Exam FM/2 Practice Practice Exam 3 Answer Key Copyright c 2013 Actuarial Investment.
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1. An n -year bond has a face amount of 1000 with annual coupons of 5%. The book value of the bond at the end of the 6 th year is 857 and the adjustment to the bond in the 6 th year is a write-up of 11. Calculate n. (A) 12 (B) 13 (C) 14 (D) 15 (E) 16
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Correct answer: (D) Solution: Since the adjustment to the bond in the 6th year is +11, then BV B V 5 = 846. Let j be B V 6 = BV B V 5 (1 + j ) 50. Thus 857 = 846(1 + j ) 50. the annual yield rate. We know that BV 907 Therefore j = 846 1 = . 0721.
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We know that the book value immediately after the 6 th payment is 857. At that time, there are n 6 payments payments remaining. remaining. Use a financial calculator calculator with I = 7.21, P V = 857, P M T = 50, and F V = 1000 to find N = 9. This This means that that there there are 9 paymen payments ts th remaining after the 6 payment, so there were 15 payments total.
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2. An investor buys a 10-year bond for X . The bond pays semiannual coupons at a rate of 5% and is priced to yield 7.5% convertible semiannually. The investor reinvests all coupons into an account earning interest at an annual effective interest rate of 6%. After 10 years, he has 5445. Calculate X . (A) 2431 (B) 2579 (C) 2696 (D) 2703 (E) 2928
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Correct answer: (C) Solution: Let F be the face amount of the bond. After 10 years, the investor has the face amount F plus the accumulated value of the coupons F . 025. The nominal rate of interest convertible semiannually at which the investsed coupons accumulate is 1.061/2 1 = . 02956. Therefore, the accumulated value of the coupons is F . 025s20.02956 . After 10 years, the investor has 5445, so we know that 5445 = F + F .025s20.02956 . Solve to find F = 3262.86.
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= .0375. Then the price of the bond is X = The semiannual yield rate of the bond is .075 2 1 20 3262.86( 1+.0375 ) + 3262.86 .025a20.0375 = 2696.
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3. Which of the following financial instruments carries the highest credit risk? (I) A forward contract purchased over-the-counter (II) A futures contract (III) A put option purchased over-the-counter (A) (I) (B) (II) (C) (III) (D) (I) and (II) both have the highest credit risk (E) (I) and (III) both have the highest credit risk
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Correct answer: (A) Solution: A forward contract purchased over-the-counter has a higher credit risk than a futures contract because a futures contract is traded on an exchange. A forward contract purchased over-the-counter has higher credit risk than a put option purchased over-the-counter because a put option may be worthless at maturity, which means that the exposure at default (the amount that could be lost if the other party defaults) may be 0. Therefore the answer is (A).
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4. A 4-year annuity makes payments at the beginning of every month starting today. The first 12 payments are 1000 each. The next 12 payments are 1100 each. The next 12 payments are 1210 each. The final 12 payments are 1331 each. What is the value of the annuity at the end of 4 years valued at an interest rate of 12% convertible monthly? (A) 43438 (B) 43872 (C) 70032 (D) 70732 (E) 89073
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Correct answer: (D) Solution: First find the accumulated value of the first 12 payments valued at the end of the s12.01 = 12809.33. Now realize that this annuity is equivalent to a geometfirst year: 1000¨ rically increasing annuity with an initial payment of 12809.33 where the payments increase )12 1 = .1268. by 10% each year. The interest rate used to value this annuity is (1 + .12 12 Thus the accumulated value of the annuity is:
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4
1.1268
· 12809.33(
9
.1 1−( 1+1+ )4 .1268 ) .1268−.1
= 70732
5. A company has a liability of 1000 due in 1.95 years. The company wants to provide Redington immunization for the liability by purchasing a combination of bond X and bond Y , which are available in any face amount. Bond X is a 2-year bond with annual coupons of 11%. Bond Y is a 2-year zero-coupon bond. The annual effective rate of interest is 5%. What amount of bond Y should be purchased to provide Redington immunization for this liability? (A) 385 (B) 434 (C) 471 (D) 518 (E) 575
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Correct answer: (C) Solution: The first two conditions for Redington immunization are: .11Xv + 1 .11Xv 2 + Y v 2 = 1000v 1.95
and 2
3
−.11Xv − 2.22Xv − 2Y v
3
=
−1950v
2.95
Solve this system of two equations with two unknowns to find Y = 471.
