BjornvanBraak/ActuarialMathBench huggingface dataset:
Question: A five-year interest-only loan in the amount of 10,000 has annual payments, and an annual effective interest rate \(i\). At the end of year 5, the borrower pays off the principal along with the last interest payment.

To finance the principal payment, the borrower buys the following two zero-coupon bonds that both mature at the end of year 5 .

\begin{tabular}{|c|c|c|c|}
\cline { 2 - 4 } \multicolumn{1}{c|}{} & Time of Purchase & Par Value & \begin{tabular}{c} 
Annual \\
Effective Yield
\end{tabular} \\
\hline Bond 1 & End of year 3 & 2000 & \(3.0 \%\) \\
\hline Bond 2 & End of year 4 & 8000 & \(2.5 \%\) \\
\hline
\end{tabular}

It costs the borrower a total of 2260.19 at the end of year 3 to pay the interest due and to buy Bond 1.

Calculate \(i\).

Options:
A
\(\quad 2.60 \%\)
B
\(\quad 3.00 \%\)
C
\(3.18 \%\)
D
\(\quad 3.75 \%\)
E
\(4.30 \%\)

Correct answer: D

Solution: The cost to purchase the bond at the end of year 3 is \(2000 \times(1.03)^{-2}=1885.19\). Subtracting this cost from 2260.19 we get 375 , the amount of interest paid which is \(3.75 \%\) of 10000 . Thus the interest rate is \(3.75 \%\).
Question: A company faces the following liabilities at the end of the corresponding years:

\begin{tabular}{|l|c|c|c|c|}
\hline Year & 1 & 2 & 3 & 4 \\
\hline Liability & 5,766 & \(X\) & 15,421 & 7,811 \\
\hline
\end{tabular}

The company can invest in the following three annual coupon bonds redeemable at par:

\begin{tabular}{|c|c|c|}
\hline & Term (in years) & Annual coupon rate \\
\hline Bond A & 1 & \(1 \%\) \\
\hline Bond B & 3 & \(5 \%\) \\
\hline Bond C & 4 & \(7 \%\) \\
\hline
\end{tabular}

The company invests in each bond so that the asset and liability cash flows are exactly matched.
Calculate \(X\).

Options:
A
1221
B
1245
C
1290
D
1318
E
1375

Correct answer: A

Solution: Let \(a, b\), and \(c\) represent the face values of the three bonds. One, two, three, and four years from now, respectively:
the 1 -year bond provides payments of \(1.01 a, 0,0,0\);
the 3 -year bond provides payments of \(0.05 b, 0.05 b, 1.05 b, 0\); and
the 4 -year bond provides payments of \(0.07 c, 0.07 c, 0.07 c, 1.07 c\).
The total payments one, two, three, and four years from now must match the liabilities. Therefore, we have
\(1.01 a+0.05 b+0.07 c=5766\)
\(0.05 b+0.07 c=X\)
\(1.05 b+0.07 c=15421\)
\(1.07 c=7811\)
Note that to find \(X\), we do not need the first equation.
Solving the fourth equation for \(c\) yields \(c=\frac{7811}{1.07}=7300\).
Substituting this value of \(c\) into the third equation and solving for \(b\) yields
\(b=\frac{15421-0.07(7300)}{1.05}=14200\).
Finally, substituting these values of \(b\) and \(c\) into the second equation yields \(X=0.05(14200)+0.07(7300)=1221\).
Question: You are given the following information regarding two annuities with annual payments:
i) Annuity X is a 20-payment annuity-immediate which provides an initial payment of 2500 and each subsequent payment is \(5 \%\) larger than the preceding payment.
ii) Annuity Y is a 30-payment annuity-due which provides an initial payment of \(k\) and each subsequent payment is \(4 \%\) larger than the preceding payment.

Using an annual effective interest rate of \(4 \%\), Annuity \(X\) and Annuity \(Y\) have the same present value.

Calculate \(k\).

