BjornvanBraak/ActuarialMathBench huggingface dataset:
Question: Wilma takes out a loan of \(5,000,000\) to start a new business. She will repay the loan with level semiannual payments of \(303,244.53\), with the first payment due in six months.

The loan payments are based on an annual nominal interest rate of \(6.5 \%\) convertible semiannually.

Immediately after the tenth payment, the outstanding loan balance is \(X\).
Calculate \(X\).

Options:
A
\(1,483,350\)
B
\(2,350,407\)
C
\(2,916,667\)
D
\(3,367,821\)
E
\(4,330,426\)

Correct answer: D

Solution: The outstanding loan balance here can be calculated using either the prospective approach or the retrospective approach, although the retrospective approach is easier, since it does not require the value of \(n\).
\(5,000,000(1.0325)^{10}-303,244.53 s_{100.0325}=3,367,821.18\).
Question: A common stock pays a constant dividend at the end of each year into perpetuity.
Using an annual effective interest rate of \(10 \%\), calculate the Macaulay duration of the stock.

Options:
A
7 years
B
9 years
C
11 years
D
19 years
E
27 years

Correct answer: C

Solution: The size of the dividend does not matter, so assume it is 1 . Then the duration is \(\frac{\sum_{t=1}^{\infty} t v^{t}}{\sum_{t=1}^{\infty} v^{t}}=\frac{(I a)_{\infty}}{a_{\infty}}=\frac{\ddot{a}_{\infty} / i}{1 / i}=\frac{1 /(d i)}{1 / i}=\frac{1}{d}=\frac{1.1}{0.1}=11\).
Question: An insurance company has an obligation to pay the medical costs for a claimant. Annual claim costs today are 5000 , and medical inflation is expected to be \(7 \%\) per year. The claimant will receive 20 payments.

Claim payments are made at yearly intervals, with the first claim payment to be made one year from today.

Calculate the present value of the obligation using an annual effective interest rate of \(5 \%\).

Options:
A
\(\quad 87,900\)
B
102,500
C
114,600
D
122,600
E
Cannot be determined

Correct answer: D

Solution: The present value is
\(5000\left[1.07 v+1.07^{2} v^{2}+\cdots+1.07^{20} v^{20}\right]\)
\(=5000 \frac{1.07 v-1.07^{21} v^{21}}{1-1.07 v}=5000 \frac{1.01905-1.48622}{1-1.01905}=122,617\).
Question: An electric company faces liabilities with present values 24,000 and 1,000 , due 7.5 years from now and 20 years from now, respectively.

The company will purchase an \(n\)-year zero-coupon bond and a \(2 n\)-year zero-coupon bond.
Determine which of the following expressions describes the set of all values of \(n\) for which the company can achieve Redington immunization.

Options:
A
\(\quad 4 \leq n \leq 8\)
B
\(\quad 5<n<7\)
C
\(\quad n=5\) or \(n=7\)
D
\(\quad 0<n<5\) or \(n>7\)
E
\(\quad 4 \leq n<5\) or \(7<n \leq 8\)

Correct answer: B

Solution: The PV and duration of the assets must match the liabilities, \(24,000+1,000=25,000\) and \([24,000(7.5)+1,000(20)] / 25,000=8\). The convexity of the assets must be greater than \(\left[24,000\left(7.5^{2}\right)+1,000\left(20_{2}\right)\right] / 25,000=70\).

Let \(f\) represent the fraction of 25,000 that is invested in the \(n\)-year bond. The duration of the assets is \(f n+(1-f) 2 n\) and the convexity is \(f n^{2}+(1-f) 4 n^{2}\).

For duration, \(8=f n+(1-f) 2 n\) which implies \(f=2-8 / n\). Since \(f\) must be between 0 and \(1, n\) must be between 4 and 8 (inclusive).

