+si} _{=a(s)}+\underbrace {1+ti} _{=a(t)}-1,$ as desired.
$\Box$

### Compound interest

For compound interest, the interest earned for each period is calculated according to the accumulated value at the beginning of that period.
To be more precise, with principal of $k$ and compound interest rate $i$, at the end of first year, the interest earned is $ki$, and thus the accumulated value is $k+ki=k(1+i)$.
Thus, at the end of second year, the interest earned is $k(1+i)i$, and so the accumulated value is $k(1+i)+k(1+i)i=k(1+i)^{2}$
Using the same argument, at the end of $n$th year, the interest received is $ki(1+i)^{n-1}$, and the accumulated value is $k(1+i)^{n}$. We obtain the accumulation function with nonnegative integer $n$ as input here, namely $a(t)=(1+i)^{n}$.
Intuitively, we may expect that the same form of accumulation function also holds for other nonnegative numbers. This motivates the definition of compound interest.
Definition. (Compound interest) The accumulation of interest according to the accumulation function $a(t)=(1+i)^{t},\quad t\geq 0$ is the compound interest, in which $i$ is the compound interest rate.
Example. Ivan invests $10000 into a bank account which pays compound interest with an annual rate of 5%. Then, the balance in Ivan's account after two years is $\$10000(1.05)^{2}=\$11025$.
Exercise.
Example. For compound interest rate $i$, the effective interest rate during the $n$th period $i_{n}=i$.
Proof. $i_{n}={\frac {(1+i)^{n}-(1+i)^{n-1}}{(1+i)^{n-1}}}={\frac {(1+i)^{n-1}{\big(1+i)-1{\big )}}{(1+i)^{n-1}}}=i.$
$\Box$
Remark.
* Because of this nice result, from now on, every interest (rate) is assumed to be compound interest (rate) unless stated otherwise.

### Effective discount rate

The effective interest rate was defined as a measure of interest paid at the end of the period. However, there are also discount rates, denoted by $d$, which is a measure of the interest paid at the beginning of the period.
Example.
* If Amy borrows $100 from a bank for a year at an effective interest rate of 5%, then the bank will give Amy $100 at the beginning of the year, and at the end of the year, Amy will repay the bank $100 plus the interest of $5, a total of $105.
* On the other hand, if Amy is charged by an effective discount rate of 5%, then Amy will need to pay the interest of $5 at the beginning of the year, so the bank will only give Amy $\$100-\$5=\$95$ at the beginning, and Amy will repay the bank $100 at the end (interest is paid already).
We can see from this example that the effective interest rate is a percentage of the principal, while the effective discount rate is a percentage of the balance at the end of the year. Thus, we can define effective discount rate more precisely as follows:
Definition. (Effective discount rate) The effective discount rate, denoted by $d$, is the ratio of the amount of interest earned during the period to the amount invested at the end of the period.
Remark.
* It follows that the effective discount rate during the $n$th period, denoted by $d_{n}$, is
$d_{n}={\frac {A(n)-A(n-1)}{A(n)}}.$
Example. In the above example, the effective discount rate is 5% since ${\frac {\$100-\$95}{\$100}}=5\%$.
Example. $d_{n}={\frac {a(n)-a(n-1)}{a(n)}}$.
Proof. $d_{n}{\overset {\text{ def }}{=}}{\frac {A(n)-A(n-1)}{A(n)}}={\frac {ka(n)-ka(n-1)}{ka(n)}}={\frac {a(n)-a(n-1)}{a(n)}}.$
$\Box$
Example. $A(0)=A(n)(1-d_{n})\dotsb (1-d_{1})$.
Proof. For each positive integer $n$, $A(n)(1-d_{n})\dotsb (1-d_{1})=A(n)\cdot {\frac {A(n-1)}{A(n)}}\cdot \dotsb \cdot {\frac {A(0)}{A(1)}}=A(0).$
$\Box$
Example. Given that $a(t)=e^{t}$, then $d_{3}={\frac {e^{3}-e^{2}}{e^{3}}}\approx 0.632.$ In contrast, $i_{3}={\frac {e^{3}-e^{2}}{e^{2}}}\approx 1.718.$
Exercise.

### Simple discount rate

For simple discount, the interest paid is calculated according to the accumulated value at the end of $n$th period. That is, the interest paid at the beginning of each period is $dA(n)$ (constant), i.e. $A(m)-A(m-1)=dA(n)$ for each positive integer $m$.
So, $A(0)=A(n)-{\bigA(n)-A(n-1){\big )}-\dotsb -{\bigA(1)-A(0){\big )}=A(n)-dnA(n)=A(n)(1-dn).$ Since $A(n)=A(0)a(n)\Leftrightarrow A(0)=A(n)a^{-1}(n)$, $a^{-1}(n)=1-dn\Rightarrow a(n)={\frac {1}{1-dn}}$ for each nonnegative integer $n$ such that $dn<1\Leftrightarrow n<1/d$ (so that the accumulation function is defined). Similarly, we may intuitively expect that the same form of accumulation function holds for other nonnegative numbers, which motivates the following definition.
Definition. (Simple discount) The accumulation of discount according to the accumulation function $a(t)={\frac {1}{1-dt}},\quad 0\leq t<1/d$ is the simple discount, in which $d$ is the simple discount rate.

### Compound discount rate

For compound discount, the interest paid at the beginning of each period is calculated according to the balance at the end of that period.
To be more precise, suppose $A(n)=k$ and the compound discount rate is $d$. At the beginning of $n$th year, the interest paid is $kd$, and so the balance at the beginning of $n$th year is $k-kd=k(1-d)$.
Since the balance at the end of $n-1$th year (which is the same as that at the beginning of $n$th year) is $k(1-d)$,
the interest paid at the beginning of $n-1$th year is $k(1-d)d$, and thus the balance at the beginning is $k(1-d)-k(1-d)d=k(1-d)^{2}$.
Using the same argument, the balance at the beginning of first year is $k(1-d)^{n}$, i.e. $A(0)=k(1-d)^{n}=A(n)(1-d)^{n}$, and we can see that $a^{-1}(n)=(1-d)^{n}\Rightarrow a(n)={\frac {1}{(1-d)^{n}}}$ similarly for each nonnegative integer $n$. This motivates the following generalized definition similarly.
Definition. (Compound discount) The accumulation of interest according to the accumulation function $a(t)={\frac {1}{(1-d)^{t}}},\quad t\geq 0$ is the compound discount, in which $d$ is the compound discount rate.
Example.
* Amy invests 1000 into a fund with compound discount rate $d$, and she receives 2000 after ten years.
* Then, we know that
$1000(1-d)^{-10}=2000\Rightarrow (1-d)^{-10}=2.$
* Then, we can calculate that the accumulated value after 100 years is
$A(100)=1000(1-d)^{-100}=1000{\big(1-d)^{-10}{\big )}^{10}=1000(2^{10})=1024000.$
Exercise.
Suppose the fund value accumulates at different rates according to the following pattern: for each nonnegative integer $n$,
* during $4n$th year, the fund values accumulate at simple interest rate of 5%;
* during $4n+1$th year, the fund values accumulate at simple discount rate of 5%;
* during $4n+2$th year, the fund values accumulate at compound interest rate of 5%;
* during $4n+3$th year, the fund values accumulate at compound discount rate of 5%.

### Equivalent rates

Definition. (Equivalent rates) Two interest or discount rates are equivalent if a given amount of principal invested for the same length of time at each of the rates produces the same accumulated value.
Remark.
* It means that the corresponding accumulation functions have the same output with the same input.
* The "$i$" and "$d$" in the exam FM questions are assumed to be equivalent, since it is assumed that $d={\frac {i}{1+i}}$, which will be shown below to be equivalent to the equivalence of $i$ and $d$.
Example. Suppose $i$ is an effective interest rate and $d$ is an effective discount rate that is equivalent to $i$. Then, $d={\frac {i}{1+i}}$.
Proof. Since they are equivalent, their accumulation functions are equal in value with the same input. In particular, with input $t=1$, $1+i={\frac {1}{1-d}}\Leftrightarrow 1-d={\frac {1}{1+i}}\Leftrightarrow d={\frac {1+i-1}{1+i}}={\frac {i}{1+i}}.$
$\Box$
Remark.
* It follows that $d=iv$ if and only if $i$ and $d$ are equivalent, since $v={\frac {1}{1+i}}$.
* Also, $v={\frac {1}{1+i}}=1-d$.
Example. Suppose $i$ is an effective interest rate and $d$ is an effective discount rate that is equivalent to $i$. Then, $i={\frac {d}{1-d}}$.
Proof. Since they are equivalent, $1+i={\frac {1}{1-d}}\Leftrightarrow i={\frac {1-(1-d)}{1-d}}={\frac {d}{1-d}}.$
$\Box$
Exercise.

### Nominal rates

We have discussed effective interest and discount rates. For those effective rates, the interest is paid exactly once per measurement period (either at the beginning (for discount rates) or at the end (for interest rates)).
However, the interest can be paid more than once per measurement period, and the interest and discount rates, for which the interest is paid more than once per measurement period, are called nominal, rather than effective, rates.
Definition. (Nominal interest and discount rates) Nominal interest and discount rates are rates for which the interest is paid more than once per measurement period.
The reason for calling those rates as "nominal" is that the notation for the nominal interest (discount) rate payable (or "convertible" or "compounded") $m$thly per measurement period is $i^{(m)}$ ($d^{(m)}$), and its value is a nominal value only, in the sense that the actual rate used in the calculation for each payment is $i^{(m)}/m$ ($d^{(m)}/m$) by definition, rather than $i^{(m)}$ ($d^{(m)}$), and this is the $m$thly rate.
Example. Amy deposits 1000 into a bank account with 24% interest compounded monthly (i.e. "12thly", compounded 12 times a year). Then,
* the nominal interest rate $i^{(12)}=0.24$;
* the accumulated value after a year (or 12 months) is $1000\left(1+{\frac {i^{(12)}}{12}}\right)^{12}\approx 1268.24$,
    by regarding "months" as the measurement period, since the monthly rate is ${\frac {i^{(12)}}{12}}$, and there are 12 measurement periods, and thus the result follows from the definition of compound interest.
* We can see that the effective interest rate is approximately ${\frac {1268.24-1000}{1000}}\approx 27\%$, which is different from the nominal rate of 24%, showing how the nominal rate is "nominal" again.
Exercise.
Example. The nominal interest rate convertible monthly, denoted by $i^{(12)}$, that is equivalent to $d^{(6)}=18\%$ is calculated by ${\begin{aligned}&&\left(1+{\frac {i^{(12)}}{12}}\right)^{12}&=\left(1-{\frac {d^{(6)}}{6}}\right)^{-6}\\&\Rightarrow &1+{\frac {i^{(12)}}{12}}&={\big(1-0.03)^{-6}{\big )}^{1/12}{\text{ or }}-{\big(1-0.03)^{-6}{\big )}^{1/12}({\text{rejected}})\\&\Rightarrow &i^{(12)}&=12(0.97^{-6/12}-1)\approx 0.184.\end{aligned}}$
Exercise.

### Force of interest

We have discussed nominal interest rates, and in this subsection, we will discuss what will happen if the compounding frequency gets higher and higher, i.e. the $m$ in "compounded $m$thly" becomes larger and larger, to the infinity. We call this "compounded continuously".
To be more precise, we would like to know the value of $\lim _{m\to \infty }i^{(m)}$ during the "infinitesimal" time interval $[t,t+1/m]$ which tends to be simply the time point $t$, and we call this force of interest at time $t$, denoted by $\delta _{t}$. Now, we would like to develop a formula for $\delta _{t}$.
For nominal interest rate $i^{(m)}$, we have the following relationship between $A(t)$ and $A(t+1/m)$ by definition (treating $1/m$ of year as measurement period, then the effective interest rate during the period $[t,t+1/m]$ is $i^{(m)}/m$ by definition): $A(t+1/m)=A(t)(1+i^{(m)}/m)\Rightarrow i^{(m)}={\frac {m{\bigA(t+1/m)-A(t){\big )}}{A(t)}}={\frac {A(t+1/m)-A(t)}{1/m}}\cdot {\frac {1}{A(t)}}$ So, taking limit, $\underbrace {\lim _{m\to \infty }i^{(m)}} _{=\delta _{t}}={\frac {1}{A(t)}}\cdot \lim _{m\to \infty }{\frac {A(t+1/m)-A(t)}{1/m}}={\frac {1}{A(t)}}\cdot \underbrace {\lim _{1/m\to 0}{\frac {A(t+1/m)-A(t)}{1/m}}} _{{\overset {\text{ def }}{=}}A'(t)}={\frac {A'(t)}{A(t)}}.$ This motivates the definition of force of interest.
Definition. (Force of interest) The force of interest at time $t$, denoted by $\delta _{t}$, is defined by $\delta _{t}={\frac {A'(t)}{A(t)}}$.
Remark.
* It follows that $\delta _{t}={\frac {a'(t)}{a(t)}}$, since $A(t)=A(0)a(t)$ and thus $A'(t)=A(0)a'(t)$.
* The expression ${\frac {A'(t)}{A(t)}}$ may be interpreted as relative rate of change of the amount function at time $t$, in the sense that it tells the rate of change at time $t$ equals what portion of $A(t)$, such that $\delta _{t}$ is independent from the value of $A(t)$.
   * Thus, the force of interest measure the "intensity" of interest at time $t$, since the only factor that change $A(t)$ is interest.
Proposition. (Expression of accumulation function in terms of force of interest) $a(t)=\exp \left(\int _{0}^{t}\delta _{s}\,ds\right).$
Proof. $\delta _{s}={\frac {a'(s)}{a(s)}}\Leftrightarrow \int _{0}^{t}\delta _{s}\,ds=\int _{0}^{t}{\frac {a'(s)}{a(s)}}\,ds\Leftrightarrow \int _{0}^{t}\delta _{s}\,ds=\int _{0}^{t}{\frac {1}{a(s)}}\,d{\biga(s){\big )}\Leftrightarrow \int _{0}^{t}\delta _{s}\,ds=\ln a(t)-\underbrace {\ln a(0)} _{=\ln 1=0}\Leftrightarrow a(t)=\exp \left(\int _{0}^{t}\delta _{s}\,ds\right).$
$\Box$
Remark.
* A corollary is that ${\frac {a(t_{2})}{a(t_{1})}}=\exp \left(\int _{0}^{t_{2}}\delta _{s}\,ds-\int _{0}^{t_{1}}\delta _{s}\,ds\right)=\exp \left(\int _{t_{1}}^{t_{2}}\delta _{s}\,ds\right)$, which is sometimes useful.
Proposition. (Constant force of interest) If and only if the force of interest over a measurement period is constant, denoted by $\delta$, and the effective interest rate during the time interval is $i$, $\delta =\ln(1+i)$.
Proof.
* Without loss of generality, suppose the measurement period is $[t,t+1]$. Then,
${\frac {a(t+1)}{a(t)}}=\exp \left(\int _{t}^{t+1}\delta \,ds\right)\Leftrightarrow {\frac {(1+i)^{t+1}}{(1+i)^{t}}}=\exp \left(\delta \right)\Leftrightarrow \delta =\ln(1+i).$
$\Box$
Remark.
* In general, if and only if the force of interest over $n$ measurement periods, say time interval $[0,n]$, is constant, the accumulation function
$a(n)=\exp \left(\int _{0}^{n}\delta \,ds\right)\Leftrightarrow a(n)=e^{n\delta }.$
Example. (Equivalency between forces of interest and discount) The force of discount can be analogously defined by $\lim _{m\to \infty }d^{(m)}$. Then, the force of discount $\lim _{m\to \infty }d^{(m)}=\delta _{t}$.
Proof.
* Similar to the motivation for force of interest, we have
$A(t)=A(t+1/m)(1-d^{(m)}/m)\Rightarrow d^{(m)}={\frac {-m{\bigA(t)-A(t+1/m{\big )})}{A(t+1/m)}}={\frac {A(t+1/m)-A(t)}{1/m}}\cdot {\frac {1}{A(t+1/m)}}.$
* Taking limit,
$\lim _{m\to \infty }d^{(m)}=\lim _{1/m\to 0}\left({\frac {A(t+1/m)-A(t)}{1/m}}\cdot {\frac {1}{A(t+1/m)}}\right)=A'(t)\cdot {\frac {1}{A(t)}}=\delta _{t}.$
$\Box$
Remark.
* Thus, it suffices to discuss force of interest, but not force of discount.
Example. Value of a fund accumulates at 10% (annual, which is assumed) force of interest, i.e. $\delta =0.1$. Amy invests 10000 into the fund. Then, the accumulated value of the investment is $10000e^{2\cdot (0.1)}\approx 12214.03$.
Exercise.
Example. Value of a fund accumulates at the annual force of interest $\delta _{t}=(2+t)^{-1}$ for each nonnegative $t$ (in years, i.e. $t=j$ means "at the end of $j$th year", since the time interval from $t=j-1$ to $t=j$ is the $j$th year.). Then, the accumulated value at $t=15$ with principal one is ${\begin{aligned}a(15)=\exp \left(\int _{0}^{15}(2+t)^{-1}\,dt\right)=\exp(\ln 17-\ln 2)=\exp \left(\ln(17/2)\right)=17/2.\end{aligned}}$
Exercise.
Example.
* Amy deposits 100 into a bank and the deposit of 100 accumulates according to the force of interest
$\delta _{t}={\begin{cases}0.01,&0\leq t<7;\\(1+t)^{-1},&7\leq t<10;\\0.0009t^{2},&t\geq 10,\\\end{cases}}$
    in which $t$ is in years.
* Then, the accumulated value of the deposit at the end of 8th year is
$100\exp \left(\int _{0}^{8}\delta _{t}\,dt\right)=100\exp \left(\int _{0}^{7}0.01\,dt+\int _{7}^{8}(1+t)^{-1}\,dt\right)=100\exp \left(0.07+\ln 9-\ln 8\right)\approx 120.66;$
* the accumulated value at the end of 11th year is
$100\exp \left(\int _{0}^{11}\delta _{t}\,dt\right)=100\exp \left(\int _{0}^{7}0.01\,dt+\int _{7}^{10}(1+t)^{-1}\,dt+\int _{10}^{11}0.0009t^{2}\,dt\right)=100\exp \left(0.07+\ln 11-\ln 8+0.0003(11^{3}-10^{3})\right)\approx 162.87.$
Exercise.

## Present, current and future values

From previous sections, we have seen that money has a time value because of the interest, in the sense that $1 today will worth more than $1 after a period of time (assuming positive interest rate).
To be more precise, an investment of $k$ will accumulate to $k(1+i)$ at the end of one period, in which $i$ is the effective interest rate during the period. In particular, the term $1+i$ is called accumulation factor, since it accumulates the investment value at the beginning to its value at the end. Graphically, it looks like the following time diagram (a graph represents statuses at different time).
We would often like to do something "reverse" to calculating the accumulated value given the principal. That is, calculating the principal given the accumulated value. Since the principal is the investment value at initial time, which is often now (or present), the "reverse" calculation is essentially calculating the present value of the investment, given its accumulated (or future) value at the future.
To be more precise, we would like to calculate the principal (or present value, denoted by $PV$) such that it accumulates to $k$ (which is future value, denoted by $FV$) at the end of one period. Using equation to describe this situation, we have $PV(1+i)=FV\Leftrightarrow PV=FV\cdot {\frac {1}{1+i}}$ in which $i$ is the effective interest rate during this period. The term ${\frac {1}{1+i}}$, which is denoted by $v$, is the discount factor, since it "discounts" the future value to the present value.
The term current value (which is "at the middle" of present and future values) is sometimes used. It means the value of the payments at a specified date, and some payments are made before that date, while some payments are made after that date.
We have discussed how to calculate the present value for one period, but we can generalize the result to more than one period. To be more precise, we would like to also calculate the present value given the future value at the end of $t$ periods. We can use the accumulation function $a(t)$ to describe this situation in general. $PVa(t)=FV\Leftrightarrow PV=FV\cdot {\frac {1}{a(t)}}=FVa^{-1}(t)$ in which $a^{-1}(t)$ is the inverse function of $a(t)$.
Also, given multiple future values, we can calculate the total present value of these future values by summing up all present values corresponding to these future values.
Example. In exchange for your car, John has promised to pay you 5,000 after one year and 10,000 after three years. Using an annual effective interest rate of 5%, the present value of these payments is $PV=5000(1.05)^{-1}+10000(1.05)^{-3}\approx 13400.28.$ Time diagram:
Exercise.
Remark.
* It is often useful to draw a rough time diagram to understand the situation given in the question.
Example. A fund provides payment of 50000 at the end of 7th year. Suppose there is annual force of interest $\delta _{t}=0.04t+0.05$ ($t$ is in years). Then, the present value of the payment is $PV=50000e^{-{\big0.02(7^{2})+0.05(7)-0.02(0^{2})-0.05(0){\big )}}\approx 13223.86.$
Proof.
* First, the inverse of accumulation function corresponding to the force of interest $\delta _{t}$ at $t=7$ is
$a^{-1}(7)=\left(\exp \left(\int _{0}^{7}0.04t+0.05\,dt\right)\right)^{-1}=\left(\exp \left([0.02t^{2}+0.05t]_{0}^{7}\right)\right)^{-1}=\exp \left(-0.02(7^{2})-0.05(7)\right).$
* The result follows from the relationship $PV=50000a^{-1}(7)$.
$\Box$
Exercise.

### Equations of value

For two or more payments at different time points, to compare them fairly, we need to accumulate or discount them to a common time point, so that the effects on payments from the time value of money are eliminated.
The equation which accumulates or discounts each payment as in above is called the equation of value.
Indeed, we have encountered equations of value in previous sections, since an example of equations of value is calculating present values of multiple payments ("present" is the common time point).
The concepts involved in equations of value have been discussed previously.

## Inflation and real interest rate

In previous sections, we have not consider the effect from inflation, and we will introduce how interest rates changes under inflation.
Because of the inflation, there are two types of interest rates, namely nominal interest rate and real interest rate.
For nominal interest rate, it is the same as the "normal" interest rate discussed previously, and thus is denoted by $i$.
Definition. (Real interest rate) The real interest rate, denoted by $r$, is the interest rate with inflation effect eliminated. To be more precise, the real interest rate is defined by $1+r={\frac {1+i}{1+j}}$ with inflation rate $j$ for the corresponding year at which interest rates are calculated..
Remark.
* By definition of (annual) inflation rate, $1+j={\frac {\text{price level at the end of the year}}{\text{price level at the beginning of the year}}}$, so ${\frac {1}{1+j}}={\frac {\text{price level at the beginning of the year}}{\text{price level at the end of the year}}}$, and thus ${\frac {1}{1+j}}$ can be interpreted as adjusting the price level at which the interest is paid (i.e. at the end of the year) to the price level at the beginning, so the change in price level in this year is eliminated.
* An approximation of real interest rate is $r\approx i-j$, since $1+i=(1+r)(1+j)=1+r+j+rj\approx 1+r+j$ ($rj$ is small).
Example.
* Suppose the annual inflation rate is 5% and the nominal interest rate is 4%.
* Then, the annual real interest rate is ${\frac {1.04}{1.05}}-1\approx -0.0095.$
* In general, the (effective) nominal interest rate during $n$ years is $1.04^{n}$, and
* the inflation rate for $n$ years is $1.05^{n}$.
* So, the real (effective) interest rate during $n$ years is ${\frac {1.04^{n}}{1.05^{n}}}-1=\left({\frac {1.04}{1.05}}\right)^{n}-1$.
* As we can see here, if the nominal interest rate is compound interest rate, the resulting real interest rate is also compound interest rate.
* We can use real interest rate for various calculations, e.g. calculating present values, i.e. use the real interest rate for the "$i$" in formulas.
Proof. Let $r$ be the real interest rate. Then, by definition, $1+r={\frac {1+4\%}{1+5\%}}={\frac {1.04}{1.05}}\Rightarrow r={\frac {1.04}{1.05}}-1\approx -0.0095.$
$\Box$
Exercise.

## References

* ↑ It is usually assumed to be the same as the end of $n-1$th period
* ↑ It is usually assumed to be the same as the start of $n+1$th period
* ↑ https://www.soa.org/globalassets/assets/Files/Edu/2019/exam-fm-notation-terminology2.pdf
* ↑ $a^{-1}(n)$ is the inverse function of $a(n)$
* ↑ It is analogous to the motivation of force of interest, in which force of interest can be defined as $\lim _{m\to \infty }i^{(m)}$.
* ↑ possibly with subscript $i$, indicating the corresponding effective interest rate
* ↑ This holds for simple and compound interests, and also other arbitrary (valid) accumulation functions whose inverse exists.
* ↑ i.e. by definition, $a^{-1}(t)a(t)=1\Rightarrow a^{-1}(t)={\frac {1}{a(t)}}$
* ↑ This phrase has different meaning compared to the same phrase in the context for "payable more than once per measurement period"
* ↑ This is known as Fisher equation.

# Annuities
## Learning objectives

The Candidate will be able to calculate present value, current value, and accumulated value for sequences of non-contingent payments.

## Learning outcomes

The Candidate will be able to:
* Define and recognize the definitions of the following terms: annuity-immediate, annuity due, perpetuity, payable m-thly or payable continuously, level payment annuity, arithmetic increasing/decreasing annuity, geometric increasing/decreasing annuity, term of annuity.
* For each of the following types of annuity/cash flows, given sufficient information of immediate or due, present value, future value, current value, interest rate, payment amount, and term of annuity, calculate any remaining item.
   * Level annuity, finite term.
   * Level perpetuity.
   * Non-level annuities/cash flows.
      * Arithmetic progression, finite term and perpetuity.
      * Geometric progression, finite term and perpetuity.
      * Other non-level annuities/cash flows.

## Geometric series formulas

Recall the following formulas, which are useful for deriving the formulas for different types of annuities.
* $a+ar+ar^{2}+\cdots {\overset {\text{ def }}{=}}\sum _{k=0}^{\infty }ar^{k}={\frac {a}{1-r}},\quad |r|<1$;
* $\underbrace {a+ar+ar^{2}+\cdots +ar^{n}} _{n+1{\text{ terms }}}{\overset {\text{ def }}{=}}\sum _{k=0}^{n}ar^{k}=a\left({\frac {1-r^{n+1}}{1-r}}\right),\quad r\neq 1$.

## Level annuities

Definition. (Annuity) An annuity is a series of payments made at equal time intervals.
Remark.
* The length of each time interval is arbitrary, but it is usually one year here.
* An annuity is level if all payments are equal in amount, non-level otherwise.
   * We will discuss mainly level annuities in this section, and some special types of non-level annuities will be discussed later.
* We will only discuss annuities with non-contingent (or certain) payments, but there exist annuities with contingent (or uncertain) payments.
Example.
* A fund provides payments at June 1st for each year.
   * Then, it is an annuity, since the time intervals for payments are all one year.
   * (Rough) time diagram for illustration:
* A fund provides payments at the end of $n$th year if $n$ is odd.
   * Then, it is an annuity, since the time intervals for payments are all two years.
   * Time diagram:
* A fund provides payments at the beginning of $k$th year if $k$ is prime.
   * Then, it is not an annuity, since the time intervals are not equal. E.g., payments are made at the beginning of 2nd, 3rd and 5th years, and time intervals vary here.
   * Time diagram:
Exercise.

