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Page 1: Exam FM Practice Exam 1 Answer Key

Exam FM/2Practice Exam 1

Answer KeyCopyright c©2013 Actuarial Investment.

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Page 2: Exam FM Practice Exam 1 Answer Key

1. A perpetuity makes level payments of 1 at the end of each year. The perpetuity’s modifiedduration is 25. Calculate the present value of the perpetuity.

(A) 16.67

(B) 22.50

(C) 24.00

(D) 24.33

(E) 25.00

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Page 3: Exam FM Practice Exam 1 Answer Key

Correct answer: (E)

Solution: The perpetuity’s volitility ModD is 25. We know that the Macaulay duration of aperpetuity is 1 + 1

i, so we use the formula ModD = MacD · v:

25 = (1 + 1i) · v

25 = ( i+1i)( 1

1+i)

25 = 1i

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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Page 4: Exam FM Practice Exam 1 Answer Key

2. Johnathan buys a 12-year bond with face amount 1000 and annual coupons of 4% priced toyield 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 ofJohnathan’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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Page 5: Exam FM Practice Exam 1 Answer Key

Correct answer: (E)

Solution: Since the price of Rachel’s bond is less than the price of Johnathan’s bond, weknow that Rachel’s bond has a higher yield than Johnathan’s bond. (She invested less moneyto 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 wasless than 1000 and we know that j > .04.

Rachel’s bond cost less than Johnathan’s bond, but we cannot determine whether Johnathan’sbond cost more or less than 1000. Therefore we do not know if Johnathan’s bond was boughtat a discount or at a premium. No other relevant facts can be determined from the giveninformation.

Therefore there is not enough information to determine that any of (A), (B), (C), or (D) istrue, so the answer is (E).

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Page 6: Exam FM Practice Exam 1 Answer Key

3. An asset’s price is 81. The price of a put option for the asset that matures in 72 days and hasa strike price of 84 is 3.85. The annual effective rate of interest is 5%. What is the price ofa call option for the asset that matures in 72 days and has a strike price of 84? (Assume a360-day year.)

(A) 0.85

(B) 1.67

(C) 2.85

(D) 4.85

(E) 6.17

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Page 7: Exam FM Practice Exam 1 Answer Key

Correct answer: (B)

Solution: The option matures in 72 days, or 72360

= 15

years. Use the put-call parity formula:C − P = S(0)− PV (K), or C − 3.85 = 81− 84(1 + .05)−(1/5). Solve to find the price ofthe call option: C = 1.67.

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Page 8: Exam FM Practice Exam 1 Answer Key

4. A stock pays annual dividends, beginning in one year with a dividend of 134. The stock paysdividends until the company goes bankrupt n years from now, at which time the stock paysthe 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 knownthat i+ .03 = r. Calculate n.

(A) 8

(B) 9

(C) 10

(D) 11

(E) 12

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Page 9: Exam FM Practice Exam 1 Answer Key

Correct answer: (B)

Solution: Use the formula for a geometric annuity:

1289 = 134 ·1− (1+r)n

(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 knowthat (1 + i)n = 248

159.86= 1.552.

1289 = 134 · 1−(1+r)n

1.552

−.03

1289 = 134 · 1−(1+r)n−1(1+r)

1.552

−.03

From time 1 to time n, the dividend increases by a factor of 1+ r exactly n− 1 times. Sincethe first dividend is 134 and the last dividend is 248, this means that 134(1 + r)n−1 = 248,or (1 + r)n−1 = 248

134= 1.851.

1289 = 134 · 1−1.851·(1+r)

1.552

−.03

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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Page 10: Exam FM Practice Exam 1 Answer Key

5. An asset is currently worth 61. Jack buys a call option for the asset with strike price 62 andmaturity in one year. Robin buys a put option for the asset with strike price 62 and maturityin one year.

After a year, the price of the underlying asset is 58. Jack’s profit is −4.41 and Robin’s profitis 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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Page 11: Exam FM Practice Exam 1 Answer Key

Correct answer: (C)

Solution: Since Jack’s call option expires out-of-the-money and his profit is −4.41, thepremium he paid for the call option must have been the present value of 4.41, which is4.41(1 + .05)−1 = 4.20.