Substituting for \(f\), for convexity:
\((2-8 / n) n^{2}+(1-2+8 / n) 4 n^{2}>70\)
\(-2 n^{2}+24 n>70\)
\(n^{2}-12 n+35<0\)
\((n-7)(n-5)<0\).
This forces \(n\) to be between 5 and 7 which is the answer.
Question: A student invested a math competition prize into a savings account. The account receives interest quarterly, and the account balance increases by \(4.5 \%\) each year.

Determine the correct term that the \(4.5 \%\) figure represents.

Options:
A
annual effective interest rate
B
force of interest
C
annual nominal interest rate, convertible quarterly
D
annual discount rate
E
annual simple interest rate

Correct answer: A

Solution: The percentage by which the balance accumulates each year is the effective interest rate.
Question: You are given the following information about a 30 -year bond:
i) The par value is 2000 .
ii) The redemption value is 2250 .
iii) Coupons are paid annually.
iv) The annual coupon rate is twice the annual yield rate.
v) The purchase price is 3609.29 .
vi) Based on the yield rate, the Macaulay duration of the bond is 14.41 years.

Calculate the modified duration of the bond, based on the yield rate.

Options:
A
12.40 years
B
13.07 years
C
13.71 years
D
14.41 years
E
15.15 years

Correct answer: C

Solution: Let \(i\) be the yield rate. Then,
\(3609.29=2000(2 i) a_{\overline{30 i}}+2250(1+i)^{-30}\)
\(=4000\left[1-(1+i)^{-30}\right]+2250(1+i)^{-30}\)
\((1+i)^{-30}=(4000-3609.29) /(4000-2250)=0.22326\)
\(i=0.22326^{-1 / 30}-1=0.051251\).
Modified duration is Macaulay duration divided by one plus the yield rate: \(14.14 / 1.051251=\) 13.71.
Question: Determine which of the following expressions represents the modified duration for a zerocoupon bond that is currently priced at an annual effective yield rate \(i\) and an \(n\)-year maturity.

Options:
A
\(n\)
B
\(n(1+i)\)
C
\(\quad n(1+i)^{-1}\)
D
\(\frac{\sum_{t=1}^{n} t(1+i)^{-t}}{\sum_{t=1}^{n}(1+i)^{t}}\)
E
\(\frac{\sum_{t=1}^{n} t(1+i)^{-t}}{\sum_{t=1}^{n}(1+i)^{-t}}\)

Correct answer: C

Solution: \(D_{\text {mod }}=-\frac{\frac{d}{d i}(1+i)^{-n}}{(1+i)^{-n}}=\frac{n(1+i)^{-n-1}}{(1+i)^{-n}}=n(1+i)^{-1}\)
Question: A trucking company with assets and liabilities needs to choose between various ten-year par value bonds each with \(8 \%\) annual effective yield rate and annual coupons. The bonds have varying face values and varying coupon rates.

The company wants to analyze the effects of face value and coupon rate changes on Macaulay duration of these bonds, in order to choose an investment strategy that immunizes its position.

Determine which of the following statements is true about the separate effects of face value and coupon rate changes on the duration of these bonds.

Options:
A
Macaulay duration increases as face value increases, and increases as coupon rate increases.
B
Macaulay duration increases as face value increases, and decreases as coupon rate increases.
C
Macaulay duration remains constant as face value increases, and increases as coupon rate increases.
D
Macaulay duration remains constant as face value increases, and remains constant as coupon rate increases.
E
Macaulay duration remains constant as face value increases, and decreases as coupon rate increases.

Correct answer: E

Solution: A change in face value multiplies all cash flows by the same amount. Therefore, there is no change in the duration. If the coupon rate increases, the coupons become larger, but the redemption value stays the same. This causes payments prior to redemption to receive more weight relative to the payment at redemption and thus the duration will decrease.
Question: Three asset-liability cash flows are given in the following table where a positive amount is an asset cash flow and a negative amount is a liability due at the corresponding time.

\begin{tabular}{|c|c|c|c|c|}
\hline\(t\) (in years) & 0 & 1 & 2 & 3 \\
\hline X & 102,400 & \(-192,000\) & 0 & 100,000 \\
\hline Y & 158,400 & \(-342,000\) & 100,000 & 100,000 \\
\hline Z & \(-89,600\) & 288,000 & 100,000 & \(-300,000\) \\
\hline
\end{tabular}

Determine which set of cash flows is Redington immunized for an annual effective interest rate of \(i=25 \%\).

