Exam FM/2 Practice Practice Exam 1 Answer Key c 2013 Actuarial Investment. Copyright
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1. A perpetuity perpetuity makes level level payments payments of 1 at the end of each year. year. The perpetuity’ perpetuity’ss modified duration is 25. Calculate the present value of the perpetuity. (A) 16.67 16.67 (B) 22.50 22.50 (C) 24.00 24.00 (D) 24.33 24.33 (E) 25.00 25.00
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Correct answer: (E) Solution: The perpetuity’s volitility ModD is 25. We know that the Macaulay duration of a perpetuity is 1 + 1i , so we use the formula formula ModD = MacD · v :
25 = (1 + 1i ) · v 1 25 = ( i+1 )( 1+i ) i 1+i
25 =
1 i
Remember that the present value of a perpetuity with level payments of 1 is 1i . Therefore the present value of this perpetuity is 25.
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2. Johnathan buys a 12 -year bond with face amount 1000 and annual coupons of 4% priced to yield i%. Rachel buys a 12 -year bond with face amount 1000 and annual coupons of 4% priced to yield j %. Rachel’s bond is bought at a discount. The price of Rachel’s bond is less than the price of Johnathan’s bond. Which of the following is true? (A) i < j < .04 (B) j < i < .04 (C) i < .04 < j (D) .04 < j < i (E) There is not enough information to determine that any of (A), (B), (C), or (D) is true.
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Correct answer: (E) Solution: Since the price of Rachel’s bond is less than the price of Johnathan’s bond, we know that Rachel’s bond has a higher yield than Johnathan’s bond. (She invested less money to get the same cashflow, so she had a higher yield.) Therefore i < j . Since Rachel’s bond was bought at a discount, we know that the price of Rachel’s bond was less than 1000 and we know that j > .04. Rachel’s bond cost less than Johnathan’s bond, but we cannot determine whether Johnathan’s bond cost more or less than 1000. Therefore we do not know if Johnathan’s bond was bought at a discount or at a premium. No other relevant facts can be determined from the given information. Therefore there is not enough information to determine that any of (A), (B), (C), or (D) is true, so the answer is (E).
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3. An asset’s price is 81. The price of a put option for the asset that matures in 72 days and has a strike price of 84 is 3.85. The annual effective rate of interest is 5%. What is the price of a call option for the asset that matures in 72 days and has a strike price of 84? (Assume a 360-day year.) (A) 0.85 (B) 1.67 (C) 2.85 (D) 4.85 (E) 6.17
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Correct answer: (B) 72 = 15 years. Use the put-call parity formula: Solution: The option matures in 72 days, or 360 C − P = S (0) − P V (K ), or C − 3.85 = 81 − 84(1 + .05) (1/5) . Solve to find the price of the call option: C = 1.67. −
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4. A stock pays annual dividends, beginning in one year with a dividend of 134. The stock pays dividends until the company goes bankrupt n years from now, at which time the stock pays the final dividend of 248. The present value of the final dividend is 159.86, and the present value of the stock is 1289. Dividends increase by r % per year and the annual effective rate of interest is i%. It is known that i + .03 = r . Calculate n . (A) 8 (B) 9 (C) 10 (D) 11 (E) 12
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Correct answer: (B) Solution: Use the formula for a geometric annuity: r
1289 = 134 ·
n
) 1− (1+ (1+i)n
i−r
Since i + .03 = r , we know that i − r = −.03. Since the final dividend is 248 and the present value of the final dividend is 159.86, we know 248 = 1.552. that (1 + i)n = 159.86 r
1289 = 134 ·
n
) 1− (1+ 1.552 −.03 n−1
1289 = 134 ·
1− (1+r)1.552(1+r) −.03
From time 1 to time n, the dividend increases by a factor of 1 + r exactly n − 1 times. Since the first dividend is 134 and the last dividend is 248, this means that 134(1 + r )n 1 = 248, = 1.851. or (1 + r)n 1 = 248 134 −
−
(1+r) 1− 1.851 1.552 −.03 ·
1289 = 134 ·
Solve to find r = . 08. This means that i = . 05. Now we know that (1+ i)n = 1.05n = 1.552. Solve to find n = 9.
