 Exam FM/2 Practice Exam 3 Answer Key Copyright c 2013 Actuarial Investment. 1
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# Exam FM Practice Exam 3

Oct 20, 2015

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Exam FM Practice Exam 3

#### accumulated value

• Exam FM/2Practice Exam 3

1

• 1. An n-year bond has a face amount of 1000 with annual coupons of 5%. The book value ofthe bond at the end of the 6th year is 857 and the adjustment to the bond in the 6th year is awrite-up of 11. Calculate n.

(A) 12

(B) 13

(C) 14

(D) 15

(E) 16

2

• Correct answer: (D)

Solution: Since the adjustment to the bond in the 6th year is +11, then BV5 = 846. Let j bethe annual yield rate. We know that BV6 = BV5(1 + j) 50. Thus 857 = 846(1 + j) 50.Therefore j = 907

846 1 = .0721.

We know that the book value immediately after the 6th payment is 857. At that time, thereare n 6 payments remaining. Use a financial calculator with I = 7.21, PV = 857,PMT = 50, and FV = 1000 to find N = 9. This means that there are 9 paymentsremaining after the 6th payment, so there were 15 payments total.

3

• 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 intoan account earning interest at an annual effective interest rate of 6%. After 10 years, he has5445. Calculate X .

(A) 2431

(B) 2579

(C) 2696

(D) 2703

(E) 2928

4

• Correct answer: (C)

Solution: Let F be the face amount of the bond. After 10 years, the investor has the faceamount F plus the accumulated value of the coupons F .025. The nominal rate of interestconvertible semiannually at which the investsed coupons accumulate is 1.061/21 = .02956.Therefore, the accumulated value of the coupons is F .025s20.02956. After 10 years, theinvestor has 5445, so we know that 5445 = F +F .025s20.02956. Solve to find F = 3262.86.The semiannual yield rate of the bond is .075

2= .0375. Then the price of the bond is X =

3262.86( 11+.0375

)20 + 3262.86 .025a20.0375 = 2696.

5

• 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

6

• Correct answer: (A)

Solution: A forward contract purchased over-the-counter has a higher credit risk than afutures 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 pur-chased over-the-counter because a put option may be worthless at maturity, which meansthat the exposure at default (the amount that could be lost if the other party defaults) may be0.

Therefore the answer is (A).

7

• 4. A 4-year annuity makes payments at the beginning of every month starting today. The first12 payments are 1000 each. The next 12 payments are 1100 each. The next 12 payments are1210 each. The final 12 payments are 1331 each. What is the value of the annuity at the endof 4 years valued at an interest rate of 12% convertible monthly?

(A) 43438

(B) 43872

(C) 70032

(D) 70732

(E) 89073

8

• Correct answer: (D)

Solution: First find the accumulated value of the first 12 payments valued at the end of thefirst year: 1000s12.01 = 12809.33. Now realize that this annuity is equivalent to a geomet-rically increasing annuity with an initial payment of 12809.33 where the payments increaseby 10% each year. The interest rate used to value this annuity is (1 + .12

12)12 1 = .1268.

Thus the accumulated value of the annuity is:

1.12684 12809.33(1(1+.1

1+.1268)4

.1268.1 ) = 70732

9

• 5. A company has a liability of 1000 due in 1.95 years. The company wants to provide Red-ington 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 of11%. 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 thisliability?

(A) 385

(B) 434

(C) 471

(D) 518

(E) 575

10

• Correct answer: (C)

Solution: The first two conditions for Redington immunization are:

.11Xv + 1.11Xv2 + Y v2 = 1000v1.95

and

.11Xv2 2.22Xv3 2Y v3 = 1950v2.95

Solve this system of two equations with two unknowns to find Y = 471.

11

• 6. A 10-year bond with annual coupons of 5% is priced to yield an effective annual interest rateof 8.3%. A 10-year bond with semiannual coupons at a rate of 5% has the same face amountand 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

12

• Correct answer: (C)

Solution: We can choose to use any face amount to find the answer. Suppose that the faceamount for both bonds is 1000. Then the price of the first bond can be calculated usinga financial calculator with N = 10, I = 8.3, PMT = 50, and FV = 1000; solve tofind PV = 781.53, so the bond price is 781.53. Then the semiannual yield rate of thesecond bond can be calculated using a financial calculator with N = 20, PV = 781.53,PMT = 25, and FV = 1000; solve to find I = 4.125. This represents the rate per couponperiod, so the annual effective interest rate i is (1 + .04125)2 1 = .0842. Thus i = 8.42.

