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The Infinite Actuary’sJoint Exam 2/FM

Sample Exam 1

by James Washer, FSA, MAAA

last updated - August 1, 2014

Take this sample exam under strict exam conditions. Start a timer for 3hours and stop immediately when the timer is done. Do not stop the clockwhen you go to the bathroom. Do not look at your notes. Do not look atthe answer key.

This exam contains 35 questions. Do not spend too much time on any onequestion. Choose the best available answer for each question.

1



1. Which of the following is not a way to create a 40-45-50 butterfly?

A. buy 40-strike call, write two 45-strike calls, buy 50-strike call

B. buy 40-strike put, write two 45-strike puts, buy 50-strike put

C. buy 40-strike put, write 45-strike call, write 45-strike put, buy 50-strike call

D. buy 40-strike call, write 45-strike call, write 45-strike put, buy 50-strike put

E. all of the above will create a 40-45-50 butterfly

2. Let δ t = 1

4 + t, 0 ≤ t ≤ 15

What is the first year for which the effective rate of discount is less than12.5%?

A. 3 B. 4 C. 5 D. 6 E. 7

3. A bond will pay a coupon of 100 at the end of each of the next three yearsand will pay the face value of 1000 at the end of the three-year period. Thebond’s modified duration when valued using an annual effective interest rateof 20% is X .

Calculate X .

A. 2.25 B. 2.61 C. 2.70 D. 2.77 E. 2.89

c2009 The Infinite Actuary, LLC 2 Joint Exam 2/FM - Sample Exam 1



4. You are given the following table of interest rates:

CalendarYear of Portfolio

Original RatesInvestment Investment Year Rates (in %) (in %)

y iy

1 iy

2 iy

3 iy

4 iy

5 iy+5

1992 8.25 8.25 8.40 8.50 8.50 8.351993 8.50 8.70 8.75 8.90 9.00 8.601994 9.00 9.00 9.10 9.10 9.20 8.851995 9.00 9.10 9.20 9.30 9.40 9.10

1996 9.25 9.35 9.50 9.55 9.60 9.351997 9.50 9.50 9.60 9.70 9.701998 10.00 10.00 9.90 9.801999 10.00 9.80 9.702000 9.50 9.502001 9.00

A person deposits 1000 on January 1, 1997. Let the following be the accu-mulated value of the 1000 on January 1, 2000.

P : under the investment year method

Q: under the portfolio yield method

Calculate P + Q.

A. 2575 B. 2595 C. 2610 D. 2655 E. 2700

5. A loan is repaid with 10 annual payments. The first payment occurs oneyear after the loan. The first payment is 100 and each subsequent paymentincreases by 10. The annual effective rate of interest is 5%. The amount of principal repaid in the 4th payment is X .

Determine X .

A. 71 B. 76 C. 80 D. 84 E. 91




6. A 1000 par value 10-year bond that pays 9% coupons semiannually is pur-

chased for X . The coupons are reinvested at a nominal rate of 7% convertiblesemiannually.

The bond investor’s nominal annual yield rate convertible semiannually overthe 10-year period is 9.2%.

Determine X .

A. 924 B. 987 C. 1024 D. 1386 E. 1442

7. Bill writes a $100-strike call option on stock XYZ with 6 months to expiration

for a premium of $7.24. The risk-free rate is 5% convertible semiannually.For what range of prices at expiration does Bill make a profit?

A. [0, 92.58) B. (92.58,∞) C. [0, 107.42) D. (107.42,∞) E. [0, 107.60)

8. 10 deposits of $2000 are made every other year with the first deposit madeimmediately. The resulting fund is used to buy a perpetuity with paymentsmade once every 3 years following the pattern X , 4X , 7X , 10X , . . . Thefirst perpetuity payment is made 3 years after the last deposit of $2000. Theannual effective rate of interest is 6%.

Determine X .

A. 408 B. 458 C. 471 D. 512 E. 603

9. John buys a perpetuity-due with annual payments that are adjusted eachyear for inflation. The first payment is 100. Inflation is 3% for years 1-5 and2% thereafter.

Calculate the price of the perpetuity if the yield rate is an effective 6% perannum.

A. 2750 B. 2760 C. 2770 D. 2780 E. 2790




10. Given the following information about the treasury market:

Term Coupon Price1 0% 96.622 0% X

3 0% 88.90

It is known that the 1-year forward rate starting in two years is 4.5%.