Now use the put-call parity formula to calculate the premium that Robin pays for her putoption:

C − P = S(0)− PV (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} − FV (Premium)

X = max{62− 58, 0} − 2.25(1 + .05)

Therefore X = 1.64.

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Page 12: Exam FM Practice Exam 1 Answer Key

6. A 14-year annuity due makes payments of 100 every year except for year 3. In year 3, theannuity makes a payment of 500. The effective annual interest rate is 4%. What is the presentvalue of the annuity?

(A) 1398

(B) 1412

(C) 1433

(D) 1454

(E) 1468

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Page 13: Exam FM Practice Exam 1 Answer Key

Correct answer: (E)

Solution: Break the annuity into two separate pieces consisting of a level annuity with pay-ments of 100 and a one-time payment of 400 during the 3rd year. Notice that because theannuity is an annuity due, the payment in year 3 is made at the beginning of the year, whichis equivalent to the end of year 2. So the present value is:

100a14.04 + 400( 11.04

)2 = 1468

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Page 14: Exam FM Practice Exam 1 Answer Key

7. The following table gives one-year forward rates for the next three years.

T (years) i(T − 1, T )1 4.62 4.33 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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Page 15: Exam FM Practice Exam 1 Answer Key

Correct answer: (A)

Solution: The swap rate is the fixed payment rate at which the present value of interestpayments using the fixed rate is equal to the present value of interest payments using thecurrent term structure.

Suppose that 1000 is borrowed and payments of only interest are made on the principal forthree years. Then the swap rate r solves the following equation:

1000·.046(1+.046)

+ 1000·.043(1+.046)(1+.043)

+ 1000·.039(1+.046)(1+.043)(1+.039)

= 1000r(1+.046)

+ 1000r(1+.046)(1+.043)

+ 1000r(1+.046)(1+.043)(1+.039)

Solve to find r = 4.28.

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Page 16: Exam FM Practice Exam 1 Answer Key

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 paymentis 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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Page 17: Exam FM Practice Exam 1 Answer Key

Correct answer: (D)

Solution: This is equivalent to a geometrically changing annuity with 13 payments, plus aballoon payment at time 14.

Since each payment is 6% more than the subsequent payment, each payment is 11.06

= .9434times the previous payment. Therefore each payment changes by −5.66% compared to theprevious payment. To find the first payment, observe that the eighth payment is X; theseventh payment is 1.061X; the sixth payment is 1.062X; and the first payment is 1.067X .

Therefore the loan amount of 1000 is equal to the present value of the geometric annuityplus the present value of the balloon payment:

1000 = 1.067X1−( 1+(−.0566)

1+.03)13

.03−(−.0566) + 2X( 11+.03

)14

Solve to find X = 76.10.

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Page 18: Exam FM Practice Exam 1 Answer Key

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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Page 19: Exam FM Practice Exam 1 Answer Key

Correct answer: (C)

Solution: Let F be the face amount of the bond. Since the bond is priced at par, F is alsothe price of the bond. Also, the yield rate is equal to the coupon rate of 10%. Thereforev = 1

1+.1.

Convexity is defined as P ′′(i)P (i)

, where P (i) is the present value of a portfolio as a function ofthe interest rate i.

The present value of the bond is P (i) = .1Fv + .1Fv2 + 1.1Fv3. Then P ′(i) = −.1Fv2 −.2Fv3 − 3.3Fv4. Also P ′′(i) = .2Fv3 + .6Fv4 + 13.2Fv5.

Since P (i) = F , the convexity is equal to P ′′(i)P (i)

= .2Fv3+.6Fv4+13.2Fv5

F= .2v3 + .6v4 +

13.2v5 = 8.76.

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Page 20: Exam FM Practice Exam 1 Answer Key

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 interestwill Mark pay by extending the loan?

(A) 52

(B) 61

(C) 69

(D) 73

(E) 82

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Page 21: Exam FM Practice Exam 1 Answer Key

Correct answer: (B)

Solution: The annual effective interest rate is given by 1000 = 120a10i. Use a financialcalculator 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 is120 · 6 − 640 = 80. Let P be the new payment after extending the loan. There were 6payments left, but Mark added an additional 5 payments, so he now has 11 payments left.Then 640 = Pa11.0346, so P = 71. The interest Mark will pay on the outstanding balance of640 is therefore 11 ·71−640 = 140. Thus the additional interest Mark will pay by extendingthe loan is 140− 80 = 61.