### Annuity-immediate

Definition. (Annuity-immediate) An annuity-immediate is an annuity under which payments are made at the end of each period for $n$ periods ($n$ is a positive integer).
Remark.
* The present value of $n$-period annuity-immediate with payments of 1 is denoted by $a_{{\overline {n}}|}$ (${\overline {n}}|$ is read "angle-n").
   * If the (effective) interest rate per period is $i$, then it can also be denoted by $a_{{\overline {n}}|i}$. This holds for other similar notations.
* The accumulated value (or future value) at time $n$ of $n$-period annuity-immediate with payments of 1 (i.e. at the end of $n$th period) is dentoed by $s_{{\overline {n}}|}$.
* Annuity-immediate is "immediate" in the sense that the payments start at the end of first year, without deferring to later year, instead of "start at the beginning of each year, without delay".
Time diagram:
Proposition. (Formula of present value of annuity-immediate) $a_{{\overline {n}}|i}={\frac {1-v^{n}}{i}}$.
Proof.
From the time diagram, we have $a_{{\overline {n}}|i}=\underbrace {v+v^{2}+\dotsb +v^{n}} _{n{\text{ terms}}}={\frac {v(1-v^{n})}{1-v}}={\frac {(1-v^{n})/1+i}{i/1+i}}={\frac {1-v^{n}}{i}}$.
$\Box$
Remark.
* We can calculate the value of $a_{{\overline {n}}|i}$ using BA II Plus.
   * If the (annual effective) interest rate is $k\%$, then press k I/Y.
   * Since the amount of each payment is 1 by definition, press 1 PMT (in general, if the amount is $m$, then press m PMT) (positive (negative) value should be inputted for cash inflow (outflow) by convention, we input payment as cash inflow from the annuity's owner's perspective, since the owner receives payments).
   * If the annuity lasts for $n$ years, then press n N.
   * Finally, press CPT PV to compute the present value, i.e. the value of $a_{{\overline {n}}|i}$. Negative value is obtained, since it tells how much cash outflow is needed in present value to exchange for the cash inflows inputted, which is the same as the present value of cash inflows by definition.
      * Alternatively, we can input negative value to PMT (treating payments as cash outflows for the cash inflow at present) and then CPT PV will yield positive value (present value is the cash inflow).
      * We can input numbers in arbitrary order before computing the present value .
   * We can also press CPT FV to compute the future value at time $n$ (negative value is obtained, since it tells cash outflow needed in future value).
   * Similarly, given information about other things, we input numbers accordingly and compute the desired thing.
   * Press 2ND CE|C 2ND FV for clearing calculator memory (memory for inputted values).
* For accumulated (or future) value at time $n$ of payments from this annuity, we have $s_{{\overline {n}}|i}=a_{{\overline {n}}|i}(1+i)^{n}$ from the relationship $FV=PVa(n),\quad a(t)=(1+i)^{t}$ (the present value of annuity accumulates with accumulation function $a(t)=(1+i)^{t}$).
   * Explicitly, it follows from this proposition that $s_{{\overline {n}}|}={\frac {(1+i)^{n}-1}{i}}$.
* This formula provides a mnemonic: for annuity-immediate, the present value $a_{{\overline {n}}|}={\frac {1-v^{n}}{\color {darkgreen}i}}$ ("$i$" in the denominator matches with "i" in immediate).
Example.
* An annuity pays 1000 at the end of each year for 10 years. The effective interest rate is 5%.
* Then, the present value of payments is $1000a_{{\overline {10}}|0.05}\approx 7721.73$ (by pressing 5 I/Y 1000 PMT 10 N CPT PV, which yields negative value, representing cash outflow).
* The present value is approximately 8110.90 if the interest rate is 4% instead (by pressing 4 I/Y CPT PV without clearing memory).
* The effective interest rate, such that the present value of payments is 4500, is approximately 17.96% (by pressing 4500 +|- PV CPT I/Y without clearing memory, and the output is the number before "%", and we input negative value into PV since it is cash outflow).
* Suppose the annuity lasts for $n$ years instead (and other conditions are the same). Then, the least value of $n$ such that the present value of payments is 4500, is 6 (by pressing 5 I/Y CPT N and get 5.22 without clearing memory, which implies the least value is 6 since $n$ is integer).
* Suppose the annuity pays $k$ at the end of each year, such that the present value of payments is 4500. Then, $k\approx 582.77$ (by pressing 10 N CPT PMT without clearing memory)
Exercise.
Example. Calculate $i$ such that $a_{{\overline {10}}|i}=a_{{\overline {5}}|0.03}$.
Solution:
* Using BA II Plus and pressing 5 N 3 I/Y 1 PMT CPT PV, $a_{{\overline {5}}|0.03}\approx 4.58$.
* After that, pressing 10 N CPT I/Y without clearing memory (so that the number stored in PV is still the previous answer) yields $i\approx 17.47\%$.
Exercise.
Example. Show that $a_{{\overline {n}}|i}$ is a decreasing function of $i$ for each nonnegative $i$.
Proof.
* It is a decreasing function of $i$ if and only if ${\frac {d}{di}}a_{{\overline {n}}|i}\leq 0$, for each nonnegative $i$.
* Since $a_{{\overline {n}}|i}=v+v^{2}+\dotsb +v^{n}$ (it is more convenient to use this form here, rather than $a_{{\overline {n}}|i}={\frac {1-v^{n}}{i}}$),
${\begin{aligned}{\frac {d}{di}}a_{{\overline {n}}|i}&={\frac {d}{di}}((1+i)^{-1}+(1+i)^{-2}+\dotsb +(1+i)^{-n})\\&=-\underbrace {(1+i)^{-2}} _{\geq 0}-2\underbrace {(1+i)^{-3}} _{\geq 0}-\dotsb -n\underbrace {(1+i)^{-n-1}} _{\geq 0}\\&=-{\big\underbrace {(1+i)^{-2}+2(1+i)^{-3}+\dotsb +n(1+i)^{n-1}} _{\geq 0}{\big )}\\&\leq 0.\end{aligned}}$
$\Box$
Exercise.

### Annuity-due

Definition. (Annuity-due) An annuity-due is an annuity under which payments are made at the beginning of each period for $n$ periods ($n$ is a positive integer).
Remark.
* The present value of $n$-period annuity-due with payments of 1 is denoted by ${\ddot {a}}_{{\overline {n}}|}$.
* The future value at time $n$ (i.e. at the end of $n$th year) of $n$-period annuity-due with payments of 1 is denoted by ${\ddot {s}}_{{\overline {n}}|}$.
* Annuity-due is "due" in the sense that payment is due as soon as the annuity starts, so payments are made at the beginning of a year.
Time diagram:
Proposition. (Relationship between present values of annuity-immediate and annuity-due) ${\ddot {a}}_{{\overline {n}}|i}=a_{{\overline {n}}|i}(1+i)$.
Proof.
* Consider the time diagram for annuity-immediate with payments of 1:
* The present value of this annuity is $a_{{\overline {n}}|i}$.
* So, the value of this annuity at $t=1$ is $a_{{\overline {n}}|i}(1+i)$.
* Then, if we regard $t=1$ as present (i.e. $t=0$) by changing time labels, the time diagram becomes:
* We can observe that this is the time diagram for annuity-due with payments of 1, and the value at $t=0$ (i.e. present value) is ${\ddot {a}}_{{\overline {n}}|i}$.
* It follows that ${\ddot {a}}_{{\overline {n}}|i}=a_{{\overline {n}}|i}(1+i)$, since these two expressions tell the value at the same time point (with different labels only), with the same series of payment.
$\Box$
Remark.
* It follows from this proposition that ${\ddot {a}}_{{\overline {n}}|}=(1+i)a_{{\overline {n}}|}=(1+i)\cdot {\frac {1-v^{n}}{i}}={\frac {1-v^{n}}{d}}$ ($d$ is equivalent to $i$).
   * This formula provides a mnemonic: for annuity-due, the present value ${\ddot {a}}_{{\overline {n}}|i}={\frac {1-v^{n}}{\color {darkgreen}d}}$ ("$d$" in the denominator matches with "d" in due).
* Because of this relationship, we can calculate the value of ${\ddot {a}}_{{\overline {n}}|i}$ using BA II Plus, by calculating the value of $a_{{\overline {n}}|i}$ first, and then divide it by $1+i$ to get ${\ddot {a}}_{{\overline {n}}|i}$.
* Alternatively, we may press 2ND PMT 2ND ENTER to change the calculation mode to "BGN" for BA II Plus (then at the top right corner, there will be a "BGN" sign), and then we can use the same keys to compute $a_{{\overline {n}}|i}$ before to compute ${\ddot {a}}_{{\overline {n}}|i}$ instead.
      * Warning: However, we should press 2ND PMT 2ND SET to change back to the default calculation mode, "END", to compute the value of $a_{{\overline {n}}|i}$, otherwise the computed value will be wrong when we are computing $a_{{\overline {n}}|i}$ even if we are using the same keys as before.
   * Thus, to avoid this, it may be better to use the first method to calculate ${\ddot {a}}_{{\overline {n}}|i}$.
* Similarly, since the future value at time $n$ is given by $FV=PVa(n),\quad a(t)=(1+i)^{t}$, we have ${\ddot {s}}_{{\overline {n}}|i}={\ddot {a}}_{{\overline {n}}|i}(1+i)^{n}$.
Example.
* A fund pays 500 at the end of each of first 5 years, and then pays 2000 at the beginning of each of next 5 years (after the end of 5th year).
* The annual effective interest rate is 10%.
* Then, the present value of payments is
$500a_{{\overline {5}}|}+2000{\ddot {a}}_{{\overline {5}}|}v^{5}\approx 6174.99.$
Proof.
* Consider the time diagram:
* For payments of 500, their present value is $500a_{{\overline {5}}|}$.
* For payments of 2000, their value at $t=5$ is $2000{\ddot {a}}_{{\overline {5}}|}$.
   * It follows that their present value (i.e. value at $t=0$) is $2000{\ddot {a}}_{{\overline {5}}|}v^{5}$.
* Then, using BA II Plus, pressing 500 PMT 5 N 10 I/Y CPT PV, and we compute that $500a_{{\overline {5}}|}\approx 1895.39$.
* Similarly, pressing 2000 PMT CPT PV ÷ 1.1 = ÷ (1.1 y^x 5) = yields $2500a_{{\overline {5}}|}v^{5}\approx 4279.60$.
* Adding these two numbers up yields the desired result.
$\Box$
Exercise.

### Annuities payable mthly

Sometimes, annuities are not payable annually, and can be payable more or less frequently than annually.
To calculate the present value of these kind of annuities, we can simply change the measurement period, and calculate the interest rate during that period that is equivalent to the given interest (or discount) rate (or force of interest), and the new term of the annuity in the new measurement period.
Using new terms and new interest rates, we can calculate the present value of these kind of annuities by applying previously discussed method.
Example.
* The annual interest rate is 12%.
* Then, the equivalent monthly interest rate is $12(1.12^{1/12}-1)$ .
* Since there are 120 months in 10 years, the present value of 10-year annuity-immediate with payments of 1 payable monthly is $a_{{\overline {120}}|12(1.12^{1/12}-1)}\approx 8.78$.
Time diagram:
Exercise.
Example.
* A fund provides payments with 4000 quarterly in arrears (i.e. at the end of each quarter) for 36 months.
* The monthly discount rate is 4%.
* Calculate the present value of these payments.
Solution:
* Let $i$ be the equivalent quarterly interest rate.
* Then, $(1+i)^{4}=(1-0.04)^{-12}\Rightarrow i\approx 0.13028$.
* Also, there are 9 quarters in 36 months.
* Thus, the present value is
$4000a_{{\overline {9}}|0.13028}\approx 20505.21.$
Exercise.

### Annuities payable continuously

* Recall from the motivation of force of interest that "payable continuously" is essentially "payable $\infty$thly (abuse of notation)"
* If an annuity pays 1 in each "infinitesimal" time interval, the present value of payments in a measurement period will be infinite, since there are infinitely many such time intervals during arbitrary measurement period.
* Because of this, it does not make sense to say an annuity has payments of xxx payable continuously.
* Instead, we should use the notion of rate to describe the behaviour of continuous payment.
Example.
* A $n$-year annuity has payment continuously at a (uniform and constant) rate of 1 per year.
* Suppose the annual constant force of interest is $\delta$ (and thus the annual interest rate is $e^{\delta }-1$).
* The present value of this type of annuity is denoted by ${\overline {a}}_{{\overline {n}}|}$, and equals $\int _{0}^{n}e^{-\delta t}\,dt={\frac {1-e^{-n\delta }}{\delta }}$.
Proof.
* The present value is $\underbrace {\int _{0}^{n}} _{\text{summing up}}\underbrace {a^{-1}(t)} _{\text{for PV}}\overbrace {(\underbrace {1} _{\text{rate}})\,\underbrace {dt} _{\text{`time'}}} ^{\text{`amount'}}$.
* Since $a^{-1}(t)=e^{-\delta t}$, we get the desired integral.
* Also, $\int _{0}^{n}e^{-\delta t}\,dt={\frac {-1}{\delta }}{\bige^{-\delta t}{\big ]}_{0}^{n}={\frac {-(e^{-n\delta }-1)}{\delta }}={\frac {1-e^{-n\delta }}{\delta }}$.
* In the integral, we can interpret $t$ is in years (so the lower limit is $t=0$ (at the beginning of first year), and the upper limit is $t=n$ (at the end of $n$th year).
$\Box$
Exercise.
Remark.
* For variable force of interest $\delta _{t}$ ($t$ is in years), the present value (with payment rate $k$) is $\int _{0}^{n}k\exp \left(\int _{0}^{t}\delta _{s}\,ds\right)\,dt$, since $a^{-1}(t)=\exp \left(\int _{0}^{t}\delta _{s}\,ds\right)$.
* In general, the present value (with payment rate $k$) is $\int _{0}^{n}ka^{-1}(t)\,dt$.
Example.
* Fund A provides payment continuously at a rate of 100 per year, at the force of interest $\delta _{t}=(1+t)^{-1}$.
* Fund B provides level payments monthly in advance, at a rate of 1200 per year, at an annual constant force of interest 0.08.
* Both funds stop payments after the end of 10th year.
* After 10 years, calculate the difference between accumulated value of fund A and fund B.
Solution:
* The accumulated value of fund A after 10 years is
$\int _{0}^{10}100\exp \left(\int _{0}^{t}(1+s)^{-1}\,ds\right)\,dt=\int _{0}^{10}100e^{\ln(1+t)}\,dt=\int _{0}^{10}100(1+t)\,dt=[50t^{2}+100t]_{0}^{10}=6000.$
* For fund B, the monthly interest rate that is equivalent to the constant force of interest is $(e^{0.08})^{1/12}-1\approx 0.0067$.
* Also, the monthly payment is $1200/12=100$ (since all monthly payments are the same), and there are 120 monthly payments in 10 years.
* So, the accumulated value of fund B after 10 years is
$100{\ddot {s}}_{{\overline {120}}|0.0067}\approx 18200.17.$
* It follows that the difference is approximately $18200.17-6000=12200.17$ .
Exercise.

### Perpetuities

Definition. (Perpetuity) A perpetuity is an annuity whose payments continue forever.
Remark.
* "continue forever" means the term of the annuity is infinite, or mathematically, tends to be infinity.
* A perpetuity-immediate is an annuity-immediate whose payments continue forever.
* A perpetuity-due is an annuity-due whose payments continue forever.
* The present value of perpetuity-immediate with payments of 1 is denoted by $a_{{\overline {\infty }}|}$, and equals $\lim _{n\to \infty }a_{{\overline {n}}|}=\lim _{n\to \infty }{\frac {1-v^{n}}{i}}={\frac {1}{i}}$ (since $v^{n}\to 0$ as $n\to \infty$).
* The present value of perpetuity-due with payments of 1 is denoted by ${\ddot {a}}_{{\overline {\infty }}|}$, and equals $\lim _{n\to \infty }{\ddot {a}}_{{\overline {n}}|}=\lim _{n\to \infty }{\frac {1-v^{n}}{d}}={\frac {1}{d}}$.
Time diagram:
Example. We can derive the formula of perpetuity-immediate alternatively as follows: $a_{{\overline {\infty }}|}=\lim _{n\to \infty }a_{{\overline {n}}|}=\lim _{n\to \infty }\sum _{j=1}^{n}v^{j}{\overset {\text{ def }}{=}}\sum _{j=1}^{\infty }v^{j}={\frac {v}{1-v}}={\frac {1/(1+i)}{i/(1+i)}}={\frac {1}{i}}$.
Example.
* Amy purchases an annual perpetuity-immediate at its present value, which is 1200.
* Suppose the annual interest rate is 5%.
* Then, the annual payment is $1200(0.05)=60$.
Proof.
* Let $P$ be the annual payment.
* Then, for this perpetuity, we have
$Pa_{{\overline {\infty }}|}={\frac {P}{0.05}}=1200\Rightarrow P=1200(0.05)=60.$
$\Box$
Exercise.
* For perpetuities payable $m$thly, the approach for calculating their present values is the same as that for annuities payable $m$thly, namely adjusting the measurement period and calculating new interest rate correspondingly.
* Also, for perpetuities payable continuously, the approach for calculating their present values is the same, except that we are dealing with improper integrals (upper limits of integrals involved are $\infty$).

## Non-level annuities

* In general, the present value of non-level annuities can be calculated by summing up the present value of each payment.
* However, this approach can take a lot of time, and thus may not be the most efficient approach.
* We will discuss several special cases of non-level annuities, for which the present value can be calculated in an efficient way.

### Arithmetic varying annuities

* For some annuities, payments vary (increase or decrease) in arithmetic sequence.
* We will develop a formula for calculating their present values in this subsection.
Theorem. (Present value of arithmetic varying annuities) Suppose payments in an $n$-period annuity-immediate begin at $P$ and increase by $D$ per period thereafter. Then, the present value of payments is $\left(P+{\frac {D}{i}}\right)a_{{\overline {n}}|i}-{\frac {Dn}{i}}\cdot v^{n}$ with effective interest rate $i$ during each of $n$ periods.
Proof.
* Consider the following time diagram:
* For payments in the $n$th row, the present value is $Pa_{{\overline {n}}|i}$;
* for payments in the $n-1$th row, the present value is $Da_{{\overline {n-1}}|i}v=Dv\cdot {\frac {1-v^{n-1}}{i}}$;
* for payments in the $n-j$th row, the present value is $Da_{{\overline {n-j}}|i}v^{j}=Dv^{j}\cdot {\frac {1-v^{n-j}}{i}}$;
* for payments in the 2nd row, the present value is $Da_{{\overline {2}}|i}v^{n-2}=Dv^{n-2}\cdot {\frac {1-v^{2}}{i}}$;
* for payments in the 1st row, the present value is $Da_{{\overline {1}}|i}v^{n-1}=Dv^{n-1}\cdot {\frac {1-v}{i}}$.
* So, the present value of all payments is
${\begin{aligned}Pa_{{\overline {n}}|i}+\sum _{k=1}^{n-1}Dv^{k}\cdot {\frac {1-v^{n-k}}{i}}&=Pa_{{\overline {n}}|i}+{\frac {D}{i}}\sum _{k=1}^{n-1}(v^{k}-v^{n})\\&=Pa_{{\overline {n}}|i}+{\frac {D}{i}}\left(\sum _{k=1}^{n-1}(v^{k})-(n-1)v^{n}\right)\\&=Pa_{{\overline {n}}|i}+{\frac {D}{i}}{\bigg\underbrace {\sum _{k=1}^{n-1}(v^{k})+v^{n}} _{=\sum _{k=1}^{\color {blue}n}v^{k}{\overset {\text{ def }}{=}}a_{{\overline {n}}|i}}-nv^{n}{\bigg )}\\&=Pa_{{\overline {n}}|i}+{\frac {D}{i}}(a_{{\overline {n}}|i}-nv^{n})\\&=\left(P+{\frac {D}{i}}\right)a_{{\overline {n}}|i}-{\frac {Dn}{i}}\cdot v^{n}.\\\end{aligned}}$
$\Box$
Remark.
* Using BA II Plus, press $P+{\frac {D}{i}}$ PMT $-{\frac {Dn}{i}}$ FV $n$ N $100i$ I/Y CPT PV.
   * In particular, $-{\frac {Dn}{i}}$ is inputted into FV, since it can be regarded as cash outflow, or negative of cash inflow, at time $n$ (because of the factor $v^{n}$), if we regard $P+{\frac {D}{i}}$ as cash inflow.
* If $D$ is negative, then payments decrease in arithmetic sequence.
* It follows from this theorem that the present value of the arithmetic varying perpetuities having the same properties (i.e. only the term of the annuity changes) is
$\lim _{n\to \infty }\left(\left(P+{\frac {D}{i}}\right)a_{{\overline {n}}|}-{\frac {Dn}{i}}\cdot v^{n}\right)=\left(P+{\frac {D}{i}}\right)\cdot {\frac {1}{i}}={\frac {P}{i}}+{\frac {D}{i^{2}}}.$
   * In particular, $\lim _{n\to \infty }nv^{n}=0$, which can be shown by L'Hospital rule. Intuitively, the limit equals zero since $n\mapsto v^{n}$ decreases "much faster" than $n\mapsto n$.
* When $P=D=1$, the annuity-immediate is called increasing annuity, whose present value is denoted by $(Ia)_{{\overline {n}}|}$.
   * Its accumulated value at the end of $n$th period is denoted by $(Is)_{{\overline {n}}|}$, which equals $(Ia)_{{\overline {n}}|}(1+i)^{n}$.
* When $P=n$ and $D=-1$, the annuity-immediate is called decreasing annuity (payments decrease from $n$ to 1 with common difference 1), whose present value is denoted by $(Da)_{{\overline {n}}|}$.
   * Its accumulated value at the end of $n$th period is denoted by $(Ds)_{{\overline {n}}|}$, which equals $(Da)_{{\overline {n}}|}(1+i)^{n}$.
* For annuity-due and annuity payable $m$thly with payments varying in arithmetic sequence, we can use similar approach discussed previously, in addition to this theorem, to calculate their present values.
Example.
* An annuity has payments of 100, 120, 140, 160, 180, 200, 200 at the end of 1st, 2nd, 3rd, 4th, 5th, 6th and 7th year respectively.
* The annual effective interest rate is 10%.
* Consider the time diagram:
* For the first six payments, they increase in arithmetic sequence with first term 100 and common difference 20, and they are made at the end of year.
* So, we can apply the above theorem ($P=100,D=20,n=6,{\text{ and }}i=0.1$) to get their present value, which is approximately 629.21.
* For the 7th payment, its present value is $200v_{0.1}^{7}\approx 102.63$.
* So, the present value of these seven payments is approximately 731.84.
Exercise.

### Geometric varying annuities

* Since the expression for present value of annuity is essentially geometric series, even with payments varying in geometric sequence, the expression is still geometric series, and thus we can use the geometric series formula to calculate the present value.
* So, in general, for geometric varying annuities, we use "first principle" to calculate their present value, in the sense that we use geometric series formula to evaluate the expanded form of the present value.
Example.
* An annuity has payments at the beginning of each year without stopping, which begin at 100 at the beginning of first year, and then increase by 10% per year.
* Suppose the interest rate is 20% payable quarterly.
* Calculate the present value of the annuity.
Solution:
* The nominal interest rate of 20% implies the quarterly effective interest rate is 5%.
* So, the annual effective interest rate is $1.05^{4}-1\approx 21.551\%$.
* Consider the time diagram:
* It follows that the present value of the annuity is
$100+100(1.1)(1.21551)^{-1}+100(1.1)^{2}(1.21551)^{-2}+\dotsb =100+100\cdot {\frac {1.1}{1.21551}}+100\cdot \left({\frac {1.1}{1.21551}}\right)^{2}+\dotsb ={\frac {100}{1-{\frac {1.1}{1.21551}}}}\approx 1052.33.$
Exercise.

# Loans
## Learning objectives

The Candidate will understand key concepts concerning loans and how to perform related calculations.

## Learning outcomes

The Candidate will be able to:
* Define and recognize the definitions of the following terms: principal, interest, term of loan, outstanding balance, final payment (drop payment, balloon payment), amortization.
* Calculate:
   * The missing item, given any four of: term of loan, interest rate, payment amount, payment period, principal.
   * The outstanding balance at any point in time.
   * The amount of interest and principal repayment in a given payment.
   * Similar calculations to the above when refinancing is involved.

## Introduction

In this chapter, two methods of repaying a loan will be discussed, namely amortization method and sinking fund method. In particular, for each of these two methods, we will discuss how to determine he outstanding loan balance at any point in time, and the amount of interest and principal repayment in each payment made by borrower.

## Amortization method

Definition. (Amortization method)
* For amortization method, the borrower repays the lender by a series of payments at regular intervals.
* Each payment is applied first to interest due on the outstanding balance at the time just before the payment is made to pay the interest, and
* after deducting the amount of interest from each payment, the amount left in each payment is going as the principal repayment to reduce the loan balance (i.e. how much the borrower owes).
* Payments are made to reduce the loan balance to exactly zero.