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6. A 10-year bond with annual coupons of 5% is priced to yield an effective annual interest rate of 8.3%. A 10-year bond with semiannual coupons at a rate of 5% has the same face amount and price and yields an annual effective interest rate of i %. Calculate i . (A) 8.25 (B) 8.30 (C) 8.42 (D) 8.51 (E) 8.57
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Correct answer: (C) Solution: We can choose to use any face amount to find the answer. Suppose that the face amount for both bonds is 1000. Then the price of the first bond can be calculated using a financial calculator with N = 10, I = 8.3, P M T = 50, and F V = 1000; solve to find P V = 781.53, so the bond price is 781.53. Then the semiannual yield rate of the second bond can be calculated using a financial calculator with N = 20, P V = 781.53, P M T = 25, and F V = 1000; solve to find I = 4.125. This represents the rate per coupon period, so the annual effective interest rate i is (1 + .04125)2 1 = . 0842. Thus i = 8.42.
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7. Betty buys a 30-year bond with annual coupons of 4% priced to yield 7% for a purchase price of X . Coupon payments from the bond are reinvested into an account earning interest at an annual effective rate of 6%. After 30 years, Betty has 100,000. Calculate X . (A) 14977 (B) 15081 (C) 15119 (D) 15228 (E) 15414
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Correct answer: (B) Solution: Let F be the face amount of the bond that Betty buys. Then the coupon payments are .04F . The coupon payments accumulate at an annual effective rate of interest of 6% and Betty will also receive F at the end of 30 years. Therefore 100 , 000 = .04F s 30.06 + F . Thus F = 24025 and so the coupon payments are 961.
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Find the purchase price X using a financial calculator with N = 30, I = 7, P M T = 961, and F V = 24025 to find X = 15081.
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8. A portfolio worth 500 produces cashflows of 150 at time 3, X at time 4, and 200 at time 6. The cashflow of X at time 4 makes up 45% of the value of the portfolio. Calculate the Macaulay duration of the portfolio. (A) 3.67 (B) 4.00 (C) 4.33 (D) 5.00 (E) 5.50
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Correct answer: (C) Solution: Since the cashflow of X makes up 45% of the value of the portfolio, the remaining cashflows of 300 makes up 55% of the value of the portfolio: 500 .55 = 150v 3 +200v 6 . Use the substitution y = v 3 to transform this into a quadratic equation and solve to find y = . 856. Therefore v = . 95.
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Then, since the cashflow of X makes up 45% of the value of the portfolio, we have 500 .45 = Xv 4 . Solve to find X = 277.
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Then the Macaulay duration of the portfolio is
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3·150·v 3 +4·277·v 4 +6·200·v 6 500
= 4.33.
9. Which of the following are true regarding swaps? (I) A prepaid swap is a swap in which the present value of all payments, calculated using the current term structure, is paid immediately (II) A deferred swap is a swap in which the future value of all payments, calculated using the current term structure, is paid at the end of the swap’s term (III) The swap spread is collected by an intermediary for the service of arranging a swap between two parties (A) (I) only (B) (II) only (C) (I) and (III) (D) (II) and (III) (E) The answer is not given by any of (A), (B), (C), or (D)
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Correct answer: (C) Solution: A prepaid swap is a swap in which the present value of all payments, calculated using the current term structure, is paid immediately. Therefore (I) is true. A deferred swap is a swap in which the conditions of the swap are agreed upon today, but the exchange of payments begins sometime in the future. Therefore (II) is false. The swap spread is collected by an intermediary for the service of arranging a swap between two parties. Therefore (III) is true. Therefore the answer is (C).