Options:
A
X only
B
Y only
C
Z only
D
\(X, Y\), and \(Z\)
E
The correct answer is not given by (A), (B), (C) or (D).

Correct answer: A

Solution: Let \(h(i)\) be the present value of the cash flows. For Redington immunization, the value of the function and its first derivative at \(25 \%\) must be zero and the second derivative must be positive. X is immunized because:
\(h(0.25)=102,400-192,000 / 1.25+100,000 / 1.25^{3}=0\)
\(h^{\prime}(0.25)=192,000 / 1.25^{2}-100,000(3) / 1.25^{4}=0\)
\(h^{\prime \prime}(0.25)=-192,000(2) / 1.25^{3}+100,000(3)(4) / 1.25^{5}=196,608>0\)
Y is not immunized because:
\(h(0.25)=158,400-342,000 / 1.25+100,000 / 1.25^{2}+100,000 / 1.25^{3}=0\)
\(h^{\prime}(0.25)=342,000 / 1.25^{2}-100,000(2) / 1.25^{3}-100,000(3) / 1.25^{4}=-6,400 \neq 0\)
Z is not immunized because
\(h(0.25)=-89,600+288,000 / 1.25+100,000 / 1.25^{2}-300,000 / 1.25^{3}=51,200 \neq 0\)
Question: An insurance company aims to structure a portfolio of assets and liabilities so that:
i) Its assets and liabilities have equal present values and equal modified durations.
ii) The convexity of its assets exceeds the convexity of its liabilities.

Determine which term precisely describes the company's position.

Options:
A
Diversified
B
Fully immunized
C
Fully leveraged
D
Exactly matched
E
Redington immunized

Correct answer: E

Solution: The conditions
1) assets and liabilities have equal present values and equal modified durations, and

2 ) the convexity of its assets exceeds the convexity of its liabilities are precisely what is required for Redington immunization.
Question: Which of the following is an expression for the present value of a perpetuity with annual payments of \(1,2,3, \ldots\), where the first payment will be made at the end of \(n\) years, using an annual effective interest rate of \(i\) ?

Options:
A
\(\frac{\ddot{a}_{n}-n v^{n}}{i}\)
B
\(\frac{n-a_{\text {п }}}{i}\)
C
\(\frac{v^{n}}{d}\)
D
\(\frac{v^{n}}{d^{2}}\)
E
\(\frac{v^{n}}{d i}\)

Correct answer: D

Solution: One way to view these payments is as a sequence of level immediate perpetuities of 1 that are deferred \(n-1, n, n+1, \ldots\) years. The present value is then
\(v^{n-1} / i+v^{n} / i+v^{n+1} / i+\cdots=\left(v^{n-2} / i\right)\left(v+v^{2}+v^{3}+\cdots\right)=v^{n-2} / i^{2}\).
Noting that only answers \(\mathrm{C}, \mathrm{D}\), and E have this form and all have the same numerator, \(v^{n-2} / i^{2}=v^{n} /(v i)^{2}=v^{n} / d^{2}\) 。
Question: A construction firm is facing three liabilities of 1000 , due at times 1,2 , and 3 in years. There are three bonds available to match these liabilities, as follows:
Bond I A bond due at the end of period 1 with a coupon rate of \(1 \%\) per year, valued at a annual effective yield rate of \(14 \%\).
Bond II A bond due at the end of period 2 with a coupon rate of \(2 \%\) per year, valued at a annual effective yield rate of \(15 \%\).
Bond III A zero-coupon bond due at time 3 valued at a periodic effective yield rate of \(18 \%\).
Calculate the face value of each bond that should be purchased to exactly match the liabilities.