Options:
A
1,758
B
1,828
C
1,901
D
\(\quad 2,078\)
E
\(\quad 2,262\)

Correct answer: A

Solution: \(P V_{X}=2500\left[\frac{1-\left(\frac{1.05}{1.04}\right)^{20}}{0.04-0.05}\right]=52,732.61\)
Annuity Y has the same increasing percentage as interest rate, so:
\(P V_{Y}=30 k\)
\(P V_{X}=P V_{Y}=>k=1757.75\)
Question: A bond is priced at 950 , giving an annual effective yield to maturity of \(9 \%\). At \(9 \%\), the derivative of the price of the bond with respect to the yield rate is -4750 .

Calculate the Macaulay duration of the bond in years.

Options:
A
4.59
B
4.62
C
5.00
D
5.41
E
\(\quad 5.45\)

Correct answer: E

Solution: The price is \(P=950\). The modified duration is \(-\frac{P^{\prime}}{P}=-\frac{-4750}{950}=5\).
Macaulay duration is \((1.09)(5)=5.45\)
Question: An individual is to receive \(1,000,000\) today and \(1,000,000\) five years from today.
These payments are to be converted to an increasing annual perpetuity, with the first payment, \(X\), paid today and each succeeding payment 1000 more than the previous payment.

At an annual effective interest rate of \(4 \%\), the present value of the two payments is equal to the present value of the perpetuity.

Calculate \(X\).

Options:
A
45,074
B
47,877
C
51,923
D
55,000
E
\(\quad 66,795\)

Correct answer: A

Solution: The present value of the two payments is:
\(1,000,000+\frac{1,000,000}{1.04^{5}}=1,821,927.07\)
The present value of the perpetuity is:
(1.04) \(\left[\frac{X}{0.04}+\frac{1000}{0.04^{2}}\right]=1,821,927.07\)
\(\left[\frac{X}{0.04}+\frac{1000}{0.04^{2}}\right]=1,751,852.99\)
\(\frac{X}{0.04}=1,126,852.99\)
\(X=45,074.12\)
Question: A loan of 100 is to have all principal and accrued interest paid at the end of five years. Interest accrues at an annual effective rate of \(5 \%\) for the first two years and at a force of interest at time \(t\) in years \((t>2)\) of \(\delta_{t}=\frac{1}{1+t}\).

Calculate the equivalent annual effective rate of discount for the five-year period.

Options:
A
\(\quad 10.9 \%\)
B
\(\quad 13.0 \%\)
C
\(14.6 \%\)
D
\(\quad 15.4 \%\)
E
\(17.1 \%\)

Correct answer: C

Solution: \(100(1+0.05)^{2} \exp \left(\int_{2}^{5} \frac{1}{1+t}\right)=100(1.1025) \exp \left[\left.\ln (1+t)\right|_{2} ^{5}\right]\)
\(=110.25\left(\frac{6}{3}\right)=220.50\)
\(\frac{100}{(1-d)^{5}}=220.50\)
\((1-d)^{5}=\frac{100}{220.50}=0.453515\)
\(1-d=0.853726\)
\(d=0.1463\)
Question: A loan of 100,000 at an annual effective rate of \(9.12 \%\) is repaid with level payments at the end of each month over 15 years. Immediately after the \(59^{\text {th }}\) payment is made, the outstanding balance is refinanced at an annual effective rate of \(5.76 \%\). The term of the refinanced loan is 20 years with level payments at the end of each month.

Calculate the interest portion of the first payment of the refinanced loan.

Options:
A
375
B
378
C
381
D
384
E
387

Correct answer: A

Solution: \(1.0912=(1+j)^{12}\)
\(j=0.0073\)
\(100,000=P a_{\overline{180} 0.0073}\)
\(P=1000.02\)
\(O B_{59}=1000.02 a_{\overline{121} 0.0073}\)
\(O B_{59}=80,174.59\)
\(1.0576=(1+j)^{12}\)
\(j=0.004678\)
Interest portion of first revised payment:
\(80,174.59(0.004678)=375.04\).
Question: Two bonds have the same annual effective yield rate, \(r\), where \(r>0\). The bonds have Macaulay duration of 5 years and 6 years with respect to \(r\). One of the bonds has modified duration of 5.76 years while the other bond has modified duration of \(d\) years.

Calculate \(d\).