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5. An asset is currently worth 61. Jack buys a call option for the asset with strike price 62 and maturity in one year. Robin buys a put option for the asset with strike price 62 and maturity in one year. After a year, the price of the underlying asset is 58. Jack’s profit is −4.41 and Robin’s profit is X . The annual effective rate of interest is 5%. Calculate X . (A) 1.42 (B) 1.54 (C) 1.64 (D) 1.78 (E) 1.91
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Correct answer: (C) Solution: Since Jack’s call option expires out-of-the-money and his profit is −4.41, the premium he paid for the call option must have been the present value of 4.41, which is 4.41(1 + .05) 1 = 4.20. −
Now use the put-call parity formula to calculate the premium that Robin pays for her put option: C − P = S (0) − P V (K )
4.20 − P = 61 − 62(1 + .05)
1
−
P = 2.25
Therefore the premium paid for the put option is 2.25. The profit from the put option is: Profit = max{K − S, 0} − F V (Premium) X = max {62 − 58, 0} − 2.25(1 + .05)
Therefore X = 1.64.
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6. A 14-year annuity due makes payments of 100 every year except for year 3. In year 3, the annuity makes a payment of 500. The effective annual interest rate is 4%. What is the present value of the annuity? (A) 1398 (B) 1412 (C) 1433 (D) 1454 (E) 1468
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Correct answer: (E) Solution: Break the annuity into two separate pieces consisting of a level annuity with payments of 100 and a one-time payment of 400 during the 3 rd year. Notice that because the annuity is an annuity due, the payment in year 3 is made at the beginning of the year, which is equivalent to the end of year 2. So the present value is: 1 2 100¨ ) = 1468 a14.04 + 400( 1.04
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7. The following table gives one-year forward rates for the next three years. T (years)
1 2 3
i(T − 1, T ) 4.6 4.3 3.9
The three-year swap rate for an interest rate swap is r %. Cacluate r . (A) 4.28 (B) 4.32 (C) 4.38 (D) 4.44 (E) 4.60
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Correct answer: (A) Solution: The swap rate is the fixed payment rate at which the present value of interest payments using the fixed rate is equal to the present value of interest payments using the current term structure. Suppose that 1000 is borrowed and payments of only interest are made on the principal for three years. Then the swap rate r solves the following equation: 1000·.046 (1+.046)
=
+
1000·.043 1000·.039 + (1+.046)(1+.043)(1+.039) (1+.046)(1+.043)
1000r 1000r 1000r + (1+.046)(1+.043) + (1+.046)(1+.043)(1+.039) (1+.046)
Solve to find r = 4.28.
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8. A loan of 1000 is repaid with 14 annual payments starting one year after the loan is made. Each of the first 12 payments is 6% more than the subsequent payment. The eighth payment is X . The final payment is 2X . The annual effective rate of interest is 3%. Calculate X . (A) 72.20 (B) 73.29 (C) 74.78 (D) 76.10 (E) 77.28
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Correct answer: (D) Solution: This is equivalent to a geometrically changing annuity with 13 payments, plus a balloon payment at time 14. 1 = . 9434 Since each payment is 6% more than the subsequent payment, each payment is 1.06 times the previous payment. Therefore each payment changes by − 5.66% compared to the previous payment. To find the first payment, observe that the eighth payment is X ; the seventh payment is 1.061 X ; the sixth payment is 1.062 X ; and the first payment is 1.067 X .
Therefore the loan amount of 1000 is equal to the present value of the geometric annuity plus the present value of the balloon payment: 1−( 1+(1+.0566) )13 03 .03−(−.0566) −.
1000 = 1.067 X Solve to find X = 76.10.
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1 + 2X ( 1+.03 )14
9. Calculate the convexity of a 3-year bond with annual coupons of 10% priced at par. (A) 3.00 (B) 8.01 (C) 8.76 (D) 9.00 (E) 10.86
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Correct answer: (C) Solution: Let F be the face amount of the bond. Since the bond is priced at par, F is also the price of the bond. Also, the yield rate is equal to the coupon rate of 10%. Therefore 1 v = 1+.1 . Convexity is defined as the interest rate i.
P (i) , P (i)
where P (i) is the present value of a portfolio as a function of
The present value of the bond is P (i) = . 1F v + . 1F v2 + 1 .1F v3 . Then P (i) = − .1F v2 − .2F v 3 − 3.3F v 4 . Also P (i) = . 2F v 3 + .6F v 4 + 13.2F v 5 .