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10. Two bonds have the same yield rate. Both bonds have a face amount of 1000, a redemption amount of 1100, and a price of 800. The first is a 15-year bond with annual coupons at a coupons rate of 5%. The second is a 20-year bond with coupons at a coupon rate of r % convertible semiannually. Calculate r . (A) 2.43 (B) 2.68 (C) 4.87 (D) 5.35 (E) 5.45
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Correct answer: (D) Solution: From the first bond, use a financial calculator with N = 15, P V = 800, P M T = 50, and F V = 1100 to find that the annual effective yield rate for both bonds is .07667. Therefore the yield rate per six months is (1 + .07667)1/2 1 = . 03763. Then for the second bond, use a financial calculator with N = 40, I = 3.763, P V = 800, and F V = 1100 to 2 = find that each semiannual coupon is 26.76. Therefore the coupon rate r is given by 26.76 1000 .0535.
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11. Which of the following are true concerning the cost of carry of a long position in a stock? (I) The cost of carry includes any dividends paid by the stock. (II) The cost of carry includes the full price at purchase of the stock. (III) The cost of carry is zero if the stock pays no dividends and if the increase in stock price exactly matches the annual effective rate of interest. (A) (I) only (B) (III) only (C) (II) and (III) (D) (I), (II), and (III) (E) The answer is not given by any of (A), (B), (C), or (D)
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Correct answer: (E) Solution: The cost of carry of a long position does not include dividends paid. (The cost of carry of a short position does include dividends paid.) Therefore (I) is false. The cost of carry includes the interest paid on a hypothetical loan for the stock’s purchase price, but does not include the purchase price. Therefore (II) is false. The cost of carry includes the interest paid on a hypothetical loan for the stock’s purchase price, regardless of whether the stock price increase matches the annual effective rate of interest. Unless the stock price is zero, the cost of carry will not be zero. Therefore (III) is false. Therefore the answer is (E).
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12. Which of the following are true about convertible bonds? (I) A convertible bond is always a zero-coupon bond. (II) A convertible bond does not have a fixed maturity date. (III) If a company’s stock value falls, the value of a convertible bond issued by the company also must fall. (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: (E) Solution: Convertible bonds can have coupons. Therefore (I) is false. Convertible bonds have maturity dates. Therefore (II) is false. If a company’s stock value is low, the value of a convertible bond will come from its coupons and redemption value. If the stock value falls, the value of convertible bonds will not fall any lower. Therefore a drop in stock value does not necessarily mean a drop in convertible bond value. Therefore (III) is false. Therefore the answer is (E).
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13. Adam and Jane both invest the same amount of money into 10-year bonds that have annual coupons and are priced to yield 6%. Adam’s bond has coupons of 5%, and Adam reinvests coupons into an account earning interest at an effective annual rate of 12%. Jane’s bond has coupons of 15%, and Jane reinvests coupons into an account earning interest at an effective annual rate of i %. After 10 years, Adam and Jane have the same amount of money. Calculate i . (A) 6.4 (B) 7.7 (C) 8.2 (D) 9.0 (E) 9.8
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Correct answer: (E) Solution: Suppose that Adam and Jane each invest 1000 into their bonds. The face amount of Adam’s bond can be calculated using the formula P = F + (r j )F anj , or 1000 = F +(.05 .06)F a10.06 . Solve to find F = 1079.45. Therefore Adam’s coupons are each 1079.45 .05 = 53.97. After 10 years, Adam receives the face amount of the bond plus the accumulated value of the coupons reinvested at 12%, or 1079.45+53.97s10.12 = 2026.60.