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Page 22: Exam FM Practice Exam 1 Answer Key

11. Abigail takes out a 48-month loan worth 132,000 with payments of 3100 at the end of eachmonth. In order to pay off the loan early, Abigail instead makes payments of 3800 at the endof each month for n months, plus a final payment of X at the end of the n+ 1st month suchthat X < 3800.

Calculate X .

(A) 893

(B) 904

(C) 918

(D) 950

(E) 954

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Page 23: Exam FM Practice Exam 1 Answer Key

Correct answer: (E)

Solution: Let j be the monthly rate of interest. Then use a financial calculator to solve theformula 132, 000 = 3100a48j to find j = .005. Then find out how many payments of 3800will be made by solving the formula 132, 000 = 3800an.005 to find n = 38.25. This meansthat Abigail will make 38 payments of 3800 plus a final payment of X . The outstandingbalance immediately after the 38th 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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Page 24: Exam FM Practice Exam 1 Answer Key

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 paymentis 10. The annual effective rate of interest is 7%. Calculate the accumulated value of theannuity immediately after the final payment.

(A) 529

(B) 541

(C) 594

(D) 604

(E) 613

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Page 25: Exam FM Practice Exam 1 Answer Key

Correct answer: (A)

Solution: The monthly rate of interest is (1 + .07)112 − 1 = .005654 and the 6-month rate

of interest is (1 + .07)12 − 1 = .03441. Now break the annuity into two separate annuities.

The first has 36 monthly payments of 10. Its accumulated value is 10s36.005654 = 398. Thesecond has 6 semi-annual payments of 20. Its accumulated value is 20s6.03441 = 131. Thetotal accumulated value is 398 + 131 = 529.

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Page 26: Exam FM Practice Exam 1 Answer Key

13. A stock currently pays no dividends, but will begin paying annual dividends in n years. Thefirst year’s dividend will be 18 and subsequent dividends will increase by 2% per year. At aprice 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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Page 27: Exam FM Practice Exam 1 Answer Key

Correct answer: (E)

Solution: Since the stock price at time 0 is 288.61, and the stock is priced to yield 5%, thenthe stock price at time n is 288.61(1 + .05)n.

At time n, the dividends form a geometrically increasing perpetuity due. The first dividendis 18. The second dividend is 18 · 1.02, and subsequent dividends increase by 2%. The valueof the perpetuity is 18 + 18 · 1.02 · ( 1

.05−.02).

Therefore 288.61(1 + .05)n = 18 + 18 · 1.02 · ( 1.05−.02). Solve to find n = 16.

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Page 28: Exam FM Practice Exam 1 Answer Key

14. An n-year bond has annual coupons of 5%. The bond is bought to yield 6.89%. The accu-mulated 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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Page 29: Exam FM Practice Exam 1 Answer Key

Correct answer: (D)

Solution: Since the accumulated value of the coupons is equal to the face amount of thebond, we know that F = .05F · sn.0689, or 20 = sn.0689. Use a financial calculator to findn = 13.

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Page 30: Exam FM Practice Exam 1 Answer Key

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 longposition 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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Page 31: Exam FM Practice Exam 1 Answer Key

Correct answer: (A)

Solution: The definition of a zero-cost collar is that it has no net premium. Therefore (I) istrue.

The payoff of a zero-cost collar is less than the payoff of a collar with positive premiumbecause 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 priceis equivalent to a long forward position, not a collar. Therefore (III) is false.

Therefore the answer is (A).

