 # Exam FM Practice Exam 1 Answer Key

Nov 26, 2015

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Exam FM Practice Exam 1 Answer Key

• Exam FM/2Practice Exam 1

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• 1. A perpetuity makes level payments of 1 at the end of each year. The perpetuitys 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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Solution: The perpetuitys 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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• 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%.

Rachels bond is bought at a discount. The price of Rachels bond is less than the price ofJohnathans 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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Solution: Since the price of Rachels bond is less than the price of Johnathans bond, weknow that Rachels bond has a higher yield than Johnathans bond. (She invested less moneyto get the same cashflow, so she had a higher yield.) Therefore i < j.

Since Rachels bond was bought at a discount, we know that the price of Rachels bond wasless than 1000 and we know that j > .04.

Rachels bond cost less than Johnathans bond, but we cannot determine whether Johnathansbond cost more or less than 1000. Therefore we do not know if Johnathans 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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• 3. An assets 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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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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• 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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Solution: Use the formula for a geometric annuity:

1289 = 134 1(1+r)n

(1+i)n

ir

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)n1(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)n1 = 248,or (1 + r)n1 = 248

134= 1.851.

1289 = 134 11.851(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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• 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. Jacks profit is 4.41 and Robins 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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Solution: Since Jacks 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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• 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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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(1

1.04)2 = 1468

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• 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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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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• 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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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(1

1+.03)14

Solve to find X = 76.10.

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• 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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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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• 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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