Options:
A
4.760
B
4.800
C
5.208
D
5.240
E
\(\quad 6.912\)

Correct answer: B

Solution: \(\bar{v}=\frac{\bar{d}}{1+r}\)
\(5.76=\frac{6}{1+r}\)
\(r=0.041667\)
\(\bar{v}=\frac{5}{1.041666}=4.8\)
Question: Annual payments of 240 are made at the end of each year to repay a loan of 3400 . The payments are based on an annual effective interest rate of \(4.5 \%\). The loan is settled with a drop payment of \(X\).

Calculate \(X\).

Options:
A
1.61
B
4.53
C
8.83
D
12.48
E
13.04

Correct answer: E

Solution: \(3400=240 a_{\bar{n} \mid 0.045}\)
\(n=23.05\)
Use 23 for the number of full payments of \(240 . X\) will be paid at time 24 .
\(3400=240 a_{\overline{230.045}}+X v^{24}\)
\(3400=3395.47+X v^{24}\)
\(4.53=X v^{24}\)
\(X=13.04\)
Question: You have decided to invest in a newly issued 20-year bond with annual coupons and the following characteristics:
i) The price at issue is 1321 .
ii) The face amount is 1111 .
iii) The annual coupon rate is \(10 \%\).
iv) The bond is callable for 1111 immediately after the payment of either the \(18^{\text {th }}\) or \(19^{\text {th }}\) coupon.

Calculate the minimum possible annual effective yield rate that you can earn.

Options:
A
\(7.881 \%\)
B
\(\quad 7.937 \%\)
C
\(7.985 \%\)
D
\(8.028 \%\)
E
\(8.065 \%\)

Correct answer: C

Solution: The minimum yield will occur at the earliest redemption date since the bond is brought at a premium. Therefore \(\mathrm{n}=18\).
\(1321=111.10 a_{18 \mid i}+1111 v^{18}\)
\(i=0.07985\)
Question: A 15-year bond with a face amount of 5000 , a redemption value of \(C\), and an annual coupon rate of \(7 \%\) paid quarterly is purchased at a price that yields an annual nominal rate of \(5.6 \%\) convertible quarterly.

The amount for amortization of premium in the third coupon payment is 6.88 .
Calculate \(C\).

Options:
A
5150
B
5165
C
5190
D
5235
E
5275

Correct answer: A

Solution: Using the basic formula:
\(B V_{2}-B V_{3}=6.88\)
\(\left(87.5 a_{\overline{5} 8 \mid}+C v^{58}\right)-\left(87.5 a_{\overline{57} \mid}+C v^{57}\right)=6.88\)
\(39.0667-0.00625 C=6.88\)
\(C=5150\)
Question: A three-year 1000 face amount bond pays coupons of \(X\) quarterly. It is bought at a price to yield an annual nominal rate of \(8 \%\) convertible quarterly. If the amount of each coupon were doubled, the purchase price would have to increase by 500 for the bond to maintain the same yield rate.

Calculate \(X\).

Options:
A
20
B
24
C
31
D
40
E
47

Correct answer: E

Solution: \(P=X a_{\overline{12 \mid 0.02}}+1000 v^{12}\)
\(500+P=2 X a_{\overline{12} 0.02}+1000 v^{12}\)
Subtract the first equation from the second.
\(500=X a_{\overline{12} \mid 0.02}\)
\(X=47.28\)
Question: An investor purchases a geometrically decreasing perpetuity-due that provides annual payments. The first payment is 1 , and each subsequent payment is \(1 \%\) less than the previous payment.

At the time of purchase, the Macaulay duration of the perpetuity-due is 12.00 years.
Calculate the modified duration of the perpetuity-due, in years, at the time of purchase.