13

• 7. Betty buys a 30-year bond with annual coupons of 4% priced to yield 7% for a purchaseprice of X . Coupon payments from the bond are reinvested into an account earning interestat 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

14

• Correct answer: (B)

Solution: Let F be the face amount of the bond that Betty buys. Then the coupon paymentsare .04F . The coupon payments accumulate at an annual effective rate of interest of 6% andBetty will also receive F at the end of 30 years. Therefore 100, 000 = .04F s30.06 + F .Thus F = 24025 and so the coupon payments are 961.

Find the purchase price X using a financial calculator with N = 30, I = 7, PMT = 961,and FV = 24025 to find X = 15081.

15

• 8. A portfolio worth 500 produces cashflows of 150 at time 3, X at time 4, and 200 at time6. The cashflow of X at time 4 makes up 45% of the value of the portfolio. Calculate theMacaulay duration of the portfolio.

(A) 3.67

(B) 4.00

(C) 4.33

(D) 5.00

(E) 5.50

16

• Correct answer: (C)

Solution: Since the cashflow of X makes up 45% of the value of the portfolio, the remainingcashflows of 300 makes up 55% of the value of the portfolio: 500 .55 = 150v3+200v6. Usethe substitution y = v3 to transform this into a quadratic equation and solve to find y = .856.Therefore v = .95.

Then, since the cashflow ofX makes up 45% of the value of the portfolio, we have 500.45 =Xv4. Solve to find X = 277.

Then the Macaulay duration of the portfolio is 3150v3+4277v4+6200v6

500= 4.33.

17

• 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 thecurrent term structure, is paid immediately

(II) A deferred swap is a swap in which the future value of all payments, calculated usingthe current term structure, is paid at the end of the swaps term

(III) The swap spread is collected by an intermediary for the service of arranging a swapbetween 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)

18

• Correct answer: (C)

Solution: A prepaid swap is a swap in which the present value of all payments, calculatedusing 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, butthe 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 betweentwo parties. Therefore (III) is true.

Therefore the answer is (C).

19

• 10. Two bonds have the same yield rate. Both bonds have a face amount of 1000, a redemptionamount of 1100, and a price of 800. The first is a 15-year bond with annual coupons at acoupons 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

20

• Correct answer: (D)

Solution: From the first bond, use a financial calculator withN = 15, PV = 800, PMT =50, and FV = 1100 to find that the annual effective yield rate for both bonds is .07667.Therefore the yield rate per six months is (1+ .07667)1/21 = .03763. Then for the secondbond, use a financial calculator with N = 40, I = 3.763, PV = 800, and FV = 1100 tofind that each semiannual coupon is 26.76. Therefore the coupon rate r is given by 26.762

1000=

.0535.

21

• 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 priceexactly 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)

22

• Correct answer: (E)

Solution: The cost of carry of a long position does not include dividends paid. (The cost ofcarry 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 stocks purchaseprice, 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 stocks purchaseprice, regardless of whether the stock price increase matches the annual effective rate ofinterest. Unless the stock price is zero, the cost of carry will not be zero. Therefore (III) isfalse.

Therefore the answer is (E).

23

• 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 companys stock value falls, the value of a convertible bond issued by the companyalso 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)

24

• Correct answer: (E)

Solution: Convertible bonds can have coupons. Therefore (I) is false.

Convertible bonds have maturity dates. Therefore (II) is false.

If a companys stock value is low, the value of a convertible bond will come from its couponsand redemption value. If the stock value falls, the value of convertible bonds will not fallany lower. Therefore a drop in stock value does not necessarily mean a drop in convertiblebond value. Therefore (III) is false.

Therefore the answer is (E).

25

• 13. Adam and Jane both invest the same amount of money into 10-year bonds that have annualcoupons and are priced to yield 6%.