### Amortization of level payment

The series of payments made by borrower is level in this subsection, and payments form annuity-immediate in our discussion. To illustrate this, consider the following diagrams.
Borrower's perspective:
Lender's perspective:
in which
* ↑ means the amount is received, ↓ means the amount is paid;
* $L$ is the amount borrowed (i.e. the amount of loan);
* $n$ is the number of payments;
* $R$ is the level payment made by the borrower (this is return from the lender's perspective).
* Let $B_{k}$ be the outstanding balance at time $k$, just after the $k$th payment ($B_{0}=L$, which is the initial balance).
* Let $i$ be the effective interest rate during each interval for payments.
Proposition. (Recursive method to determine outstanding balance (level payment)) $B_{k+1}=(1+i)B_{k}-R$.
Proof.
* First, $B_{k}$ will accumulate to $B_{k}(1+i)$ from time $k$ to $k+1$.
* The interest due on $B_{k}$ is $iB_{k}$.
* So, the reduction of outstanding balance from the payment of $R$ at $t=k+1$ is $R-iB_{k}$.
* It follows that the outstanding balance at time $k+1$ is $B_{k}-(R-iB_{k})=B_{k}(1+i)-R$.
$\Box$
Proposition. (Fundamental relationship between amount of loan and payments) $L=Ra_{{\overline {n}}|i}$.
Proof.
* Using the above recursive method, $B_{1}=B_{0}(1+i)-R=L(1+i)-R$;
* $B_{2}=B_{1}(1+i)-R=L(1+i)^{2}-R(1+i)-R$;
* $B_{3}=L(1+i)^{3}-R(1+i)^{2}-R(1+i)-R$;
* $B_{n}=L(1+i)^{n}-R(1+i)^{n-1}-R(1+i)^{n-2}-\dotsb -R$.
* Since $B_{n}=0$ for amortization method (the loan balance is reduceed to zero at the end by definition), we have
${\begin{aligned}&&L(1+i)^{n}-R(1+i)^{n-1}-R(1+i)^{n-2}-\dotsb -R&=0\\&\Rightarrow &L(1+i)^{n}&=R(1+i)^{n-1}+R(1+i)^{n-2}+\dotsb +R\\&\Rightarrow &L(1+i)^{n}v^{n}&=R(1+i)^{n-1}v^{n}+R(1+i)^{n-2}v^{n}+\dotsb +Rv^{n}\\&\Rightarrow &L&=Rv+Rv^{2}+\dotsb +Rv^{n}{\overset {\text{ def }}{=}}Ra_{{\overline {n}}|i}.\\\end{aligned}}$
$\Box$
Remark.
* Because of this relationship, we can calculate $L$ by pressing $-R$ PMT $n$ N $100i$ I/Y CPT PV to get $L$.
   * In particular, $-R$ is inputted since payments are cash outflow (we use borrower's perspective here).
Proposition. (Prospective method to determine outstanding balance (level payment)) $B_{k}=Ra_{{\overline {n-k}}|i}$.
Proof.
* From the proof of fundamental relationship between $L$ and $R$, we have
${\begin{aligned}B_{k}&=L(1+i)^{k}-R(1+i)^{k-1}-R(1+i)^{k-2}-\dotsb -R\\&=(1+i)^{k}(Rv+Rv^{2}+\dotsb +Rv^{n})-R(1+i)^{k-1}-R(1+i)^{k-2}-\dotsb -R\\&={\cancel {R(1+i)^{k-1}+R(1+i)^{k-2}+\dotsb +R(1+i)^{k-(k-1)}+R(1+i)^{k-k}}}+R(1+i)^{k-(k+1)}\dotsb +R(1+i)^{k-n}{\cancel {-R(1+i)^{k-1}-R(1+i)^{k-2}-\dotsb -R}}\\&=R(1+i)^{-1}+R(1+i)^{-2}+\dotsb +R(1+i)^{-(n-k)}\\&=Rv+Rv^{2}+\dotsb +Rv^{n-k}\\&=Ra_{{\overline {n-k}}|i}.\end{aligned}}$.
$\Box$
Proposition. (Retrospective method to determine outstanding balance (level payment)) $B_{k}=L(1+i)^{k}-Rs_{{\overline {k}}|i}.$
Proof.
* From the proof of fundamental relationship between $L$ and $R$, we have
${\begin{aligned}B_{k}&=L(1+i)^{k}-R(1+i)^{k-1}-R(1+i)^{k-2}-\dotsb -R\\&=L(1+i)^{k}-(1+i)^{k}(Rv+Rv^{2}+\dotsb +Rv^{k})\\&=L(1+i)^{k}-Rs_{{\overline {k}}|i}.\end{aligned}}$
$\Box$
* Another method to determine outstanding balance (and also principal and interest paid in different payments) is using BA II Plus.
* Procedure:
* Input $-R$ into PMT (if $R$ is unknown, it should be determined first).
* Input $L$ into PV
   * We can also compute PMT or PV given sufficient information.
* Press 2ND PV
* Press the starting payment number (for $k$th payment, press $k$) and press ENTER ↓.
* Press the ending payment number (for $k$th payment, press $k$) and press ENTER ↓ (press the same number as the starting payment number for selecting exactly one payment).
* Then, outstanding balance just after the selected payment(s) is displayed (BAL=... is displayed).
* Press ↓ and loan (or "principal") paid in the selected payment(s) is displayed (PRN=... is displayed).
* Press ↓ and interest paid in the selected payment(s) is displayed (INT=... is displayed).
Example.
* Suppose a loan of 2000 is repaid by 15 annual payments of $R$ in arrears.
* With annual interest rate of 10%, $R={\frac {2000}{a_{{\overline {15}}|0.1}}}\approx 262.95$.
Exercise.
Example.
* Suppose a loan of $L$ is repaid by 20 monthly payments of $0.1L$ in arrears.
* Calculate the annual effective interest rate $i$.
Solution:
* The monthly effective interest rate $j$ is calculated by $L=0.1La_{{\overline {20}}|j}\Rightarrow a_{{\overline {20}}|j}=10\Rightarrow j\approx 7.75\%$ (using BA II Plus).
* So, the annual effective interest rate $i=(1+j)^{12}-1\approx 145.04\%$.
Exercise.
Example.
* A loan of 1000 is repaid by 12 annual level payments in arrears.
* The annual interest rate is 8%.
* Then, the outstanding balance at the end of 5th year is 690.86 by pressing 1000 PV 12 N 8 I/Y CPT PMT 2ND PV 5 ENTER ↓ 5 ENTER ↓, and outstanding balance is displayed.
Exercise.
Now, we consider the amount of interest and principal repayment in each payment made by borrower.
Proposition. (Splitting an installment into principal and interest repayments (level payment)) Let $P_{k}$ be the principal repaid in the $k$th installment (i.e. $k$th payment made by borrower), and $I_{k}$ be the amount of interest paid in the $k$th installment. (This notation has different meaning in the context of Chapter 1.) Suppose the installments made by borrower is level, and each of them equals $R$. Then, $P_{k}=Rv^{n-k+1},\quad I_{k}=R-P_{k}.$
Proof.
* First, by definitions, $R=P_{k}+I_{k}$ because the installment is first deducted by the interest due ($I_{k}$), and the remaining amount ($R-I_{k}$) is used to repay principal. Therefore, $I_{k}=R-P_{k}$.
* It remains to prove the formula for $P_{k}$. By definition, $I_{k}=iB_{k-1}$ because the interest is due on the outstanding balance (before the $k$th installment).
$P_{k}=R-I_{k}=R-iB_{k-1}=R-i\underbrace {Ra_{{\overline {n-k+1}}|}} _{\text{prospective}}=R\left({\cancel {1}}-{\cancel {i}}\left({\frac {{\cancel {1}}-v^{n-k+1}}{\cancel {i}}}\right)\right)=Rv^{n-k+1}$
$\Box$
Remark.
* We can use BA II Plus to determine $P_{k}$ and $I_{k}$, which is discussed before.
After splitting each installment, we can make an amortization schedule which illustrates the splitting of each repayment in a tabular form. An example of amortization schedule is as follows:
Amortization schedule for a loan of $a_{{\overline {n}}|}$ repaid over $n$ periods at rate $i$ :
| Period | Payment | Interest paid | Principal repaid | Outstanding loan balance |
| 0 | 0 | 0 | 0 | $a_{{\overline {n}}|}$ (prospective) |
| 1 | 1 | $i\underbrace {a_{{\overline {n}}|}} _{B_{0}}=\underbrace {1} _{R}-\underbrace {v^{n}} _{P_{1}}$ | $\underbrace {v^{n}} _{1(v^{n-1+1})}$ | $\underbrace {a_{{\overline {n}}|}} _{B_{0}}-\underbrace {v^{n}} _{P_{1}}=\underbrace {a_{{\overline {n-1}}|}} _{\text{prospective}}$ |
| 2 | 1 | $ia_{{\overline {n-1}}|}=1-v^{n-1}$ | $v^{n-1}$ | $a_{{\overline {n-1}}|}-v^{n-1}=a_{{\overline {n-2}}|}$ |
| $k$ | 1 | $ia_{{\overline {n-k+1}}|}=1-v^{n-k+1}$ | $v^{n-k+1}$ | $a_{{\overline {n-k+1}}|}-v^{n-k+1}=a_{{\overline {n-k}}|}$ |
| $n-1$ | 1 | $ia_{{\overline {2}}|}=1-v^{2}$ | $v^{2}$ | $a_{{\overline {2}}|}-v^{2}=a_{{\overline {1}}|}$ |
| $n$ | 1 | $ia_{{\overline {1}}|}=1-v$ | $v$ | $a_{{\overline {1}}|}-v=0$ |
| Total | $n$ | $n-a_{{\overline {n}}|}$ | $a_{{\overline {n}}|}$ | not important |
(You may verify the recursive method to determine outstanding balance using this table, e.g. $a_{{\overline {n}}|}(1+i)-1={\ddot {a}}_{{\overline {n}}|}-1=a_{{\overline {n-1}}|}$)
It can be seen that total payment ($n$) equals total interest paid ($n-a_{{\overline {n}}|}$) plus total principal repaid ($a_{{\overline {n}}|}$), and each payment equals the interest paid plus principal repaid in the corresponding period (read horizontally), as expected, because the payment is either used for paying interest, or used for repaying principal.
It can also be seen that the total principal repaid equals the amount of loan (i.e. outstanding loan balance in period 0) ($a_{{\overline {n}}|}$), as expected, because the whole loan is repaid by the payments in $n$ periods.
Example.
* A loan of 1000 is repaid by seven annual payments of $R$ in arrears.
* The annual interest rate is 5%.
* The principal repaid in 3rd payment is approximately 135.41 (press 1000 PV 7 N 5 I/Y CPT PMT 2ND PV 3 ENTER ↓ 3 ENTER ↓ ↓);
* the interest paid in 3rd to 6th payment is approximately 107.65 (press ↑ ↑ 6 ENTER ↓ ↓ ↓, continuing from the above pressing sequence).

### Amortization of non-level payment

In this subsection, we will consider amortization of non-level payment. The ideas and concepts involved are quite similar to the amortization of non-level payment. Borrower's perspective:
Lender's perspective:
in which $R_{1},R_{2},\ldots ,R_{n}$ are non-level payments, and the other relevant notations used in amortization of level payment have the same meaning.
Because the payments are now non-level, we need formulas different from that for the amortization of level payment to determine amount of loan and outstanding balance at different time, and to split the payment into interest payment and principal repayment. They are listed in the following.
Proposition. (Relationship between amount of loan and payments (non-level payment)) $L=R_{1}v+R_{2}v^{2}+\cdots +R_{n}v^{n}.$
Proof. Omitted since the main idea is identical to the proof for the level payment version.
$\Box$
Proposition. (Prospective method to determine outstanding balance (non-level payment)) $B_{k}=R_{k+1}v+R_{k+2}v^{2}+\cdots +R_{n}v^{n-k}.$
Proof. Omitted since the main idea is identical to the proof for the level payment version.
$\Box$
Proposition. (Retrospective method to determine outstanding balance (non-level payment)) $B_{k}=L(1+i)^{k}-R_{1}(1+i)^{k-1}-R_{2}(1+i)^{k-2}-\cdots -R_{k}.$
Proof. Omitted since the main idea is identical to the proof for the level payment version.
$\Box$
Proposition. (Recursive method to determine outstanding balance (non-level payment)) $B_{k}=B_{k-1}(1+i)-R_{k}.$
Proof. Omitted since the main idea is identical to the proof for the level payment version.
$\Box$
Remark. Recursive method is more useful for non-level payment than level payment.
Proposition. (Splitting an installment into principal and interest repayments (non-level payment)) $I_{k}=iB_{k-1},\quad P_{k}=R_{k}-I_{k}.$
Proof.
* $I_{k}=iB_{k-1}$: It follows from definition of $I_{k}$.
* $P_{k}=R_{k}-I_{k}$: It follows from $R_{k}=P_{k}+I_{k}$ which is true by definition.
$\Box$

### Amortization of payments that are made at a different frequency than interest is convertible

In this situation, we can obtain the amount of loan, outstanding balance, and principal repaid and interest paid in a payment, by calculating the equivalent interest rate that is convertible at the same frequency at which payments are made. Then, the previous formulas can be used directly, at this equivalent interest rate. This method is analogous to the method for calculating the annuity with payments made at a different frequency than interest is convertible.

## Sinking fund method

After discussing amortization method, we discuss another way to repay a loan, namely sinking fund method.
Definition. (Sinking fund method) For sinking fund method, all principal (i.e. amount of loan) is repaid by the borrower in a single payment at maturity. Interest due on the principal is paid at the end of each period and a deposit is made into a sinking fund at the end of each period (same amount of deposit for each of the time points), so that the accumulated value of sinking fund equals the amount of principal at maturity.
Borrower's perspective:
Lender's perspective: (Lender do not know how the borrower repays the loan, so sinking fund is not shown)
in which
* $L$ is the amount borrowed
* $n$ is the number of payment periods
* $i$ is the effective interest rate paid by borrower to lender
* $j$ is the effective interest rate earned on the sinking fund (which is usually strictly less than $i$ in practice)
* $D$ is the level sinking fund deposit
Let $R$ is the level payment made by borrower at the end of each period, which equals $D+$ interest paid to lender, i.e. $R=Li+D$.
By definition of sinking fund method, $L=Ds_{{\overline {n}}|j}$ because the accumulated value of sinking fund equals amount of loan at maturity.
Using these two equation, we can have the following theorem.
Proposition. (Relationship between each payment made by borrower and amount of loan in sinking fund method) $R=L\left(i+{\frac {1}{s_{{\overline {n}}|j}}}\right).$
Proof. Because $L=Ds_{{\overline {n}}|j}\Rightarrow D={\frac {L}{s_{{\overline {n}}|j}}}$ $R=Li+D=Li+{\frac {L}{s_{{\overline {n}}|j}}}=L\left(i+{\frac {1}{s_{{\overline {n}}|j}}}\right).$
$\Box$
Recall that ${\frac {1}{a_{{\overline {n}}|i}}}=i+{\frac {1}{s_{{\overline {n}}|i}}}$. We can observe that a similar expression compared with the right hand side appears in above equation ($i+{\frac {1}{s_{{\overline {n}}|j}}}$). In view of this, we define ${\frac {1}{a_{{\overline {n}}|i\&j}}}=i+{\frac {1}{s_{{\overline {n}}|j}}}.$ (we use '$i\&j$' because the right hand side involves both $i$ and $j$.) Then, if the amount of loan is 1, then the payment made by borrower at the end of each period is ${\frac {1}{a_{{\overline {n}}|i\&j}}}$.
Naturally, we would like to know what $a_{{\overline {n}}|i\&j}$ equals. We can determine this as follows: ${\begin{aligned}{\frac {1}{a_{{\overline {n}}|i\&j}}}&=i+{\frac {1}{s_{{\overline {n}}|j}}}\\&=\left({\frac {1}{a_{{\overline {n}}|j}}}-j\right)+i\qquad {\text{because }}{\frac {1}{a_{{\overline {n}}|j}}}={\frac {1}{s_{{\overline {n}}|j}}}+j\\&={\frac {1}{a_{{\overline {n}}|j}}}+(i-j)\\&={\frac {1+(i-j)a_{{\overline {n}}|j}}{a_{{\overline {n}}|j}}}\\\Rightarrow a_{{\overline {n}}|i\&j}&={\frac {a_{{\overline {n}}|j}}{1+(i-j)a_{{\overline {n}}|j}}}.\end{aligned}}$ (The right hand side also involve $i$ and $j$, as expected, because the reciprocal of an expression involving $i$ and $j$ should also involve $i$ and $j$) In particular, if $i=j$, $a_{{\overline {n}}|i\&j}=a_{{\overline {n}}|i}=a_{{\overline {n}}|j}$ as expected, and $R=Li+D=L\left(i+{\frac {1}{s_{{\overline {n}}|i}}}\right)={\frac {L}{a_{{\overline {n}}|i}}}.$ Therefore, each level payment made by borrower in the sinking fund method is the same as the levelpayment in the amortization method, because $L=Ra_{{\overline {n}}|i}$ in amortization method of level payment.
Using this notation, we can express the relationship between $R$ and $L$ as follows: $R={\frac {L}{a_{{\overline {n}}|i\&j}}}={\frac {L(1+(i-j)a_{{\overline {n}}|j})}{a_{{\overline {n}}|j}}}$
If we assume that the balance in the sinking fund could be used to reduce the amount of loan, then net amount of loan after $k$th payment is $L-Ds_{{\overline {k}}|j},$ net amount of interest paid in the $k$th period is $Li-j(Ds_{{\overline {k-1}}|j}),$ and principal repaid in the $k$th period is $Ds_{{\overline {k}}|j}-Ds_{{\overline {k-1}}|j}=D(1+j)^{k-1}.$

# Bonds
## Learning objectives

The Candidate will understand key concepts concerning bonds, and how to perform related calculations.

## Learning outcomes

The Candidate will be able to:
* Define and recognize the definitions of the following terms: price, book value, amortization of premium, accumulation of discount, redemption value, par value/face value, yield rate, coupon, coupon rate, term of bond, callable/non-callable.
* Given sufficient partial information about the items listed below, calculate any of the remaining items
   * Price, book value, amortization of premium, accumulation of discount. (Note that valuation of bonds between coupon payment dates will not be covered).
   * Redemption value, face value.
   * Yield rate.
   * Coupon, coupon rate.
   * Term of bond, point in time that a bond has a given book value, amortization of premium, or accumulation of discount.

## Bond

A bond is a debt security, in which the issuer, usually a corporation or public institution, owes the holders a debt and is obliged to pay interest (the coupon) and to repay the principal at a later date. A bond is a formal contract to repay borrowed money with interest at fixed intervals. There are two main kinds of bonds: accumulation bonds (zero coupon bonds) and bonds with coupons. An accumulation bond is where the issuer of the bond agrees to pay the face value at a later redemption date, but they are sold at a discount.
Example: a 20 year $1000 face value bond with a 3.5% nominal annual yield would have a price of $502.56.
Bonds with coupons are more common and it's where the issuer of the bond makes period payments (coupons) and a final payment.
Example: a 10 year $1000 par value bond with a 8% coupon convertible semiannually would pay $40 coupons every 6 months and then $1000 at the end of the 10 years.

## Terminology and variable naming convention

* $P$ is the price of a bond. The price of a bond P is the amount that the lender, the person buying the bond, pays to the government or corporation issuing the bond.
* $A$ is the price per unit nominal, i.e. $A:={\frac {P}{F}}$.
* $F$, is the face amount, face value, par value, or nominal value of a bond which is the amount by which the coupons are calculated, and is printed on the front of the bond.
* $C$ is the redemption value of a bond which is the amount of money paid to the bond holder at the redemption date.
* $R$ is the redemption value per unit nominal, i.e. $R:={\frac {C}{F}}$.
   * If $R=1$, the bond is redeemable at par
   * If $R>1$, the bond is redeemable above par
   * If $R<1$, the bond is redeemable below par
* $r$ is coupon rate (or nominal yield) which is the rate per coupon payment period used in determining the amount of coupon.
* $Fr$ is the amount of the coupon
* $g$ is modified coupon rate which is defined by $g:={\frac {Fr}{C}}$, i.e. the coupon rate per unit of redemption value instead of per unit of par value (which is the case for $r$).
* $i$ is the yield rate or yield to maturity of a bond, which is the interest rate earned by the investor (i.e. the effective interest rate), assuming the bond is held until it is redeemed.
* $n$ is the number of coupon payment periods from the date of calculation to redemption date.
* $K$ is the present value, that is calculated at the yield rate, of the redemption value of a bond at redemption date, i.e. $K=Cv^{n}$ in which $v:={\frac {1}{1+i}}$ ($i$ is the yield rate of the bond.
* $G$ is the base amount of a bond which is defined by $Gi:=Fr$, i.e. it is the amount of which an investment at yield rate $i$ produces periodic interest payment that equals the amount of every coupon of the bond.
From now on, unless otherwise specified, the redemption value ($C$) of a bond equals the face amount (par value) ($F$) of the bond. This is also true in the SOA FM Exam. Therefore, we have $g:={\frac {Fr}{C}}={\frac {Fr}{F}}=r$, i.e. the modified coupon rate is not 'modified' unless otherwise specified.

## Four formulas to calculate the price of a bond
### Situation in which there are no taxes

In this subsection, we discuss the calculation of price of a bond when there are no taxes. We will discuss the calculation of price of a bond when there are some taxes, namely income tax and capital gains tax.
We will obtain the same price no matter we use which of the following four formulas, because we can use basic formula to derive all other three formulas. The choice of formula is mainly based on what information is given, and we choose the formula for which we can use it most conveniently.
Proposition. (Basic formula) $P=Fra_{{\overline {n}}|i}+\underbrace {Cv^{n}} _{K}.$
Illustration:
Proof. It follows from the fact that the price is set to equal the present value of future coupons plus present value of the redemption value (i.e. all future payments), so that the price is a 'fair price'. (We treat the time at which the bond price is calculated as present.)
$\Box$
Remark.
* this is intuitive because how much the bond worth (bond price) should depend on its coupons and redemption value, and we consider their value at the time at which the bond price is calculated (i.e. at present)
* you may observe that this formula is in a similar form when compared with the formula for prospective method for determining the outstanding balance of loan
   * actually, this is expected, since bond is essentially a loan from the buyer to the seller, and $P$ can be viewed as the loan given to seller, $Fr$ can be viewed as the installment from the seller on the loan, and $C$ can be viewed as the single payment at maturity
   * you may compare this with sinking fund method
Proposition. (Premium/Discount formula) $P=C+C(g-i)a_{{\overline {n}}|i}.$
Proof. By basic formula, ${\begin{aligned}P&=Fra_{{\overline {n}}|i}+Cv^{n}\\&={\color {darkgreen}Fra_{{\overline {n}}|i}}+{\color {darkgreen}C}(1-{\color {darkgreen}ia_{{\overline {n}}|i}})\qquad {\text{by }}a_{{\overline {n}}|}={\frac {1-v^{n}}{i}}\Leftrightarrow v^{n}=1-ia_{{\overline {n}}|}\\&={\color {darkgreen}(\underbrace {Fr} _{=Cg}-Ci)a_{{\overline {n}}|i}}+C\qquad {\text{by }}g:={\frac {Fr}{C}}\Leftrightarrow Fr=Cg\\&=C+C(g-i)a_{{\overline {n}}|i}\\\end{aligned}}$
$\Box$
Remark.
* this formula is equivalent to $P-C=C(g-i)a_{{\overline {n}}|i}$
   * $P-C$ is the premium if $P>C\Leftrightarrow P-C>0$
   * $C-P=-(P-C)$ is the discount if $C>P\Leftrightarrow P-C<0$
   * premium and discount are both positive
* alternative form: (a step in the proof) $P=C+(Fr-Ci)a_{{\overline {n}}|i}$
Proposition. (Base amount formula) $P=G+(C-G)v^{n}.$
Proof. ${\begin{aligned}P&=\underbrace {Fr} _{G{\color {blue}i}}{\color {blue}a_{{\overline {n}}|i}}+Cv^{n}\\&={\color {darkgreen}G}(1{\color {darkgreen}-v^{n}})+{\color {darkgreen}Cv^{n}}\qquad {\text{by }}a_{{\overline {n}}|i}={\frac {1-v^{n}}{i}}\Leftrightarrow {\color {blue}ia_{{\overline {n}}|i}}=1-v^{n}\\&=G+{\color {darkgreen}(C-G)v^{n}}\end{aligned}}$
$\Box$
Proposition. (Makeham's formula) $P=K+{\frac {g}{i}}(C-K).$
Proof. ${\begin{aligned}P&=\underbrace {Fr} _{Cg}a_{{\overline {n}}|}+\underbrace {Cv^{n}} _{K}\\&=Cg\left({\frac {1-v^{n}}{i}}\right)+K\\&={\frac {g}{i}}(C-\underbrace {Cv^{n}} _{K})+K\\&=K+{\frac {g}{i}}(C-K)\end{aligned}}$
$\Box$
Remark.
* Makeham is a British actuary of the 19th century.
In practice, stocks are generally quoted as 'percent'. For example, we buy a quantity of a stock at 80% redeemable at 100% (at par), or at 105% (above par). Sometimes, the nominal value of bond is not specified. In this case, we should express our answer in percent (without the % sign), or equivalently, price per 100 nominal, e.g. price percent is 110 is equivalent to price is 110 per 100 nominal.
Example. (Bond without coupons) The price ($P$) of a 10-year zero coupon ($n=0$) bond with a par value $1000 ($F=1000$)and a yield rate ($i$) of 4% convertible semiannually (redeemable at par) is $P=\underbrace {C} _{=F}(1+4\%/2)^{-2\times 10}=(1000)(1.02)^{-20}\approx 672.971333$
Example. (Bond with coupons) The price of a 20-year bond, with a par value of $5,000 and a coupon rate of 8%, convertible semiannually, at a yield of 6%, convertible semiannually (redeemable at par) is $P=Fra_{{\overline {n}}|i}+Cv^{n}=5000(8\%/2)(a_{{\overline {40}}|6\%/2})+5000(1.03)^{-40}=200\cdot {\frac {1-(1.03)^{-40}}{0.03}}+5000(1.03)^{-40}\approx 6155.738599$ Or, on your calculator press: 2ND FV; 5000 FV; 200 PMT; 3 I/Y; 40 N; CPT PV
The first command clears the financial calculator. The second command enters 5000 as the future value (FV). The third command enters 200 as the coupon payment (PMT). The fourth command enters 3 as the interest rate (I/Y). The final command computes (CMPT) the present value (PV).
Exercise.

### Situation in which there are income tax and capital gains tax

When there is income tax, the four formulas to calculate price of a bond ($P$) are slightly modified in a similar way. Suppose the income tax rate is $t_{1}$. By definition, $Fr$ is counted as income, and $C$ is not counted as income (the gain due to the difference between $P$ and $C$ is taxed by capital gains tax instead). So, under income tax, the bond price is computed with $Fr$ multiplied by $1-t_{1}$ ($t_{1}Fr$ is the income tax paid per coupon payment). Also, we consider the present value of the tax payment to compute the present bond price. Hence, we have the following modified formulas under income tax:
The basic formula becomes $P=Fr(1-{\color {darkgreen}t_{1}})a_{{\overline {n}}|i}+\underbrace {Cv^{n}} _{K}.$ The premium/discount formula becomes $P=\underbrace {Cg} _{Fr}(1-{\color {darkgreen}t_{1}})a_{{\overline {n}}|i}+Cv^{n}=C+C(g(1-{\color {darkgreen}t_{1}})-i)a_{{\overline {n}}|i}.$
* $P-C$ is premium if $P>C\Leftrightarrow g(1-{\color {darkgreen}t_{1}})>i$
* $P-C$ is discount if $P<C\Leftrightarrow g(1-{\color {darkgreen}t_{1}})<i$ (and there is also capital gain).
   * this gives us a convenient way to check that whether there is capital gain, and we should be careful that the time period for which $g$ and $i$ is computed must be the same, so that the comparison is fair, and valid (the time period is not necessarily one year)
The base amount formula becomes $P=\underbrace {Gi} _{Fr}(1-{\color {darkgreen}t_{1}})+Cv^{n}=G(1-{\color {darkgreen}t_{1}})+(C-G(1-{\color {darkgreen}t_{1}}))v^{n}.$ The Makeham's formula becomes $P=\underbrace {Cg} _{Fr}(1-{\color {darkgreen}t_{1}})a_{{\overline {n}}|i}+Cv^{n}=K+{\frac {g(1-{\color {darkgreen}t_{1}})}{i}}(C-K)$
On the other hand, capital gains tax is a tax levied on difference between the redemption value of a stock (or other asset) and purchase price if and only if it is strictly lower than redemption value. When there is capital gains tax, say the rate is $t_{2}$, we need to subtract the purchase price by present value of $t_{2}(C-P)$ at the redemption date (the present value of the capital gain tax paid for the bond).
Example. (Bond subject to income tax, but not capital gain tax) Recall a bond from previous example: a 20-year bond, with a par value of $5,000 and a coupon rate of 8%, convertible semiannually, at a yield of 6%, convertible semiannually. Suppose there is 20% income tax rate, and 30% capital gain tax rate now.
First, we determine whether the bond is subject to capital gain tax. Since (the bond is assumed to be redeemable at par) $g(1-20\%)=r(0.8)=(8\%/2)(0.8)=0.032>0.03=i$ (we compare half-year modified coupon rate with half-year yield rate). So, there is no capital gain for this bond, and therefore it is not subject to capital gain tax.
Then, considering the income tax, the bond price becomes $P=Fr(1-20\%)a_{{\overline {n}}|i}+Cv^{n}=5000(8\%/2)(0.8)(a_{{\overline {40}}|6\%/2})+5000(1.03)^{-40}\approx 5231.147720,$ which is lower than the price in previous example, as expected (since it 'worth less' under income tax).
Example. (Bond subject to both income tax and capital gain tax) Consider a 5-year bond with redemption value $10000$, coupon rate of $2\%$ compounded quarterly, and yield rate of $3\%$ compounded semiannually. Suppose there is 30% income tax rate and 35% capital gain tax rate.
Since $g(1-30\%)=2\%(0.7)=0.014<(1.03)^{1/2}-1\approx 0.014889\approx i$, (comparing quarterly coupon rate with quarterly yield rate, and we only have quarterly coupon rate) purchasing the bond has capital gain, and thus is subject to capital gain tax.
Then, considering the income tax and also the capital gain tax, the bond price is ${\begin{aligned}&&P&=C+C(g(1-0.3)-i)a_{{\overline {n}}|i}{\color {darkgreen}-0.35(C-P)(1.03)^{-20}}\\&\Rightarrow &(1-0.35(1.03^{-20}))P&=10000+10000(0.02(0.7)-0.014889)\cdot {\frac {(1+0.014889)^{-20}-1}{0.014889}}-0.35(10000)(1.03)^{-20}\\&\Rightarrow &P&\approx 9810.476577\end{aligned}}$ If there are no capital gain tax, then the bond price is $P=10000+10000(0.02(1-0.3)-0.014889)\cdot {\frac {1-(1+0.014889)^{-20}}{0.014889}}\approx 9847.203661$
Exercise.

### Serial bond

Definition. (Serial bond) A serial bond is a bond that is redeemable by installments, i.e. there are multiple redemptions at multiple redemption dates.
Then, for serial bond, we have the following equation. $F=F_{1}+F_{2}+\cdots +F_{k}$ in which $F_{i}$ is the nominal amount that will be redeemed after $n_{i}$ years, and other notations with subscript $i$ corresponds to this nominal amount. Also, by definition, $C_{j}=RF_{j},\quad C=RF=R\sum _{j}F_{j}=\sum _{j}C_{j},$ and $K_{j}=C_{j}v^{n_{j}},\quad K=Cv^{n_{j}}=\sum _{j}C_{j}v^{n_{j}}.$ In this situation, Makeham's formula is quite useful, and ease calculation. Its usefulness is illustrated in the following.
When there are no taxes, $P_{j}=K_{j}+{\frac {g}{i}}(C_{j}-K_{j})\Leftrightarrow \sum _{j}P_{j}=\sum _{j}K_{j}+{\frac {g}{i}}\left(\sum _{j}(C_{j})-\sum _{j}(K_{j})\right)=K+{\frac {g}{i}}(C-K)=P.$
Example. (Serial bond without taxes) Consider a 10-year serial bond of nominal value $10000$ that is redeemable at $110\%$ (i.e. redeemable above par) by 5 equal installments at the end of 2nd, 4th, 6th, 8th, and 10th year, and with annual coupon rate $3\%$ . An investor, who is not liable to both income tax and capital gains tax, purchases this serial bond at the price $P$ such that he obtains an effective semiannual yield rate of $10\%$. Compute $P$.
Solution: We will use Makeham's formula. Based on the given information, ${\begin{aligned}C&=10000(110\%)=11000\\K&=2000(1.1)(v^{2}+v^{4}+\cdots +v^{10})\qquad v{\text{ is computed at }}1.1^{2}-1=21\%\\&=2000(1.1)(1.21^{-2}+\cdots +1.21^{-10})\\&=2000(1.1)((1.21^{2})^{-1}+\cdots +(1.21^{2})^{-5})\\&=2000(1.1)(a_{{\overline {5}}|1.21^{2}-1})\\&=4035.733718\\g&={\frac {10000(0.03)}{C}}={\frac {300}{11000}}\approx 0.0272727\qquad {\text{(annual)}}\end{aligned}}$ Thus, by Makeham's formula, $P\approx 4035.733718+{\frac {0.0272727}{0.21}}(11000-4035.733718)\approx 4940.182980.$
Let $P'$ be the price of serial bond when there is income tax. When there is income tax, say the rate is $t_{1}$, $P'_{j}=K_{j}+{\frac {g(1-t_{1})}{i}}(C_{j}-K_{j})\Leftrightarrow \sum _{j}P'_{j}=\sum _{j}K_{j}+{\frac {g(1-t_{1})}{i}}\left(\sum _{j}(C_{j})-\sum _{j}(K_{j})\right)=K+{\frac {g(1-t_{1})}{i}}(C-K)=P'.$
Example. (Serial bond under income tax only) Recall the serial bond in previous example: a 10-year serial bond of nominal value $10000$ that is redeemable at $110\%$ (redeemable above par) by 5 equal installments at the end of 2nd, 4th, 6th, 8th, and 10th year, and with annual coupon rate $3\%$.
Now, suppose that another investor, who is liable to income tax of $15\%$ only, purchases this serial bond at the price $P$ such that he obtains an effective semiannual yield rate of $10\%$. Compute $P$.
Solution: Based on the results in previous example, $P\approx 4035.733718+{\frac {0.0272727(1-0.15)}{0.21}}(11000-4035.733718)\approx 4804.515591$
Let $P''$ be the price of serial bond when there are both income and capital gains tax. If the bond is sold at discount (and there is income tax), i.e. $g(1-t_{1})<i$, there is capital gain. Assume that the capital gain at time $n_{j}$ is $C_{j}-\left({\frac {F_{j}}{F}}\right)P'',$ ($F_{j}/F$ is the portion of bond, in terms of nominal value, redeemed, corresponding to the redemption value $C_{j}$) and thus the total present value of the capital gains tax (say at rate $t_{2}$) payable is $\sum _{j=1}^{k}\left(t_{2}\left(\underbrace {RF_{j}} _{C_{j}}-{\frac {F_{j}}{F}}P''\right)v^{n_{j}}\right)=t_{2}\left(1-{\frac {P''}{FR}}\right)\sum _{j=1}^{k}RF_{j}v^{n_{j}}=t_{2}\left(1-{\frac {P''}{C}}\right)K=t_{2}\left(C-P''\right)v^{n},$ which is same as how the capital gain tax for normal bond with single redemption is computed.
Example. (Serial bond under both income tax and capital gain tax) Recall the serial bond in previous example: a 10-year serial bond of nominal value $10000$ that is redeemable at $110\%$ (redeemable above par) by 5 equal installments at the end of 2nd, 4th, 6th, 8th, and 10th year, and with annual coupon rate $3\%$.
Now, suppose that another investor, who is liable to income tax of $15\%$ and capital gain tax of $20\%$, purchases this serial bond at the price $P$ such that he obtains an effective semiannual yield rate of $10\%$. Compute $P$.
Solution: Based on the results in previous example, since $g(1-0.1)\approx 0.0245454<0.21=i$, the investor has capital gain, and thus investor is liable to capital gain tax. So, $P\approx 4035.733718+{\frac {0.0272727(1-0.15)}{0.21}}(11000-4035.733718)-(1-0.2)(11000-P)(1.21)^{-10}\Rightarrow P\approx 3456.373106.$
Exercise.