Since P (i) = F , the convexity is equal to 13.2v 5 = 8.76.
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P (i) P (i)
=
.2F v 3 +.6F v 4 +13.2F v 5 F
= .2v 3 + . 6v 4 +
10. Mark takes out a 10-year loan worth 1000 with payments of 120 at the end of every year. After 4 years, he extends the loan by an additional 5 years. How much additional interest will Mark pay by extending the loan? (A) 52 (B) 61 (C) 69 (D) 73 (E) 82
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Correct answer: (B) Solution: The annual effective interest rate is given by 1000 = 120a10i . Use a financial calculator to find i = .0346. After 4 years, the outstanding balance is 120a6.0346 = 640. The interest Mark would pay on this outstanding balance if he did not extend the loan is 120 · 6 − 640 = 80. Let P be the new payment after extending the loan. There were 6 payments left, but Mark added an additional 5 payments, so he now has 11 payments left. Then 640 = P a11.0346 , so P = 71. The interest Mark will pay on the outstanding balance of 640 is therefore 11 · 71 − 640 = 140. Thus the additional interest Mark will pay by extending the loan is 140 − 80 = 61.
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11. Abigail takes out a 48-month loan worth 132,000 with payments of 3100 at the end of each month. In order to pay off the loan early, Abigail instead makes payments of 3800 at the end of each month for n months, plus a final payment of X at the end of the n + 1st month such that X < 3800. Calculate X . (A) 893 (B) 904 (C) 918 (D) 950 (E) 954
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Correct answer: (E) Solution: Let j be the monthly rate of interest. Then use a financial calculator to solve the formula 132 , 000 = 3100a48 j to find j = .005. Then find out how many payments of 3800 will be made by solving the formula 132 , 000 = 3800an.005 to find n = 38.25. This means that Abigail will make 38 payments of 3800 plus a final payment of X . The outstanding balance immediately after the 38 th payment is 132, 000(1 + . 005)38 − 3800s38.005 = 950. The final payment is made one month after this, so the final payment is 950(1+ .005)1 = 954.
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12. A 3-year annuity immediate with monthly payments makes its first payment on January 31. In each July and in each December, the payment is 30. In all other months, the payment is 10. The annual effective rate of interest is 7%. Calculate the accumulated value of the annuity immediately after the final payment. (A) 529 (B) 541 (C) 594 (D) 604 (E) 613
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Correct answer: (A) 1
Solution: The monthly rate of interest is (1 + . 07) 12 − 1 = .005654 and the 6-month rate 1 of interest is (1 + . 07) 2 − 1 = .03441. Now break the annuity into two separate annuities. The first has 36 monthly payments of 10. Its accumulated value is 10 s36.005654 = 398. The second has 6 semi-annual payments of 20. Its accumulated value is 20 s6.03441 = 131. The total accumulated value is 398 + 131 = 529.
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13. A stock currently pays no dividends, but will begin paying annual dividends in n years. The first year’s dividend will be 18 and subsequent dividends will increase by 2% per year. At a price of 288.61, the stock is priced to yield 5%. Calculate n . (A) 12 (B) 13 (C) 14 (D) 15 (E) 16
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Correct answer: (E) Solution: Since the stock price at time 0 is 288.61, and the stock is priced to yield 5%, then the stock price at time n is 288.61(1 + .05)n . At time n , the dividends form a geometrically increasing perpetuity due. The first dividend is 18. The second dividend is 18 · 1.02, and subsequent dividends increase by 2%. The value of the perpetuity is 18 + 18 · 1.02 · ( .05 1 .02 ). −
Therefore 288.61(1 + .05)n = 18 + 18 · 1.02 · ( .05 1 .02 ). Solve to find n = 16. −
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14. An n -year bond has annual coupons of 5%. The bond is bought to yield 6.89%. The accumulated value of the coupons is equal to the face amount of the bond. Calculate n . (A) 10 (B) 11 (C) 12 (D) 13 (E) 14
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Correct answer: (D) Solution: Since the accumulated value of the coupons is equal to the face amount of the bond, we know that F = .05F · s n.0689 , or 20 = sn.0689 . Use a financial calculator to find n = 13.