Options:
A
10.97
B
11.11
C
11.19
D
12.81
E
12.88

Correct answer: C

Solution: The price of the geometric perpetuity-due, as a function of the annual effective interest rate \(i\), is given by
\[
P(i)=1+\frac{0.99}{1+i}+\left(\frac{0.99}{1+i}\right)^{2}+\left(\frac{0.99}{1+i}\right)^{3}+\ldots=\frac{1}{1-\frac{0.99}{1+i}}=\frac{1+i}{0.01+i}=1+\frac{0.99}{0.01+i}
\]

Therefore, the modified duration is given by
\[
-\frac{P^{\prime}(i)}{P(i)}=-\frac{-\frac{0.99}{(0.01+i)^{2}}}{\frac{1+i}{0.01+i}}=\frac{0.99}{(0.01+i)(1+i)}
\]

Since Macaulay duration is modified duration times \((1+i)\), the Macaulay duration is
\(\frac{0.99}{0.01+i}=12\). Solving for \(i\) gives \(i=0.0725\).
Therefore, the modified duration is \(\frac{12}{1+i}=\frac{12}{1+0.0725}=11.18881\) years.
Question: Payments of 200 are invested at the beginning of each year for five years. The payments earn interest at an annual effective interest rate of \(i\). The interest is reinvested at the end of each year at an annual effective rate of \(6.00 \%\). At the end of the five-year period, the value of the fund including both the sum of the annual payments and the accumulated value of the interest is 1260 .

Calculate \(i\).

Options:
A
\(\quad 8.00 \%\)
B
\(\quad 8.75 \%\)
C
\(10.00 \%\)
D
\(11.25 \%\)
E
\(12.25 \%\)

Correct answer: A

Solution: \(1260=200(5)+200 i(I s)_{\left.5\right|_{0.06}}=1000+200 i\left[\frac{\ddot{S}_{\left.5\right|_{0.06}}-5}{0.06}\right]\)
\[
\begin{aligned}
= & 1000+200 i(16.2553) \\
i & =0.08
\end{aligned}
\]
Question: The present value of a series of payments of 1 made every six months forever, with the first payment made immediately, is equal to 23 .

Calculate the annual effective rate of interest.

Options:
A
4.3\%
B
\(\quad 4.5 \%\)
C
\(8.9 \%\)
D
\(9.1 \%\)
E
\(\quad 9.3 \%\)

Correct answer: E

Solution: If \(j\) is the semi-annual effective interest rate, then
\(1+(1 / j)=23\)
\(j=1 / 22\)
Then the annual effective interest rate is \((1+1 / 22)^{2}-1=0.093\).
Question: A perpetuity-immediate has annual payments starting at 1 , increasing by 1 each year until it reaches 10 , then decreasing by 1 each year until it reaches 5 , and remaining at 5 thereafter.

The present value of this perpetuity at an annual effective interest rate of \(9 \%\) is \(X\).
Calculate \(X\).

Options:
A
51
B
53
C
55
D
58
E
61

Correct answer: D

Solution: \((I a)_{\infty}=\frac{1}{i}+\frac{1}{i^{2}}=\frac{1}{0.09}+\frac{1}{0.09^{2}}=134.5679\)
\(\mathrm{X}=(I a)_{\text {ळ } \mid}-2 v^{10}(I a)_{\text {ळ } \mid}+v^{15}(I a)_{\text {ळ } \mid}\)
\(=134.5679(1-2 \times 0.42241+0.274538)\)
\(=57.83\)
Question: A person borrows 150,000 to purchase a house. The borrower makes interest-only payments at the end of each month for five years, then makes payments of 1000 at the end of each month.

Interest accrues at a nominal annual rate of \(4 \%\) convertible monthly for the first five years and then at a nominal annual rate of \(6 \%\) convertible monthly.

Determine the total number of payments it will take to repay the loan.

Options:
A
209
B
269
C
278
D
329
E
338

Correct answer: E

Solution: At the end of 5 years and 60 payments, the borrower still owes 150,000 .
It will take \(n\) monthly payments of 1000 to retire the loan where
\(1000 a_{\bar{n} 0.005}=150,000\)
\(n=278\)
Thus, it will take 338 total payments.
Question: Bond A and bond B are both annual-coupon bonds.
Bond A is an \(m\)-year bond with face amount 1000 and price 800 , where \(m\) is a whole number. The coupon rate is half of the yield rate.

Bond B is a \(3 m\)-year bond with the same face amount, coupon rate, and yield rate as bond A .
Calculate the price for bond B .