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The face amount of Jane’s bond can be calculated in the same way using the formula 1000 = F + (.15 .06)F a10.06 . This yields F = 601.54. Therefore Jane’s coupons are each 90.23. After 10 years, Jane has the face amount plus the accumulated value of the coupons reinvested at i%. We know that after 10 years, Jane has 2026.60. Therefore 2026.60 = 601.54 + 90.23s10i . Solve using a financial calculator to find i = 9.8.
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14. At time t = 0, Roger deposits $100 into an account earning interest at a force of interest of 1 . At time t = 4, he transfers all of the money to a second account earning interest at a 1+3t rate of 12% compounded monthly. How much money does Roger have at time t = 8? (A) 200 (B) 379 (C) 422 (D) 429 (E) 560
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Correct answer: (B) Correct answer: 379 Solution: At time t = 4, Roger will have 4
100e
0
(1+3t)dt
1
4
= 100e 3 ln(1+3t)]0 = 100e.855 = 235 At time t = 8, Roger will have 235(1 +
.12 12·4 ) 12
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= 379.
15. A 30-year bond has a face amount of 10,000 and a redemption amount of 12,000. It has annual coupons of 5% and the purchase price is 10,000. Calculate the adjustment to book value in the 5th year. (A) Write-down of 103.87 (B) Write-down of 35.21 (C) No adjustment (D) Write-up of 35.21 (E) Write-up of 103.87
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Correct answer: (D) Solution: Use a financial calculator to find the yield rate using N = 30, P V = 10, 000, P M T = 500, and F V = 12, 000 to find I = 5.287. The adjustment to book value in the 5th year is the price immediately after the 5th payment minus the price immediately after the 4th payment. The value immediately after the 5th payment can be computed using a financial calculator with N = 25, I = 5.287, P MT = 500, and F V = 12, 000 to find the value is 10,159.23. The value immediately after the 4th payment can be computed in the same way with N = 26 to find the value is 10,124.02. Therefore the adjustment was 10, 159.23 10, 124.02 = 35.21. This is a write-up of 35.21.
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16. Which of the following positions could result in unlimited gain? (I) Short position in a call ratio (II) Long position in a put option (III) Short position in a strangle (A) (I) only (B) (III) only (C) (I) and (II) (D) (I), (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: As the price of the underlying asset increases, a long position in a call ratio could result in unlimited loss, and a short position in a call ratio could result in unlimited gain. Therefore (I) could provide unlimited gain. As the price of the underlying asset decreases, a long position in a put option will provide more gain. However, once the asset price reaches 0, the gain can no longer increase. Therefore (II) could not provide unlimited gain. As the price of the underlying asset increases or decreases, a long position in a strangle will provide increasing gain, but a short position in a strangle will provide increasing loss. Therefore (III) could not provide unlimited gain. Therefore the answer is (A).
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17. At an annual effective interest rate of 7%, the accumulated value of an annuity due with n annual payments of 10 is 110. At an annual effective rate of i%, the accumulated value of an annuity immediate with n annual payments of 13 is 140. Calculate i . (A) 5.1 (B) 5.8 (C) 6.5 (D) 7.3 (E) 8.3
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Correct answer: (E) Solution: Use the first annuity to find n using the formula 110 = 10¨ sn.07 . This shows that n = 8. The second annuity gives a formula for i : 140 = 13s8i . Use a financial calculator to find i = . 083.
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18. Daniel writes a covered call with maturity in six months, a premium of 4, and a strike price of 38. He also buys a put option for the same underlying asset with maturity in six months, a premium of 4, and a strike price of 38. After six months, the underlying asset’s price is 44. The annual effective interest rate is 11.8%. What is Daniel’s profit? (A) -10 (B) 0 (C) 10 (D) 12 (E) 18
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Correct answer: (B) Solution: By drawing profit diagrams for the covered written call and the purchased put, we can see that the profit is 0. We can also calculate this using profit formulas. Writing a covered call consists of writing a call option and purchasing the underlying asset. Daniel’s profit from writing the call option is max 44 38, 0 + F V (4). His profit from owning the underlying asset is 44 38. His profit from buying the put option is max 38 F V (4). 44, 0
}−
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The profit from the entire portfolio is F V (4) = 0. 44, 0
}−
− { −
}
{ −
−max{44 − 38, 0} + F V (4) + 44 − 38 + max{38 −
Notice that we did not need the annual effective interest rate because we did not have to calculate the future value of the premiums.