\begin{tabular}{lrrr} 
& Bond I & Bond II & Bond III \\

Options:
A
& 970.68 & 980.39 & 1000.00 \\
B
& 970.68 & 1000.00 & 980.39 \\
C
& 980.39 & 970.68 & 1000.00 \\
D
& 1000.00 & 980.39 & 970.68 \\
E
& 1000.00 & 1000.00 & 1000.00
\end{tabular}

Correct answer: A

Solution: Only Bond III can match the liability at time 3. The bond must mature for 1000. Only Bond II can match the liability at time 2 . The face value and coupon must total 1000 . If \(X\) is the face value, then \(X+0.02 X=1000\) and thus \(X=980.39\). Only Answer A has these to values. To check, Bond II also provides a coupon of \(0.02(980.39)=19.61\) at time 1. Therefore, Bond I must provide the remaining 980.39 from its coupon and redemption value. If \(Y\) is the face value, then \(Y+0.01 Y=980.39\) for \(Y=970.68\).
Question: Seth has two retirement benefit options.
His first option is to receive a lump sum of 374,500 at retirement.
His second option is to receive monthly payments for 25 years starting one month after retirement. For the first year, the amount of each monthly payment is 2000 . For each subsequent year, the monthly payments are \(2 \%\) more than the monthly payments from the previous year.

Using an annual nominal interest rate of \(6 \%\), compounded monthly, the present value of the second option is \(P\).

Determine which of the following is true.

Options:
A
\(\quad P\) is 323,440 more than the lump sum option amount.
B
\(\quad P\) is 107,170 more than the lump sum option amount.
C
The lump sum option amount is equal to \(P\).
D
The lump sum option amount is 60 more than \(P\).
E
The lump sum option amount is 64,090 more than \(P\).

Correct answer: D

Solution: The accumulated value of the first year of payments is \(2000 s_{\overline{12} \mid 0.005}=24,671.12\). This amount increases at \(2 \%\) per year. The effective annual interest rate is \(1.005^{12}-1=0.061678\). The present value is then
\(P=24,671.12 \sum_{k=1}^{25} 1.02^{k-1}(1.061678)^{-k}=24,671.12 \frac{1}{1.02} \sum_{k=1}^{25}\left(\frac{1.02}{1.061678}\right)^{k}\)
\(=24,187.37 \frac{0.960743-0.960743^{26}}{1-0.960743}=374,444\).
This is 56 less than the lump sum amount.
Question: At an annual effective interest rate of \(10.9 \%\), each of the following are equal to \(X\) :
- The accumulated value at the end of \(n\) years of an \(n\)-year annuity-immediate paying 21.80 per year.
- The present value of a perpetuity-immediate paying 19,208 at the end of each \(n\)-year period.

Calculate X.

Options:
A
1555
B
1750
C
1960
D
2174
E
There is not enough information given to calculate X .

Correct answer: C

Solution: From the first annuity, \(X=21.8 s_{\bar{n} 0.109}=21.8 \cdot \frac{1.109^{n}-1}{0.109}=200\left[1.109^{n}-1\right]\).
From the second annuity, \(X=19,208\left(v^{n}+v^{2 n}+\cdots\right)=19,208 \frac{v^{n}}{1-v^{n}}=19,208 \frac{1}{1.109^{n}-1}\).
Hence,
\(200\left[1.109^{n}-1\right]=19,208 \frac{1}{1.109^{n}-1}\)
\(\left[1.109^{n}-1\right]^{2}=19,208 / 200=96.04\)
\(1.109^{n}-1=9.8\)
\(X=200(9.8)=1960\).
Question: Determine which of the following statements regarding asset-liability management techniques is true.