## Book value

From the previous section, we can see that $P$ of a bond is generally different from $C$. This implies that the value of the bond is adjusted from the purchasing price to the redemption value of the bond, during the period lasted by the bond.
The reason for this adjustment is that there are coupon payments, and also there are change in value caused by the interest rate. In the previous section, we have determined that the initial value ($P$) of the bond and also the ending value ($C$) of the bond. In this section, we will also determine the value between the start and the end, which is adjusted by the coupon payment and interest rate, and we call these adjusted values book values.
Definition. (Book value) Book value of a bond at time $k$ is the value of the bond at time $k$, adjusted by the coupon payments and interest rates.
Remark.
* in this book, we will mainly discuss the book value of a bond at a time where there is coupon payment, the book value between coupon payments will not be covered much
* this is analogous to outstanding balance of the loan (it is adjusted by the principal repaid and interest rates)
Since book value measures the value of a bond, we use a formula for computing it which is similar to the basic formula (which measures the value of the bond, to determine a fair price), as follows:
Proposition. (Basic formula of book value) Book value at time $k$ in which $k$ is a nonnegative integer is $B_{k}=Fra_{\overline {n-k|}}+Cv^{n-k},$ assuming there are $n-k$ future coupon payments.
Proof. It follows from the fact that book value at time $k$ is measuring the adjusted value at time $k$.
$\Box$
Remark.
* this formula is also similar to the formula for the prospective method for computing the outstanding balance of loan.
* we can see from this formula that the book value at time $0$ is the purchase price of the bond (when $k=0$, this formula becomes $Fra_{{\overline {n}}|i}+Cv^{n}$, which is the same as the basic formula
Then, we can have the following recursive formula for computing book value, using this basic formula.
Proposition. (Recursive formula of book value) Book value at time $k$ in which $k$ is a nonnegative integer is computed by $B_{k+1}=B_{k}(1+i)-Fr.$
Proof. First, we claim that $a_{{\overline {n-k}}|}=v+va_{{\overline {n-k-1}}|},$ which is true since $a_{{\overline {n-k}}|}={\frac {1-v^{n-k}}{i}}={\frac {1-v+v-v^{n-k}}{i}}={\frac {(1{\cancel {+i}}-1)/(1+i)}{\cancel {i}}}+v\cdot {\frac {1-v^{n-k-1}}{i}}=v+va_{{\overline {n-k-1}}|}.$ Then, ${\begin{aligned}&&B_{k}&=Fra_{\overline {n-k}}+Cv^{n-k}\\&&&=Fr(v+va_{{\overline {n-k-1}}|})+Cv^{n-k}\\&&&=v(Fr+\underbrace {Fra_{{\overline {n-(k+1}}|}+Cv^{n-(k+1)}} _{=B_{k+1}})\\&\Leftrightarrow &B_{k}(1+i)&=Fr+B_{k+1}\\\end{aligned}}$
$\Box$
Remark.
* it follows that $\Delta B_{k}:=B_{k+1}-B_{k}=iB_{k}-Fr$
* (terminology) if $\Delta B_{k}>0\Leftrightarrow B_{k+1}>B_{k}$, then there is writing up or accumulation of discount (recall that $C-P$ is discount if $P<C$)
   * actually, there is accumulation of discount if and only if $P<C$
* (terminology) if $\Delta B_{k}<0\Leftrightarrow B_{k+1}<B_{k}$, then there is writing down or accumulation of premium (recall that $P-C$ is premium if $P>C$)
   * actually, there is accumulation of premium if and only if $P>C$
Example. Consider a bond with $P=1000,F=1500,C=908.65,r=5\%,i=3\%$. Then, since $B_{0}=P=1000,B_{1}=B_{0}(1+i)-Fr=1000(1.03)-1500(0.05)=955,B_{2}=955(1.03)-1500(0.05)=908.65=C$, we know that $n=2$. Also, since $\Delta B_{1}<0$ and $\Delta B_{2}<0$, there is writing down, or accumulation of premium at time $1$ and $2$.
Alternatively, we can use the basic formula to determine $n$: ${\begin{aligned}&&1000&=P=B_{0}=1500(0.05)a_{{\overline {n}}|}+908.65(1.03)^{-n}\\&\Rightarrow &1000&=75(1-1.03^{-n})/0.03+908.65(1.03)^{-n}\\&\Rightarrow &1591.35(1.03)^{-n}&=1500\\&\Rightarrow &-n&={\frac {\ln(1500/1591.35)}{\ln 1.03}}\\&\Rightarrow &n&=2.\end{aligned}}$
Exercise.

### Bond amortization schedule

Since the nature of a bond is quite similar to that of a loan, we can construct a bond amortization schedule, which is similar to the loan amortization schedule.
Recall that in the loan amortization schedule, the columns correspond to payment (or installment), interest paid, principal repaid, and outstanding balance. So, to construct a similar amortization schedule, we need to determine which term of bonds correspond to these terms.
* for the outstanding balance, we have mentioned that a corresponding term of bond is book value ($B_{k}$)
* for the installment, we have mentioned that a corresponding term of bond is coupon payment ($Fr$)
* for the principal repaid, a similar term is $\Delta B_{k}:=B_{k+1}-B_{k}$, but since we are constructing amortization schedule, the book value is expected to be decreasing (premium bond), and so $\Delta B_{k}<0$, and we often do not want to deal with the negative sign, so we define an alternative term as follows:
Definition. (Principal adjustment) The principal adjustment is the decrease (or amortization) of book value occured at the end of coupon payment period. That is, the principal adjustment at the end of the $k$th coupon payment period, denoted by $P_{k}$, is $P_{k}=B_{k-1}-B_{k}=-\Delta B_{k-1}.$
Then, for the interest paid, we can determine it in a similar way compared with that in loan (multiplying the outstanding balance from the previous end of period by interest rate), namely multiplying the book value from the previous end of period by interest rate, i.e.
Proposition. (Interest paid on the book value) The interest paid on the book value at the beginning of $k$th period is $I_{k}=iB_{k-1}.$
Proof. It follows from the definition of interest.
$\Box$
Then, we have an similar formula which links $P_{k}$ and $I_{k}$, compared to the situation of loan.
Proposition. (Principal adjustment plus interest paid equals coupon) The $k$th coupon ($k$ is a number such that $k$th coupon exists) is $k{\text{th coupon}}=P_{k}+I_{k}.$
Proof. Since $P_{k}=-\Delta B_{k-1}=Fr-iB_{k-1}$ by recursive formula of book value, and $I_{k}=iB_{k-1}$, $P_{k}+I_{k}=Fr{\cancel {-iB_{k-1}+iB_{k-1}}}=Fr=k{\text{th coupon}},$ since each coupon is of the same amount $Fr$.
$\Box$
Now, we proceed to construct the amortization schedule.
To increase the usefulness of the amortization schedule, we would like to determine a formula for the book value, principal adjustment, etc. at different period.
To do this, we start from $B_{0}$. Recall the basic formula of book value. Since it is in the same form compared to the basic formula of bond price, we have an analogous premium/discount formula for book value, as follows:
Proposition. (Premium/discount formula of book value) Book value at time $k$ in which $k$ is a nonnegative integer is $B_{k}=C+C(g-i)a_{{\overline {n-k}}|i}$
Proof. The proof is identical to the proof for premium/discount formula of bond price, except that $n$ is replaced by $n-k$.
$\Box$
By definition, the coupon payment is $Cg=Fr$.
Then, using these, we can determine $I_{k}$ and $P_{k}$ as follows:
Corollary. (Determining interest paid and principal adjustment of each coupon) The interest paid in the $k$th coupon, denoted by $I_{k}$, is $iB_{k-1}=i\left(C+C(g-i)a_{{\overline {n-k+1}}|i}\right),$ and the principal adjustment (or amount of premium/discount (depending on whether the bond is premium or discount bond) accumulated) in the $k$th coupon, denoted by $P_{k}$ is $C(g-i)v^{n-k+1}.$
Proof. For the formula for $I_{k}$, by the proposition about formula of $I_{k}$ and premium/discount formula of book value, we have $I_{k}=iB_{k-1}=i\left(C+C(g-i)a_{{\overline {n-k+1}}|i}\right).$
For the formula of $P_{k}$, by the proposition about relationship between $P_{k}$ and $I_{k}$, we have ${\begin{aligned}P_{k}&={\text{coupon}}-I_{k}=Cg-i\left(C+C(g-i)a_{{\overline {n-k+1}}|i}\right)\\&=Cg-Ci-Ci(g-i)a_{{\overline {n-k+1}}|i}\\&=(g-i)\left(C-Cia_{{\overline {n-k+1}}|i}\right)\\&=(g-i)\left(C-C{\cancel {i}}\cdot {\frac {1-v^{n-k+1}}{\cancel {i}}}\right)\\&=C(g-i)\left(1-(1-v^{n-k+1})\right)\\&=C(g-i)v^{n-k+1}.\end{aligned}}$
$\Box$
After that, we can construct the amortization schedule as follows:
Bond amortization schedule :
| $k$ | Coupon | $I_{k}=iB_{k-1}$ | $P_{k}={\text{coupon}}-I_{k}$ | $B_{k}$ |
| $0$ | N/A | N/A | N/A | $C+C(g-i)a_{{\overline {n}}|i}$ |
| $1$ | $Cg$ | $iB_{0}=i(C+C(g-i))a_{{\overline {n}}|i}$ | $Cg-I_{1}=C(g-i)v^{n}$ | $C+C(g-i)a_{{\overline {n-1}}|i}$ |
| $2$ | $Cg$ | $iB_{1}=i(C+C(g-i))a_{{\overline {n-1}}|i}$ | $Cg-I_{2}=C(g-i)v^{n-1}$ | $C+C(g-i)a_{{\overline {n-2}}|i}$ |
| $\vdots$ |
| $m$ | $Cg$ | $iB_{m-1}=i(C+C(g-i))a_{{\overline {n-m+1}}|i}$ | $Cg-I_{m}=C(g-i)v^{n-m+1}$ | $C+C(g-i)a_{{\overline {n-m}}|i}$ |
| $\vdots$ |
| $n-1$ | $Cg$ | $iB_{n-1}=i(C+C(g-i))a_{{\overline {2}}|i}$ | $Cg-I_{n-1}=C(g-i)v^{2}$ | $C+C(g-i)a_{{\overline {1}}|i}$ |
| $n$ | $Cg$ | $iB_{n}=i(C+C(g-i))a_{{\overline {1}}|i}$ | $Cg-I_{n}=C(g-i)v$ | $C+\underbrace {C(g-i)a_{{\overline {0}}|i}} _{0}$ |
| Total | $nCg$ | ${\text{total coupon}}-{\text{total principal adjustment}}=nCg-C(g-i)a_{{\overline {n}}|i}$ | $C(g-i)a_{{\overline {n}}|i}$ |
Remark.
* we can see that when a bond is bought at premium (i.e. $P>C\Rightarrow C(g-i)a_{{\overline {n}}|i}>0$), the book value will be gradually adjusted downward since each principal adjustment is positive, i.e. there is decrease (amortization) in book value from one period to another period
* we can see that when a bond is bought at discount (i.e. $P<C\Rightarrow C(g-i)a_{{\overline {n}}|i}<0$), the book value will be gradually adjusted upward since each principal adjustment is negative, i.e. there is increase in book value from one period to another period
Example. Consider a 5-year bond with $3\%$ semiannual coupon rate, and $F=150$. It is bought at a premium with yield rate $10\%$ compounded semiannually. Suppose $C=100$.
Compute the sum of principal adjustments at the end of 1st, 2nd and 3rd half-years, i.e. $P_{1}+P_{2}+P_{3}$. Hence, compute the book value at half-year 3 using $P_{1},P_{2},P_{3}$.
Solution: The effective semiannual yield rate $i=5\%$. Also, 5 years $\Rightarrow$ 10 semiannual coupons, and each semiannual coupon is of $Fr=Cg=150(0.03)=4.5\Rightarrow g=4.5/100=0.045$. So, ${\begin{aligned}P_{1}+P_{2}+P_{3}&=100(0.045-0.05)(1.05)^{-10}+100(0.045-0.05)(1.05)^{-9}+100(0.045-0.05)(1.05)^{-8}&\approx -0.96768.\end{aligned}}$ Thus, $B_{3}=B_{0}-P_{1}-P_{2}-P_{3}\approx 100+100(0.045-0.05)a_{{\overline {10}}|0.05}-(-0.96768)\approx 97.10681.$
Example. (Bond amortization using indirect information) Consider a 10-year bond with annual coupons. It is bought at a premium to yield $10\%$ annually. It is given that the accumulation of premium (i.e. amount written down) in the last coupon is $10$. Compute the accumulation of premium from 2nd year to 9th year, i.e. $P_{2}+\cdots +P_{9}$.
Solution: There are 10 coupons. Since the amount written down in the last (i.e. 10th) coupon is $10$, $C(g-i)v_{0.1}=10\Rightarrow C(g-i)=10(1.1)=11$ It is more convenient to compute the total accumulation of premium of 1st year to 10th year and then subtract the unwanted accumulations of premium: $P_{2}+\cdots +P_{9}=P_{1}+\cdots +P_{10}-P_{1}-P_{10}=C(g-i)a_{{\overline {10}}|0.1}-C(g-i)v_{0.1}^{10}-10=11a_{{\overline {10}}|0.1}-11(1.1)^{-10}-10\approx 53.349264.$
Exercise.
We will introduce a very simple way for bond amortization: straight line method. In this method (assuming there are $n$ coupons paid), the principal adjustment for each coupon is constant, namely ${\frac {P-C}{n}}$ which is positive for premium bonds (bonds for which $P>C$), and is negative for discount bonds (bonds for which $P<C$). The interest earned in each coupon is also constant, namely $Fr-{\frac {P-C}{n}}$. So, essentially, the book value of the bond changes in a straight line (upward sloping for discount bonds, downward sloping for premium bonds) in this method, and hence the name 'straight line method'.
Example. Consider a 8-year bond with $P=1000,C=1300$, with $g=10\%$. It is a discount bond, and using straight line method, the principal adjustment for each coupon is ${\frac {1000-1300}{8}}=-37.5$, and the interest paid for each coupon is $1300(0.1)-(-37.5)=167.5$.
Remark.
* it gives a very rough number for the principal adjustment and interest paid, and the numbers obtained are often not meaningful, and thus it is rarely used
Treasury coupon securities are quoted on a dollar price basis in price units of ${\frac {1}{32}}$ of $1\%$ of par, with par taken to be 100.
Example.
* a quote of $94{\text{-}}16$ means a price of $94$ and ${\frac {16}{32}}$, i.e. $94.5$
   * when there is a '$+$' sign placed after the quotation, a half unit, i.e. ${\frac {1}{64}}$ is added, e.g. $94{\text{-}}16+$ means a price of $94$ and ${\frac {33}{64}}$, i.e. $94.515625$
Remark. Using this way of quotation, less space is occupied.
An investor may purchase a Treasury coupon bond between coupon payments. Then, we cannot use the previously mentioned way to determine the purchasing price, since the time is located between two coupon payments, and we have not discussed how the price should be calculated in this context. There are different ways to deal with this, and we will discuss the way for Treasury coupon bond.
For Treasury coupon bond, the investor needs to compensate the seller of the bond for the coupon interest earned from the time of the last coupon payment to the settlement date (i.e. the date at which the transaction is done) of the bond, and this amount is called accrued interest, computed by ${\text{coupon}}\cdot {\frac {\text{no. of days from last coupon payment to settlement date}}{\text{no. of days in coupon period}}}.$ Thus, the total amount that the buyer of the bond pays the seller equals the price agreed upon the buyer and the seller (it may be the 'fair price' determined previously, or may be not) plus the accrued interest.
Example. Suppose coupons of a Treasury coupon bond are paid annually, and the amount of each coupon is $100$. Assuming each year has 360 days, and each month has 30 days, if the buyer purchase the bond from the seller at the price of $2000$, and the settlement date is on March 1, then the total price that the buyer need to pay is $2000+100\cdot {\frac {2\cdot 30}{360}}\approx 2016.667.$

## Callable bond

Definition. (Callable bond) A callable bond is a bond for which the borrower (or issuer) has an option to redeem prior to its maturity date.
Remark.
* we say that a borrower call a bond when he redeems it prior to its maturity date, and hence the name 'callable bond'
* 'call' has similar meaning in some of other contexts, e.g. 'call' a loan by bank
Illustration of callable bond:
Because of the callable nature of this kind of bond, the term of the bond is uncertain. So, there is problem in computing prices, yield rates, etc.
To solve this, we assume the worst-case scenario to the investor. That is, the borrower will choose the option such that the investor has most disadvantage, as follows:
* if the redemption value ($C$) on all redemption dates are equal, then if $i<g\Leftrightarrow P>C$, then assume that the redemption date will be the earliest possible date, otherwise (i.e. $i>g\Leftrightarrow P<C$), assume that the redemption date will be the latest possible date
   * this is because if $i<g\Leftrightarrow P>C$, the bond was bought at a premium, and thus there will be a loss at redemption, so the most unfavourable condition to the investor occurs when the loss happens at the earliest time
   * if $i>g\Leftrightarrow P<C$, the bond was bought at a discount, and thus there will be a gain at redemption, so the most unfavourable condition to the investor occurs when the gain happens at the latest time
* if the redemption value ($C$) on all the redemption dates including the maturity date are not equal, then we need to compute the bond price at different possible redemption dates to check which is the lowest , and thus is most unfavourable to the investor
In particular, it is common for a callable bond to have redemption values that decrease as the term of the bond increases, i.e. the later the redemption, the lower the redemption value, and if the bond is not called, the bond is redeemed at the redemption value. We have a special name for the difference between the redemption value (through call) and the par value:
Definition. (Call premium) The call premium is the excess of the redemption value when calling the bond over the par value.
Example. (Callable premium bond) An investor purchases a $100$ par-value 5-year bond with $5\%$ annual coupon rate, at the bond price $P$. The bond is callable at $90$ (which equals its redemption value) any coupon payment date from the end of 3rd year to the end of 5th year. Given that the annual interest rate is $10\%$, compute $P$.
Solution: since $g={\frac {100(0.05)}{90}}\approx 0.05555<i=0.1$, the bond is a discount bond, and so we assume that the redemption date will be the latest possible date, i.e. at the end of 5th year.
So, $P=Fra_{{\overline {5}}|0.1}+Cv_{0.1}^{5}=5a_{{\overline {5}}|0.1}+90(1.1)^{-5}\approx 74.836849.$
Remark. If the redemption date is the earliest possible date instead, i.e. at the end of 3rd year, then $P=5a_{{\overline {3}}|0.1}+90(1.1)^{-3}\approx 80.05259.$
Example. (Callable discount bond with callable premium) An investor purchases a $1000$ par-value 10-year bond with $10\%$ semiannual coupon rate, at the bond price $P$. The bond is callable at $1100$ from any coupon payment date from the 10th coupon to 15th coupon, and at $1050$ from the 16th coupon to 20th coupon. Given that the annual interest rate is $10.25\%$, compute $P$.
Solution: The semiannual interest rate is $i=1.1025^{1/2}-1=0.05$. Since the redemption values on different redemption dates are different, we need to compute the bond price at different redemption date to check that which is the lowest.
When $10\leq n\leq 15$ ($n$ means the $n$th coupon payment date), the bond price is $P=1100+1100(1000(0.1)/1100-0.05)a_{{\overline {n}}|0.05}=1100+45a_{{\overline {n}}|0.05}$ by premium/discount formula. When $16\leq n\leq 20$, $P=1050+1050(1000(0.1)/1050-0.05)a_{{\overline {n}}|0.05}=1050+47.5a_{{\overline {n}}|0.05}$ by premium/discount formula.
Since the larger the $n$, the larger the value of $a_{{\overline {2n}}|0.05}$, the price is the lowest at $n=10$ when $10\leq n\leq 15$, and the price is the lowest at $n=16$ when $16\leq n\leq 20$.
It remains to compare these two prices to determine which is the lowest price when $10\leq n\leq 20$.
* when $n=10$, $P=1100+45a_{{\overline {10}}|0.05}\approx 1447.478$
* when $n=16$, $P=1050+47.5a_{{\overline {16}}|0.05}\approx 1564.794$
Thus, the lowest price occurs at $n=10$, and is $1447.478$.
Exercise.

## Optional topic

Inflation can be measured with reference to an index representing the cost of some goods and services. One of common index is Consumer Price Index (CPI).
Real return rate, as opposed to the internal rate of return, or yield rate, takes into account the changes in the value of money, measured by the CPI or another suitable index.
If we let $i$ be the internal rate of return, $r$ be the real rate of return, and $j$ be the constant inflation rate, then we have $(1+i)=(1+r)(1+j)$ Let $j$ the inflation rate from year $A$ to year $B$. Then, ${\text{CPI at year }}B={\text{CPI at year }}A(1+j)\Rightarrow 1+j={\frac {{\text{CPI at year }}B}{{\text{CPI at year }}A}}$ Thus, we have $(1+i)=(1+r)\cdot {\frac {{\text{CPI at year }}B}{{\text{CPI at year }}A}}$ for rates of return from year $A$ to year $B$.
Every computation about price, rate of return, etc., we cover in this book can technically be carried out using the unit of real purchasing power rather than the unit of currencies, given that the index used for such computations. Yet, we will not do this in this book, and the following are some examples of how we can compute different things in real terms.
Example. (Real yield rate of bond) Suppose there is a 2-year bond with par value $100$, and has a annual coupon rate of $10\%$, and is redeemable at par.
An investor purchases the bond at the beginning of the 2020 at a price of $100$. Given that the CPI in 2020, 2021, 2022 are $230,250,260$ respectively, compute the real yield rate $r$.
${\begin{aligned}&&100&=100(0.1)\left(\left((1+r)\cdot {\frac {250}{230}}\right)^{-1}+\left((1+r)\cdot {\frac {250}{230}}(1+r){\frac {260}{250}}\right)^{-1}\right)+100\left((1+r)\cdot {\frac {260}{250}}\right)^{-1}\left((1+r)\cdot {\frac {250}{230}}\right)^{-1}\\&\Rightarrow &1&=0.1\left({\frac {230}{250}}(1+r)^{-1}+{\frac {230}{260}}(1+r)^{-2}\right)+{\frac {230}{260}}(1+r)^{-2}\\&\Rightarrow &(1+r)^{-1}&\approx 0.9676{\text{ or }}(1+r)^{-1}\approx -1.0621{\text{ (rejected)}}\\&\Rightarrow &r&\approx 0.033485.\end{aligned}}$
For some securities, they are index-linked, e.g. index-linked coupon gives varying coupons, depending on the CPI.
Example. (Index-linked bond) Several index-linked bonds of 200 nominal each was issued on June 1, 2018, and were redeemed at par on June 1, 2020. Each bond had a nominal coupon rate of $5\%$ per annum, payable semiannually in arrears (i.e. at the end of each half-year period). The actual coupon and redemption value payments were indexed according to the change in CPI from 1 year before the bond issue date and 6 months before the coupon or redemption payment dates (i.e. the coupon and redemption are multiplied by the latter and divided by the former). Some CPIs are given in the following:
* Jun, 2017: 95
* Dec, 2017: 99
* Jun, 2018: 100
* Dec, 2018: 105
* Jun, 2019: 107
* Dec, 2019: 103
* Jun, 2020: 110
* Dec, 2020: 106
* Jun, 2021: 112
* Dec, 2021: 115
Compute the purchase price $P$ of $10$ bonds to obtain a $5\%$ semiannual real yield, given that the bonds are held until redemption.
Solution: $P=200(10)(0.05)\left({\frac {100}{95}}\cdot {\frac {100}{105}}(1.05)^{-1}+{\frac {105}{95}}\cdot {\frac {100}{107}}(1.05)^{-2}+{\frac {107}{95}}\cdot {\frac {100}{103}}(1.05)^{-3}+{\frac {103}{95}}\cdot {\frac {100}{110}}(1.05)^{-4}\right)+200(10){\frac {103}{95}}\cdot {\frac {100}{110}}(1.05)^{-4}\approx 1986.507$
Exercise.
Remark. The fractions with $95$ as denominator are the index-linked adjustments.

# General Cash Flows and Portfolios
## Learning objectives

The Candidate will understand key concepts concerning yield curves, rates of return, and measures of duration and convexity, and how to perform related calculations.

## Learning outcomes

The Candidate will be able to:
* Define and recognize the definitions of the following terms: yield rate/rate of return, dollar-weighted rate of return, time-weighted rate of return, current value, duration (Macaulay and modified), convexity (Macaulay and modified), portfolio, spot rate, forward rate, yield curve, stock price, stock dividend.
* Calculate:
   * The dollar-weighted and time-weighted rate of return.
   * The duration and convexity of a set of cash flows.
   * Either Macaulay or modified duration given the other.
   * The approximate change in present value due to a change in interest rate,
      * The approximate change in present value due to a change in interest rate,
      * Using 1st-order approximation based on Macaulay duration.
   * The price of a stock using the dividend discount model.
   * The present value of a set of cash flows, using a yield curve developed from forward and spot rates.