Options:
A
400
B
512
C
608
D
671
E
716

Correct answer: C

Solution: Let \(i=\) yield rate. Then, for bond A :
\(800=1000(i / 2) a_{\overline{m i}}+1000 v^{m}=500\left(1-v^{m}\right)+1000 v^{m} \Rightarrow v^{m}=0.6\).
The price of bond \(B\) is:
\(1000(i / 2) a_{3 m i}+1000 v^{3 m}=500\left(1-v^{3 m}\right)+1000 v^{3 m}=500\left(1+v^{3 m}\right)=500\left(1+0.6^{3}\right)=608\).
Question: A perpetuity-due with annual payments is priced at \(X\) based on an annual effective interest rate of \(7 \%\). The amount of the first payment is 350 . Each payment, from the second through the fifth, is \(3 \%\) larger than the previous payment. Each of the sixth and subsequent payments is equal to the fifth payment.

Calculate \(X\).

Options:
A
5249
B
5530
C
5617
D
5917
E
6046

Correct answer: D

Solution: \(X=350(1.07)\left[\frac{1-\left(\frac{1.03}{1.07}\right)^{5}}{0.07-0.03}\right]+350(1.03)^{4} v^{4}\left(\frac{1}{0.07}\right)\)
\(X=1623.96+4293.23=5917.19\).
Question: An insurance company plans to sell an annual perpetuity-immediate, where each payment after the first payment is \(2 \%\) less than the previous payment. The company has chosen a price for this annuity and needs to calculate the amount of the first payment.

The annual effective yield rate is \(7.8 \%\), but the company mistakenly uses \(8.7 \%\) and therefore incorrectly calculates 1500 for the first payment.

Calculate the correct first payment.

Options:
A
1299
B
1309
C
1345
D
1374
E
1385

Correct answer: D

Solution: \(P V=1500\left[\frac{1}{0.087-(-0.02)}\right]\)
\(P V=14,018.69\)
\(14,018.69=P\left[\frac{1}{0.078-(-0.02)}\right]\)
\(P=1373.83\)
Question: An investor has purchased an 18-year 7000 face amount bond with an annual coupon rate of \(6 \%\) paid annually. The redemption value of the bond is 7500 . The purchase price, \(P\), results in an annual effective yield of \(i\).

Upon receipt, each coupon payment is immediately reinvested in an account earning an annual effective rate of \(5.70 \%\). Immediately following the receipt of the last payment from the 18 -year bond, the investor has earned an annual effective yield of \(5.18 \%\) on the original investment of \(P\).

Calculate \(i\).

Options:
A
\(\quad 4.54 \%\)
B
\(\quad 4.56 \%\)
C
\(4.79 \%\)
D
\(4.91 \%\)
E
\(5.13 \%\)

Correct answer: D

Solution: Each coupon payment is \(7000(0.06)=420\).
The accumulated value of the reinvested coupons after 18 years is: \(420 s_{{ }_{180.057}}=12,617.50\).
Therefore,
\(P(1.0518)^{18}=12,617.50+7,500 \Rightarrow P=8105.45\)
Finally,
\(P=8105.45=420 a_{18 i}+7500 v^{18}=>i=4.91 \%\)
Question: You are given the following information regarding two perpetuities:
i) Perpetuity X provides payments of 3000 every six months with the first payment due immediately.
ii) Perpetuity Y provides payments of \(k\) every two years with the first payment due one year from now.
iii) Based on an annual effective interest rate of \(4.94 \%\), Perpetuity X and Perpetuity Y have the same present value.

Calculate \(k\).

Options:
A
12,150
B
12,450
C
12,750
D
13,050
E
13,380

Correct answer: A

Solution: The \(P V\) of perpetuity X is \(\frac{3000}{j_{1}}+3000=125,939.41\), where \((1.0494)=\left(1+j_{1}\right)^{2}\)
The \(P V\) of perpetuity Y is \(\frac{1.0494 k}{j_{2}}=10.36543134 k\), where \(1.0494^{2}=1+j_{2}\)
Setting \(P V\) 's equal gives \(k=12,149.94\).
Question: A ten-year 1000 face amount bond has an annual coupon rate of \(6 \%\) paid semiannually. The bond is purchased at a premium, to yield an annual effective interest rate of \(5 \%\).

Calculate the amount of the amortization of premium in the 12th coupon.