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15. Which of the following are true about zero-cost collars? (I) A zero-cost collar has no net premium. (II) The payoff of a long position in a zero-cost collar is greater than the payoff of a long position in a collar with a positive premium paid. (III) Buying a call option and writing a put option with the same maturity date, strike price, and premium creates a zero-cost collar. (A) (I) only (B) (III) only (C) (I) and (II) (D) (II) and (III) (E) The answer is not given by any of (A), (B), (C), or (D)
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Correct answer: (A) Solution: The definition of a zero-cost collar is that it has no net premium. Therefore (I) is true. The payoff of a zero-cost collar is less than the payoff of a collar with positive premium because the premium paid allows for a larger payoff. Therefore (II) is false. A long position in a call option and a short position in a put option with the same strike price is equivalent to a long forward position, not a collar. Therefore (III) is false. Therefore the answer is (A).
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16. An investor sells 1000 barrels of oil at a forward price of 60 per barrel with maturity in six months. At expiration, the price of oil is 52 per barrel. There is a 3% commission on the futures contract. Calculate the investor’s net gain from the futures contract. (A) -9800 (B) -9560 (C) -8240 (D) 6200 (E) 7760
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Correct answer: (D) Solution: At expiration, the investor sells 1000 barrels of oil for 60 per barrel, and must buy 1000 barrels of oil to offset this sale. Therefore the payoff (not profit) from the contract is 1000(60 − 52) = 8000. Then he must pay a commission of 3% on the futures contract, or .03 · 1000 · 60 = 1800. Therefore his net gain is 8000 − 1800 = 6200.
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17. Tabitha buys a 20-year bond purchased to yield 8% convertible semiannually with semiannual coupons at a rate of 4% convertible semiannually. Coupons are reinvested into an account earning interest at a nominal rate of 6% convertible semiannually. John buys a 20-year bond purchased to yield j % convertible semiannually with semiannual coupons at a rate of 12% convertible semiannually. Coupons are reinvested into an account earning interest at a nominal rate of 6% convertible semiannually. The rate of return of John’s investment is the same as the rate of return of Tabitha’s investment. Calculate j . (A) 8.29 (B) 8.42 (C) 8.53 (D) 8.69 (E) 8.98
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Correct answer: (C) Solution: Suppose that Tabitha purchases a bond with a face amount of 1000. Then the price of the bond can be found using a financial calculator with N = 40, I = 4, P MT = 20, and F V = 1000 to find P V = − 604.14. Then the amount she has after 20 years is the face amount plus the accumulated value of the reinvested coupons, or 1000+ 20s40.03 = 2508.03. Therefore, the rate of return of Tabitha’s investment can be found by solving the equation 604.14(1 + i)20 = 2508.03 to find i = . 07377. Suppose that at the end of 20 years, John has 10,000. Since the rate of return of John’s investment is also .07377, this means that he invests 2408.83. The portion of the 10,000 that he earns from accumulation of coupons is .06F s40.03 . The final amount of 10,000 is composed of the face amount of the bond plus the accumulated value of the reinvested coupons. Therefore 10, 000 = F + .06F s40.03 . Solve this to find F = 1810.26. Then, since he invested 2408.83, use a financial calculator with N = 40, P V = −2408.83, P M T = .06 · 1810.26 = 108.62, and F V = 1810.26 to find I = 4.26. This is John’s bond yield per coupon period. Therefore his yield rate is j = 4.26 · 2 = 8.53.
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18. The six-month forward price for a stock is 35.94. The prepaid six-month forward price for the stock is 34.05. Assume that the stock pays no dividends and that there are no opportunities for arbitrage. The implied annual effective rate of interest is i%. Calculate i . (A) 2.7 (B) 5.3 (C) 5.6 (D) 11.1 (E) 11.4
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Correct answer: (E) Solution: Based on the two available options, the stock can be purchased in two ways: it can either be purchased now for 34.05 or in six months for 35.94. Therefore the sixmonth interest rate is 35.94 − 1 = .0555. Therefore the annual effective rate of interest is 34.05 2 (1 + .0555) = 11.4%.