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Page 32: Exam FM Practice Exam 1 Answer Key

16. An investor sells 1000 barrels of oil at a forward price of 60 per barrel with maturity in sixmonths. At expiration, the price of oil is 52 per barrel. There is a 3% commission on thefutures 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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Page 33: Exam FM Practice Exam 1 Answer Key

Correct answer: (D)

Solution: At expiration, the investor sells 1000 barrels of oil for 60 per barrel, and must buy1000 barrels of oil to offset this sale. Therefore the payoff (not profit) from the contract is1000(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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Page 34: Exam FM Practice Exam 1 Answer Key

17. Tabitha buys a 20-year bond purchased to yield 8% convertible semiannually with semi-annual coupons at a rate of 4% convertible semiannually. Coupons are reinvested into anaccount earning interest at a nominal rate of 6% convertible semiannually.

John buys a 20-year bond purchased to yield j% convertible semiannually with semiannualcoupons at a rate of 12% convertible semiannually. Coupons are reinvested into an accountearning 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 invest-ment. Calculate j.

(A) 8.29

(B) 8.42

(C) 8.53

(D) 8.69

(E) 8.98

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Page 35: Exam FM Practice Exam 1 Answer Key

Correct answer: (C)

Solution: Suppose that Tabitha purchases a bond with a face amount of 1000. Then the priceof the bond can be found using a financial calculator with N = 40, I = 4, PMT = 20,and FV = 1000 to find PV = −604.14. Then the amount she has after 20 years is the faceamount 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 equation604.14(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’sinvestment is also .07377, this means that he invests 2408.83. The portion of the 10,000that he earns from accumulation of coupons is .06Fs40.03. The final amount of 10,000is composed of the face amount of the bond plus the accumulated value of the reinvestedcoupons. Therefore 10, 000 = F + .06Fs40.03. Solve this to find F = 1810.26. Then,since he invested 2408.83, use a financial calculator with N = 40, PV = −2408.83,PMT = .06 · 1810.26 = 108.62, and FV = 1810.26 to find I = 4.26. This is John’sbond yield per coupon period. Therefore his yield rate is j = 4.26 · 2 = 8.53.

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Page 36: Exam FM Practice Exam 1 Answer Key

18. The six-month forward price for a stock is 35.94. The prepaid six-month forward price forthe stock is 34.05. Assume that the stock pays no dividends and that there are no opportuni-ties 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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Page 37: Exam FM Practice Exam 1 Answer Key

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 six-month interest rate is 35.94

34.05− 1 = .0555. Therefore the annual effective rate of interest is

(1 + .0555)2 = 11.4%.

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Page 38: Exam FM Practice Exam 1 Answer Key

19. The current price of a stock is 55 and the annual effective rate of interest is 8%. The profitearned by a call option with maturity in one year and strike price of 62 is 5.48. The profitearned by a put option with maturity in one year and strike price of 62 is -3.80. What is theprice 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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Page 39: Exam FM Practice Exam 1 Answer Key

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} − FV (Premium)

Profitcall = S −K − FV (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:

Profitput = max{K − S, 0} − FV (Premium)

Profitput = 0− FV (Premium)

−3.80 = 0− P (1 + .08)1

Since we are given information about call options and put options with the same strike priceand maturity, we can use the put-call parity formula. If S(0) is the current price of the stock,then:

C − P = S(0)− PV (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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Page 40: Exam FM Practice Exam 1 Answer Key

20. Ron’s account earns interest at a force of interest of 1+kt. Money deposited into the accountwill double after n years.

Jane’s account earns interest at a force of interest of kt + t2. Money deposited into theaccount 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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Page 41: Exam FM Practice Exam 1 Answer Key

Correct answer: (A)

Solution: Money in Ron’s account doubles after n years:

2 = exp(∫ n

0(1 + kt)dt)

ln(2) = n+ 12kn2

Money in Jane’s account doubles after n years:

2 = exp(∫ n

0(kt+ t2)dt)

ln(2) = 12kn2 + 1

3n3

Solve this system of two equations with two unknowns to find k = −0.69.

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Page 42: Exam FM Practice Exam 1 Answer Key

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 monthlyof 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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Page 43: Exam FM Practice Exam 1 Answer Key

Correct answer: (C)

Solution:

Solve the following equation for t:

100(1− .0477)−t = 88(1 + .06512

)12t

Guess-and-check may be appropriate.