Adams bond has coupons of 5%, and Adam reinvests coupons into an account earninginterest at an effective annual rate of 12%.

Janes bond has coupons of 15%, and Jane reinvests coupons into an account earning interestat 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

26

• Correct answer: (E)

Solution: Suppose that Adam and Jane each invest 1000 into their bonds.

The face amount of Adams bond can be calculated using the formula P = F +(rj)Fanj ,or 1000 = F+(.05.06)Fa10.06. Solve to find F = 1079.45. Therefore Adams coupons areeach 1079.45 .05 = 53.97. After 10 years, Adam receives the face amount of the bond plusthe accumulated value of the coupons reinvested at 12%, or 1079.45+53.97s10.12 = 2026.60.

The face amount of Janes bond can be calculated in the same way using the formula1000 = F + (.15 .06)Fa10.06. This yields F = 601.54. Therefore Janes couponsare each 90.23. After 10 years, Jane has the face amount plus the accumulated value ofthe coupons reinvested at i%. We know that after 10 years, Jane has 2026.60. Therefore2026.60 = 601.54 + 90.23s10i. Solve using a financial calculator to find i = 9.8.

27

• 14. At time t = 0, Roger deposits \$100 into an account earning interest at a force of interest of1

1+3t. At time t = 4, he transfers all of the money to a second account earning interest at a

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

28

• Correct answer: (B)

Solution: At time t = 4, Roger will have

100e 40 (1+3t)dt

= 100e13ln(1+3t)]40

= 100e.855

= 235

At time t = 8, Roger will have 235(1 + .1212)124 = 379.

29

• 15. A 30-year bond has a face amount of 10,000 and a redemption amount of 12,000. It hasannual coupons of 5% and the purchase price is 10,000. Calculate the adjustment to bookvalue in the 5th year.

(A) Write-down of 103.87

(B) Write-down of 35.21

(D) Write-up of 35.21

(E) Write-up of 103.87

30

• Correct answer: (D)

Solution: Use a financial calculator to find the yield rate using N = 30, PV = 10, 000,PMT = 500, and FV = 12, 000 to find I = 5.287. The adjustment to book value in the5th year is the price immediately after the 5th payment minus the price immediately afterthe 4th payment. The value immediately after the 5th payment can be computed using afinancial calculator with N = 25, I = 5.287, PMT = 500, and FV = 12, 000 to findthe value is 10,159.23. The value immediately after the 4th payment can be computed inthe same way with N = 26 to find the value is 10,124.02. Therefore the adjustment was10, 159.23 10, 124.02 = 35.21. This is a write-up of 35.21.

31

• 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)

32

• Correct answer: (A)

Solution: As the price of the underlying asset increases, a long position in a call ratio couldresult 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 providemore gain. However, once the asset price reaches 0, the gain can no longer increase. There-fore (II) could not provide unlimited gain.

As the price of the underlying asset increases or decreases, a long position in a stranglewill 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).

33

• 17. At an annual effective interest rate of 7%, the accumulated value of an annuity due with nannual payments of 10 is 110. At an annual effective rate of i%, the accumulated value ofan 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

34

• Correct answer: (E)

Solution: Use the first annuity to find n using the formula 110 = 10sn.07. This shows thatn = 8. The second annuity gives a formula for i: 140 = 13s8i. Use a financial calculator tofind i = .083.

35

• 18. Daniel writes a covered call with maturity in six months, a premium of 4, and a strike priceof 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 assets price is 44.

The annual effective interest rate is 11.8%. What is Daniels profit?

(A) -10

(B) 0

(C) 10

(D) 12

(E) 18

36

• Correct answer: (B)

Solution: By drawing profit diagrams for the covered written call and the purchased put, wecan 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.

Daniels profit from writing the call option is max{44 38, 0} + FV (4). His profit fromowning the underlying asset is 44 38. His profit from buying the put option is max{38 44, 0} FV (4).The profit from the entire portfolio is max{44 38, 0} + FV (4) + 44 38 + max{3844, 0} FV (4) = 0.Notice that we did not need the annual effective interest rate because we did not have tocalculate the future value of the premiums.