## Discounted cash flow analysis

Definition. (Net cash flow) Net cash flow at time $t$ equals cash inflow at time $t$ minus cash outflow at time $t$.
Remark. We use $C_{t}$ to denote net cash flow at time $t$.
Then, we can express a series of cash flows as follows:
| Net cash flow | $C_{0}$ | $C_{1}$ | $C_{2}$ | $\cdots$ | $C_{n}$ |
| Time | $0$ | $1$ | $2$ | $\cdots$ | $n$ |
After defining net cash flows, we can also define net present value.
Definition. (Net present value) The net present value (NPV), denoted by $P(i)$, is sum of present value of every net cash flow, i.e. $P(i)=\sum _{t}{C_{t}v^{t}}.$
Remark.
* Net present value is a function of $i$.
* An investment is profitable if and only if its net present value is positive, i.e. the series of cash flows in the investment is equivalent to one cash inflow at time 0.
* The present value of cash flow can be interpreted as discounted cash flow: cash flow at time $t$ is discounted by the factor $v^{t}$.
An example to illustrate how we can use net present value to determine profitability of investment is in the following.
Apart from determining whether the investment is profitable, we sometimes also want to know how profitable the investment is. A natural way to measure how profitable the investment is using its rate of return which can be measured by internal rate of return.
Definition. (Internal rate of return) The internal rate of return (IRR) of an investment is the effective interest rate per measurement period at which the net present value of the investment equals zero.
Remark.
* It is also called yield rate.
* Equivalently, the internal rate of return is the interest rate at which the initial investment, i.e. cash flow at time zero, equals the net present value of the future net cash flows. Therefore, we may split the initial investment into several portions, and each of them will accumulate to the amount of each of the future cash flow correspondingly at the internal rate of return, at the corresponding time of the cash flow. (See the exercise below.)
* Internal rate of return is also the interest rate at which the total present value of cost (i.e. cash outflow(s)) equals total present value of benefit (i.e. cash inflow(s)).
* It is internal in the sense that the calculation excludes external factors, e.g. financial risk, inflation, etc.
* In general, it is very difficult to calculate internal rate of return directly by hand. Therefore, financial calculator, e.g. BA II Plus, is usually used to calculate it. We may also use linear interpolation in the interest table to approximate it.
* It may not exist and may not be unique. There can be no internal rate of return, or multiple internal rates of return, although it is uncommon in practice.
Let's illustrate the split of initial investment mentioned in the remark in the following exercise.
Consider the following cash flows for an investment.
| Time | Net cash Flow |
| 0 | -500 |
| 1 | 100 |
| 2 | 0 |
| 3 | 300 |
| 4 | 0 |
| 5 | 500 |

### BA II Plus calculator usage

The TI BA II plus has a cash flow function to deal with uneven cash flow streams. Here is an example of cash flows.
| Time | Net cash Flow |
| 0 | -500 |
| 1 | 100 |
| 2 | 0 |
| 3 | 300 |
| 4 | 0 |
| 5 | 500 |
kk
You can use the following command to calculate the net present value at effective interest rate of 10% per measurement period.
    2ND CE|C CF 500 +|- ENTER ↓ 100 ENTER ↓ ↓ 0 ENTER ↓ ↓ 300 ENTER ↓ ↓ 0 ENTER ↓ ↓ 500 ENTER NPV 10 ENTER ↓ CPT
You should get NPV=126.76. The following are some explanations on the command.
* 2ND CE|C will clear any previous work saved in the calculator.
* CF Enters into cash flow mode.
* Next, you will see CF0= for which you need to enter the first cash flow of -500 which is made at time zero. You should enter negative value for cash outflow. This is the convention used in BAII Plus calculator.
* After entering -500 and then pressing ↓, you will see C01, for which you need to enter the second cash flow of 100 which is made at time one.
* After pressing ↓ again, you will see F1, for which you need to enter the frequency of the cash flow, i.e. how many times this cash flow is repeated. The default value is 1, which is correct in this case. We will later see an exercise in which the frequency is higher than 1.
* Then, we enter the remaining cash flows. We need to enter all zero cash flows to ensure that the later nonzero cash flow is entered at a correct time.
* After entering the last cash flow of 500, press NPV instead of ↓. Then you will see I, and need to enter the effective interest rate (only the number in the percentage expression). You should enter 10 in this case.
* After pressing ↓ and CPT, you should see "NPV=126.76" which is the net present value.
To calculate internal rate of return, press IRR and then CPT after entering the last cash flow. You should get IRR=17.05, i.e. the internal rate of return is 17.05%. The calculator will only display the number before the % sign of percentage expression. (This result is used in exercise about spliting initial investment.)
Here is another example of cash flows.
| Time | Net cash Flow |
| 0 | -500 |
| 1 | 200 |
| 2 | 200 |
| 3 | 200 |
| 4 | 200 |
| 5 | 200 |
Press 2ND CE|C CF 500 +|- ENTER ↓ 200 ENTER ↓ 5 ENTER NPV 10 ENTER ↓ CPT to calculate net present value at effective interest rate of 10% per measurement period. To calculate internal rate of return, press IRR CPT after 5 ENTER instead. You should get NPV=258.16 for net present value, and IRR=28.65 for internal rate of return.
After pressing 200 ENTER ↓, we will see F01 for which we need to enter the frequency of this cash flow. Because this cash flow repeats five times, we enter 5.
In the later commands, 2ND CE|C will be omitted, and it is assumed that you always press this sequence of commands before entering the cash flows.
Let's practice the calculator usage in the following exercise.

### Idealized practical situation

In this subsection, we assume that we can borrow and lend money at a fixed interest rate freely. Practically, although we can borrow and lend money, we may not be allowed to borrow and lend money freely, and we typically cannot borrow and lend money at a fixed interest rate for a long time, because interest rate will change, and interest rate for borrowing and lending are usually different (and the former is usually higher than latter).
Under this assumption we can calculate 'net accumulated value', which is analogous to net present value. This term is nonstandard and rarely used. We can connect the net cash flows from one project with another project in which interest is payable (when net cash flow is positive, we can put the money in, just like lending money) or credited (when net cash flow is negative, we need to take the money out, just like borrowing money) at a fixed rate $i$. Let's also assume we can put money in and take money out freely, in arbitrary amount (and of course they are subject to the interest payable or credited). When the another project ends (say at time $T$), the accumulated value will be
    $\sum _{t\leq T}C_{t}(1+i)^{T-t}.$
in which $C_{t}$ is the net cash flow from the first project at time $t$. If the first project ends before the another project ends, then we can remove the '$\leq T$' because $t$ is always smaller than or equal to $T$, i.e. the accumulated value will be
    $\sum _{t}C_{t}(1+i)^{T-t}.$
If the another project continues indefinitely, this value is undefined (because it tends to infinity). However, for a project that continues indefinitely, in which there are net cash flows, its net present value may be defined, just like perpetuity.
In this subsection, let's also assume that the internal rate of return exists and $P(i)$ is positive at the interest rate that is strictly smaller than the internal rate of return, and negative at the interest rate that is strictly greater than the internal rate of return in this subsection. Actually, this is usually the case in practice, unless there are multiple internal rates of return. Then, we have the following proposition.
Proposition. (Necessary and sufficient condition for a project to be profitable) Assume that the internal rate of return exists and $P(i)$ is positive at the interest rate that is strictly smaller than the internal rate of return, and negative at the interest rate that is strictly greater than the internal rate of return. A project is profitable if and only if the effective interest rate per measurement period is strictly smaller than internal rate of return.
Proof. It follows from the assumption that $P(i)$ is positive at the interest rate that is strictly smaller than the internal rate of return, and the fact that a project is profitable if and only if $P(i)$ is positive.
$\Box$
Remark.
* An equivalent condition is that the internal rate of return is strictly greater than the interest rate at which the investor may lend
or borrow money.
   * Indeed, this idea is also mentioned in the explanation of answer in the exercise about calculating internal rate of return.

### Comparison of two investment projects

Sometimes, we want to compare two investment projects to decide which one has a higher profitability, and then we invest in the project with higher profitability. Naturally, we may think that the project with higher internal rate of return should always have a higher profitability. However, this is not always the case.
To decide the profitability of each project, we should compare the profit at time $T$, which is the date at which the later of the two projects ends, of each project. Equivalently, this is the net present value calculated at the rate of interest $i_{1}$ at which the investor may lend or borrow money. If the net present value of project A, $P_{A}(i_{1})$, is strictly greater than that of project B, $P_{B}(i_{1})$, project A is more profitable than project B.
Because the internal rate of return of project B ($i_{B}$) is strictly higher than project A ($i_{A}$) may not imply $P_{B}(i_{1})>P_{A}(i_{1})$ (whether the inequality holds depends on value of $i_{1}$), project with strictly higher internal rate of return does not necessarily have a strictly higher profitability.

### Different interest rates for lending and borrowing

In the subsection of idealized practical situation, we assume that the investor may borrow or lend money at the same rate of interest. However, in practice, the investor may need to pay a higher interest rate on borrowing than the rate earned on money invested, e.g. interest rate earned in deposit account. (When we deposit money, the money in the account may be used for lending.)
In these circumstances, the concepts of net present value and yield are not meaningful anymore in general. We must calculate the accumulation of net cash flow from first principles. Let's illustrate how the calculation looks like in the following exercise.

### Discounted payback period

In practice, the net cash flow usually changes sign only once, and this change is from negative to positive. In these circumstances, the balance in the investor's account will change from negative to positive at a unique time $t^{*}$. If the balance will always be negative, the project is always not profitable, and thus no such time exists. If such time $t^{*}$ exists, it is the time at the end of discounted payback period. We define it more formally as follows:
Definition. (Discounted payback period) Discounted payback period is the smallest positive integer $t$ in the unit of time such that
    $A(t)=\sum _{s\leq t}C_{s}(1+i)^{t-s}\geq 0$
in which $i$ is the effective rate of interest at which the net cash flows accumulate.
Remark. Assume that a project ends at time $T$. If $A(T)<0$, the project has no discounted payback period and is not profitable. This is because the balance in the investor's account will only change from negative to positive at a unique time. If the balance is still negative at the end of the project, it shows that the balance is negative throughout the project, i.e. the net present value is negative.
In particular, we may need to calculate discounted payback period for project in which we borrow money to invest, when given the effective interest rate of borrowing money, say $j_{1}$, which may not be the same as the effective interest rate of depositing money, say $j_{2}$ as suggested in the subsection of different interest rates for lending and borrowing. However, $j_{2}$is not involved in and does not affect our calculation.
Naturally, we may think that we can just replace every $i$ by $j_{1}$ in the definition of discounted payback period and use it to calculate the discounted payback period. However, there is a problem. Although we can borrow money for cash outflow, so it makes sense to accumulate that amount of money ,which is negative for net cash flow, at $j_{1}$, we cannot 'borrow money for cash inflow'.
Instead, we can only use the cash inflow to repay loan, i.e. reduce the amount we borrow. In view of this, we need to have an assumption, namely assuming that repayment can be made at arbitrary time. (The longer the time the payment is borrowed, the higher the amount of interest accumulated. Therefore, to minimize the interest payment and thus minimize the time needed for the accumulated value of the net cash flow to be nonnegative, we should repay the loan as soon as possible, when we receive some cash inflow.)
Then, we can also accumulate the cash inflow, or positive net cash flow, at $j_{1}$, because they can be treated as the reduction of accumulated value of loan at effective interest rate of $j_{1}$ (the accumulated value includes both the amount of loan that is not repaid, and the interest accumulated). To be more precise, for each positive net cash flow $C_{s}$, it is used to repay the loan at time $s$ and then the loan repaid will stop accumulating interest after time $s$. Therefore, it reduces the loan by $C_{s}$ at time $s$, and also reduces the future interest that will be kept accumulating if the loan is not repaid until the time $t$, to zero, i.e. reduces by $C_{s}\left((1+j_{1})^{t-s}-1\right)$. Therefore, the total amount of reduction of loan and interest is
    $\underbrace {C_{s}} _{\text{loan}}+\underbrace {C_{s}\left((1+j_{1})^{t-s}-1\right)} _{\text{interest}}=\underbrace {C_{s}(1+j_{1})^{t-s}} _{\text{total}}.$
Then, at a certain cash inflow, its amount is sufficient for the loan to be completely repaid (the interest is also paid in the progress of repaying the loan using cash inflow), and then at that time point, the accumulated value of sum of each net cash flow will be greater than or equal to zero. Therefore, that cash flow is the last repayment of loan, and the time at which that cash flow is made is time $t$ (so the future interest equals zero, because there is no time for accumulating).
As a result, we can replace every $i$ with $j_{1}$ in the definition of discounted payback period, with the assumption that repayment can be made at arbitrary time.
If the project is profitable, the accumulated profit when the project ends at time $T$ is
    $\underbrace {A(t^{*})} _{{\text{balance at }}t^{*}}\overbrace {(1+j_{2})^{T-t^{*}}} ^{{\text{ depositing }}A(t^{*})}+\underbrace {\sum _{t>t^{*}}C_{t}(1+j_{2})^{T-t}} _{\text{net cash flows after discounted payback period}}.$
in which $t^{*}$ is the time at the end of discounted payback period.
If the project is not profitable, then the accumulated profit is negative (i.e. we have loss) or zero (then we are indifferent in investing in the project), and we cannot use the above formula.

### Reinvestment rates

In the section of idealized practical situation, we examine what happens if we connect the cash inflow to another project. The main idea in this subsection is similar to that case, and can be interpreted as the generalized version of this case, because we discuss some more ways of reinvestment in this subsection, not limited to connect to another project.
Suppose we invest 1 in a project, and the project pays interest at the effective interest rate of $i$ at the end of each period for $n$ periods, and the interest received is reinvested at effective interest rate of $j$. Then, accumulated value at the end of $n$ periods is
    $\underbrace {1} _{\text{principal}}+\underbrace {is_{{\overline {n}}|j}} _{\text{accumulated value of interest}}.$
In particular, it equals $(1+i)^{n}$ if $i=j$, which is the same as the case in compound interest because this is equivalent to the definition of compound interest: interest earned is automatically reinvested back to the project at the same rate $i$.
Suppose we invest 1 at the end of each period for $n$ periods at the effective interest rate of $i$, and the interest is reinvested at the effective interest rate of $j$. Then, the accumulated value at the end of $n$ periods is
    $n+i(Is)_{{\overline {n-1}}|j}=n+i\left({\frac {{\ddot {s}}_{{\overline {n-1}}|j}-(n-1)}{j}}\right)=n+i\left({\frac {s_{{\overline {n}}|}-n}{j}}\right).$
In particular, it equals $s_{{\overline {n}}|i}$ if $i=j$.

### Interest measurement of a fund

To calculate the yield rate earned by an investment fund, we can use its earned effective interest rate. Recall that the definition of effective interest rate assumes that principal remains constant throughout the period, all interest earned throughout the period is paid at the end of the period. These assumption are usually not satisfied in practice, because there are usually irregular principal deposits and withdrawals (they are net cash flows), and irregular interest earning (possibly at different effective rate for each interval) throughout the period. (There may also be some time periods at which no interest is earned, i.e. the effective rate is zero.) To illustrate this, consider the following figure.
There are two ways to calculate a reasonable effective interest rate for such complicated situation, namely dollar-weighted rate of return (or interest) and time-weighted rate of return (or interest). These two ways can simplify the calculations involved.

#### Dollar-weighted rate of return

The aim is finding the effective interest rate $i$ earned by a fund over one measurement period. For simplicity, let us use the following notations:
* $A$ is the amount in fund at the start of the period
* $B$ is the amount in fund at the end of the period
* $I$ is the amount of interest earned during the period
* $C_{t}$ is the net amount of principal contributed at time $t\in [0,1]$. This value can be positive, negative or zero. (This is net cash flow)
* $C$ is the total net amount of principal contributed during the period, i.e. $\sum _{t}C_{t}$.
* $_{a}i_{b}$ is the amount of interest earned by 1 invested at time $b$ over the following period of length $a$, i.e. to time $a+b$, in which $a,b$ are positive real numbers such that $a+b\leq 1$ (because we are considering one measurement period).
Then, by definitions,
    $\underbrace {B} _{\text{end}}=\underbrace {A} _{\text{start}}+\underbrace {C} _{\text{contribution}}+\underbrace {I} _{\text{interest}},$
and if we assume that all the interest earned, $I$, is received at the end of the period to be consistent to the definition of effective interest rate, the exact equation of $I$ is
    $I=\underbrace {i} _{_{1}i_{0}}A+\sum _{t}{C_{t}}(_{1-t}i_{t}).$
To solve for $i$ in the above equation, the term for which we need to calculate in a more complicated way is $_{1-t}i_{t}$. Without any assumption, it is very hard or even impossible to calculate it directly. Therefore, we need to have an assumption to simplify the calculation and approximate its value.
If we assume compound interest throughout every period from time $t$ to $1$ (each period corresponds to one net cash flow.),
    $_{1-t}i_{t}=(1+i)^{1-t}-1$
because the length of time involved is $1-t$. If we put this into the exact equation of $I$, the equation can be solved by iteration using computer or financial calculator. This is not the main focus in this subsection. Instead, the following is the main focus.
If we want to simplify our calculation, we can assume simple interest throughout every period from time $t$ to $1$, (each period corresponds to one net cash flow.) then
    $_{1-t}i_{t}=\underbrace {(1-t)} _{\text{time length}}i.$
Putting it into the exact equation of $I$, we can solve for $i$ and
    $I=iA+\sum _{t}C_{t}(1-t)i\iff i={\frac {I}{A+\sum _{t}C_{t}(1-t)}}$.
This is referred as the dollar-weighted rate of return (or interest). Let's define it formally as follows:
Definition. (Dollar-weighted rate of return) Dollar-weighted rate of return is the simple interest rate approximation to the yield rate (or the effective interest rate), and is given by
    $i^{DW}={\frac {I}{A+\sum _{t}C_{t}(1-t)}}.$
Remark.
* This approximation is close to the exact effective interest rate if $C_{t}$'s are small compared to $A$.
   * This is because when $C_{t}$'s are small, then $\sum _{t}^C_{t}(1-t)\approx 0$, and thus $i^{DW}\approx {\frac {I}{A}}$.
   * On the other hand, the effective interest rate $i={\frac {I}{A}}$ by definition. So, $i^{DW}\approx i$.
* It is dollar-weighted in the sense that the terms involved are mostly related to dollar ($A,C_{t},I$)
Because the calculation of the numerator term can be tedious, we may further assume that every net principal contribution occurs at time $t=0.5$, then we have
    $i^{DW}={\frac {I}{A+\sum _{t}C_{t}(1-0.5)}}={\frac {I}{A+0.5C}}={\frac {I}{A+0.5(B-A-I)}}={\frac {2I}{A+B-I}}$
Apart from the advantage of simpler calculation, another advantage is that we can calculate $i$ using $A,B,I$ only, and we do not need to know the values of $C_{t}$'s.
Exercise. Prove that the dollar-weighted rate of return assuming that every net principal contribution occurs at time $t=k\in [0,1]$ is
    ${\frac {I}{kA+(1-k)B-(1-k)I}}.$
The dollar-weighted rate of return with such assumption is
    $i^{DW}={\frac {I}{A+\sum _{t}C_{t}(1-k)}}={\frac {I}{A+(1-k)C}}={\frac {I}{A+(1-k)(B-A-I)}}={\frac {I}{A+(1-k)B-(1-k)A+(1-k)I}}={\frac {I}{kA+(1-k)B+(1-k)I}}.$$\Box$

#### Time-weighted rate of return

For dollar-weighted rate of return, it is sensitive to the amount of money invested during different subperiods. To see this, consider the following situations for a fund.
Situation 1:
Dollar-weighted rate of return for the entire period is
    ${\frac {\overbrace {200} ^{B}-\overbrace {50} ^{C}-\overbrace {100} ^{A}}{\underbrace {100} _{A}+\underbrace {50} _{C_{0.5}}(1-\underbrace {0.5} _{t})}}=40\%.$
Situation 2:
Dollar-weighted rate of return for the entire period is $0\%$ because $I=\underbrace {100} _{B}-\underbrace {100} _{A}-\underbrace {0} _{C}=0$.
Situation 3:
Dollar-weighted rate of return for the entire period is
    ${\frac {\overbrace {50} ^{B}-(\overbrace {-25} ^{C})-\overbrace {100} ^{A}}{\underbrace {100} _{A}\underbrace {-25} _{C_{0.5}}(1-\underbrace {0.5} _{t})}}\approx -28.57\%.$
The dollar weighted rate of return for each of the situation is very different. However, the effective interest rate of the fund at each subperiod is the same in every situation, and the effective interest rate for the entire period is $(1-50\%)(1+100\%)-1=0\%$ which measures the 'performance' of the fund.
In two of the situations, the dollar-weighted rate of return differ from $0\%$ a lot. Therefore, this is inaccurate to measure the performance of the fund. A more accurate way is using the time-weighted rate of return which is not affected by the amount of contribution and balance.
Definition. (Time-weighted rate of return) Consider the following diagram.
in which
* $C_{t_{n}}$ is the net contribution to the fund at time $t_{n}\in [0,1]$
* $C_{t_{0}}=C_{t_{m}}=0$ because there is no net contribution at time $t_{0}$ and $t_{m}$.
* $B'_{n}$ is the fund value just before the $n$th contribution, and $n\in [1,m-1]$ because there are only $m-1$ net contributions. For $B'_{0},B'_{m}$, they are defined as follows to make the following formula of time-weighted rate of return valid for each $k\in 1,2,\ldots ,m$
* $B'_{0}$ is the fund value at the beginning of the period.
* $B'_{m}$ is the fund value at the end of the period.
* $j_{n}$ is the time-weighted rate of return over the $n$th subinterval.
Assume that there are $m-1$ net contributions to the fund in total, and they do not occur exactly at the beginning or the end of the period. Then there are $m$ subinterval corresponding to the $m-1$ net contributions, as shown in the above diagram. The time-weighted rate of return over the $k$th subinterval (i.e. from time $t_{k-1}$ to time $t_{k}$), is defined by
    $1+j_{k}={\frac {\overbrace {B'_{k}} ^{{\text{Balance just before }}k{\text{th contribution}}}}{\underbrace {B'_{k-1}+C_{t_{k-1}}} _{{\text{Balance at time }}t_{k-1}}}},$
and the yield rate over the whole period ($i^{TW}$) by time-weighted rate of return is defined by
    $1+i^{TW}=(1+j_{1})(1+j_{2})\cdots (1+j_{m}).$
Remark.
* This rate is not consistent with the assumption of compound interest, because the amount of principal changes throughout the period.
* It is time-weighted in the sense that the yield rate by time-weighted rate of return is affected by the time points at which the net contributions are made.
* Time-weighted rate of return over the $k$th subinterval is the effective interest rate over the $k$th subinterval, because the balance at time $t_{k-1}$ accumulates at the effective interest rate to the balance just before the $k$th net contribution, and there are no net contributions during the accumulation process.
* The yield rate over the whole period is the effective interest rate over the whole period.
It can be observed that the time-weighted rate of return for each of three previous situation is
    $(1-50\%)(1+100\%)-1=0\%$
which is not affected by the balance and contribution. Therefore, it is more accurate than dollar-weighted rate of return in this situation. Although it is more accurate, it requires more information than dollar-weighted rate of return, because we need the information about contributions at different time points and the corresponding balance at each time point to calculate the overall yield rate. On the other hand, only information about the balance at the beginning and the end, and the contributions are needed for dollar-weighted rate of return for dollar-weighted rate of return Therefore, it is sometimes impossible to calculate the overall yield rate using time-weighted rate of return, but possible when we use the dollar-weighted rate of return

## Stocks

We will discuss two types of stock: preferred stock (or preference share) and common stock (or ordinary share) in this section.
In general, preferred stock is a fixed-income security which provides fixed dividends periodically. However, it differs from bond, which also gives fixed income, since preferred stock is an ownership security instead of a debt security (bond is a debt security), and also preferred stock ranks behind bonds and other debt instructments in terms of the degree of security. This is because all payments on indebtedness must be made before preferred stock receives a dividend.
Proposition. (Price of preferred stock without maturity date) The price of preferred stock without maturity date, with amount of each periodic dividend equal $D$ is $P={\frac {D}{i}}.$
Proof. To set a 'fair' price for the stock that reflects its value, the price is set to be the present value of all periodic future dividends, which last forever, i.e. $P=Da_{{\overline {\infty }}|i}={\frac {D}{i}}.$
$\Box$
Common stock is also a type of ownership security. However, it does not earn a fixed dividend rate. Instead, common stock dividends are paid only after interest payments on all bonds and other debt and dividends on preferred stock are paid. Thus, common stock has an even lower degree of security than preferred stock. Also, the common stock price is usually quite volatile.
We can compute the theoretical price (which is generally not the price in reality) of common stock based on the dividend discount model, i.e. the price should represent the present value of future dividends, similar to that of preferred stock.
Proposition. (Price of common stock with dividends varying in geometric sequence) Let $D$ be the common stock dividend at the end of current period. Suppose the dividends are expected to change in a geometric sequence with common ratio $1+k$ indefinitely, in which $-1<k<i$. Then, the theoretical price is ${\frac {D}{i-k}}.$
Proof. The theoretical price is $Dv+\underbrace {D(1+k)v^{2}} _{Dv[v(1+k)]}+\cdots ={\frac {Dv}{1-v(1+k)}}={\frac {Dv(1+i)}{1+i-v(1+i)(1+k)}}={\frac {D}{i-k}}.$
$\Box$
Illustration of the dividends paid:
Example. Suppose a common stock pay annual dividends with amount $10$ at the end of each year forever. The common stock dividend is expected to increase by $5\%$ per year.
An investor purchases this common stock at the price of $200$. Compute the yield rate $i$.
Solution: $200=10v+10(1.05)v^{2}+\cdots \Rightarrow 200={\frac {10v}{1-1.05v}}\Rightarrow 200-210v=10v\Rightarrow v\approx 0.90909\Rightarrow i=0.1$
Exercise.
Some investors in securities use short sales (or short transactions) when they think that the price of a security is likely to decline. When there is a short sale, sale occurs first and the purchase occurs later (ideally at a price that is lower than selling price).
Because of this terminology, we say the normal transaction (purchase occurs first, then sale occurs later), as long transactions.
Typically, with a short sale, the investor borrows (not purchases) the security from a 2nd party, and sells it in the market to a 3rd party. Later, the investor buys back the security from the market (ideally at a lower price than selling price), to return the security to the 2nd party. The process of buying back the security is called 'covering the short'.
Practically, when the sale is made, the short seller need to make a deposit of a percentage of the price, and the percentage is called margin and cannot be recovered by the short seller, until the short position is covered (i.e. 'covering the short' is done). Short seller will be credited interest on the margin deposit.
Also, if the security pays dividends, then the short seller need to pay these to the purchaser of the security, without receiving them from the seller of security, e.g. company. That is, the short seller plays the role of the company selling the security temporarily, and need to pay the dividend to the 3rd party.
Example. An investor sold a stock short for $10000$ and bought back for $9500$ at the end of 1st year. The stock paid $700$ in dividend during the year. Given that the margin requirement is $30\%$, and the interest rate on the margin is $10\%$. Compute the yield rate ofthe investor (which is ${\frac {\text{net profit}}{\text{margin deposit}}}$, since the margin deposit can be viewed as the amount of investment, and the yield rate is the rate of return on the investment).
Solution:
* gain on short sale is $10000-9500=500$
* interest on margin is $0.1(10000(0.3))=300$
* dividend on stock is $700$
So, the net profit is $500+300-700=100.$ The yield rate is ${\frac {100}{10000(0.3)}}\approx 0.033333.$
Exercise.

## Term structure of interest rate

The effective interest rates vary according to the term of investment, which is shown by yield curve.
Definition. (Yield curve) Yield curve is a curve showing the relationship between the effective interest rate and term of investment.
Remark.
* Typically, the effective interest rate increases when the term is longer.
* A yield curve is normal or upward-sloping if the interest rate increases with term.
* A yield curve is inverted or downward-sloping if the interest rate decreases with term.
* A yield curve is bowed if the interest rate increases and then decreases, or decreases and then increases with term</ref>.
* A yield curve is flat if the interest rate is the same for each term.
There are some theories explaining the change of effective interest rate when the term changes in length, which are explained in the Financial Math FM/Determinants of Interest Rates chapter.

