Exam FM/2Practice Exam 1

Copyright c©2013 Actuarial Investment.

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

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.

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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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

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

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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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

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

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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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

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

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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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

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

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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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

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

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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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

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

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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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

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

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

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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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

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

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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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

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

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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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

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

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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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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