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19. The current price of a stock is 55 and the annual effective rate of interest is 8%. The profit earned by a call option with maturity in one year and strike price of 62 is 5.48. The profit earned by a put option with maturity in one year and strike price of 62 is -3.80. What is the price of the stock one year from now? (A) 67.01 (B) 67.80 (C) 68.68 (D) 68.87 (E) 69.31
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Correct answer: (C) Solution: Since the call option has a positive profit and the put option has a negative profit, we know that S > K . Let C be the premium paid for the call option. The formula for the profit of a call option is: Profitcall = max{S − K, 0} − F V (Premium) Profitcall = S − K − F V (Premium)
5.48 = S − 62 − C (1 + .08)1 . Let P be the premium paid for the put option. The formula for the profit of a put option is: Profit put = max{K − S, 0} − F V (Premium) Profit put = 0 − F V (Premium)
−3.80 = 0 − P (1 + .08)1 Since we are given information about call options and put options with the same strike price and maturity, we can use the put-call parity formula. If S (0) is the current price of the stock, then: C − P = S (0) − P V (K ) C − P = 55 − 62(1 + .08)
1
−
This is a system of three equations with three unknowns. Solve to find S = 68.68.
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20. Ron’s account earns interest at a force of interest of 1+ kt . Money deposited into the account will double after n years. Jane’s account earns interest at a force of interest of kt + t2 . Money deposited into the account will double after n years. Calculate k . (A) -0.69 (B) -0.39 (C) 0.26 (D) 0.84 (E) 1.21
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Correct answer: (A) Solution: Money in Ron’s account doubles after n years:
n
2 = exp ( 0 (1 + kt )dt) ln(2) = n + 12 kn 2
Money in Jane’s account doubles after n years:
n
2 = exp ( 0 ( kt + t2 )dt) ln (2) = 12 kn2 + 13 n3
Solve this system of two equations with two unknowns to find k = − 0.69.
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21. At time 0, Sally deposits $100 into an account bearing an annual rate of discount of 4.77%, and Alan deposits $88 into an account bearing a nominal rate of interest convertible monthly of 6.5%. At time t, the values of the accounts are equal. Find t. (A) 6 (B) 7 (C) 8 (D) 9 (E) 10
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Correct answer: (C) Solution: Solve the following equation for t :
100(1 − .0477)
t
−
Guess-and-check may be appropriate.
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= 88(1 +
.065 12t ) 12
22. The annual effective rate of interest is i. It is known that (Is)15i = 136 and s15i = 18. Calculate (Ds)15i . (A) 134 (B) 140 (C) 144 (D) 149 (E) 152
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Correct answer: (E) Solution: Recognize that (Is)15i + ( Ds)15i = 16s15i . Plug in the information that is already known to get 136 + (Ds)15i = 16 · 18. Then (Ds)15i = 152.
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23. A company has provided Redington immunization for its liability of L in 1 year. The company has two assets: a five-year zero-coupon bond with face amount of L , and current cash on hand of X . The values of X and L are related such that X = rL. Calculate r . (A) .298 (B) .376 (C) .432 (D) .457 (E) .535
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Correct answer: (E) Solution: The first two conditions for Redington immunization are: X + Lv5 = Lv
and
0 − 5Lv6 = − Lv2 . Using the second condition, we see that 5 v 6 = v 2 or v 4 = 15 . Therefore v = .6687. Then, using the first condition, we see that X + .1337L = . 6687L or X = . 535L.
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24. Andrew takes out a 5-year loan worth 10,000 with payments of 197 at the end of each month. After 2 years, Andrew decides to pay the loan off faster by paying an additional P per month. This allows him to pay the loan off 8 months early. Calculate P . (A) 20 (B) 31 (C) 34 (D) 40 (E) 51
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Correct answer: (E) Solution: Find the monthly interest rate j by using a financial calculator to solve the equation 10, 000 = 197a60 j 60 j to find j = . 005654. The outstanding balance after 2 years is 10, 000(1 + 24 6400. Now .005654) − 197s24. Now there are 36 months months remainin remaining g until the loan loan 24 .005654 = 6400 would be paid off with payments of 197, but there are only 28 months remaining until the loan will be paid off with payments of 197 + P . Find Find P by solving the equation 6400 = (197 + P )a28. 28 .005654 to find P = 51.