### Spot rates

When we are computing yield rate of arbitrary group of fixed interest securities at arbitrary given date, the interest rates vary according to the term of the investment, as shown in the previous subsection. So, we need to take this variation into consideration.
If there are factors other than term that vary, e.g. frequency of coupon payments, then it makes the comparison between different groups of the securities complicated. So, to avoid complications, we compare short-term and long-term interest rates with reference to zero-coupon bonds, by considering each security as a combination of (notional) zero-coupon bond(s), if we assume that there is no arbitrage (i.e. risk-free trading profit) (this is called no-arbitrage assumption).
After assuming there is no arbitrage, it is impossible for the fixed-interest security and the combination of zero-coupon bonds that replicates the security to have two different prices (this is called the law of one price), otherwise, investor may be able to gain a risk-free trading profit using the price difference. We will not discuss the strategy to obtain arbitrage in this book. However, such arbitrage opportunity is rare in modern financial market, and also, even if it exists, it will be quickly eliminated after being spotted by some investors, and exploited by them.
Actually, the spot rate of interest is quite related to the zero-coupon bond.
Definition. (Spot rate of interest) The n-year spot rate of interest is the annual yield rate of zero-coupon bonds of term $n$, denoted by $i_{n}$.
Remark.
* since interest rate varies according to the term of investment, $i_{n}$ has different values for different value of $n$ generally.
* the term structure of interest rates means the spot rate vector $(i_{1},i_{2},\ldots ,i_{n})$ (the numbers inside are spot rates)
* $n$ is often positive integer, with year as unit, but in some rare cases, it may be positive decimal number, with different unit
Example. (Considering a fixed-interest bond as a combination of zero-coupon bonds) Consider a 10-year bond that pays coupons with amount $20$ at the end of each year and has the redemption value of $500$ at the end of 10th year.
Given that the $k$-year spot rate is $0.01k$, compute $P$.
Then, we can consider this bond as a combination of zero coupon bonds as follows:
* for the ten annual coupons, we replicate them by ten zero coupon bonds, with redemption value $20$ at the end of 1st, 2nd, ... , 10th year
* for the redemption payment, we replicate it by a zero coupon bond, with redemption value $500$ at the end of 10th year
Thus, the bond price is $P=20(v_{i_{1}}+v_{i_{2}}^{2}+\cdots +v_{i_{10}}^{10})+500v_{i_{10}}^{10}=20(1.01^{-1}+1.02^{-2}+\cdots +1.1^{-10})+500(1.1)^{-10}\approx 644.759{\text{ (by direct computation)}}$

### Forward rates

Definition. (Forward rates) The forward rate $f_{t\to t+r}$ represents the annual interest rate payable on a risk-free investment made at time $t$ and maturing at time $t+r$.
The forward rates can be computed from the spot rates, vice versa. This is because an investment of $k$ in a $t$-year zero-coupon bond followed by an investment of the redemption value (which is $k(1+i_{t})^{t}$, since $k=Cv_{i_{t}}^{t}$) from this bond in an $r$-year zero-coupon bond is worth the same as an investment of $k$ in a $(t+r)$-year zero-coupon bond. That is, $k(1+i_{t})^{t}(1+f_{t\to t+r})^{r}=k(1+i_{t+r})^{t+r}\Rightarrow (1+i_{t})^{t}(1+f_{t\to t+r})^{r}=(1+i_{t+r})^{t+r}.$
Example. Consider a 3-year bond, with coupon of amount $10$ paid at the end of year 1 and year 3, and its redemption value is $50$. Given that the 1-year spot rate is $5\%$ and 1-year deferred 2-year forward rate (i.e. $f_{1\to 3}$) is $7\%$,
compute $P$. Solution: The 3-year spot rate is $\left((1.05)(1.07)^{2}\right)^{1/3}-1\approx 0.063291360$. $P=10(1.05)^{-1}+10(1.063291360)^{-3}+50(1.063291360)^{-3}\approx 59.43459.$

### Par yields

Definition. (Par yield) The $n$-year par yield, denoted by $i_{p_{n}}$
is the coupon per unit nominal (i.e. $Fr/F=r$) that would be payable on a bond with a term of $n$ years, that would give the bond a current price under current term structure of $1$ per unit nominal (i.e. $A=P/F$), assuming the bond is redeemed at par (i.e. $C=1$). That is, $i_{p_{n}}\left(v_{i_{1}}+v_{i_{2}}^{2}+\cdots +v_{i_{n}}^{n}\right)+v_{i_{n}}^{n}=1.$ (by the formula $P/F=Fr/F\left(v_{i_{1}}+v_{i_{2}}^{2}+\cdots +v_{i_{n}}^{n}\right)+Cv_{i_{n}}^{n}$)
Remark.
* the par yields given an alternative measure of the relationship between the yield and the term of investments
* the difference between $n$-year par yield and $n$-year spot rate is called the $n$-year coupon bias
Illustration of par yield:
Example. Given that the annual term structure of interest rate is $(0.01,0.02,0.03,\ldots ,)$, compute the 3-year par yield.
Solution: Since ${\begin{aligned}&&i_{p_{3}}(v_{i_{1}}+v_{i_{2}}^{2}+v_{i_{3}}^{3})+v_{i_{3}}^{3}&=1\\&\Rightarrow &i_{p_{3}}(1.01^{-1}+1.02^{-2}+1.03^{-3})&=1\\&\Rightarrow &i_{p_{3}}\approx 0.3488685.\end{aligned}}$
Exercise.

## Macaulay duration, modified duration, modified convexity, and Macaulay convexity

Macaulay duration measures the average length or term of a financial transaction.
There is another way to measure the average term to maturity, namely the method of equated time. The index is computed as the weighted average of different payments in which the weights are the amount paid. E.g., if the payment is $s_{k}$ at time $t=k$, then the average term to maturity is ${\overline {t}}={\frac {\sum _^ts_{t}}{\sum _^s_{t}}}.$ However, it does not consider the effect of interest rate, and Macaulay duration considers the effect, and thus is generally a better index.
Definition. (Macaulay duration) The Macaulay duration $\operatorname {D} _{\text{mac}}(i)$ is ${\frac {t_{1}v_{i}^{t_{1}}C_{t_{1}}+\cdots +t_{n}v_{i}^{t_{n}}C_{t_{n}}}{C_{t_{1}}v_{i}^{t_{1}}+\cdots +C_{t_{n}}v_{i}^{t_{n}}}}$ in which cash flow $C_{t_{k}}$ is made at time $t_{k}$.
Illustration:
Remark.
* The Macaulay duration is the mean term of cash flows weighted by their present values (instead of simply amount, which is used in method of equated time)
* The Macaulay duration is a function of $i$, i.e. it changes as $i$ changes
* Alternative names of Macaulay duration include discounted mean term.
* Unless otherwise specified, "duration" means Macaulay duration.
Example. A 20-year bond with par value $10$ has a $3\%$ semiannual coupon rate. Suppose its redemption value is $20$, and the semiannual yield rate is $i=5\%$. Compute its Macaulay duration.
Solution: Its Macaulay duration is ${\frac {10(0.03)(Ia)_{{\overline {40}}|0.05}+40(20)(1.05)^{-40}}{10(0.03)a_{{\overline {40}}|0.05}+20(1.05)^{-40}}}\approx 23.4893.$
Exercise.
To measure the sensitivity of a series of cash flows to movements in the interest rates is the volatility.
Definition. (Modified duration) Consider a series of cash flows $C_{t_{1}},C_{t_{2}},\ldots ,C_{t_{n}}$. Let $P(i)$ be the present value of the cash flows at rate $i$, i.e. $P(i)=C_{t_{1}}v_{i}^{t_{1}}+\cdots +C_{t_{n}}v_{i}^{t_{n}}.$ The modified duration is $\operatorname {D} _{\text{mod}}(i)={\color {darkgreen}-}{\frac {P'(i)}{P(i)}}={\frac {C_{t_{1}}t_{1}v_{i}^{t_{2}}+\cdots +C_{t_{n}}t_{n}v_{i}^{t_{n+1}}}{C_{t_{1}}v_{i}^{t_{1}}+\cdots +C_{t_{n}}v_{i}^{t_{n}}}}={\frac {\operatorname {D} _{\text{mac}}(i)}{1+i}}$
Remark.
* It measures the rate of change of $P(i)$ with $i$ and is independent of the size of the present value (because of the denominator)
* It is analogous to force of interest, and the negative sign is added to ensure that the volatility is positive, since $P'(i)<0$, considering that ${\frac {d}{di}}v^{t}<0$ for each $t$
* There is close relationship between Macaulay duration and modified duration, and hence the name modified duration
* Alternative name includes volatility.
Definition. (Modified convexity) Consider a series of cash flows $C_{t_{1}},C_{t_{2}},\ldots ,C_{t_{n}}$. Let $P(i)$ be the present value of the cash flows at rate $i$, i.e. $P(i)=C_{t_{1}}v_{i}^{t_{1}}+\cdots +C_{t_{n}}v_{i}^{t_{n}}.$ The modified convexity is $\operatorname {C} _{\text{mod}}(i)={\frac {P''(i)}{P(i)}}={\frac {C_{t_{1}}(t_{1})(t_{1}+1)v_{i}^{t_{3}}+\cdots +C_{t_{n}}(t_{n})(t_{n}+1)v_{i}^{t_{n+2}}}{C_{t_{1}}v_{i}^{t_{1}}+\cdots +C_{t_{n}}v_{i}^{t_{n}}}}.$
Remark.
* Unless otherwise specified, "convexity" refers to modified convexity.
* In mathematics, whether a function is "convex" is related to its second derivative, and hence the name "convexity".
   * The convexity is modified when compared to the Macaulay convexity.
Definition. (Macaulay convexity) Consider a series of cash flows $C_{t_{1}},C_{t_{2}},\ldots ,C_{t_{n}}$. The Macaulay convexity is $\operatorname {C} _{\text{mac}}(i)={\frac {t_{1}^{2}C_{t_{1}}v^{t_{1}}+\dotsb +t_{n}^{2}C_{t_{n}}v^{t_{n}}}{C_{t_{1}}v^{t_{1}}+\dotsb +C_{t_{n}}v^{t_{n}}}}$
Remark.
* Since the expression for this convexity is similar to the one for Macaulay duration (each time in the numerator is squared only) and it is also similar to the one for (modified) convexity, it is named Macaulay convexity.
The usage of convexity is as follows: consider a small change in interest from $i$ to $i+\epsilon$. By Taylor series expansion, $P(i+\epsilon )=P(i)+\epsilon P'(i)+{\frac {\epsilon ^{2}}{2}}P''(i+\delta )$ in which $0<|\delta |<|\epsilon |$. Thus, using both volatility and convexity gives an approximation to the change in $P(i)$ by a small change in interest rates. This will be quite useful in Immunization chapter.
Example. Suppose there are cash inflows of $100$, $200$ and $300$ at the end of year 2,3 and 4 respectively, and that the annual interest rate is $5\%$. Its duration is $\operatorname {D} _{\text{mac}}(i)={\frac {100(2)v_{0.05}^{2}+200(3)v_{0.05}^{3}+300(4)v_{0.05}^{4}}{100v_{0.05}^{2}+200v_{0.05}^{3}+300v_{0.05}^{3}}}={\frac {200(1.05)^{-2}+600(1.05)^{-3}+1200(1.05)^{-4}}{100(1.05)^{-2}+200(1.05)^{-3}+300(1.05)^{-4}}}\approx 3.30593.$ Its modified duration is $\operatorname {D} _{\text{mod}}(i)={\frac {d(i)}{1.05}}\approx 3.1485.$ Its convexity is $\operatorname {C} _{\text{mod}}(i)={\frac {100(2)(3)v_{0.05}^{4}+200(3)(4)v_{0.05}^{5}+300(4)(5)v_{0.05}^{6}}{100v_{0.05}^{2}+200v_{0.05}^{3}+300v_{0.05}^{4}}}\approx 13.4267$
Exercise. Let $P(i)$ be the net present value of the cash flows in this example.

### Approximation for change in present value due to a small change in interest rate

We can use Macaulay duration or modified duration to approximate the change in present value from a small change in interest rate.
Proposition. (First-order Macaulay approximation of change in present value) Let $P(i)$ be the present value under interest rate $i$. Then, when interest rate changes slightly from $i_{0}$ to $i$, the new present value is $P(i)\approx P(i_{0})\left({\frac {1+i_{0}}{1+i}}\right)^{\operatorname {D} _{\text{mac}}(i_{0})}$ given that the change in interest rate is small.
Proof.
* First, $P(i)\approx P(i_{0})\left({\frac {1+i_{0}}{1+i}}\right)^{\operatorname {D} _{\text{mac}}(i_{0})}\Leftrightarrow P(i)(1+i)^{\operatorname {D} _{\text{mac}}(i_{0})}\approx P(i_{0})(1+i_{0})^{\operatorname {D} _{\text{mac}}(i_{0})}$.
* Define function $Q_{t}(i)=P(i)(1+i)^{t}$.
* Then, since for each $i$,
${\begin{aligned}&&Q_{t}'(i)&=0\\&\Leftrightarrow &P(i)t(1+i)^{t-1}+(1+i)^{t}P'(i)&=0\\&\Leftrightarrow &t&={\frac {-P'(i)(1+i)^{t}}{P(i)(1+i)^{t-1}}}\\&&&=\underbrace {\frac {-P'(i)}{P(i)}} _{=\operatorname {D} _{\text{mod}}(i)}(1+i)\\&&&=\operatorname {D} _{\text{mac}}(i),\end{aligned}}$
* using the first-order Taylor approximation (because the change in interest rate is small) to $Q_{t}(i)$ about $i_{0}$ with $t=\operatorname {D} _{\text{mac}}(i_{0})$, we have
${\begin{aligned}&&Q_{t}(i)&\approx Q_{t}(i_{0})+(i-i_{0})\underbrace {Q_{t}'(i_{0})} _{=0{\text{ when }}t=\operatorname {D} _{\text{mac}}(i_{0})}\\&&&=Q_{t}(i_{0})\\&\Leftrightarrow &P(i)(1+i)^{D_{\text{mac}}(i_{0})}&\approx P(i_{0})(1+i_{0})^{D_{\text{mac}}(i_{0})},\end{aligned}}$
* and the result follows.
$\Box$
Proposition. (First-order modified approximation of change in present value) Let $P(i)$ be the present value under interest rate $i$. Then, when interest rate changes from $i_{0}$ to $i$, the new present value is $P(i)\approx P(i_{0}){\big1-(i-i_{0})\operatorname {D} _{\text{mod}}(i_{0}){\big )}$ given that the change in interest rate is small.
Proof.
* Using first -order Taylor approximation (because the change in interest rate is small) to $P(i)$ about $i_{0}$,
$P(i)\approx P(i_{0})+(i-i_{0})P'(i_{0})=P(i_{0}){\color {darkgreen}-}{\color {blue}P(i_{0})}(i-i_{0})\cdot \underbrace {\frac {{\color {darkgreen}-}P'(i_{0})}{\color {blue}P(i_{0})}} _{\operatorname {D} _{\text{mod}}(i_{0})}=P(i_{0}){\big1-(i-i_{0})\operatorname {D} _{\text{mod}}(i_{0}){\big )},$ as desired.
$\Box$

# Immunization
## Learning objectives

The Candidate will understand key concepts concerning cash flow matching and immunization, and how to perform related calculations.

## Learning outcomes

The Candidate will be able to:
* Define and recognize the definitions of the following terms: cash flow matching, immunization (including full immunization), Redington immunization.
* Construct an investment portfolio to:
   * Redington immunize a set of liability cash flows.
   * Fully immunize a set of liability cash flows.
   * Exactly match a set of liability cash flows.

## Redington immunization

Definition. (Immunization) Immunization is a technique to reduce or even eliminate the impact of interest rate movements.
Remark. There are different ways of immunization. We will discuss several immunizations in this chapter, and will discuss Redington immunization in this section.
Consider a fund with asset cash flows and liability cash flows. We use the following notations:
* $P_{A}(i)$: the present value of the assets at the effective interest rate $i$
* $P_{L}(i)$: the present value of the liabilities at the effective interest rate $i$
* ${\overline {v}}_{A}(i)$: the volatility of the asset cash flows
* ${\overline {v}}_{L}(i)$: the volatility of the liability cash flows
* ${\overline {c}}_{A}(i)$: the convexity of the asset cash flows
* ${\overline {c}}_{L}(i)$: the convexity of the liability cash flows
We have the following definition of immunized conditions:
Definition. (Redington immunization conditions) At an interest rate $i_{0}$, the fund is immunized against small movements in interest rate $\epsilon$ if and only if $P_{A}(i_{0})=P_{L}(i_{0})\quad {\text{and}}\quad P_{A}(i_{0}+\epsilon )\geq P_{L}(i_{0}+\epsilon ).$
Remark.
* i.e. small changes in interest rate ($\epsilon$) in either direction ($\epsilon >0$ or $\epsilon <0$) will increase $P$
* in practice, there are difficulties in implementing this immunization strategy, since, e.g., continuous rebalancing of portfolios to keep the asset and liability volatilities equal is needed, which is infeasible
   * although there are some problems, immunization can still improve investment strategies
In practice, we use some other equivalent to conditions to check that whether the fund is immunized under Redington immunization. We have such equivalent conditions as follows.
Proposition. (Alternative conditions for Redington immunization) Let $P(i)$ be the surplus $P_{A}(i)-P_{L}(i)$. At a interest rate $i_{0}$, the fund is immunized against small movements in interest rate $\epsilon$ if and only if the following three conditions are met:
* (correct amount of assets to support liabilities) $P(i_{0})=0$ or $P_{A}(i_{0})=P_{L}(i_{0})$
* (same interest sensitivity for assets and liabilities) $P'(i_{0})=0$, $P'_{A}(i_{0})=P'_{L}(i_{0})$(, or ${\overline {v}}_{A}(i_{0})={\overline {v}}_{L}(i_{0})$ (result from satisfying both $P_{A}(i_{0})=P_{L}(i_{0})$ and $P'_{A}(i_{0})=P'_{L}(i_{0})$))
* $P''(i_{0})>0$, $P''_{A}(i_{0})>P''_{L}(i_{0})$(, or ${\overline {c}}_{A}(i_{0})>{\overline {c}}_{L}(i_{0})$ (result from satisfying both $P_{A}(i_{0})=P_{L}(i_{0})$ and $P''_{A}(i_{0})>P''_{L}(i_{0})$))
Proof. By Taylor series expansion, $P(i_{0}+\epsilon )=P(i_{0})+\epsilon P'(i_{0})+{\frac {\epsilon ^{2}}{2}}P''(i_{0})+\cdots$
* by definition, $P_{A}(i_{0})=P_{L}(i_{0})\Leftrightarrow P(i_{0})=0.$
* the 2nd term $\epsilon P'(i_{0})=0$ for each $\epsilon$ if and only if $P'(i_{0})=0\Leftrightarrow P'_{A}(i_{0})=P'_{L}(i_{0})$
   * Also, $\underbrace {P'_{A}(i_{0})/P_{A}(i_{0})=P'_{L}(i_{0})/P_{L}(i_{0})} _{\because P_{A}(i_{0})=P_{L}(i_{0})}\Leftrightarrow v_{A}(i_{0})=v_{L}(i_{0})$
* Since $\epsilon ^{2}>0$, the 3rd term is always positive if and only if $P''(i_{0})>0\Leftrightarrow P''_{A}(i_{0})>P''_{L}(i_{0})$.
   * Also, $\underbrace {P''_{A}(i_{0})/P_{A}(i_{0})>P''_{L}(i_{0})/P_{L}(i_{0})} _{\because P_{A}(i_{0})=P_{L}(i_{0})>0}\Leftrightarrow c_{A}(i_{0})>c_{L}(i_{0})$
* The 4th and subsequent terms ($\underbrace {\epsilon ^{3}} _{\text{small}}\left(P'''(i_{0})\right)/3!,\ldots$) in the expansions are very small and negligible since $|\epsilon |$ is small.
$\Box$
Remark.
* the 3rd condition assures that a $\downarrow (\uparrow )$ in interest rate will cause asset values to $\uparrow (\downarrow )$ by more (less) than the $\uparrow (\downarrow )$ in liability values (can be observed from Taylor series expansion)

## Full immunization

Full immunization is an even stronger immunization technique than Redington immunization, in the sense that if a fund is fully immunized, then it is Redington immunized, but the converse may not be true. In particular, Redington immunization only works for small changes of interest rate, but full immunization works for changes of interest rate with arbitrary magnitude.
Definition. (Full immunization) Let $P(i)$ be the present value of cash flows from a fund. The fund is fully immunized if $P(i)>0,\quad P(i_{0})=0{\text{ and }}i\neq i_{0}$.
Remark.
* That is, except the interest rate $i_{0}$ at which the present value is zero the present value is positive for each other interest rate. This means that the present value is always nonnegative.
Proposition. (Conditions for full immunization) Let $P(i)$ be the present value of cash flows from a fund. The fund is fully immunized if the following conditions are satisfied:
* for each liability cash outflow, there are two corresponding assets providing cash inflows, one of which is made before the liability cash flow, and another one is made after the liability cash flow;
* $P(i)=0$;
* $P'(i)=0$.
Proof.
* With $P(i)=0$ as one of the conditions, it suffices to prove that $P(i')>0,\quad i'\neq i$ from the remaining two conditions.
* First, consider one of the liability cash outflows with amount $L$ , and suppose a cash inflow of $A$ is made at $a$ units of time before the liability cash outflow, and another cash inflow of $B$ is made at $b$ units after the liability cash outflow.
* Then, $P(i)=0\Rightarrow {\color {blue}A(1+i)^{a}+B(1+i)^{-b}-L=0}$.
* Also, $P'(i)=0\Rightarrow Aa(1+i)^{a}\ln(1+i)-Bb(1+i)^{-b}\ln(1+i)=0\Rightarrow Aa(1+i)^{a}=Bb(1+i)^{-b}$.
* Then, for each $i'\neq i$,
${\begin{aligned}P(i')&=A(1+i')^{a}+B(1+i')^{-b}-{\color {blue}L}\\&=A(1+i')^{a}+B(1+i')^{-b}-({\color {blue}A(1+i)^{a}+B(1+i)^{-b}})\\&=A(1+i')^{a}-A(1+i)^{a}-B(1+i)^{-b}+B(1+i)^{-b}\cdot {\frac {(1+i')^{-b}}{(1+i)^{-b}}}\\&=A(1+i)^{a}\left({\frac {(1+i')^{a}}{(1+i)^{a}}}-1\right)+B(1+i)^{-b}\left({\frac {(1+i')^{-b}}{(1+i)^{-b}}}-1\right)\\&=A(1+i)^{a}\left({\frac {(1+i')^{a}}{(1+i)^{a}}}-1\right)+{\frac {Aa}{b}}(1+i)^{a}\left({\frac {(1+i')^{-b}}{(1+i)^{-b}}}-1\right)\\&=A(1+i)^{a}\left(\left({\frac {1+i'}{1+i}}\right)^{a}-1+{\frac {a}{b}}\cdot \left({\frac {1+i'}{1+i}}\right)^{-b}-{\frac {a}{b}}\right).\\\end{aligned}}$
* Let $x={\frac {1+i'}{1+i}}$. Then, $P(i')=A(1+i)^{a}\left(x^{a}+{\frac {a}{b}}x^{-b}-1-{\frac {a}{b}}\right)$.
* Since $A(1+i)^{a}>0$ ($A$ and $(1+i)^{a}$ are both positive), to determine whether $P(i')>0$, it suffices to only consider the function $f(x)=x^{a}+{\frac {a}{b}}x^{-b}-1-{\frac {a}{b}}$.
* Since $f'(x)=ax^{a-1}-ax^{-b-1}=a(x^{a-1}-x^{-b-1})$, and $a-1>-b-1$ (because $a>-b,\quad a,b>0$),
* $f(x){\begin{cases}>0,&x>1;\\=0,&x=1;\\<0,&x<1.\end{cases}}$.
* This is because $f(y)=x^{y}$ is strictly increasing when $x>1$, always equals one when $x=1$, and strictly decreasing when $x<1$.
* This shows that $f(x)$ has the global minimum at $x=1$ with zero value (by first derivative test), and thus $f(x)>0$ when $x\neq 1$, meaning that $P(i')>0$ when $i'\neq i$ (which is equivalent to $x\neq 1$).
$\Box$

## Exact matching

Exact matching of cash flows is a simple immunization strategy. As suggested by the name, in this strategy, each of the liability cash outflows are exactly matched by cash inflow(s), in the sense that the amount of the cash inflow(s) equals that of the liability cash outflow, and the cash inflow(s) is (are) made at the same time as that of the liability cash outflow.
A common way for the exact matching is using suitable zero-coupon bond(s) to exactly match the liabilities. However, this is not the only way, and sometimes suitable zero-coupon bond(s) is (are) unavailable. An alternative way for the exact matching is using suitable coupon bond(s).
As a result of exact matching, the present value of the cash inflow(s) used for exact matching equals that of the liability cash outflows.

# Interest Rate Swaps
## Learning objectives

The Candidate will understand key concepts concerning interest rate swaps, and how to perform related calculations.

## Learning outcomes

The Candidate will be able to:
* Define and recognize the definitions of the following terms: swap rate, swap term or swap tenor, notional amount, market value of a swap, settlement dates, settlement period, counterparties, deferred swap, amortizing swap, accreting swap, interest rate swap net payments.
* Given sufficient information, calculate the market value, notional amount, spot rates or swap rate of an interest rate swap, deferred or otherwise, with either constant or varying notional amount.

## Introduction and motivation

As suggested by the name "interest rate swap", there is some sort of "swap" between interest rates. Indeed, the interest rates to be swapped are fixed interest rate, which do not change during the term of the loan, and variable interest rate, which changes based on the market interest rate.
Since the future market interest rate is uncertain, there are some uncertainties involved for the variable interest rate. Since some may want to avoid such uncertainties, they may want to have fixed, rather than variable, interest rates for their loans. This motivates interest rate swap.
Definition. (Interest rate swap) Interest rate swap is an agreement between two counterparties to exchanges payments during a specified period (called swap term or swap tenor) in the future based on interest rates applied to a specified principal amount (called notional (principal) amount).
Remark.
* The principal amount is notional in the sense that this amount is never paid by either of the two counterparties.
* The fixed interest rate for the interest rate swap is called swap rate, denoted by $R$.

## Terminologies

For the variable interest rate, it is often linked to a specified index, and common indexes include the London Inter-Bank Offered Rate (LIBOR) and the prime interest rate. By linking to an index, it does not mean the variable interest rate is exactly the same as the rate from that index. There can be some modifications on it. Typically, the variable interest rate equals an index plus some basis point(s) (bp(s)). The basis point(s) added is call "spread". For the two counterparties, one is payer and another is receiver.
Definition. (Payer) Payer under the interest rate swap is the counterparty who pays payment based on fixed interest rate.
Remark.
* As a result, payer receives payment based on variable interest rate.
Definition. (Receiver) Receiver under the interest rate swap is the counterparty who receives payment based on fixed interest rate.
Remark.
* As a result, receiver pays payment based on variable interest rate.
There are some terminologies related to the exchanges of payments during the swap term.
Definition. (Settlement dates) Settlement dates is the dates specified in the interest rate swap at which the exchanges of payments occur.
Definition. (Settlement period) Settlement period is the time between settlement dates.
Definition. (Net swap payment) Net swap payment is the amount of money paid by one counterparty to another counterparty on each settlement date.
Remark.
* On each settlement date, only one counterparty pays a certain amount of money to another counterparty, depending on the difference between the payment based on fixed interest rate and that based on variable interest rate.
Sometimes, one counterparty has a loan from a third party for which interest needs to be paid to the third party. In this case, in addition to the net swap payment (if needed), additional payment is also needed.
Definition. (Net interest payment) Net interest payment is the total net payment (interest paid on the loan plus net swap payment (if needed)) made by the counterparty with a loan.
Most interest rate swaps have a level notional amount over the swap term, but this is not a must, and the notional amount can change over the swap term.
Definition. (Accreting swap) Accreting swap is an interest rate swap for which the notional amount increases over the swap term.
Definition. (Amortizing swap) Amortizing swap is an interest rate swap for which the notional amount decreases over the swap term.
Typically, the first settlement period of an interest rate swap starts at time zero, but again, this is not a must, and the first settlement period can start at later time.
Definition. (Deferred swap) Deferred swap is an interest rate swap for which the first settlement period starts at a time that is later than time zero.

## Swap rate

The fundamental principle for determining the swap rate here is that the swap rate is set such that there is no cost to either counterparty to enter into an interest rate swap (assume the transaction cost is zero from now on), i.e. neither of the counterparties are better off or worse off from entering into the interest rate swap. To do this, the present value of expected future payments for each counterparty should be the same. As a result, the expected future net payments paid (or received) by each counterparty should be zero.
Definition. (Determining swap rate) Under the swap rate, the present value of the payment based on the fixed interest rate equals that based on the variable interest rate.
Remark.
* With a suitable equation, we can solve for the swap rate.
Example. For an interest rate swap, the (level) notional amount is 10000, and the one-year and two-year spot rate is 3% and 5% respectively. The one-year deferred one-year forward rate $f_{1\to 2}=1.05^{2}/1.03-1\approx 0.0704$ (the payment for the second year based on the variable interest rate is based on this forward rate). Then, the swap rate $R$ is determined by $10000R(1.03^{-1})+10000R(1.05^{-2})=10000(0.05)(1.05^{-1})+10000(0.0704)(1.06^{-2})\Rightarrow R={\frac {0.05/1.05+0.0704/1.06^{2}}{1.03^{-1}+1.05^{-2}}}\approx 5.87\%.$
Remark.
* The amount of the level notional amount does not matter for calculating the swap rate.
* For each year, the payment based on the fixed interest rate is $10000R$ for each of the fist and second year, while the payment based on the variable interest rate is $10000(0.03)$ for the first year, and $10000f_{1\to 2}$ for the second year.

## Market value

An interest rate swap can be sold or closed by a counterparty at a time during the swap term. To sold the swap, the seller receives the market value of the swap at the time point, calculated using the spot rates at that time. To close the swap, a possible way is that one counterparty pays the market value to another counterparty at that time.
Definition. (Market value of interest rate swap) The market value of interest rate swap is the present value of the expected future cash flows under the swap, calculated using the spot rates at the time of determining the market value.
Remark.
* The market value can be positive, negative or zero.
* Since the swap is essentially exchange of payment between two counterparties, the market value of the swap for one counterparty is negative of that for another counterparty, at the same time point.
* The future cash flows are the future net swap payments.