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19. A stock costs 54 and will pay a dividends of 6 this year. The annual effective rate of interest is 6%. Let X be the one-year cost of carry of a long position in the stock, Y be the one-year cost of carry of a short position in the stock, and Z be the one-year cost of carry of a long position in a futures contract in the stock. What is the relationship between X , Y , and Z ? (A) X > Y > Z (B) X > Z > Y (C) Y > X > Z (D) Y > Z > X (E) Z > X > Y
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Correct answer: (C) Solution: The one-year cost of carry of a long position in one share of the stock is 54 .06 = 3.24 because 3.24 is the amount of interest that could be earned at the risk-free rate if the stock were sold and the money used to buy bonds. Therefore X = 3.24.
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The cost of carry of a short position in one share of the stock is 6 because the stock will pay a dividend of 6 this year. Therefore Y = 6. The cost of carry of a long position in a futures contract is 0 because no investment is required and the position will not be short any dividends. Therefore Z = 0. Thus Y > X > Z , so the answer is (C).
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20. A loan with amount A makes n level annual payments, where n is even, beginning 1 year after the loan is taken out. The annual effective interest rate is i such that 0 < i < 1 . Which of the following scenarios are possible? (I) The interest paid in year n2 equals the principal paid in year n2 . (II) The principal paid in year 1 equals the interest paid in year n . (III) Immediately after payment n2 , the oustanding balance on the loan is 12 A. (A) (I) only (B) (II) only (C) (I) and (III) (D) (II) and (III) (E) The answer is not given by any of (A), (B), (C), or (D)
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Correct answer: (E) Solution: (I) is possible where n = 2 and i =
√ 2 − 1.
(II) is possible where n = 2 and i = . 618 (the golden ratio minus one). (III) is not possible because the majority of principal on a loan is paid during the second half of the loan. Therefore the answer is (E).
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21. A 10-year zero-coupon bond has a face value of 1000. Let i be the annual effective interest rate such that i > 0, MacD be the bond’s Macaulay duration, and ModD be the bond’s modified duration. Which of the following relationships is true? (A) MacD < ModD < 10 (B) MacD < ModD = 10 (C) ModD < MacD < 10 (D) ModD < MacD = 10 (E) It cannot be shown that any of (A), (B), (C), or (D) is true.
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Correct answer: (D) Solution: The Macaulay duration of a single cashflow is equal to the time at which the cashflow is made. Therefore MacD = 10. The modified duration ModD is related to the Macaulay duration MacD according to the following formula: ModD = MacD v . Since i > 0, we know that v < 1, and therefore ModD < MacD. (Note that if the interest rate i were 0, then ModD = MacD.)
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Therefore ModD < MacD = 10.
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22. It is known that K 1 < K 2 . An investor has purchased a combination of options that result in a position equivalent to a K 1 -K 2 bear spread. What transaction will change his position to be equivalent to a long position in a ratio spread? (A) Buy a call option with strike price K 1 (B) Buy a call option with strike price K 2 (C) Write two call options with strike price K 2 (D) Buy a put option with strike price K 1 (E) The answer is not given by any of (A), (B), (C), or (D).
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Correct answer: (E) Solution: Since the investor already has a bear spread, he will change his position to a long position in a put ratio spread. He needs to change his payoff on the interval (0 , K 1 ). He can accomplish this by writing a put option with strike price K 1 . (Buying a call option with strike price K 2 creates a short position in a ratio spread.) The correct transaction is not given among (A), (B), (C), or (D), so the answer is (E).