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Page 44: Exam FM Practice Exam 1 Answer Key

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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Page 45: Exam FM Practice Exam 1 Answer Key

Correct answer: (E)

Solution: Recognize that (Is)15i + (Ds)15i = 16s15i. Plug in the information that is alreadyknown to get 136 + (Ds)15i = 16 · 18. Then (Ds)15i = 152.

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Page 46: Exam FM Practice Exam 1 Answer Key

23. A company has provided Redington immunization for its liability of L in 1 year. The com-pany has two assets: a five-year zero-coupon bond with face amount of L, and current cashon 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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Page 47: Exam FM Practice Exam 1 Answer Key

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 5v6 = v2 or v4 = 15. Therefore v = .6687. Then,

using the first condition, we see that X + .1337L = .6687L or X = .535L.

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Page 48: Exam FM Practice Exam 1 Answer Key

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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Page 49: Exam FM Practice Exam 1 Answer Key

Correct answer: (E)

Solution: Find the monthly interest rate j by using a financial calculator to solve the equation10, 000 = 197a60j to find j = .005654. The outstanding balance after 2 years is 10, 000(1 +.005654)24 − 197s24.005654 = 6400. Now there are 36 months remaining until the loanwould be paid off with payments of 197, but there are only 28 months remaining until theloan will be paid off with payments of 197 + P . Find P by solving the equation 6400 =(197 + P )a28.005654 to find P = 51.

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Page 50: Exam FM Practice Exam 1 Answer Key

25. A 20-year loan has level annual payments of 500 at the end of each year. The interest paidin the 6th year is 150. Calculate the total amount of interest paid during the loan.

(A) 2136

(B) 2394

(C) 2610

(D) 2871

(E) 3109

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Page 51: Exam FM Practice Exam 1 Answer Key

Correct answer: (A)

Solution: The interest paid during the 6th year is 150 = 500(1 − v20−6+1). Solve thisequation to calculate v = .9765, so i = .02406. Then the loan amount A is given by A =500a20.02406 = 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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Page 52: Exam FM Practice Exam 1 Answer Key

26. A perpetuity due has a payment of P at time 0. Payments are annual and increase by 3% peryear. 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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Page 53: Exam FM Practice Exam 1 Answer Key

Correct answer: (C)

Solution: This perpetuity is equivalent to a perpetuity immediate that has an initial paymentof 1.03P and is worth 4000 − P . Then we can use the formula for the present value of ageometrically increasing perpetuity: 4000 − P = 1.03P 1

.05−.03 . Solve this equation to findP = 76.

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Page 54: Exam FM Practice Exam 1 Answer Key

27. A 30-year bond has a face value of 50,000 with semiannual coupons at a rate of 8% convert-ible semiannually and a price of 48,936. What is the adjustment to book value in the 16th

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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Page 55: Exam FM Practice Exam 1 Answer Key

Correct answer: (E)

Solution: First we note that since the bond is priced at a discount, the adjustment to bookvalue will be a write-up.

Let j be the interest rate per coupon period. Then j can be found using a financial calculatorwith N = 60, PV = −48, 936, PMT = 2000, and FV = 50, 000; solve on I to findj = .04096.

To find the adjustment to book value in the 16th year, we will find the book value of the bondimmediately after the 32nd payment and the book value of the bond after the 30th 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, PMT =2000, and FV = 50, 000 to find PV = −49, 211, so the book value is 49,211. The bookvalue after the 30th payment can be found the same way with N = 30, which gives a bookvalue of 49,181. The difference is 29.31, so the adjustment to book value is a write-up of29.31.

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Page 56: Exam FM Practice Exam 1 Answer Key

28. It is known that K1 < K2 < K3. Let X be the premium paid for a straddle around K2, Ybe the premium paid for a K1-K2 strangle, and Z be the premium paid for a call option withstrike price K2. 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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Page 57: Exam FM Practice Exam 1 Answer Key

Correct answer: (A)

Solution: The premium paid for a straddle is more than the premium paid for a stranglebecause 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 thestrangle 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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Page 58: Exam FM Practice Exam 1 Answer Key

29. An investor purchases a call ratio spread. The strike price of the purchased call option is230 and the strike price of the written call options is 245. The following table shows theinvestor’s profit for several spot prices of the underlying asset at maturity.