# Determinants of Interest Rates
## Learning objectives

The Candidate will understand key concepts concerning the determinants of interest rates, the components of interest, and how to perform related calculations.

## Learning outcomes

The Candidate will be able to:
* Define and recognize the components of interest rates including: real risk-free rate, inflation rate, default risk premium, liquidity premium, and maturity risk premium.
* Explain how the components of interest rates apply in various contexts, such as commercial loans, mortgages, credit cards, bonds, and government securities.
* Explain the roles of the Federal Reserve and the FOMC in carrying out fiscal policy and monetary policy and the tools used by the Federal Reserve and the FOMC including targeting the Federal Funds rate, setting reserve requirements, and setting the discount rate.
* Explain the theories of why interest rates differ by term, including liquidity preference (opportunity cost), expectations, preferred habitat, and market segmentation.
* Explain how interest rates differ from one country to another (e.g., U.S. vs. Canada).
* Identify the real interest and the nominal interest rate in the context of loans with and without inflation protection and calculate the effect of changes in inflation on loans with inflation protection.

## Components of interest rates
### Components related to inflation

In this subsection, we assume there is no risk (and there may be inflation). Under this assumption, components of interest rates include real (risk-free) rate and inflation rate. We have discussed these in the Financial Math FM/Time Value of Money chapter.

### Components related to risk

In reality, there is risk for loans, particularly, some borrowers may not fully repay the amount lent and the required interest on time. This behaviour is called default. Because of the possibility of default, the interest rate incorporates default risk premium. So, with annual inflation rate $j$, annual default risk premium $\beta$, and annual real risk-free rate $r$, the annual nominal interest rate $i=(1+r)(1+\beta )(1+j)-1\Rightarrow i\approx r+\beta +j$.
If the rates and premium are compounded continuously, the nominal interest rate $i$ is calculated by $e^{i}=e^{r}e^{\beta }e^{j}\Rightarrow i=r+\beta +j$. In this case, the approximation sign is replaced by equal sign.
Apart from default risk premium, there is also maturity risk premium (or term premium), this premium usually exists for long-term securities because of the uncertainty for holding the securities for long time. Similarly, if this premium is also incorporated by the interest rate, then it is added to the expression of the nominal interest rate.

### Liquidity premium

Apart from the risk and inflation, there is also a premium for liquidity, since the security may sometimes have low liquidity, in the sense that it cannot be bought or sold easily, and thus compensation for this low liquidity is demanded. Similarly, this premium is added to the expression of the nominal interest rate if incorporated.

## The (simplified) role of the U.S. central bank

The discussion here is simplified, since the actual operation of the U.S. central bank is quite complicated, and is out of scope of this book.
First, since banks are required to deposit a certain amount of money to the central bank (because of the reserve requirement), even if some party is unable to fulfill its obligation in a transaction, the central bank can step in and allow the transaction. So, the central bank is essential for the operation of payment system.
Second, the central bank can act as lender of last resort, in the sense that banks can borrow money from the central bank in case of emergency (e.g. many customers withdraw their money, and the reserve in the bank cannot cope with the withdrawal amount). Apart from the central bank, banks can also borrow money from their peers if needed, but the peers may not willing and able to do so, and in this case, banks can still borrow money from the central bank, and that is why it is referred as the lender of last resort.
Third, the U.S. central bank consists of a Board of Governors, regional reserve banks and the Federal Open Market Committee (FOMC). In particular, the FOMC sets (monetary) policy with respect to the central bank's capital market activities. Common tools used by such policy include targeting the federal fund rate, setting reserve requirement and setting the discount rate (this is different from the discount rate in the Financial Math FM/Time Value of Money).
Definition. (Federal fund rate) The federal fund rate is the interest rate at which banks with excess reserve funds at their District's Federal Reserve Bank lend these funds to other banks.
Definition. (Discount rate) The discount rate is the rate at which banks can borrow money directly from the Federal Reserve.
For the reserve requirement, it indicates the minimum amount of reserve that each bank need to maintain.

### Effects of monetary policies

When the (targeted) federal fund rate or discount rate increases (decreases), it is more (less) costly for banks to run a shortfall in their reserves, and thus they are incentivized to keep more (less) money in the reserve. As a result, less (more) loan is provided, decreasing (increasing) the money supply. With the decrease (increase) of money supply (and assuming money demand is the same), the interest rate charged on loans increases (decreases) (knowledge of Macroeconomics can help understanding this reasoning).
When the reserve requirement decreases (increases), it is more (less) costly for banks to run a shortfall in their reserves, and thus they are incentivized to keep more (less) money in the reserve. As a result, less (more) loan is provided, decreasing (increasing) the money supply. With the decrease (increase) of money supply (and assuming money demand is the same), the interest rate charged on loans increases (decreases) (knowledge of Macroeconomics can help understanding this reasoning).

## Theories explaining why interest rate differ by term
### Expectation theory

According to expectation theory, the relative attraction of short and longer term investments will vary according to the expectation of future movements in interest rates.
An expectation of decrease (increase) in interest rate at the future makes short-term investment less (more) attractive, and longer term investments more (less) attractive.
Thus, because of the change in demand, the yields on short-term investments will increase (decrease) and that on long-term investment will decrease (increase).

### Liquidity preference theory

According to liquidity preference theory (or opportunity cost theory), longer dated bonds are more sensitive to interest rate movements than short dated bonds, since investors prefer liquidity, and thus will be more sensitive about the return of long-term bonds before locking their money in long-term bonds. Also, it is assumed that risk averse investors will require compensation (in the form of higher yields) for the greater risk of loss on longer bonds. It may explain some of the excess return offered on long-term bonds over short-term bonds.

### Market segmentation theory

According to market segmentation theory, bonds of different terms are attractive to different investors, who choose assets that are similar in term to their liabilities. That is, if liabilities of an investor are mainly short-term (long-term), then short-term (long-term) bonds will be chosen by the investor.
Thus, the demand for bonds will differ in different terms. Also, the supply of bonds will also vary by term, because of the strategies used by the seller of bonds, e.g. companies and government.
As a result, due to different forces of supply and demand, the effective interest rate will vary according to the term.

### Preferred habitat theory

This theory is similar to the market segmentation theory, but for this theory, each lender and borrower in the market just prefer bonds with a certain term,and is willing to buy or sell bonds with other term, given that there are sufficient incentive to do so. and
Exercise. We have mentioned that the effective interest rate typically increases when the term is longer.

## Difference between interest rates of U.S. and Canada
### Quotation bases

There are different conventional quotation bases for U.S. treasury bills and Canada treasury bills.
Definition. (Quoted rate on U.S. treasury bills) The quoted rate on U.S. treasury bills is ${\frac {360}{\text{days to maturity}}}\times {\frac {\text{dollar amount of interest}}{\text{maturity value of the treasury bill}}}.$
Remark.
* "days to maturity" refers to the remaining number of days until maturity. In particular, for $n$-day treasury bill, days to maturity for calculating the quoted rate at issue is $n$ days.
Definition. (Quoted rate on Canada treasury bills) The quoted rate on Canada treasury bills is ${\frac {365}{\text{days to maturity}}}\times {\frac {\text{dollar amount of interest}}{\text{current price of the treasury bill}}}.$

## Inflation protection

Inflation protection for a loan is to deal with the uncertainty in inflation for the lender of the loan. For inflation protection, the amount repaid by the borrower depend on the actual inflation over the loan term, through a certain index of inflation (e.g. consumer price index, or CPI).
For this kind of loan, rates before the inflation adjustment (say $i$) are specified for the loan. For the adjustment, the rate is multiplied by the inflation rate (say $j$) from the beginning to the time at which repayment is made, with reference to the index. As an approximation, the rates after the inflation adjustment $i'\approx i+j$, and $i'=i+j$ if all rates involved are continuous.
For the loan with inflation protection, some specified rates before the adjustment may be negative, and thus if the inflation rate turns out to be zero, then there is a loss for the lender of loan, which show that there is a cost for inflation protection. However, if the inflation rate is high, then there is a protection against a potential great loss of purchasing power.

# Stochastic Interest
## Stochastic interest

In this book, we have mainly discussed deterministic (i.e. non-random) interest, and we will briefly introduce stochastic (i.e. random) interest, by regarding the interest rate as a random variable. We use the following notations:
* $I_{t}$: interest rate random variable for the period $t-1$ to $t$
* $\mu _{t}$: mean of $I_{t}$
* $\sigma _{t}^{2}$: variance of $I_{t}$

### Accumulation of single investment

Example. (some simple formulas) For the accumulation of a unit sum of money over the period $t-1$ to $t$,
* the mean is $\mathbb {E} [1+I_{t}]=1+\mathbb {E} [I_{t}]=1+\mu _{t}$ (for non-random interest, it is $1+i_{t}$)
* the variance is $\operatorname {Var} (1+I_{t})=\operatorname {Var} (I_{t})=\sigma _{t}^{2}$
* the 2nd moment is $\mathbb {E} [(1+I_{t})^{2}]=\operatorname {Var} (1+I_{t})+(\mathbb {E} [1+I_{t}])^{2}=\sigma _{t}^{2}+(1+\mu _{t})^{2}$

### Accumulation of single payment over several time periods

Assume that $I_{t}$ are independent for $t=1,\ldots ,n$. Let $S_{n}$ be the accumulation of a single unit sum of money invested for $n$ years, i.e. $S_{n}=(1+I_{1})\cdots (1+I_{n}).$ Then, by independence, ${\begin{aligned}\mathbb {E} [S_{n}]&=(1+\mu _{1})\cdots (1+\mu _{n})\\\mathbb {E} [S_{n}^{2}]&=\mathbb {E} [(1+I_{1})^{2}\cdots (1+I_{n})^{2}]\\&=\mathbb {E} [(1+I_{1})^{2}]\cdots \mathbb {E} [(1+I_{n})^{2}]\\&=\left(\sigma _{1}^{2}+(1+\mu _{1})^{2}\right)\cdots \left(\sigma _{n}^{2}+(1+\mu _{n})^{2}\right)\\\operatorname {Var} (S_{n})&=\mathbb {E} [S_{n}^{2}]-(\mathbb {E} [S_{n}])^{2}\\&=\left(\sigma _{1}^{2}+(1+\mu _{1})^{2}\right)\cdots \left(\sigma _{n}^{2}+(1+\mu _{n})^{2}\right)-(1+\mu _{1})^{2}\cdots (1+\mu _{n})^{2}\end{aligned}}$ For simplicity, further assume that $I_{t}$'s are i.i.d. (identically and independently distributed), with mean $\mu$ and variance $\sigma ^{2}$. Then, ${\begin{aligned}\mathbb {E} [S_{n}]&=\underbrace {(1+\mu )\cdots (1+\mu )} _{n{\text{ copies}}}=(1+\mu )^{n}\\\mathbb {E} [S_{n}^{2}]&=(\sigma ^{2}+(1+\mu )^{2})^{n}=(1+2\mu +\mu ^{2}+\sigma ^{2})^{n}\\\operatorname {Var} (S_{n})&=\mathbb {E} [S_{n}^{2}]-(\mathbb {E} [S_{n}])^{2}=(1+2\mu +\mu ^{2}+\sigma ^{2})^{n}-(1+\mu )^{2n}\end{aligned}}$

### Accumulation of investments with log-normal distribution
#### Some information about log-normal distribution

If $Y=\ln X$ has a normal distribution with mean $\mu$ and variance $\sigma ^{2}$, then $X$ has a log-normal distribution with parameters (not mean/variance generally) $\mu$ and $\sigma ^{2}$. The following are some properties of random variables following log-normal distribution with parameters $\mu$ and $\sigma ^{2}$:
* probability density function (pdf): $f(x)={\frac {1}{x\sigma {\sqrt {2\pi }}}}\exp \left(-{\frac {1}{2}}\left({\frac {\ln x-\mu }{\sigma }}\right)^{2}\right)$
* mean: $\mathbb {E} [X]=e^{\mu +{\frac {\sigma ^{2}}{2}}}$
* variance: $\operatorname {Var} (X)=e^{2\mu +\sigma ^{2}}\left(e^{\sigma ^{2}}-1\right)=(\mathbb {E} [X])^{2}\left(e^{\sigma ^{2}}-1\right)$

#### Motivation of using log-normal distribution

Let's apply log-normal distribution to stochastic interest. If $1+I_{t}$ follows a log-normal distribution with parameters $\mu$ and $\sigma ^{2}$, then $\ln(1+I_{t})$ will be normally distributed with mean $\mu$ and variance $\sigma ^{2}$.
Then, considering the natural logarithm of accumulation of a single investment of one unit for a period of $n$ time units, we have $\ln S_{n}=\ln((1+I_{1})\cdots (1+I_{n}))=\ln(1+I_{1})+\cdots +\ln(1+I_{n}).$ Assuming $I_{t}$'s are independent, $\ln(1+I_{t})$ will also be independent. If we further assume that $1+I_{t}$'s are also log-normally distributed with parameters $\mu _{t}$ and $\sigma _{t}^{2}$, then $\ln(1+I_{t})$'s are normally distributed with mean $\mu$ and variance $\sigma ^{2}$x, and the sum of independent normal random variables $\ln(1+I_{1})+\cdots +\ln(1+I_{n})$ is normally distributed with mean $\mu _{1}+\cdots +\mu _{n}$ and variance $\sigma _{1}^{2}+\cdots +\sigma _{n}^{2}$ (which is a well-known result about normal distribution). That is, $\ln S_{n}=\ln(1+I_{1})+\cdots +\ln(1+I_{n})\sim N(\mu _{1}+\cdots +\mu _{n},\sigma _{1}^{2}+\cdots +\sigma _{n}^{2}).$ Thus, if we apply log-normal distribution to stochastic interest, we can obtain this nice result ($\ln S_{n}$ follows a simple normal distribution).

#### Examples

Example. (a) It is given that $1+I$ follows log-normal distribution with parameters $\mu$ and $\sigma ^{2}$, $\mathbb {E} [I]=0.1$ and $\operatorname {Var} (I)=0.005$. Compute $\mu$ and $\sigma ^{2}$.
Solution: (a) Based on the given mean and variance, we have $\mathbb {E} [1+I]=1+0.1=1.1$ and $\operatorname {Var} (1+I)=0.005.$ So, ${\begin{aligned}&&&{\begin{cases}e^{\mu +{\frac {\sigma ^{2}}{2}}}=1.1\\e^{2\mu +\sigma ^{2}}\left(e^{\sigma ^{2}}-1\right)=0.005\\\end{cases}}\\&\Rightarrow &1.1^{2}\left(e^{\sigma ^{2}}-1\right)&=0.005\\&\Rightarrow &\sigma ^{2}&=\ln \left(1+{\frac {0.005}{1.1^{2}}}\right)\approx 0.004124\\&\Rightarrow &e^{\mu +0.004124/2}&=1.1\\&\Rightarrow &\mu &\approx \ln(1.1)-0.004124/2\approx 0.09325.\end{aligned}}$
(b) It is further given that $I_{t}$ is the annual yield rate for the $t$th year, $I_{t}$'s are i.i.d., and follow the distribution mentioned above. Compute the mean and variance of the accumulation of $100$ for $10$ years, and the probability that its accumulated value will be less than $220$.
Solution: (b) Since $\ln(1+I_{t})\sim N(0.09325,0.004124)$, $\ln(S_{1}0)\sim N(10(0.09325)+10(0.004124))=N(0.9325,0.04124).$ So, $S_{1}0$ follows log-normal distribution with parameters $\mu =0.9325$ and $\sigma ^{2}=0.04124$. Thus, its mean and variance are as follows:
* mean: $\mathbb {E} [100S_{10}]=100e^{0.9325+0.04124/2}\approx 259.3790$
* variance: $\operatorname {Var} (100S_{10})=100^{2}e^{2(0.9325+0.04124)}(e^{0.04124}-1)\approx 2832.5273$
Then, we can compute the probability by ${\begin{aligned}\mathbb {P} (100S_{10}<220)&=\mathbb {P} (\ln S_{10}<\ln 2.2)\\&=\mathbb {P} \left(Z<{\frac {\ln 2.2-0.9325}{\sqrt {0.04124}}}\right)\qquad {\text{in which }}Z\sim N(0,1)\\&=\mathbb {P} (Z<-0.709303)\\&\approx 0.23885\qquad {\text{from standard normal table}}\end{aligned}}$

# Derivatives
## Derivative

A derivative is a contract whose payoff depends on the behavior of some benchmark, which is known as the underlying asset. The underlying is typically a tradable asset, for example, a stock or commodity, but can be a non-tradable such as the weather (in the case of weather derivatives). Derivatives can be used to manage financial risk.
Derivatives are used as a means of hedging. Derivatives are used to reduce the likelihood of bankruptcy.
Derivatives are used to reduce transaction costs. Derivatives are used as a form of insurance.

### Over-the-counter market

Over-the-counter (OTC) derivatives are contracts that are traded (and privately negotiated) directly between two parties, without going through an exchange or other intermediary. Products such as swaps, forward rate agreements, and exotic options are almost always traded in this way.

### Bid-ask spread

The bid-ask spread (also known as bid/offer or buy/sell spread) for securities is the difference between the prices quoted for an immediate sale (ask) and an immediate purchase (bid). The size of the bid-offer spread in a security is one measure of the liquidity of the market and of the size of the transaction cost.

## Short selling

Short selling (also known as shorting or going short) is the practice of borrowing an asset and selling it immediately with the intention of buying identical assets back at a later date to return to the lender.

### Example

Shares in XYZ currently trade at $10 per share.
* Sarah the short seller borrows 10 shares of XYZ at a 5% annual interest rate.
* Sarah immediately sells the shares for $100.
* After one year the stock price drops to $9 per share.
* Sarah buys 10 shares of XYZ for $90 and returns them to the lender along with the $5 of interest.
* Sarah sees a net profit of $5.

### Short position

Represents either a right or an obligation to sell an underlying asset. Examples of short positions: Short Forward, Written Call, Purchased Put

### Long position

Represents either a right or an obligation to buy an underlying asset. Examples of short positions: Long Forward, Purchased Call, Written Put

## Stock index

Measure of the aggregate performance of predetermined group of stocks.

## Spot price

The price paid for the immediate delivery of an asset

## Credit risk

Credit risk: the risk that money owing will not be paid by an obligor. Since credit risk is the natural business of banks, but an unwanted risk for commercial traders, an early market developed between banks and traders that involved selling obligations at a discounted rate.

## Margin
### Maintenance margin
### Margin call
# Options
## The basic trades of traded stock options (American style)

These trades are described from the point of view of a speculator. If they are combined with other positions, they can also be used in hedging. An option contract in US markets usually represents 100 shares of the underlying security.

### Long call

A trader who believes that a stock's price will increase might buy the right to purchase the stock, which is called a call option, rather than just purchase the stock itself. He would have no obligation to buy the stock, only the right to do so until the expiration date. If the stock price at expiration is above the exercise price by more than the premium (price) paid, he will profit. If the stock price at expiration is lower than the exercise price, he will let the call contract expire since it is worthless, and only lose the amount of the premium. A trader might buy the option instead of shares, because for the same amount of money, he can control a much larger number of shares.
Let Call(K,T) be the premium paid for a call option with strike price K and expiration time T, let $S_{T}$ be the stock price at time T and let i be the risk-free effective interest rate. The call option holder's profit per unit is
$\Pi =\left\{{\begin{array}{c l}-Call(K,T)(1+i)^{T}&S_{T}<K,\\(S_{T}-K)-Call(K,T)(1+i)^{T}&S_{T}>=K.\end{array}}\right.$
We can also write this as
    $\Pi =max((S_{T}-K),0)-Call(K,T)(1+i)^{T}$.

### Long put

A trader who believes that a stock's price will decrease can buy the right to sell the stock at a fixed price a put option. He will be under no obligation to sell the stock, but has the right to do so until the expiration date. If the stock price at expiration is below the exercise price by more than the premium paid, he will profit. If the stock price at expiration is above the exercise price, he will let the put contract expire worthless and only lose the premium paid. The put option holder's profit per unit is
$\Pi =\left\{{\begin{array}{c l}-Put(K,T)(1+i)^{T}&S_{T}>K,\\(K-S_{T})-Put(K,T)(1+i)^{T}&S_{T}<=K.\end{array}}\right.$
We can also write this as
    $\Pi =max((K-S_{T}),0)-Put(K,T)(1+i)^{T}$.

### Short call

A trader who believes that a stock price will decrease, can sell the stock short or instead sell, or "write," a call. The trader selling a call has an obligation to sell the stock to the call buyer at the buyer's option. If the stock price decreases, the short call position will make a profit in the amount of the premium. If the stock price increases over the exercise price by more than the amount of the premium, the short will lose money, with the potential loss unlimited. The profit for the writer of a call option is
$\Pi =\left\{{\begin{array}{c l}Call(K,T)(1+i)^{T}&S_{T}<K,\\-(S_{T}-K)+Call(K,T)(1+i)^{T}&S_{T}>=K.\end{array}}\right.$
We can also write this as
    $\Pi =-max((S_{T}-K),0)+Call(K,T)(1+i)^{T}$.

### Short put

A trader who believes that a stock price will increase can buy the stock or instead sell, or "write", a put. The trader selling a put has an obligation to buy the stock from the put buyer at the put buyer's option. If the stock price at expiration is above the exercise price, the short put position will make a profit in the amount of the premium. If the stock price at expiration is below the exercise price by more than the amount of the premium, the trader will lose money, with the potential loss being up to the full value of the stock. A benchmark index for the performance of a cash-secured short put option position is the CBOE S&P 500 PutWrite Index (ticker PUT). The profit for the writer of a put option is $\Pi =\left\{{\begin{array}{c l}Put(K,T)(1+i)^{T}&S_{T}>K,\\-(K-S_{T})+Put(K,T)(1+i)^{T}&S_{T}<=K.\end{array}}\right.$
We can also write this as
    $\Pi =-max((K-S_{T}),0)+Put(K,T)(1+i)^{T}$.

## Option Styles
### European option

A European option may be exercised only at the expiry date of the option, i.e. at a single pre-defined point in time.

### American option

An American option on the other hand may be exercised at any time before the expiry date.

### Bermudan option

A Bermudan option is an option where the buyer has the right to exercise at a set (always discretely spaced) number of times. This is intermediate between a European option—which allows exercise at a single time, namely expiry—and an American option, which allows exercise at any time (the name is a pun: Bermuda is between America and Europe).

## Value of an Option
### ATM: At-the-money

An option is at-the-money if the strike price is the same as the spot price of the underlying security on which the option is written. An at-the-money option has no intrinsic value, only time value.

### ITM: In-the-money

An in-the-money option has positive intrinsic value as well as time value. A call option is in-the-money when the strike price is below the spot price. A put option is in-the-money when the strike price is above the spot price.

### OTM: Out-of-the-money

An out-of-the-money option has no intrinsic value. A call option is out-of-the-money when the strike price is above the spot price of the underlying security. A put option is out-of-the-money when the strike price is below the spot price.

#### Example

Brian buys a call option on XYZ at a strike price of $10 a share. If on the expiration date the share price is $12 the option is In-the-money. Brian will exercise his option to buy at $10 a share. If the share price is $10 the option is At-the-money. Finally, if the share price is $8 his option is Out-of-the-money. He will not exercise his option which is now worthless.

## Covered call

A covered call is a financial market transaction in which the seller of call options owns the corresponding amount of the underlying instrument, such as shares of a stock or other securities. If the trader buys the underlying instrument at the same time as he sells the call, the strategy is often called a "buy-write" strategy. In equilibrium, the strategy has the same payoffs as writing a put option.
Naked writing Dividends

## Put-call parity

Call(K,T)-Put(K,T) = S_T - Ke^T

## References

* ↑ For simplicity, assume there are always 365 days in on year, i.e. February 29th does not exist.
* ↑ On the other hand, "quarterly in advance" means at the beginning of each quarter.
* ↑ If we simply write "rate of $n$", then it implicitly shows that the rate is uniform and constant
* ↑ $D$ stands for "difference"
* ↑ For example, $a_{{\overline {n}}|}=v+v^{2}+\dotsb +v^{n}$, and ${\ddot {a}}_{{\overline {\infty }}|}=1+v+v^{2}+\dotsb$, which are geometric series
* ↑ For annuity-due, a payment is made immediately after receiving the loan, which is unusual. Even if this is the case, the situation is the same as that for annuity-immediate, except that the amount of loan is $L-R$ and payments last for $n-1$ periods (see the following for explanation of notations).
* ↑ you may press other numbers for selecting multiple payments.
* ↑ https://www.soa.org/globalassets/assets/Files/Edu/2019/exam-fm-notation-terminology2.pdf
* ↑ the bond price determined under this assumption is defensive pricing
* ↑ also, when the redemption happens at the earliest time, then the investor cannot enjoy all coupons with large amount, in the sense that the modified coupon rate exceeds the interest rate, so some gains are not captured
* ↑ also, when the redemption happens at the latest time, then the investor is forced to receive all coupons with small amount, in the sense that the modified coupon rate is lower than the interest rate, so all losses are captured
* ↑ the situation which makes the price lowest is the most unfavourable to the investor, since under the most unfavourable situation, the 'worth' of the bond is the lowest
* ↑ it measures the rate of return. See General Cash Flows and Portfolios for more discussion
* ↑ https://www.soa.org/globalassets/assets/Files/Edu/2019/exam-fm-notation-terminology2.pdf
* ↑ Graphically, the yield curve looks like a bow in this situation)
* ↑ Graphically, the yield curve is a flat line in this situation.
* ↑ In particular, $x^{a-1}>x^{-b-1}$ since $a-1>-b-1$, and thus $f(x)>0$. The results for other cases have similar reasoning.
* ↑ It is the average interest rate estimated by leading banks in London that the average leading bank would be charged from borrowing from other banks.
* ↑ It is the rate at which banks in the United States will lend money to their most favoured costumers.
* ↑ A basis point is 0.01%, or 0.0001.
* ↑ since investors can take out money from short-term investment more quickly and invest at a higher future interest rate at the future
* ↑ since investors can enjoy the current interest rate (which is expected to be higher than future interest rates) for a long time if they invest in longer term investments
* ↑ If there is not inflation protection for the loan, the rates specified for the loan may be lower than the rates after the adjustment for the loan with inflation protection.
* ↑ invest-faq or Law & Valuation for typical size of option contract

# Options Strategies

In finance an option strategy is the purchase and/or sale of one or various option positions and possibly an underlying position.
Options strategies can favor movements in the underlying that are bullish, bearish or neutral. In the case of neutral strategies, they can be further classified into those that are bullish on volatility and those that are bearish on volatility. The option positions used can be long and/or short positions in calls and/or puts at various strikes.

## Directional strategies

Bullish options strategies are employed when the options trader expects the underlying stock price to move upwards. Bearish options strategies are the mirror image of bullish strategies. They are employed when the options trader expects the underlying stock price to move downwards.

### Spreads
#### Bull call spread

A bull call spread is constructed by buying a call option with a low exercise price (K), and selling another call option with a higher exercise price.
Often the call with the lower exercise price will be at-the-money while the call with the higher exercise price is out-of-the-money. Both calls must have the same underlying security and expiration month.

##### Example

Take an arbitrary stock XYZ currently priced at $100. Furthermore, assume it is a standard option, meaning every option contract controls 100 shares.
Assume that for next month, a call option with a strike price of $100 costs $3 per share, or $300 per contract, while a call option with a strike price of $115 is selling at $1 per share, or $100 per contract.
A trader can then buy a long position on the $100 strike price option for $300 and sell a short position on the $115 option for $100. The net debit for this trade then is $300 - 100 = $200.
This trade results in a profitable trade if the stock closes on expiry above $102. If the stock's closing price on expiry is $110, the $100 call option will end at $10 a share, or $1000 per contract, while the $115 call option expires worthless. Hence a total profit of $1000 - 200 = $800.
The trade's profit is limited to $13 per share, which is the difference in strike prices minus the net debit (15 - 2). The maximum loss possible on the trade equals $2 per share, the net debit.

#### Bull put spread

A bull put spread is constructed by selling higher striking in-the-money put options and buying the same number of lower striking in-the-money put options on the same underlying security with the same expiration date. The options trader employing this strategy hopes that the price of the underlying security goes up far enough such that the written put options expire worthless.