Options:
A
Redington immunization requires that the convexity of the liabilities is greater than the convexity of the assets.
B
An advantage of the Redington immunization technique over the cash-flow matching technique is that the portfolio manager has more investment choices available.
C
Both Redington immunization and full immunization are based on the assumption that the yield curve has higher yields for longer term investments.
D
A fully immunized portfolio ensures that the present value of assets will exceed the present value of liabilities with non-parallel shifts in the yield curve.
E
A cash-flow matched portfolio requires less rebalancing than a Redington immunized portfolio, but more rebalancing than a fully immunized portfolio.

Correct answer: B

Solution: Cash-flow matching limits the number of investment choices available to the portfolio manager to a subset of the choices available for immunization.
Question: A company owes 500 and 1000 to be paid at the end of year one and year four, respectively. The company will set up an investment program to match the duration and the present value of the above obligation using an annual effective interest rate of \(10 \%\).

The investment program produces asset cash flows of \(X\) today and \(Y\) in three years.
Calculate \(X\) and determine whether the investment program satisfies the conditions for Redington immunization.

Options:
A
\(X=75\) and the Redington immunization conditions are not satisfied.
B
\(X=75\) and the Redington immunization conditions are satisfied.
C
\(X=1138\) and the Redington immunization conditions are not satisfied.
D
\(X=1138\) and the Redington immunization conditions are satisfied.
E
\(\quad X=1414\) and the Redington immunization conditions are satisfied.

Correct answer: A

Solution: The present value function and its derivatives are
\(P(i)=X+Y(1+i)^{-3}-500(1+i)^{-1}-1000(1+i)^{-4}\)
\(P^{\prime}(i)=-3 Y(1+i)^{-4}+500(1+i)^{-2}+4000(1+i)^{-5}\)
\(P^{\prime \prime}(i)=12 Y(1+i)^{-5}-1000(1+i)^{-3}-20,000(1+i)^{-6}\).
The equations to solve for matching present values and duration (at \(i=0.10\) ) and their solution are
\(P(0.1)=X+0.7513 Y-1137.56=0\)
\(P^{\prime}(0.1)=-2.0490 Y+2896.91=0\)
\(Y=2896.91 / 2.0490=1413.82\)
\(X=1137.56-0.7513(1413.82)=75.36\).
The second derivative is
\(P^{\prime \prime}(0.1)=12(1413.82)(1.1)^{-5}-1000(1.1)^{-3}-20,000(1.1)^{-6}=-1506.34\).
Redington immunization requires a positive value for the second derivative, so the condition is not satisfied.
Question: A company is considering a project where it expects to pay and receive the following cash flows:

\begin{tabular}{c|l}
\begin{tabular}{c} 
End of \\
year
\end{tabular} & Cash flow \\
\hline 2 & Receives \(X\) \\
4 & Pays \(Y\) \\
6 & Receives \(X\)
\end{tabular}

When calculated using an annual effective rate of \(i(i>0)\), the present value of this project is equal to 0 .

Determine \(i\).

Options:
A
\(\frac{Y \pm \sqrt{Y^{2}-4 X^{2}}}{2 X}\)
B
\(\frac{Y \pm \sqrt{Y^{2}-4 X^{2}}}{2 X}-1\)
C
\(\sqrt{\frac{-Y \pm \sqrt{Y^{2}-4 X^{2}}}{2 X}}-1\)
D
\(\sqrt{\frac{2 X}{-Y \pm \sqrt{Y^{2}-4 X^{2}}}}-1\)
E
\(\quad \sqrt{\frac{2 X}{Y \pm \sqrt{Y^{2}-4 X^{2}}}}-1\)

Correct answer: E

Solution: \(X v^{2}-Y v^{4}+X v^{6}=0\)
\(X v^{4}-Y v^{2}+X=0\)
\(v^{2}=\frac{Y \pm \sqrt{Y^{2}-4 X^{2}}}{2 X}\)
\((1+i)^{2}=\frac{2 X}{Y \pm \sqrt{Y^{2}-4 X^{2}}}\)
\((1+i)=\sqrt{\frac{2 X}{Y \pm \sqrt{Y^{2}-4 X^{2}}}}\)
\(i=\sqrt{\frac{2 X}{Y \pm \sqrt{Y^{2}-4 X^{2}}}}-1\)
Question: A two-year annuity-immediate with level annual payments has a Macaulay duration of \(D\) using a force of interest of \(\delta\).