Spot price at maturity Profit220 −5240 5260 −20280 X

Calculate X .

(A) -20

(B) -30

(C) -40

(D) -50

(E) -60

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Page 59: Exam FM Practice Exam 1 Answer Key

Correct answer: (E)

Solution: Purchasing a call ratio spread consists of buying one call option and writing n calloptions at a higher strike price. Let P be the future value of the net premium paid for the callratio 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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Page 60: Exam FM Practice Exam 1 Answer Key

30. It is known that the interest rate i is equal to the present value factor v. Calculate the discountrate d.

(A) .089

(B) .244

(C) .382

(D) .618

(E) .733

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Page 61: Exam FM Practice Exam 1 Answer Key

Correct answer: (C)

Solution: The present value factor v is given by v = 11+i

. Since i = v, this gives i = 11+i

.This is a quadratic equation; solve to find i = −1.618, .618. Therefore i = .618. (The otherchoice is not logical.) Then the discount rate d is given by d = i

1+i= .618

1+.618= .382.

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Page 62: Exam FM Practice Exam 1 Answer Key

31. Option X is an out-of-the-money put option. Option Y is a call option for the same assetwith the same strike price as option X . Under which of the following conditions will optionY’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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Page 63: Exam FM Practice Exam 1 Answer Key

Correct answer: (D)

Solution: Option X, which is a put option, is out-of-the-money. Since option X and option Yhave the same strike price, option Y is therefore in-the-money. This means that if it expireswith 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 assetincreases. Therefore the answer is (D).

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Page 64: Exam FM Practice Exam 1 Answer Key

32. The annual effective interest rate is 8%. A 36-month annuity due makes monthly paymentsof 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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Page 65: Exam FM Practice Exam 1 Answer Key

Correct answer: (C)

Solution: The monthly rate of interest is (1 + .08)112 − 1 = .006434. Then calculate the

accumulated value of the annuity after 36 months: 50s36.006434 = 2031. However, since thisis an annuity due, the final payment occurs at the end of the 35th month. Thus we need todiscount this result by one month, so the answer is 2031

1+.006434= 2018.

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Page 66: Exam FM Practice Exam 1 Answer Key

33. A company has liabilities of L in 1 year and 1000 in 2 years. Bond X is a one-year bondwith annual coupons of 4% and is priced at par. Bond Y is a two-year bond with annualcoupons of 8% and is priced at par.

The company has created a portfolio using bond X and bond Y that exactly matches itsliabilities. The present value of this portfolio is 2000. Calculate L.

(A) 1080

(B) 1120

(C) 1191

(D) 1220

(E) 1278

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Page 67: Exam FM Practice Exam 1 Answer Key

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 bondX , 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 bondY . Therefore Y + .08Y = 1000.

Since both bonds are priced at par, their price is equal to their face amount. ThereforeX + Y = 2000.

Solve this system of three equations with three unknowns to find L = 1191.

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Page 68: Exam FM Practice Exam 1 Answer Key

34. An investor wants to buy 20-year bonds. Current interest rates are 8%, but the investorbelieves that interest rates will rise within the next 10 years. The following 20-year bondsare 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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Page 69: Exam FM Practice Exam 1 Answer Key

Correct answer: (C)

Solution: Since the investor believes that interest rates will rise in the future, she shouldinvest in bonds that have higher coupon rates so that she can reinvest more money sooner. (Ifshe thought interest rates would go down, she should invest in zero-coupon bonds becauseher money would be locked into the rate of 8% for longer.) Therefore the investor shouldinvest solely in bond (III) so that she can reinvest as much money as possible at the higherrate in the future.

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Page 70: Exam FM Practice Exam 1 Answer Key

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. Thedollar-weighted rate of return is 25%. Calculate X .

(A) 322

(B) 344

(C) 349

(D) 362

(E) 390

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Page 71: Exam FM Practice Exam 1 Answer Key

Correct answer: (B)

Solution: The net gain is 1400−X+200−1000. The exposure is 1000 · 1212−200 · 9

12+X · 6

12.

Since the dollar-weighted rate of return is .25, we have:

.25 = 1400−X+200−10001000· 12

12−200· 9

12+X· 6

12

Solve to find X = 344.

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