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23. On January 1, a fund has a balance of 1000. At the end of each month for 12 months, interest is credited to the account at a rate of 12% convertible monthly, and a deposit of 100 is made into the account. Calculate the time-weighted rate of return of the fund. (A) 12.0% (B) 12.7% (C) 17.3% (D) 29.5% (E) 39.5%
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Correct answer: (B) Solution: Using the definition of time-weighted rate of return, we can calculate the answer: 1010 1000
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1121.1 1110
· ... ·
2295.08 2272.35
− 1 = 12.7%
However, it is much simpler to recognize that at the end of each month, the account accrues to 101% of its previous value. Therefore the time-weighted rate of return is 1.0112 1 = 12.7%. Note that this is equal to the annual effective rate of interest.
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24. A zero-coupon bond has a modified duration of 8. Calculate the convexity of the zero-coupon bond assuming an annual effective rate of interest of 12.5%. (A) 56.9 (B) 64.0 (C) 71.1 (D) 85.3 (E) 91.1
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Correct answer: (C) Solution: Solution: We know that MacD = ModD(1 + i ). Therefore Therefore the Macaulay Macaulay duration duration of the zero-coupon bond is 8(1 + . 125) = 9. Remember Remember that the Macaulay Macaulay duration duration of a single single cashflow cashflow is equal to the time at which that cashflow cashflow occurs. Therefore Therefore the redemption redemption date is t = 9.
P (i) P (i) is the price as a function of Remember that convexity is defined as C = P ( P (i) , where the interest rate i. Let F be the face amount of the bond. Then P (i) = F (1 + i) 9. Also, P (i) = 9F (1 + i ) 10 and P (i) = 90F (1 + i ) 11 . Then Then the convexi convexity ty of the bond is 11 P (i) 90F (1+ (1+ii) C = P ( = 90F = 71.1. P (i) F (1+ F (1+ii) 9 −
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25. A loan is to to be repaid repaid in n years, where n is divisible by 3, with level annual payments at the end of each year. The principal paid during year 13 n is one-half the principal paid during year 23 n. The annual effective interest rate is 8%. Calculate n . (A) 15 (B) 18 (C) 21 (D) 24 (E) 27
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Correct answer: (E) Solution: Solution: Since i = .08, then v = .9259. Let P be the annual annual payment payment.. Then Then the prin1 n n/3+1 n/3+1 ) and the principal paid during year 23 n is cipal paid during year 3 n is P (.9259 n/3+1 ). Since P (.9259n 2n/3+1 Since the principa principall paid during during year year 13 n is one-half the principal paid during year 23 n, we know that: −
−
P (.9259n
n/3+1 n/3+1
−
) = 12 P (.9259n
.92592n/3+1 .9259n/3+1 n/3 n/3
.9259
= =
n = 27
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1 2 1 2
2n/3+1 n/3+1
−
)
26. The following chart shows prices for zero-coupon bonds with face amount 1000. It is known that the one-year forward rate two years from now is 6.1%. Time to maturity (years)
1 2 3
Zero-coupon bond price 930.00 X X 50.26
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Calculate X . (A) 756.57 (B) 791.05 (C) 840.05 (D) 874.19 (E) 892.42
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Correct answer: (D) Solution: Let v (t, u) be the forward discount factor from time t to time u. (In this way, 1 v (t, u) = 1+i(t,u) .) Since the one-year forward rate two years from now is 6.1%, we know 1 = . 9425. that v (2, 3) = 1+.061 The price of a zero-coupon bond is its face amount multiplied by the forward discount factors from now until its maturity date. Therefore:
930 = 1000 v (0, 1)
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X = 1000 v (0, 1) v(1, 2)
· · X − 50.26 = 1000 · v (0, 1) · v (1, 2) · v (2, 3) The value of v(2, 3) is already known, so this is a system of three equations with three unknowns. Solve to find X = 874.19.