Determine \(D\).

Options:
A
\(\frac{1+2 e^{-\delta}}{1+e^{-\delta}}\)
B
\(\frac{1+2 e^{-2 \delta}}{1+e^{-\delta}}\)
C
\(\frac{1+e^{-2 \delta}}{1+e^{-\delta}}\)
D
\(\frac{1+e^{-\delta}}{1+e^{-2 \delta}}\)
E
\(\frac{1+2 e^{-\delta}}{1+e^{-2 \delta}}\)

Correct answer: A

Solution: The Macaulay duration is
\(\frac{e^{-\delta}+2 e^{-2 \delta}}{e^{-\delta}+e^{-2 \delta}}=\frac{1+2 e^{-\delta}}{1+e^{-\delta}}\).
Question: A company is required to pay 500,000 ten years from now and 500,000 fifteen years from now. The company needs to create an investment portfolio using 5 -year and 20 -year zero-coupon bonds, so that, using a \(7 \%\) annual force of interest, the present value and Macaulay duration of its assets match those of its liabilities.

Calculate the amount invested today in each bond.

Options:
A
211,631 for the 5-year bond and 211,631 for the 20-year bond
B
217,699 for the 5-year bond and 217,699 for the 20-year bond
C
223,852 for the 5-year bond and 199,410 for the 20-year bond
D
229,857 for the 5-year bond and 205,540 for the 20-year bond
E
248,293 for the 5-year bond and 174,969 for the 20-year bond

Correct answer: C

Solution: Let \(x\) and \(y\) be the amount invested in the five and twenty year bonds respectively. To match the present values: \(x+y=500,000 e^{-0.07(10)}+500,000 e^{-0.07(15)}=423,262\). To match the durations, noting that the denominators of the durations for assets and liabilities are the same, \(5 x+20 y=500,000(10) e^{-0.07(10)}+500,000(15) e^{-0.07(15)}=5,107,460\). Subtracting five times the first equation from the second one gives \(15 y=2,991,150\) for \(y=199,410\) and \(x=423,262- 199,410=223,852\).
Question: A 15-year bond is purchased at a premium. The bond has an annual effective yield rate of \(7.4 \%\) and pays semi-annual coupons.

Determine which of the following statements about this bond must be true.

Options:
A
The face amount is greater than the purchase price.
B
The redemption value is greater than the purchase price.
C
The redemption value is greater than the face amount.
D
The face amount is greater than the redemption value.
E
The purchase price is greater than the redemption value.

Correct answer: E

Solution: The definition of "purchased at premium" is \(\mathrm{P}>\mathrm{C}\); purchase price greater than redemption value.
Question: A company has liabilities of 402.11 due at the end of each of the next three years. The company will invest 1000 today to fund these payouts. The only investments available are one-year and three-year zero-coupon bonds, and the yield curve is flat at a \(10 \%\) annual effective rate. The company wishes to match the duration of its assets to the duration of its liabilities.

Determine how much the company should invest in each bond.

Options:
A
366 in the one-year bond and 634 in the three-year bond.
B
484 in the one-year bond and 516 in the three-year bond.
C
500 in the one-year bond and 500 in the three-year bond.
D
532 in the one-year bond and 468 in the three-year bond.
E
634 in the one-year bond and 366 in the three-year bond.