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25. A 20-year loan has level level annual annual payments of 500 at the end of each year. The interest interest paid th in the 6 year is 150. Calculate the total amount of interest paid during the loan. (A) 2136 2136 (B) 2394 2394 (C) 2610 2610 (D) 2871 2871 (E) 3109 3109
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Correct answer: (A)
150 = 500( 500(11 − v 20 6+1 ). Solv Soluti Solution: on: The interest interest paid during during the 6th year is 150 Solvee thi thiss equation to calculate v = .9765, so i = .02406. Then the loan amount A is given by A = 500a20. 20 .02406 = 7864, but the total number of dollars paid over the loan is 20 · 500 = 10000. Therefore the total amount of interest paid during the loan is 10000 − 7864 = 2136. −
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26. A perpetuity due has a payment of P at time 0. Payments are annual and increase by 3% per year. The annual effective rate of interest is 5%. The present value of the perpetuity is 4000. Calculate P . (A) 72 (B) 74 (C) 76 (D) 78 (E) 80
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Correct answer: (C) Solution: This perpetuity is equivalent to a perpetuity immediate that has an initial payment of 1 .03P and is worth 4000 − P . Then we can use the formula for the present value of a geometrically increasing perpetuity: 4000 − P = 1.03P .05 1 .03 . Solve this equation to find P = 76. −
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27. A 30-year bond has a face value of 50,000 with semiannual coupons at a rate of 8% convertible semiannually and a price of 48,936. What is the adjustment to book value in the 16 th year? (A) Write-down of 25.19 (B) Write-down of 27.05 (C) Write-down of 27.30 (D) Write-up of 27.30 (E) Write-up of 29.31
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Correct answer: (E) Solution: First we note that since the bond is priced at a discount, the adjustment to book value will be a write-up. Let j be the interest rate per coupon period. Then j can be found using a financial calculator with N = 60, P V = −48, 936, P M T = 2000, and F V = 50, 000; solve on I to find j = . 04096. To find the adjustment to book value in the 16th year, we will find the book value of the bond immediately after the 32 nd payment and the book value of the bond after the 30 th payment; their difference is the adjustment. After the 32nd payment, there are 28 payments remaining. The book value can be found using a financial calculator with N = 28, I = 4.096, P M T = 2000, and F V = 50, 000 to find P V = −49, 211, so the book value is 49,211. The book value after the 30 th payment can be found the same way with N = 30, which gives a book value of 49,181. The difference is 29.31, so the adjustment to book value is a write-up of 29.31.
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28. It is known that K 1 < K 2 < K 3 . Let X be the premium paid for a straddle around K 2 , Y be the premium paid for a K 1 -K 2 strangle, and Z be the premium paid for a call option with strike price K 2 . What is the relationship between X , Y , and Z ? (A) X > Y > Z (B) Y > X > Z (C) Y > Z > X (D) Z > X > Y (E) Z > Y > X
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Correct answer: (A) Solution: The premium paid for a straddle is more than the premium paid for a strangle because a straddle has more spot prices which result in a positive payoff. Therefore X > Y . The premium paid for a strangle is more than the premium paid for a call option because the strangle has a positive payoff when the spot price decreases whereas the call option does not. Therefore Y > Z . Therefore the answer is (A).
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29. An investor purchases a call ratio spread. The strike price of the purchased call option is 230 and the strike price of the written call options is 245. The following table shows the investor’s profit for several spot prices of the underlying asset at maturity. Spot price at maturity
Profit
220 240 260 280
−5 5 −20 X
Calculate X . (A) -20 (B) -30 (C) -40 (D) -50 (E) -60
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Correct answer: (E) Solution: Purchasing a call ratio spread consists of buying one call option and writing n call options at a higher strike price. Let P be the future value of the net premium paid for the call ratio spread. Then, if S is the spot price at maturity, the call ratio spread’s profit function is: Profit = max{S − 230, 0} − n · max{S − 245, 0} − P If the spot price at maturity is 220, the net profit is −5:
−5 = max{220 − 230, 0} − n · max{220 − 245, 0} − P −5 = − P Therefore the premium paid is 5. If the spot price at maturity is 260, the net profit is −20:
−20 = max {260 − 230, 0} − n · max{260 − 245, 0} − 5 −20 = 30 − n · 15 − 5 Solve to find n = 3. Therefore the investor purchased a 3:1 call ratio. If the spot price at maturity is 280, the net profit is X : X = max {280 − 230, 0} − 3 · max{280 − 245, 0} − 5
Solve to find X = − 60.