Calculate X .

A. 87.65 B. 89.70 C. 92.90 D. 93.45 E. 95.50

11. A 20-year bond is priced at par and pays R% coupons semiannually. Thebond’s duration is 13.95 years.

Determine R.

A. 2 B. 3 C. 4 D. 5 E. 6

12. Which of the following is not true?

A. An asset insured with a floor is equivalent to investing in a zero-coupon bondand buying a call option on the asset.

B. A short position insured with a cap is equivalent to writing a zero-couponbond and buying a put option on the asset.

C. A covered written call is equivalent to investing in a zero-coupon bond andwriting a put option on the asset.

D. A covered written put is equivalent to writing a zero-coupon bond and writinga call option on the asset.

E. All of the above are true.




13. A fairly smart actuary (also know as an FSA) is offered the following rates

on a loan:

1. X % nominal annual rate of interest compounded monthly

2. X % nominal annual rate of discount compounded monthly

3. X % annual effective rate of interest

4. X % annual effective rate of discount

5. X % constant force of interest

Which rate does the FSA take?

A. 1 B. 2 C. 3 D. 4 E. 5

14. An annuity pays 1 at the beginning of each year for n years. Using anannual effective interest rate of i, the present value of the annuity at time 0is 8.55948. It is also known that (1 + i)n = 3.172169.

Find the accumulated value of the annuity immediately after the last pay-

ment.

A. 27.152 B. 28.456 C. 29.324 D. 30.765 E. 31.973

15. Deposits are made at the beginning of every month into a fund earning anominal annual rate of 6% convertible monthly. The first deposit is 100and deposit increase 2% every year. In other words, deposits 1-12 are 100,deposits 13-24 are 100×1.02 = 102, deposits 25-36 are 100×1.022 = 104.04,and so on.

Calculate the fund balance at the end of 10 years.

A. 16,569 B. 16,893 C. 17,257 D. 17,770 E. 17,859




16. On January 1 a fund has a balance of $100. Sometime during the year a

withdrawal of $20 is made. Immediately before the withdrawal the fundbalance is $110. At year-end the balance is $95.

If the time weighted and dollar weighted rates for the year are equal, then inwhat month was the $20 withdrawal made?

A. June B. July C. August D. September E. October

17. A common stock pays annual dividends at the end of each year. The earningsper share in the year just ended were J . Earnings are assumed to grow 10%per year in the future. The percentage of earnings paid out as a dividendwill be 0% for the next 5 years and 50% thereafter.

Calculate the theoretical price of the stock to yield the investor 21%.

A. 5J

(1.1)4 B.

5J

(1.1)5 C.

5J

(1.1)6 D.

10J

(1.1)5 E.

10J

(1.1)6

18. You are the CFO of Infinite Life. Infinite Life only has one liability of $5000due in 8 years. Infinite Life uses a nominal rate of 6% convertible semiannualto discount all liability cash flows.

You call up your favorite bond broker and ask him what bonds he has forsale today. Your broker says he has 5-year and 10-year bonds. Both bondsare priced to yield 6% convertible semiannually. The 5-year bond pays 6%coupons semiannually and the 10-year bond is a zero-coupon bond. Thebonds can be bought in any face amount.

What face amount of the 5-year bond should you buy in order to meet thefirst two conditions of immunization?

A. 777 B. 888 C. 999 D. 1111 E. 2222




19. You are given the following prices for $100 zero-coupon bonds:

Term Price1 95.242 89.003 81.63

R is the swap rate in a 3-year swap contract.

Determine R.

A. 5.7% B. 6.1% C. 6.5% D. 6.9% E. 7.2%

20. A fund earns simple interest of i. Which of the following are true?

1. The effective rate of interest is constant for each period.

2. The amount of interest is constant for each period.

3. The equivalent force of interest is ln(1 + i).

A. 1 only B. 2 only C. 3 only D. 2 and 3 only

E. The correct answer is not given by A, B, C or D.

21. If δ t = t

1 + t2, t ≥ 0 and

I a

n

= 13.03567.

Determine n.

A. 10 B. 11 C. 12 D. 13 E. 14




22. You are given the following information about 3 funds:

Fund Effective Rate

Interest Paymentsper Year

A j% paid directly to investorB 5% paid directly to investorC 4% automatically reinvested

An investor deposits $10,000 into fund A. He reinvests fund A’s interestpayments into fund B. He reinvests fund B’s interest payments into fund C.The total value of all three funds after 10 years is $20,000.