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27. A company has a liability of 800 in 6 years. The company wants to provide Redington immunization for the liability by purchasing two zero-coupon bonds are available for purchase at any face amount. Bond X matures in 4 years and bond Y matures in 8 years. The annual effective rate of interest is 6%. What amount of bond X should be purchased to provide Redington immunization for this liability? (A) 255 (B) 276 (C) 309 (D) 327 (E) 356
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Correct answer: (E) Solution: Let X be the face amount of bond X purchased and let Y be the face amount of bond Y purchased. The first two conditions for Redington immunization are: Xv4 + Y v 8 = 800v 6
and 5
9
7
−4Xv − 8Y v = −4800v . Solve this system of two equations with two unknowns to find X = 356.
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28. The price of a 10-year bond, which is 100 less than its face amount, is equal to the present value of the face amount. It is known that j = r + .03, where j is the annual yield rate and r is the annual coupon rate. Calculate the face amount. (A) 291 (B) 301 (C) 332 (D) 391 (E) 440
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Correct answer: (D) Solution: Since the price of the bond is equal to the present value of its face amount, this must be a zero-coupon bond. (This is because none of the interest is paid during the life of the bond, so the coupons must be 0. This can also be proved by using the identity P = K + r (F K ), where K is the present value of the face amount. This gives us P = P + jr (F P ) j or equivalently 0 = jr (100). Therefore r = 0.)
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Since r = 0, we know that j = .03. Therefore ( F face amount. This gives F = 391.
10
− 100)(1 + .03)
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= F , where F is the
29. A company has a liability of 1000 due in 2.6 years. The company purchases a 3-year bond with face amount F and annual coupons of r % to provide Redington immunization for the liability. The annual effective rate of interest is 6%. Calculate r . (A) 12.6 (B) 14.7 (C) 16.4 (D) 18.1 (E) 19.7
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Correct answer: (E) Solution: The first two conditions for Redington immunization are:
(F r)v + ( F r)v 2 + (F + F r)v 3 = 1000v 2.6 and 2
3
4
−(F r)v − 2(F r)v − 3(F + F r)v = −2600v
3.6
.
Solve this system of two equations with two unknowns to find r = 19.7.
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30. It is known that K 1 < K 2 . Which of the following positions create a bear spread? (I) Buying a call with strike K 1 and selling a call with strike K 2 (II) Buying a call with strike K 2 and selling a call with strike K 1 (III) Selling a covered written call with strike K 1 and selling a put with strike K 2 (IV) Buying a put with strike K 1 and selling a put with strike K 2 (A) (I) only (B) (I) and (IV) (C) (II) only (D) (II) and (III) (E) The answer is not given by any of (A), (B), (C), or (D)
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Correct answer: (C) Solution: Combinations (I), (III), and (IV) create bull spreads. Combination (II) creates a bear. (Remember that a covered written call has the same profit as a purchased put.) The answer is (C).
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31. ABC Co. takes out a loan for 10,000. For the first 19 years, ABC Co. makes payments at the end of each year of 120% of the interest due. At the end of the 20 th year, ABC Co. makes a final balloon payment of X . The annual effective interest rate is 8%. Calculate X . (A) 7243 (B) 7360 (C) 7497 (D) 7822 (E) 7949
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Correct answer: (E) Solution: At the end of each year, ABC Co. pays 120% of the interest due, so the outstanding balance is reduced by 20% of the interest due. The interest due is 8% of the outstanding balance. Therefore, at the end of each year, the outstanding balance is reduced by 20% of 8% of the outstanding balance. Thus OBt = (1 . 2 . 08) OBt 1 = .984 OBt 1 . Equivalently, OB t = 10, 000 .984t .
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The outstanding balance at time 19 is 10 , 000 . 98419 = 7360. At time 20, this has accumulated to 7360(1 + . 08) = 7949. Then ABC Co. pays off the entire remaining balance of 7949.