Correct answer: D

Solution: The present value of the liabilities is 1000 , so that requirement is met. The duration of the liabilities is \(402.11\left[1.1^{-1}+2(1.1)^{-2}+3(1.1)^{-3}\right] / 1000=1.9365\). Let \(X\) be the investment in the oneyear bond. The duration of a zero-coupon is its term. The duration of the two bonds is then \([X+ (1000-X)(3)] / 1000=3-0.002 X\). Setting this equal to 1.9365 and solving yields \(X=531.75\).
Question: Ron is repaying a loan with payments of 1 at the end of each year for n years. The annual effective interest rate on the loan is \(i\). The amount of interest paid in year \(t\) plus the amount of principal repaid in year \(t+1\) equals \(X\).

Determine which of the following is equal to \(X\).

Options:
A
\(1+\frac{v^{n-t}}{i}\)
B
\(1+\frac{v^{n-t}}{d}\)
C
\(1+v^{n-t} i\)
D
\(1+v^{n-t} d\)
E
\(1+v^{n-t}\)

Correct answer: D

Solution: Year \(t\) interest is \(i a_{\overline{n-t+1} i}=1-v^{n-t+1}\).
Year \(t+1\) principal repaid is \(1-\left(1-v^{n-t}\right)=v^{n-t}\).
\(X=1-v^{n-t+1}+v^{n-t}=1+v^{n-t}(1-v)=1+v^{n-t} d\).
Question: The prices for four 1000 face amount zero-coupon bonds are as follows:

\begin{tabular}{c|c} 
Price & \begin{tabular}{c} 
Term \\
(in years)
\end{tabular} \\
\hline 943.40 & 1 \\
747.26 & 5 \\
558.39 & 10 \\
311.80 & 20
\end{tabular}

Determine which of the following statements describes the yield curve underlying these prices.

Options:
A
The yield curve is increasing with constant slope.
B
The yield curve is flat.
C
The yield curve is inverted with constant slope.
D
The yield curve is concave upward.
E
The yield curve is concave downward.

Correct answer: B

Solution: All prices imply an interest rate of \(6 \%\) indicating that the yield rate does not change with duration, that is. the yield curve is flat.
Question: Company ABC is considering the following three investments:
i) Investment X is a 50-year zero-coupon bond
ii) Investment Y is a perpetuity with annual payments
iii) Investment Z is a geometrically increasing perpetuity with annual payments that increase at a rate of \(3 \%\) per year

For both perpetuities, the first payment will occur one year from now.
The convexities of Investments \(\mathrm{X}, \mathrm{Y}\), and Z are \(x, y\), and \(z\), respectively, based on an annual effective interest rate of \(5 \%\).

Determine which expression represents the relationship between the convexity of the three investments.

Options:
A
\(x<z<y\)
B
\(y<x<z\)
C
\(y<z<x\)
D
\(z<x<y\)
E
\(z<y<x\)

Correct answer: B

Solution: For all 3 investments, convexity is \(P^{\prime \prime}(i) / P(i)\). Let \(C\) be the redemption value of the bond and the initial payments of the perpetuities (it turns out the value is irrelevant).

For i)
\(P(i)=C(1+i)^{-50}, P^{\prime}(i)=-50 C(1+i)^{-51}, P^{\prime \prime}(i)=2550 C(1+i)^{-52}\)
Convexity \(=2550(1.05)^{-52} / 1.05^{-50}=2550 / 1.05^{2}=2313\).
For ii)
\(P(i)=C i^{-1}, P^{\prime}(i)=-C i^{-2}, P^{\prime \prime}(i)=2 C i^{-3}\)
Convexity \(=2(0.05)^{-3} / 0.05^{-1}=2 / 0.05^{2}=800\).
For iii)
\(P(i)=C(i-0.03)^{-1}, P^{\prime}(i)=-C(i-0.03)^{-2}, P^{\prime \prime}(i)=2 C(i-0.03)^{-3}\)
Convexity \(=2(0.05-0.03)^{-3} /(0.05-0.03)^{-1}=2 / 0.02^{2}=5000\).
Thus, convexity when ranked from lowest to highest is:
\(800<2,313<5,000\), or \(y<x<z\).