Options:
A
3.85
B
\(\quad 3.95\)
C
4.00
D
4.10
E
4.26

Correct answer: E

Solution: \(F r=1000(0.03)=30\)
\(j=(1.05)^{1 / 2}-1=0.024695\)
\(B_{11}=1000 v_{j}^{9}+30 a_{9 j}=1042.35\)
\(B_{12}=1000 v_{j}^{8}+30 a_{\overline{8} \mid j}=1038.09\)
\(1042.35-1038.09=4.26\)
Question: You are given the following information about two 30 -year 1000 face amount bonds, A and B, each yielding an annual nominal interest rate of \(i \%\) convertible semiannually.
i) Bond A has an annual coupon rate of \(9 \%\) paid semiannually and has a price of 1249.45.
ii) Bond B has an annual coupon rate of \(8 \%\) for the first 15 years and \(10 \%\) for the last 15 years, all paid semiannually.

Calculate the price of Bond B.

Options:
A
1150
B
1160
C
1170
D
1180
E
1190

Correct answer: E

Solution: First, use bond A to determine the interest rate. Because the equation for the interest rate cannot be solved algebraically, use a financial calculator to get \(i / 2\), which yields \(i / 2=0.035\), or \(i^{(2)}=0.07\).

For bond B ,
\[
\begin{aligned}
P & =1000(0.08 / 2) a_{\overline{30} 0.035}+1000(0.10 / 2) 1.035^{-30} a_{\overline{30} 0.035}+1000(1.035)^{-60} \\
& =735.68+327.63+126.93=1190.24,
\end{aligned}
\]
Question: The full price of orthodontic treatment is \(X\). The patient must pay at least a portion of the balance by Day 15 (fifteen days following the beginning of the work). The two payment options are:
i) Pay the full balance, less a \(2.5 \%\) discount, on Day 15 .
ii) Pay 250 on Day 15 and the remaining balance, without interest, on Day 60 .

Assume a 365-day year.
The implied effective annual yield rate the patient is charged for delaying full payment from Day 15 to Day 60 is \(24.2 \%\).

Calculate \(X\).

Options:
A
697
B
875
C
1675
D
3295
E
4827

Correct answer: E

Solution: \(0.975 X=250+(X-250) v^{(60-15) / 365}\)
\(0.975 X=250+(X-250)(1.242)^{-45 / 365}\)
\(X=4827.18\)
Question: At Jack's death, his estate of 298,000 is to be divided between Charity A and Charity B.
Charity A will receive 198,000 and use it to purchase a perpetuity-due that makes a payment of \(X\) every month.

Charity B will receive 100,000 and use it to purchase a perpetuity-due that makes a payment of \(X\) every two months.

The payments of \(X\) for each charity are based on an annual effective interest rate of \(i\).
Calculate \(i\).

Options:
A
\(\quad 21.5 \%\)
B
\(\quad 24.0 \%\)
C
\(\quad 24.5 \%\)
D
\(\quad 26.8 \%\)
E
\(\quad 27.4 \%\)

Correct answer: E

Solution: Let \(\boldsymbol{v}\) denote the discount factor applicable to a period of one month and let \(X\) be the constant periodic amount received by both charities. For Charity A, \(198,000=\frac{X}{1-v}\).
Since \(\boldsymbol{v}^{\mathbf{2}}\) is the discount factor applicable to a two-month period, for Charity B, \(100,000=\frac{X}{1-v^{2}}\).
Dividing the first equation by the second gives \(\mathbf{1}+\boldsymbol{v}=\mathbf{1 . 9 8}\) so that \(\boldsymbol{v}=0.98\). The yearly discount factor is \(0.98^{12}=0.78472\) so that the annual effective interest rate is
\(\frac{1}{0.78472}-1=0.274=27.4 \%\).
Question: Kelly sets up a savings account and makes a deposit at the beginning of each month starting today. The following information is known about the account:
i) Within each year, the amounts of monthly deposits are all equal.
ii) For any given year, the monthly deposits are \(10 \%\) greater than the monthly deposits of the previous year.
iii) The savings account earns an annual nominal interest rate of \(2.4 \%\) compounded monthly.
iv) The accumulated amount of the savings account at the end of year 10 is 21,234.05.
v) The amount of the monthly deposit in year 5 is \(K\).