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30. It is known that the interest rate i is equal to the present value factor v. Calculate the discount rate d . (A) .089 (B) .244 (C) .382 (D) .618 (E) .733
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Correct answer: (C) 1 1 Solution: The present value factor v is given by v = 1+i . Since i = v , this gives i = 1+i . This is a quadratic equation; solve to find i = − 1.618, .618. Therefore i = . 618. (The other i .618 = 1+.618 = . 382. choice is not logical.) Then the discount rate d is given by d = 1+i
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31. Option X is an out-of-the-money put option. Option Y is a call option for the same asset with the same strike price as option X . Under which of the following conditions will option Y’s payoff be positive? (I) A large decrease in the price of the underlying asset (II) No change in the price of the underlying asset (III) A large increase in the price of the underlying asset (A) (I) only (B) (I) and (II) (C) (II) only (D) (II) and (III) (E) (III) only
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Correct answer: (D) Solution: Option X, which is a put option, is out-of-the-money. Since option X and option Y have the same strike price, option Y is therefore in-the-money. This means that if it expires with no change in the price of the underlying asset, option Y will have a positive payoff. Since it is a call option, it will also have a positive payoff if the price of the underlying asset increases. Therefore the answer is (D).
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32. The annual effective interest rate is 8%. A 36-month annuity due makes monthly payments of 50. Calculate the accumulated value of the annuity immediately after the final payment. (A) 1992 (B) 2005 (C) 2018 (D) 2031 (E) 2042
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Correct answer: (C) 1
Solution: The monthly rate of interest is (1 + . 08) 12 − 1 = .006434. Then calculate the accumulated value of the annuity after 36 months: 50 s¨36.006434 = 2031. However, since this is an annuity due, the final payment occurs at the end of the 35 th month. Thus we need to 2031 = 2018. discount this result by one month, so the answer is 1+.006434
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33. A company has liabilities of L in 1 year and 1000 in 2 years. Bond X is a one-year bond with annual coupons of 4% and is priced at par. Bond Y is a two-year bond with annual coupons of 8% and is priced at par. The company has created a portfolio using bond X and bond Y that exactly matches its liabilities. The present value of this portfolio is 2000. Calculate L . (A) 1080 (B) 1120 (C) 1191 (D) 1220 (E) 1278
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Correct answer: (C) Solution: Let X be the face amount of bond X and let Y be the face amount of bond Y . To exactly match the liabilities, after one year the company receives the face amount of bond X , one coupon from bond X , and one coupon from bond Y . Therefore X + .04X + .08Y = L. After two years the company receives the face amount of bond Y and one coupon from bond Y . Therefore Y + .08Y = 1000. Since both bonds are priced at par, their price is equal to their face amount. Therefore X + Y = 2000. Solve this system of three equations with three unknowns to find L = 1191.
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34. An investor wants to buy 20-year bonds. Current interest rates are 8%, but the investor believes that interest rates will rise within the next 10 years. The following 20-year bonds are available in any face amount. Which type of bond should the investor choose? (I) Zero-coupon bond (II) Bond with a coupon rate of 5% (III) Bond with a coupon rate of 10% (A) The investor should invest solely in bond (I) (B) The investor should invest solely in bond (II) (C) The investor should invest solely in bond (III) (D) The investor should invest in a combination of bond (I) and bond (II) (E) The investor should invest in a combination of bond (I) and bond (III)
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Correct answer: (C) Solution: Since the investor believes that interest rates will rise in the future, she should invest in bonds that have higher coupon rates so that she can reinvest more money sooner. (If she thought interest rates would go down, she should invest in zero-coupon bonds because her money would be locked into the rate of 8% for longer.) Therefore the investor should invest solely in bond (III) so that she can reinvest as much money as possible at the higher rate in the future.
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35. On January 1, a fund has a balance of 1000. On March 31, a withdrawal is made of 200. On June 30, a deposit is made of X . On December 31, the fund has a balance of 1400. The dollar-weighted rate of return is 25%. Calculate X . (A) 322 (B) 344 (C) 349 (D) 362 (E) 390
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