# Financial Math FM/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 occurred 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.
* ↑ 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