37

• 19. A stock costs 54 and will pay a dividends of 6 this year. The annual effective rate of interestis 6%. Let X be the one-year cost of carry of a long position in the stock, Y be the one-yearcost of carry of a short position in the stock, and Z be the one-year cost of carry of a longposition 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

38

• 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 thestock were sold and the money used to buy bonds. Therefore X = 3.24.

The cost of carry of a short position in one share of the stock is 6 because the stock will paya 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 requiredand the position will not be short any dividends. Therefore Z = 0.

Thus Y > X > Z, so the answer is (C).

39

• 20. A loan with amount A makes n level annual payments, where n is even, beginning 1 yearafter the loan is taken out. The annual effective interest rate is i such that 0 < i < 1. Whichof 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 1

2A.

(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)

40

• 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 halfof the loan.

Therefore the answer is (E).

41

• 21. A 10-year zero-coupon bond has a face value of 1000. Let i be the annual effective interestrate such that i > 0, MacD be the bonds Macaulay duration, and ModD be the bondsmodified 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.

42

• Correct answer: (D)

Solution: The Macaulay duration of a single cashflow is equal to the time at which thecashflow is made. Therefore MacD = 10.

The modified duration ModD is related to the Macaulay duration MacD according to thefollowing formula: ModD = MacD v. Since i > 0, we know that v < 1, and thereforeModD < MacD. (Note that if the interest rate i were 0, then ModD = MacD.)

Therefore ModD < MacD = 10.

43

• 22. It is known that K1 < K2. An investor has purchased a combination of options that result ina position equivalent to a K1-K2 bear spread. What transaction will change his position tobe equivalent to a long position in a ratio spread?

(A) Buy a call option with strike price K1(B) Buy a call option with strike price K2(C) Write two call options with strike price K2(D) Buy a put option with strike price K1(E) The answer is not given by any of (A), (B), (C), or (D).

44

• Correct answer: (E)

Solution: Since the investor already has a bear spread, he will change his position to a longposition in a put ratio spread. He needs to change his payoff on the interval (0, K1). Hecan accomplish this by writing a put option with strike price K1. (Buying a call option withstrike price K2 creates a short position in a ratio spread.) The correct transaction is not givenamong (A), (B), (C), or (D), so the answer is (E).

45

• 23. On January 1, a fund has a balance of 1000. At the end of each month for 12 months, interestis credited to the account at a rate of 12% convertible monthly, and a deposit of 100 is madeinto 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%

46

• Correct answer: (B)

Solution: Using the definition of time-weighted rate of return, we can calculate the answer:

10101000 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 to101% of its previous value. Therefore the time-weighted rate of return is 1.01121 = 12.7%.Note that this is equal to the annual effective rate of interest.

47

• 24. A zero-coupon bond has a modified duration of 8. Calculate the convexity of the zero-couponbond 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

48

• Correct answer: (C)

Solution: We know that MacD = ModD(1 + i). Therefore the Macaulay duration of thezero-coupon bond is 8(1 + .125) = 9. Remember that the Macaulay duration of a singlecashflow is equal to the time at which that cashflow occurs. Therefore the redemption dateis t = 9.

Remember that convexity is defined as C = P(i)P (i)

, where P (i) is the price as a function ofthe 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 the convexity of the bond isC = P

(i)P (i)

= 90F (1+i)11

F (1+i)9 = 71.1.

49

• 25. A loan is to be repaid in n years, where n is divisible by 3, with level annual payments atthe end of each year. The principal paid during year 1

3n is one-half the principal paid during

year 23n. The annual effective interest rate is 8%. Calculate n.

(A) 15

(B) 18

(C) 21

(D) 24

(E) 27

50

• Correct answer: (E)

Solution: Since i = .08, then v = .9259. Let P be the annual payment. Then the prin-cipal paid during year 1

3n is P (.9259nn/3+1) and the principal paid during year 2

3n is

P (.9259n2n/3+1). Since the principal paid during year 13n is one-half the principal paid

during year 23n, we know that:

P (.9259nn/3+1) = 12P (.9259n2n/3+1)

.92592n/3+1

.9259n/3+1= 1

2

.9259n/3 = 12

n = 27

51

• 26. The following chart shows prices for zero-coupon bonds with face amount 1000. It is knownthat the one-year forward rate two years from now is 6.1%.