Determine j.

A. 6 B. 7 C. 8 D. 9 E. 10

23. Which of the following are true?

1. Modified duration is greater for bonds with higher par values.

2. Modified duration is greater for yields compound more frequently.

3. The modified duration of a single cash flow is the time remaining untilthe cash flow.



24. An annuity-immediate has the following payments:

10, 20, 30, 40, 50, 40, 30, 20, 10

Which of the following expressions represents the present value of the annu-ity?

A. 10a5 a5 B. 10a5 a5 C. 10a5 a4 D. 10a5 a4

E. 10 (Ia)5

+ 10 (Da)4




25. You are given the following term structure of interest rates.

TermSpot Rate

(in years)1 5.00%2 5.75%3 6.25%4 6.50%

Calculate 1000 (Da)4

rounded to the nearest 100.

A. 8700 B. 8800 C. 8900 D. 9000 E. 9100

26. Suppose you desire to short-sale 200 shares of ABC stock, which has a bidprice of $24.82 and an ask price of $25.01. You cover the short position threemonths later when the bid price is $21.45 and the ask price is $21.64.

Your broker charges a 0.3% commission to engage in the short-sale and a0.3% commission to close the position. He also requires a 50% haircut onthe net proceeds of the short-sale. The market rate of interest is 6% con-vertible continuously and the short-rebate on the haircut is 4% convertiblecontinuously. No dividends were paid on the stock during the three months.

Calculate the profit on the short-sale.

A. $595.60 B. $608.13 C. $652.33 D. $671.50 E. $684.12




27. Which of the following are true?

1. The payoff on a short forward is the spot price at expiration minusforward price.

2. Zero coupon bonds can be used to shift profit diagrams.

3. For forward contracts, cash settlements often reduce transaction costs.



28. A $10,000 loan is repaid with level annual payments. The first payment ismade one year after taking out the loan.

The outstanding balance just after the 8th payment is $7,112.10. The out-standing balance just after the 16th payment is $2,845.36.

Calculate the outstanding balance just after the 17th payment.

A. $1,185 B. $1,485 C. $1,885 D. $1,985 E. $2,185

29. A 10-year 1000 par value bond pays 4% coupons semiannually and is re-deemable at 1200. The bond is priced to yield an annual nominal rate of 6%convertible semiannually.

Calculate the adjustment to the book value during the 6th year.

A. 12.26 write-upB. 24.17 write-downC. 24.17 write-upD. 35.73 write-down

E. 35.73 write-up




30. The present value of a 25-year annuity-immediate with a first payment of

$2500 and decreasing by X each year thereafter is $15,923. The annualeffective rate of interest is 10%.

Determine X .

A. 70 B. 80 C. 90 D. 100 E. 110

31. Which of the following expression does not represent a definition for m|an ?

A. vm an B. am+n − am C. vm+n sn D. vm + vm+1 + . . . + vm+n

E. All of the above represent a definition for m|

an

32. At an annual effective interest rate i:

(i) 1 grows to 5 in x years

(ii) 2 grows to 8 in y years

(iii) 3 grows to 20 in z years

Calculate 12(1 + i)3x

−2y+z

.

A. 625 B. 650 C. 675 D. 700 E. 725

33. Jane takes out a $100,000 loan at a nominal annual rate of 12% convertiblemonthly. Jane makes monthly interest only payments for the first 3 yearsand then she pays $1500 per month until the loan is paid off. How manytotal payments does Jane make including the final smaller payment?

A. 110 B. 111 C. 142 D. 146 E. 147




34. A corporation borrows 10,000 for 25 years, at an annual effective interest

rate of 5%. A sinking fund is used to accumulate the principal by means of 25 annual deposits earning an effective annual interest rate of 4%.

Calculate the sum of the net amount of interest paid in the 12th installmentand the increment in the sinking fund for the 8th year.

A. 676 B. 686 C. 696 D. 706 E. 716

35. You are given the following prices for 6-month options on S&R:

Type Strike PremiumCall 1000 93.809Put 1000 74.201

The risk-free rate is 4% convertible semiannually. Calculate the six-monthforward price for S&R.

A. 980 B. 1000 C. 1020 D. 1040 E. 1060


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