Calculate \(K\).

Options:
A
133.02
B
144.68
C
146.41
D
146.70
E
147.12

Correct answer: C

Solution: Let \(X\) be the monthly deposit of year 1 . Then determine the present value of the first year's monthly payments and then use geometric formula to accumulate to year 10 .
\(i=\left[1+\frac{0.024}{12}\right]^{12}-1=0.024266\).
\(X \ddot{a}_{12 \mid 0.002}\left[\frac{1-\left(\frac{1.10}{1.024266}\right)^{10}}{0.024266-0.10}\right](1.024266)(1.024266)^{10}=21,234.05\)
\(X(11.869136)[13.74278]=16,311.46\)
\(X=100\)
The payments in year 5 will be \(100(1.1)^{4}=146.41\).
Question: Bond X , Bond Y , and Bond Z are each \(n\)-year bonds with face amount 1000. All three bonds are priced to yield a nominal annual interest rate of \(6.5 \%\) convertible semiannually.

Bond X provides annual coupons of \(R\), Bond Y provides quarterly coupons of 20.00, and Bond Z provides semiannual coupons of 32.50 .

The price of Bond Y exceeds the price of Bond X by 401.50 and the price of Bond Y exceeds the price of Bond Z by 169.30.

Calculate \(R\).

Options:
A
41.79
B
42.03
C
43.78
D
44.25
E
45.52

Correct answer: D

Solution: The yield rate is equal to the coupon rate for Bond Z , so the price of Bond Z is 1000 . This implies the price of Bond Y is \(1,169.30\).
\(1169.30=20 a_{4 n j j}+1000 v_{j}^{4 n}\), where \(1.0325=(1+j)^{2}=>4 n=76=>n=19\)
\(1169.30-401.50=R a_{19 i}+1000 v_{i}^{19}\), where \(1.0325^{2}=1+i=>R=44.25\)
Question: A twelve-year 2500 face amount bond has an annual coupon rate of \(4.84 \%\) paid quarterly and is redeemable for \(C\). The book values of the bond following the \(4^{\text {th }}\) and \(19^{\text {th }}\) coupons are 2092.88 and 2316.30, respectively.

Calculate the amount for accumulation of discount in the last coupon.

Options:
A
7.77
B
13.10
C
22.97
D
30.91
E
31.55

Correct answer: D

Solution: Using the financial calculator to solve for the yield rate with
\(N=15, P V=2092.88, P M T=-30.25\) [because 2500(.0484) / 4=30.25], FV=-2316.20, \(i= 2.06 \%\) is the nominal quarterly yield rate.
\(2316.3(0.0206)-30.25=17.46578\) is the amount for accumulation of discount in the \(20^{\text {th }}\) coupon and \(17.46578(1.0206)^{28}=30.91\) is the amount for accumulation of discount in the last coupon.
Question: A 20 -year bond has an annual coupon rate of \(3 \%\) paid semiannually. The bond is priced using an annual nominal yield rate of \(2 \%\) convertible semiannually. The cumulative amount of premium amortization exceeds \(50 \%\) of the premium at purchase, for the first time, immediately after the \(n^{\text {th }}\) coupon is paid.

Calculate \(n\).

Options:
A
18
B
19
C
20
D
21
E
22

Correct answer: E

Solution: Assume the face amount is 1 . The price of the bond is \(0.015 a_{\overline{400.01}}+1.01^{-40}=1.1642\).
At the given coupon and return rates, the premium is \(1.1642-1=0.1642\).
We are looking for the time at which the book value is \(1+0.1642 / 2=1.0821\).
Then,
\[
\begin{aligned}
1.0821 & =0.015 a_{\overline{40-k} \mid 0.01}+1.01^{-(40-k)} \\
& =0.015 \frac{1-1.01^{-(40-k)}}{0.01}+1.01^{-(40-k)} \\
& =1.5-0.5(1.01)^{-(40-k)} \\
0.8328 & =1.01^{-(40-k)} \\
40-k & =-\frac{\ln 0.8328}{\ln 1.01}=18.03 \\
k & =21.97
\end{aligned}
\]

For the book value to exceed 1.0821 , round up to \(k=22\).