Time to maturity (years) Zero-coupon bond price1 930.002 X3 X 50.26

Calculate X .

(A) 756.57

(B) 791.05

(C) 840.05

(D) 874.19

(E) 892.42

52

• Correct answer: (D)

Solution: Let v(t, u) be the forward discount factor from time t to time u. (In this way,v(t, u) = 1

1+i(t,u).) Since the one-year forward rate two years from now is 6.1%, we know

that v(2, 3) = 11+.061

= .9425.

The price of a zero-coupon bond is its face amount multiplied by the forward discount factorsfrom now until its maturity date. Therefore:

930 = 1000 v(0, 1)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 threeunknowns. Solve to find X = 874.19.

53

• 27. A company has a liability of 800 in 6 years. The company wants to provide Redington im-munization for the liability by purchasing two zero-coupon bonds are available for purchaseat any face amount. Bond X matures in 4 years and bond Y matures in 8 years. The annualeffective rate of interest is 6%. What amount of bond X should be purchased to provideRedington immunization for this liability?

(A) 255

(B) 276

(C) 309

(D) 327

(E) 356

54

• Correct answer: (E)

Solution: Let X be the face amount of bond X purchased and let Y be the face amount ofbond Y purchased. The first two conditions for Redington immunization are:

Xv4 + Y v8 = 800v6

and

4Xv5 8Y v9 = 4800v7.

Solve this system of two equations with two unknowns to find X = 356.

55

• 28. The price of a 10-year bond, which is 100 less than its face amount, is equal to the presentvalue of the face amount. It is known that j = r + .03, where j is the annual yield rate andr is the annual coupon rate. Calculate the face amount.

(A) 291

(B) 301

(C) 332

(D) 391

(E) 440

56

• Correct answer: (D)

Solution: Since the price of the bond is equal to the present value of its face amount, thismust be a zero-coupon bond. (This is because none of the interest is paid during the life ofthe bond, so the coupons must be 0. This can also be proved by using the identity P = K +rj(FK), where K is the present value of the face amount. This gives us P = P+ r

j(FP )

or equivalently 0 = rj(100). Therefore r = 0.)

Since r = 0, we know that j = .03. Therefore (F 100)(1 + .03)10 = F , where F is theface amount. This gives F = 391.

57

• 29. A company has a liability of 1000 due in 2.6 years. The company purchases a 3-year bondwith face amount F and annual coupons of r% to provide Redington immunization for theliability. 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

58

• Correct answer: (E)

Solution: The first two conditions for Redington immunization are:

(Fr)v + (Fr)v2 + (F + Fr)v3 = 1000v2.6

and

(Fr)v2 2(Fr)v3 3(F + Fr)v4 = 2600v3.6.

Solve this system of two equations with two unknowns to find r = 19.7.

59

• 30. It is known that K1 < K2. Which of the following positions create a bear spread?

(I) Buying a call with strike K1 and selling a call with strike K2(II) Buying a call with strike K2 and selling a call with strike K1(III) Selling a covered written call with strike K1 and selling a put with strike K2(IV) Buying a put with strike K1 and selling a put with strike K2

(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)

60

• Correct answer: (C)

Solution: Combinations (I), (III), and (IV) create bull spreads. Combination (II) creates abear. (Remember that a covered written call has the same profit as a purchased put.) Theanswer is (C).

61

• 31. ABC Co. takes out a loan for 10,000. For the first 19 years, ABC Co. makes payments at theend of each year of 120% of the interest due. At the end of the 20th year, ABC Co. makes afinal balloon payment of X .

The annual effective interest rate is 8%. Calculate X .

(A) 7243

(B) 7360

(C) 7497

(D) 7822

(E) 7949

62

• Correct answer: (E)

Solution: At the end of each year, ABC Co. pays 120% of the interest due, so the outstandingbalance is reduced by 20% of the interest due. The interest due is 8% of the outstandingbalance. 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) OBt1 = .984 OBt1.Equivalently, OBt = 10, 000 .984t.The outstanding balance at time 19 is 10, 000 .98419 = 7360. At time 20, this has accu-mulated to 7360(1 + .08) = 7949. Then ABC Co. pays off the entire remaining balance of7949.