##### Example

Take an arbitrary stock ABC currently priced at $100. Furthermore, assume again that it is a standard option, meaning every option contract controls 100 shares.
Assume that for next month, a put option with a strike price of $105 costs $8 per share, or $800 per contract, while a put option with a strike price of $125 is selling at $27 per share, or $2700 per contract.
A trader can then open a long position on the $105 strike put option for $800 and open a short position on the $125 put option for $2700. The net credit for this trade then is $2700 - 800 = $1900.
This trade will be profitable if the stock closes on expiry above $106. If the stock's closing price on expiry is $110, the $105 put option will expire worthless while the $125 put option will end at $15 a share, or $1500 per contract. Hence a total profit of $1900 - 1500 = $400.
The trade's profit is limited to $19 per share, which is equal to the net credit. The maximum loss on the trade is $1 per share which is the difference in strike prices minus the net credit (20 - 19).

## Neutral or non-directional strategies

Neutral strategies in options trading are employed when the options trader does not know whether the underlying stock price will rise or fall. Also known as non-directional strategies, they are so named because the potential to profit does not depend on whether the underlying stock price will go upwards or downwards. Rather, the correct neutral strategy to employ depends on the expected volatility of the underlying stock price.
Examples of neutral strategies are:
* Guts - sell in the money put and call
* Butterfly - buy in the money and out of the money call, sell two at the money calls, or vice versa
* Straddle - holding a position in both a call and put with the same strike price and expiration. If the options have been bought, the holder has a long straddle. If the options were sold, the holder has a short straddle. The long straddle is profitable if the underlying stock changes value in a significant way, either higher or lower. The short straddle is profitable when there is no such significant move.
* Strangle - the simultaneous buying or selling of out-of-the-money put and an out-of-the-money call, with the same expirations. Similar to the straddle, but with different strike prices.
* Risk Reversal

### Bullish on volatility

Neutral trading strategies that are bullish on volatility profit when the underlying stock price experiences big moves upwards or downwards. They include the long straddle, long strangle, short condor and short butterfly.

### Bearish on volatility

Neutral trading strategies that are bearish on volatility profit when the underlying stock price experiences little or no movement. Such strategies include the short straddle, short strangle, ratio spreads, long condor and long butterfly.
In finance, a butterfly is a limited risk, non-directional options strategy that is designed to have a large probability of earning a small limited profit when the future volatility of the underlying is expected to be different from the implied volatility.

## Long butterfly

A long butterfly position will make profit if the future volatility is lower than the implied volatility.
A long butterfly options strategy consists of the following options:
* Long 1 call with a strike price of (X − a)
* Short 2 calls with a strike price of X
* Long 1 call with a strike price of (X + a)
where X = the spot price and a > 0.
Using put–call parity a long butterfly can also be created as follows:
* Long 1 put with a strike price of (X + a)
* Short 2 puts with a strike price of X
* Long 1 put with a strike price of (X − a)
where X = the spot price and a > 0.
All the options have the same expiration date.
At expiration the value (but not the profit) of the butterfly will be:
* zero if the price of the underlying is below (X − a) or above (X + a)
* positive if the price of the underlying is between (X - a) and (X + a)
The maximum value occurs at X (see diagram).

## Short butterfly

A short butterfly position will make profit if the future volatility is higher than the implied volatility.
A short butterfly options strategy consists of the same options as a long butterfly. However all the long option positions are short and all the short option positions are long.

## Variations of the butterfly

The double option position in the middle is called the body, while the two other positions are called the wings.
The option strategy where the middle two positions have different strike price is known as an Iron condor.
In an unbalanced butterfly the variable "a" has two different values.

## References

* McMillan, Lawrence G. (2002). Options as a Strategic Investment (4th ed. ed.). New York : New York Institute of Finance. ISBN 0-7352-0197-8. {{cite book}}: |edition= has extra text (help)

## External links

* Butterfly Screener
* Long and Short Butterflies graphically illustrates component options in long and short butterflies.
* Butterfly Spreads - Spread Your Wings & Profit things you should know about Butterfly Spreads
* - Know More about Butterfly Option Trading Strategy via real trade example

## References

* McMillan, Lawrence G. (2002). Options as a Strategic Investment (4th ed. ed.). Prentice Hall. ISBN 0-7352-0197-8. {{cite book}}: |edition= has extra text (help)

## External links

* Chicago Board Options Exchange
* Option Trading Strategies

# Hedging and Investment Strategies
## Arbitrage

Arbitrage is the practice of taking advantage of a price difference between two or more markets: striking a combination of matching deals that capitalize upon the imbalance, the profit being the difference between the market prices. Arbitrage is a transaction that involves no negative cash flow at any probabilistic or temporal state and a positive cash flow in at least one state; in simple terms, it is the possibility of a risk-free profit at zero cost.
In finance an option strategy is the purchase and/or sale of one or various option positions and possibly an underlying position.
Options strategies can favor movements in the underlying that are bullish, bearish or neutral. In the case of neutral strategies, they can be further classified into those that are bullish on volatility and those that are bearish on volatility. The option positions used can be long and/or short positions in call and/or put option at various strike.

## Bullish strategies

Bullish options strategies are employed when the options trader expects the underlying stock price to move upwards. It is necessary to assess how high the stock price can go and the time frame in which the rally will occur in order to select the optimum trading strategy.
The most bullish of options trading strategies is the simple call buying strategy used by most novice options traders.
Stocks seldom go up by leaps and bounds. Moderately bullish options traders usually set a target price for the bull run and utilize bull spreads to reduce cost. (It does not reduce risk because the options can still expire worthless.) While maximum profit is capped for these strategies, they usually cost less to employ for a given nominal amount of exposure. The bull call spread and the bull put spread are common examples of moderately bullish strategies.
Mildly bullish trading strategies are options strategies that make money as long as the underlying stock price does not go down by the option's expiration date. These strategies may provide a small downside protection as well. Writing out-of-the-money covered calls is a good example of such a strategy.

## Bearish strategies

Bearish options strategies are the mirror image of bullish strategies. They are employed when the options trader expects the underlying stock price to move downwards. It is necessary to assess how low the stock price can go and the time frame in which the decline will happen in order to select the optimum trading strategy.
The most bearish of options trading strategies is the simple put buying strategy utilized by most novice options traders.
Stock prices only occasionally make steep downward moves. Moderately bearish options traders usually set a target price for the expected decline and utilize bear spreads to reduce cost. While maximum profit is capped for these strategies, they usually cost less to employ. The bear call spread and the bear put spread are common examples of moderately bearish strategies.
Mildly bearish trading strategies are options strategies that make money as long as the underlying stock price does not go up by the options expiration date. These strategies may provide a small upside protection as well. In general, bearish strategies yield less profit with less risk of loss.

## Neutral or non-directional strategies

Neutral strategies in options trading are employed when the options trader does not know whether the underlying stock price will rise or fall. Also known as non-directional strategies, they are so named because the potential to profit does not depend on whether the underlying stock price will go upwards or downwards. Rather, the correct neutral strategy to employ depends on the expected volatility of the underlying stock price.
Examples of neutral strategies are:
* Guts - sell in the money put and call
* Butterfly - buy in the money and out of the money call, sell two at the money calls, or vice versa
* Straddle - holding a position in both a call and put with the same strike price and expiration. If the options have been bought, the holder has a long straddle. If the options were sold, the holder has a short straddle. The long straddle is profitable if the underlying stock changes value in a significant way, either higher or lower. The short straddle is profitable when there is no such significant move.
* Strangle - the simultaneous buying or selling of out-of-the-money put and an out-of-the-money call, with the same expirations. Similar to the straddle, but with different strike prices.
* Risk Reversal

### Bullish on volatility

Neutral trading strategies that are bullish on volatility profit when the underlying stock price experiences big moves upwards or downwards. They include the long straddle, long strangle, short condor and short butterfly.

### Bearish on volatility

Neutral trading strategies that are bearish on volatility profit when the underlying stock price experiences little or no movement. Such strategies include the short straddle, short strangle, ratio spreads, long condor and long butterfly.

## References

* McMillan, Lawrence G. (2002). Options as a Strategic Investment (4th ed. ed.). Prentice Hall. ISBN 0-7352-0197-8. {{cite book}}: |edition= has extra text (help)

## External links

* Chicago Board Options Exchange
* Option Trading Strategies

# Sharpe ratio
## Sharpe ratio

Sharpe ratio is defined as follows:
    $S={\frac {{\bar {r_{p}}}-r_{f}}{\sigma }}$
where
    ${\bar {r_{p}}}$: Expected portfolio return
    $r_{f}$:Risk free return rate (such as that of the 10-year US treasury bond)
    $\sigma$:Portfolio standard deviation
Sharpe ratio is the equation above developed by Nobel laureate William F. Sharpe to measure risk-adjusted performance. The Sharpe ratio tells us whether a investment is smart or not. The higher the sharpe ratio is, the more return the asset gives for the same risk. In other words, the higher the sharpe ratio is, the better the investment becomes.
Graphically, you're on the straight line joining cash to the Efficient Frontier, and moving to the left.

# Glossary
## Notation and terminology

See here for notation and terminology used on the FM exam.

## Glossary and index of terms

Accumulation function Accumulation of discount American option Amortization Annuity-due Annuity-immediate Arbitrage Arithmetic increasing/decreasing payment annuity Ask price At-the-money Bermudan option Bid price Bid-ask spread Book value Call option Callable Cash-flow matching Collars (including zero-cost collars) Compound interest Convertible bond Convertible m-thly Convexity Cost of carry Coupon Coupon rate Covered call Credit risk Current value Deferred swap Derivative Discount factor Discount rate (rate of discount) Diversifiable risk Dividends Dollar-weighted rate of return Duration (Macaulay modified and effective) Effective rate Equation of value European option Expiration Expiration date Face value Final payment Force of interest Forward contract Forward rate Fully leveraged purchase Future value Futures contract Geometric increasing/decreasing payment annuity Hedging Immunization Implied repo rate In-the-money Inflation and real rate of interest Interest Interest rate (rate of interest) Interest rate swap Lease rate Level payment annuity Long position Maintenance margin Mandatorily convertible bond Margin Margin call Marking-to-market Naked writing Net profit/payoff Nominal rate Nondiversifiable risk Notional Amount Out-of-the-money Outright purchase Outstanding balance Over-the-counter market Par Value Payable continuously Payable m-thly Paylater strategy Perpetuity Portfolio and investment year allocation methods Prepaid forward contract Prepaid swap Present value/net present value Price Principal Put option Put-call parity Redemption value Redington immunization Short position Short selling Simple commodity swap Simple interest Sinking fund Spot price Spot rate Spreads Stock index Stock price Straddles (including strangles written straddles and butterfly spreads) Strike price/Exercise price Swap Swap spread Swap term Synthetic forwards Term of annuity Term of bond Term of loan Time-weighted rate of return Underlying asset Yield curve Yield rate Yield rate balloon payment bear box bull drop payment full immunization future value non-callable present value rate of return ratio spreads stock dividend

# Formulas
## Basic Formulas

    $\ a(t)$ : Accumulation function. Measures the amount in a fund with an investment of 1 at time 0 at the end of period t.
    $\ a(t)-a(t-1)$ :amount of growth in period t.
    $\ s_{t}={\frac {a(t)-a(t-1)}{a(t-1)}}$ : rate of growth in period t, also known as the effective rate of interest in period t.
    $\ A(t)=k\cdot a(t)$ : Amount function. Measures the amount in a fund with an investment of k at time 0 at the end of period t. It is simply the constant k times the accumulation function.

### Common Accumulation Functions

    $\ a(t)=1+i\cdot t$ : simple interest.
    $\ a(t)=\prod _{j=1}^{t}(1+i_{j})$ : variable interest
    $\ a(t)=(1+i)^{t}$ : compound interest.
    $\ a(t)=e^{t\cdot i}$ : continuous interest.

### Present Value and Discounting

    $\ PV={\frac {1}{a(t)}}={\frac {1}{(1+i)^{t}}}=(1+i)^{-t}=v^{t}$
    $\ d_{t}={\frac {a(t)-a(t-1)}{a(t)}}$ effective rate of discount in year t.
    $\ 1-d=v$
    $\ d={\frac {i}{1+i}}=i\cdot v$
    $\ i={\frac {d}{1-d}}$

### Nominal Interest and Discount

    $i^{(m)}$ and $d^{(m)}$ are the symbols for nominal rates of interest compounded m-thly.
    $1+i=\left(1+{\frac {i^{(m)}}{m}}\right)^{m}$
    $i^{(m)}=m\left((1+i)^{\frac {1}{m}}-1\right)$
    $1-d=\left(1-{\frac {d^{(m)}}{m}}\right)^{m}$
    $d^{(m)}=m(1-(1-d)^{\frac {1}{m}})$

### Force of Interest

    $\delta _{t}={\frac {1}{a(t)}}{\frac {d}{dt}}a(t)={\frac {d}{dt}}\ln a(t)$ : definition of force of interest.
    $a(t)=e^{\int _{0}^{t}\delta _{r}\,dr}$
If the Force of Interest is Constant: $a(t)=e^{\delta t}$
    $PV=e^{-\delta t}$
    $\delta =\ln(1+i)$

### Annuities

    $a_{\overline {n|}}={\frac {1-v^{n}}{i}}=v+v^{2}+\cdots +v^{n}$ : PV of an annuity-immediate.
    ${\ddot {a}}_{\overline {n|}}={\frac {1-v^{n}}{d}}=1+v+v^{2}+\cdots +v^{n-1}$ : PV of an annuity-due.
    ${\ddot {a}}_{\overline {n|}}=(1+i)a_{\overline {n|}}=1+a_{\overline {n-1|}}$
    $s_{\overline {n|}}={\frac {(1+i)^{n}-1}{i}}=(1+i)^{n-1}+(1+i)^{n-2}+\cdots +1$ : AV of an annuity-immediate (on the date of the last deposit).
    ${\ddot {s}}_{\overline {n|}}={\frac {(1+i)^{n}-1}{d}}=(1+i)^{n}+(1+i)^{n-1}+\cdots +(1+i)$ : AV of an annuity-due (one period after the date of the last deposit).
    ${\ddot {s}}_{\overline {n|}}=(1+i)s_{\overline {n|}}=s_{\overline {n+1|}}-1$
    $a_{\overline {mn|}}=a_{\overline {n|}}+v^{n}a_{\overline {n|}}+v^{2n}a_{\overline {n|}}+\cdots +v^{(m-1)n}a_{\overline {n|}}$

#### Perpetuities

    $\lim _{n\to \infty }a_{\overline {n|}}=\lim _{n\to \infty }{\frac {1-v^{n}}{i}}={\frac {1}{i}}=v+v^{2}+\cdots =a_{\overline {\infty |}}$ : PV of a perpetuity-immediate.
    $\lim _{n\to \infty }{\ddot {a}}_{\overline {n|}}=\lim _{n\to \infty }{\frac {1-v^{n}}{d}}={\frac {1}{d}}=1+v+v^{2}+\cdots ={\ddot {a}}_{\overline {\infty |}}$ : PV of a perpetuity-due.
    ${\ddot {a}}_{\overline {\infty |}}-a_{\overline {\infty |}}={\frac {1}{d}}-{\frac {1}{i}}=1$

### m-thly Annuities & Perpetuities

$a_{\overline {n|}}^{(m)}={\frac {1-v^{n}}{i^{(m)}}}={\frac {i}{i^{(m)}}}a_{\overline {n|}}=s_{\overline {1|}}^{(m)}a_{\overline {n|}}$ : PV of an n-year annuity-immediate of 1 per year payable in m-thly installments.
${\ddot {a}}_{\overline {n|}}^{(m)}={\frac {1-v^{n}}{d^{(m)}}}={\frac {i}{d^{(m)}}}a_{\overline {n|}}={\ddot {s}}_{\overline {1|}}^{(m)}a_{\overline {n|}}$ : PV of an n-year annuity-due of 1 per year payable in m-thly installments.
$s_{\overline {n|}}^{(m)}={\frac {(1+i)^{n}-1}{i^{(m)}}}$ : AV of an n-year annuity-immediate of 1 per year payable in m-thly installments.
${\ddot {s}}_{\overline {n|}}^{(m)}={\frac {(1+i)^{n}-1}{d^{(m)}}}$ : AV of an n-year annuity-due of 1 per year payable in m-thly installments.
$\lim _{n\to \infty }a_{\overline {n|}}^{(m)}=\lim _{n\to \infty }{\frac {1-v^{n}}{i^{(m)}}}={\frac {1}{i^{(m)}}}=a_{\overline {\infty |}}^{(m)}$ : PV of a perpetuity-immediate of 1 per year payable in m-thly installments.
$\lim _{n\to \infty }{\ddot {a}}_{\overline {n|}}^{(m)}=\lim _{n\to \infty }{\frac {1-v^{n}}{d^{(m)}}}={\frac {1}{d^{(m)}}}={\ddot {a}}_{\overline {\infty |}}^{(m)}$ : PV of a perpetuity-due of 1 per year payable in m-thly installments.
${\ddot {a}}_{\overline {\infty |}}^{(m)}-a_{\overline {\infty |}}^{(m)}={\frac {1}{d^{(m)}}}-{\frac {1}{i^{(m)}}}={\frac {1}{m}}$

#### Continuous Annuities

Since $\lim _{m\to \infty }i^{(m)}=\lim _{m\to \infty }d^{(m)}=\delta$,
$\lim _{m\to \infty }a_{\overline {n|}}^{(m)}=\lim _{m\to \infty }{\frac {1-v^{n}}{i^{(m)}}}={\frac {1-v^{n}}{\delta }}={\overline {a}}_{\overline {n|}}={\frac {i}{\delta }}a_{\overline {n|}}$ : PV of an annuity (immediate or due) of 1 per year paid continuously.
Payments in Arithmetic Progression: In general, the PV of a series of $n$ payments, where the first payment is $P$ and each additional payment increases by $Q$ can be represented by: $A=Pa_{\overline {n|}}+Q{\frac {a_{\overline {n|}}-nv^{n}}{i}}=Pv+(P+Q)v^{2}+(P+2Q)v^{3}+\cdots +(P+(n-1)Q)v^{n}$
Similarly: ${\ddot {A}}=P{\ddot {a}}_{\overline {n|}}+Q{\frac {a_{\overline {n|}}-nv^{n}}{d}}$
$S=Ps_{\overline {n|}}+Q{\frac {s_{\overline {n|}}-n}{i}}$ : AV of a series of $n$ payments, where the first payment is $P$ and each additional payment increases by $Q$.
${\ddot {S}}=P{\ddot {s}}_{\overline {n|}}+Q{\frac {s_{\overline {n|}}-n}{d}}$
$(Ia)_{\overline {n|}}={\frac {{\ddot {a}}_{\overline {n|}}-nv^{n}}{i}}$ : PV of an annuity-immediate with first payment 1 and each additional payment increasing by 1; substitute $d$ for $i$ in denominator to get due form.
$(Is)_{\overline {n|}}={\frac {{\ddot {s}}_{\overline {n|}}-n}{i}}$ : AV of an annuity-immediate with first payment 1 and each additional payment increasing by 1; substitute $d$ for $i$ in denominator to get due form.
$(Da)_{\overline {n|}}={\frac {n-{a}_{\overline {n|}}}{i}}$ : PV of an annuity-immediate with first payment $n$ and each additional payment decreasing by 1; substitute $d$ for $i$ in denominator to get due form.
$(Ds)_{\overline {n|}}={\frac {n(1+i)^{n}-{s}_{\overline {n|}}}{i}}$ : AV of an annuity-immediate with first payment $n$ and each additional payment decreasing by 1; substitute $d$ for $i$ in denominator to get due form.
$(Ia)_{\overline {\infty |}}={\frac {1}{id}}={\frac {1}{i}}+{\frac {1}{i^{2}}}$ : PV of a perpetuity-immediate with first payment 1 and each additional payment increasing by 1.
$(I{\ddot {a}})_{\overline {\infty |}}={\frac {1}{d^{2}}}$ : PV of a perpetuity-due with first payment 1 and each additional payment increasing by 1.
$(Ia)_{\overline {n|}}+(Da)_{\overline {n|}}=(n+1)a_{\overline {n|}}$
Additional Useful Results: ${\frac {P}{i}}+{\frac {Q}{i^{2}}}$ : PV of a perpetuity-immediate with first payment $P$ and each additional payment increasing by $Q$.
$(Ia)_{\overline {n|}}^{(m)}={\frac {{\ddot {a}}_{\overline {n|}}-nv^{n}}{i^{(m)}}}$ : PV of an annuity-immediate with m-thly payments of ${\frac {1}{m}}$ in the first year and each additional year increasing until there are m-thly payments of ${\frac {n}{m}}$ in the nth year.
$(I^{(m)}a)_{\overline {n|}}^{(m)}={\frac {{\ddot {a}}_{\overline {n|}}^{(m)}-nv^{n}}{i^{(m)}}}$ : PV of an annuity-immediate with payments of ${\frac {1}{m^{2}}}$ at the end of the first mth of the first year, ${\frac {2}{m^{2}}}$ at the end of the second mth of the first year, and each additional payment increasing until there is a payment of ${\frac {mn}{m^{2}}}$ at the end of the last mth of the nth year.
$({\overline {I}}{\overline {a}})_{\overline {n|}}={\frac {{\overline {a}}_{\overline {n|}}-nv^{n}}{\delta }}$ : PV of an annuity with continuous payments that are continuously increasing. Annual rate of payment is $t$ at time $t$.
$\int _{0}^{n}f(t)v^{t}dt$ : PV of an annuity with a continuously variable rate of payments and a constant interest rate.
$\int _{0}^{n}f(t)e^{-\int _{0}^{t}\delta _{r}dr}dt$ : PV of an annuity with a continuously variable rate of payment and a continuously variable rate of interest.

### Payments in Geometric Progression

${\frac {1-({\frac {1+k}{1+i}})^{n}}{i-k}}$ : PV of an annuity-immediate with an initial payment of 1 and each additional payment increasing by a factor of $(1+k)$. Chapter 5

### General Definitions

$R_{t}$ : payment at time $t$. A negative value is an investment and a positive value is a return.
$P(i)=\sum {v^{t}R_{t}}$ : PV of a cash flow at interest rate $i$. Chapter 6
$R_{t}=I_{t}+P_{t}$ : payment made at the end of year $t$, split into the interest $I_{t}$ and the principal repaid $P_{t}$.
$I_{t}=iB_{t-1}$ : interest paid at the end of year $t$.
$P_{t}=R_{t}-I_{t}=(1+i)P_{t-1}+(R_{t}-R_{t-1})$ : principal repaid at the end of year $t$.
$B_{t}=B_{t-1}-P_{t}$ : balance remaining at the end of year $t$, just after payment is made.
On a Loan Being Paid with Level Payments:
$I_{t}=1-v^{n-t+1}$ : interest paid at the end of year $t$ on a loan of $a_{\overline {n|}}$.
$P_{t}=v^{n-t+1}$ : principal repaid at the end of year $t$ on a loan of $a_{\overline {n|}}$.
$B_{t}=a_{\overline {n-t|}}$ : balance remaining at the end of year $t$ on a loan of $a_{\overline {n|}}$, just after payment is made.
For a loan of $L$, level payments of ${\frac {L}{a_{\overline {n|}}}}$ will pay off the loan in $n$ years. To scale the interest, principal, and balance owed at time $t$, multiply the above formulas for $I_{t}$, $P_{t}$, and $B_{t}$ by ${\frac {L}{a_{\overline {n|}}}}$, ie $B_{t}={\frac {L}{a_{\overline {n|}}}}a_{\overline {n-t|}}$ etc.

## Yield Rates

    $I=B-A-C$
    $i={\frac {I}{A+\sum _{t_{k}}C_{t_{k}}(1-t_{k})}}$ : dollar-weighted
    $(1+i)=\prod _{t_{k}}^{t}\left({\frac {B_{t_{k}}}{B_{t_{k-1}}+C_{t_{k-1}}}}\right)$ : time-weighted

## Sinking Funds

$PMT=Li+{\frac {L}{s_{{\overline {n|}}j}}}$ : total yearly payment with the sinking fund method, where $Li$ is the interest paid to the lender and ${\frac {L}{s_{{\overline {n|}}j}}}$ is the deposit into the sinking fund that will accumulate to $L$ in $n$ years. $i$ is the interest rate for the loan and $j$ is the interest rate that the sinking fund earns.
$L=(PMT-Li)s_{{\overline {n|}}j}$

## Bonds

Definitions: $P$ : Price paid for a bond.
$F$ : Par/face value of a bond.
$C$ : Redemption value of a bond.
$r$ : coupon rate for a bond.
$g={\frac {Fr}{C}}$ : modified coupon rate.
$i$ : yield rate on a bond.
$K$ : PV of $C$.
$n$ : number of coupon payments.
$G={\frac {Fr}{i}}$ : base amount of a bond.
$Fr=Cg$

### Determination of Bond Prices

$P=Fra_{{\overline {n|}}i}+Cv^{n}=Cga_{{\overline {n|}}i}+Cv^{n}$ : price paid for a bond to yield $i$.
$P=C+(Fr-Ci)a_{{\overline {n|}}i}=C+(Cg-Ci)a_{{\overline {n|}}i}$ : Premium/Discount formula for the price of a bond.
$P-C=(Fr-Ci)a_{{\overline {n|}}i}=(Cg-Ci)a_{{\overline {n|}}i}$ : premium paid for a bond if $g>i$.
$C-P=(Ci-Fr)a_{{\overline {n|}}i}=(Ci-Cg)a_{{\overline {n|}}i}$ : discount paid for a bond if $g<i$.
Bond Amortization: When a bond is purchased at a premium or discount the difference between the price paid and the redemption value can be amortized over the remaining term of the bond. Using the terms from chapter 6: $R_{t}$ : coupon payment.
$I_{t}=iB_{t-1}$ : interest earned from the coupon payment.
$P_{t}=R_{t}-I_{t}=(Fr-Ci)v^{n-t+1}=(Cg-Ci)v^{n-t+1}$ : adjustment amount for amortization of premium ("write down") or
$P_{t}=I_{t}-R_{t}=(Ci-Fr)v^{n-t+1}=(Ci-Cg)v^{n-t+1}$ : adjustment amount for accumulation of discount ("write up").
$B_{t}=B_{t-1}-P_{t}$ : book value of bond after adjustment from the most recent coupon paid.
Price Between Coupon Dates: For a bond sold at time $k$ after the coupon payment at time $t$ and before the coupon payment at time $t+1$: $B_{t+k}^{f}=B_{t}(1+i)^{k}=(B_{t+1}+Fr)v^{1-k}$ : "flat price" of the bond, ie the money that actually exchanges hands on the sale of the bond.
$B_{t+k}^{m}=B_{t+k}^{f}-kFr=B_{t}(1+i)^{k}-kFr$ : "market price" of the bond, ie the price quoted in a financial newspaper.
Approximations of Yield Rates on a Bond: $i\approx {\frac {nFr+C-P}{{\frac {n}{2}}(P+C)}}$ : Bond Salesman's Method.
Price of Other Securities: $P={\frac {Fr}{i}}$ : price of a perpetual bond or preferred stock.
$P={\frac {D}{i-k}}$ : theoretical price of a stock that is expected to return a dividend of $D$ with each subsequent dividend increasing by $(1+k)$, $k<i$. Chapter 9
Recognition of Inflation: $i'={\frac {i-r}{1+r}}$ : real rate of interest, where $i$ is the effective rate of interest and $r$ is the rate of inflation.

## Method of Equated Time and (Macaulay) Duration

${\overline {t}}={\frac {\sum _{t=1}^{n}tR_{t}}{\sum _{t=1}^{n}R_{t}}}$ : method of equated time.
${\overline {d}}={\frac {\sum _{t=1}^{n}tv^{t}R_{t}}{\sum _{t=1}^{n}v^{t}R_{t}}}$ : (Macauley) duration.

#### Duration

$P(i)=\sum {v^{t}R_{t}}$ : PV of a cash flow at interest rate $i$.
${\overline {v}}=-{\frac {P'(i)}{P(i)}}=v{\overline {d}}={\frac {\overline {d}}{1+i}}$ : volatility/modified duration.
${\overline {d}}=-(1+i){\frac {P'(i)}{P(i)}}$ : alternate definition of (Macaulay) duration.

## Convexity and (Redington) Immunization

${\overline {c}}={\frac {P''(i)}{P(i)}}$ convexity
To achieve Redington immunization we want: $P'(i)=0$ $P''(i)>0$

## Options

Put–Call parity
    $C(t)-P(t)=S(t)-K\cdot B(t,T)\,$
where
    $C(t)$ is the value of the call at time $t$,
    $P(t)$ is the value of the put,
    $S(t)$ is the value of the share,
    $K$ is the strike price, and
    $B(t,T)$ value of a bond that matures at time $T$. If a stock pays dividends, they should be included in $B(t,T)$, because option prices are typically not adjusted for ordinary dividends.

# Problems
## Problem 1

    Drago deposits 100 into an account at the beginning of each 2-year period for 40 years. The account credits interest at an interest rate of i.
The accumulated amount in the account at the end of the 40 years is 3 times the accumulated amount in the account at the end of 20 years.
Calculate i.