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32. A 12-year loan makes 11 annual payments of 100 at the end of each year followed by a balloon payment of 400 at the end of year 12. The annual effective interest rate is i . Which of the following represents the interest paid in the 4th payment? 1 9 ) 1+i 1 9 ) (B) 100 + 400( 1+i 1 8 1 9 ) 400( 1+i ) (C) 100( 1+i 1 8 1 9 ) 200( 1+i ) (D) 300( 1+i 1 8 1 9 ) 400( 1+i ) (E) 100 + 300( 1+i
(A) 100
− 100(
− −
−
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Correct answer: (E) Solution: Treat this as a 12-year loan with level payments of 100 plus a lump sum with a future value of 300. The interest paid in the 4th year on the loan with level payments is 1 12 4+1 1 9 100(1 ( 1+i ) ) or equivalently 100 100( 1+i ) . The interest paid on the lump sum in 1 8 1 9 1 8 1 9 th ) 400( 1+i ). the 4 year is 300( 1+i ) 300( 1+i ) . Thus the answer is 100 + 300( 1+i
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−
−
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33. A geometric annuity immediate with 24 monthly payments is worth 3000. A geometric annuity immediate with 12 monthly payments is worth 1000. The first payment of both annuities is the same, and the payments of the annuities increase at the same rate. The annual effective rate of interest is 10%. What is the amount of the final payment of the 24-month annuity? (A) 253.20 (B) 271.65 (C) 290.10 (D) 308.55 (E) 327.00
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Correct answer: (B) Solution: The monthly interest rate is 1.11/12
− 1 = . 00797.
r 1−( 1.1+ )24 00797 Let X be the first payment of each annuity. The 24-month annuity is worth 3000 = X .00797−r . r 1−( 1.1+ )12 1+r 00797 )12 . The 12-month annuity is worth 1000 = X .00797 . Make the substitution y = ( 1.00797 −r
Then:
2
1 y 3000 = X .00797 −
1
1 y and 3000 = 3X .00797 −
r 2
−
r
−
− y = 3(1 − y) y − 3y + 2 = 0 2
y = 1 or y = 2
1+r )12 to find r = .0679. (If y = 1 is used, there is no possible value Use y = 2 = ( 1.00797 1−(
1.0679
)24
1.00797 of X that satisfies the problem description.) Then use the equation 3000 = X .00797 .0679 to find X = 59.94. Therefore the first payment is 59.94 1 .06790 , the second payment is 59.94 1.06791 , and the 24th payment is 59.94 1.067923 = 271.65.
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34. Order the following quantities from smallest to greatest valued at the same annual effective interest rate. I. The present value of an annuity immediate with 12 annual payments of P . II. The present value of an annuity due with 12 annual payments of P . III. The present value of an annuity immediate with 12 monthly payments of P . IV. The present value of an annuity due with 12 monthly payments of P . V. The present value of an 12-month continuously payable annuity paying a total of 12P during the annuity. (A) I, II, III, IV, V (B) I, II, III, V, IV (C) I, III, II, V, IV (D) I, III, V, II, IV (E) The answer is not given by any of (A), (B), (C), or (D).
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Correct answer: (B) Solution: One solution is to calculate the present value of each annuity. Another approach is to recognize that the present value of sooner payments is greater. This means that payments far into the future are worth less. The 12-year annuities have smaller present values than the 12-month annuities. Furthermore, the 12-year annuity immediate has a smaller present value than the 12-year annuity due because the payments are delayed by 1 year. Therefore, the smallest quantity is (I) followed by (II). The 12-month continuously payable annuity is payed evenly over 12-months, while the payments of the 12-month annuity immediate are skewed toward the end of the year and the payments of the 12-month annuity due are skewed toward the beginning of the year. Therefore, the 12-month annuities are ordered as follows: (III), (V), (IV). The answer is (B).
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35. An investor creates a 2:1 ratio spread by buying one call with strike price 105 and selling two calls with strike price X . If the spot price at maturity were 130, the investor’s payoff would be 5. What is the investor’s maximum possible payoff?
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(A) 10 (B) 15 (C) 20 (D) 25 (E) 30
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