63

• 32. A 12-year loan makes 11 annual payments of 100 at the end of each year followed by aballoon payment of 400 at the end of year 12. The annual effective interest rate is i. Whichof the following represents the interest paid in the 4th payment?

(A) 100 100( 11+i

)9

(B) 100 + 400( 11+i

)9

(C) 100( 11+i

)8 400( 11+i

)9

(D) 300( 11+i

)8 200( 11+i

)9

(E) 100 + 300( 11+i

)8 400( 11+i

)9

64

• Correct answer: (E)

Solution: Treat this as a 12-year loan with level payments of 100 plus a lump sum with afuture value of 300. The interest paid in the 4th year on the loan with level payments is100(1 ( 1

1+i)124+1) or equivalently 100 100( 1

1+i)9. The interest paid on the lump sum in

the 4th year is 300( 11+i

)8 300( 11+i

)9. Thus the answer is 100 + 300( 11+i

)8 400( 11+i

)9.

65

• 33. A geometric annuity immediate with 24 monthly payments is worth 3000. A geometricannuity immediate with 12 monthly payments is worth 1000. The first payment of bothannuities is the same, and the payments of the annuities increase at the same rate. Theannual effective rate of interest is 10%. What is the amount of the final payment of the24-month annuity?

(A) 253.20

(B) 271.65

(C) 290.10

(D) 308.55

(E) 327.00

66

• Correct answer: (B)

Solution: The monthly interest rate is 1.11/12 1 = .00797.LetX be the first payment of each annuity. The 24-month annuity is worth 3000 = X 1(

1+r1.00797

)24

.00797r .

The 12-month annuity is worth 1000 = X 1(1+r

1.00797)12

.00797r . Make the substitution y = (1+r

1.00797)12.

Then:

3000 = X 1y2

.00797r and 3000 = 3X1y

.00797r1 y2 = 3(1 y)y2 3y + 2 = 0y = 1 or y = 2

Use y = 2 = ( 1+r1.00797

)12 to find r = .0679. (If y = 1 is used, there is no possible value

of X that satisfies the problem description.) Then use the equation 3000 = X 1(1.06791.00797

)24

.00797.0679to find X = 59.94. Therefore the first payment is 59.94 1.06790, the second payment is59.94 1.06791, and the 24th payment is 59.94 1.067923 = 271.65.

67

• 34. Order the following quantities from smallest to greatest valued at the same annual effectiveinterest 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 12Pduring 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).

68

• Correct answer: (B)

Solution: One solution is to calculate the present value of each annuity. Another approach isto recognize that the present value of sooner payments is greater. This means that paymentsfar 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 duebecause the payments are delayed by 1 year. Therefore, the smallest quantity is (I) followedby (II).

The 12-month continuously payable annuity is payed evenly over 12-months, while the pay-ments of the 12-month annuity immediate are skewed toward the end of the year and thepayments of the 12-month annuity due are skewed toward the beginning of the year. There-fore, the 12-month annuities are ordered as follows: (III), (V), (IV).

The answer is (B).

69

• 35. An investor creates a 2:1 ratio spread by buying one call with strike price 105 and sellingtwo calls with strike price X . If the spot price at maturity were 130, the investors payoffwould be 5. What is the investors maximum possible payoff?

(A) 10

(B) 15

(C) 20

(D) 25

(E) 30

70

• Correct answer: (A)

Solution: The payoff for the ratio spread is max{S 105, 0} 2 max{S X, 0}. Ifthe investors payoff is negative, we know that all three options were exercised, so 5 =(130 105) 2(130X). Therefore X = 115.The investors maximum payoff occurs when the spot price is as large as possible (to max-imize payoff from the bought call) but when the sold calls are not exercised. This occurswhen S = 115. The payoff for this price is max{115 105, 0} 2 max{115 115, 0} =115 105 = 10.

71

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