FM1 Solutions

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

(A) buy (A) buy 40-strike call, write two 45-strike calls, buy 50-strike 50-strike call (B) buy (B) buy 40-strike put, write two 45-strike puts, buy 50-strike 50-strike put (C) buy (C) buy 40-strike put, write 45-strike call, write write 45-strike put, buy 50-strike call (D) buy (D) buy 40-strike call, write 45-strike call, write write 45-strike put, buy 50-strike put (E) all of the above will create a 40-45-50 butterfly

© 2007 The Infinite Actuary, LLC

1

Joint Exam 2/FM (Sample Exam)

2. Let

 t 

1 4  t

, 0  t  15

What is the first year for which the effective rate of discount is less than 12.5? (A) 3 (B) 4 (C) 5 (D) 6 (E) 7


2


2. Let

 t 

1 4  t

, 0  t  15

What is the first year for which the effective rate of discount is less than 12.5? (A) 3 (B) 4 (C) 5 (D) 6 (E) 7


2


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

Calculate X. (A) 2.25 (B) 2.61 (C) 2.70 (D) 2.77 (E) 2.89


3


4. You are given the following table of interest rates: Calendar Year of Original Investment y 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001

Portfolio Rates (in %)

Investment Year Rates (in %) y

i1

8.25 8.50 9.00 9.00 9.25 9.50 10.00 10.00 9.50 9.00

y

i2

8.25 8.70 9.00 9.10 9.35 9.50 10.00 9.80 9.50

y

y

y

y  5

i3

i4

i5

i

8.40 8.75 9.10 9.20 9.50 9.60 9.90 9.70

8.50 8.90 9.10 9.30 9.55 9.70 9.80

8.50 9.00 9.20 9.40 9.60 9.70

8.35 8.60 8.85 9.10 9.35

A person deposits 1000 on January 1, 1997. Let the following be the accumulated 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


4


5. A loan is repaid with 10 annual payments. The first payment occurs one year after the loan. The first payment is 100 and each subsequent payment increases 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


5


6. A 1000 par value 10-year bond that pays 9% coupons semiannually is purchased for X. The coupons are reinvested at a nominal rate of 7% convertible semiannually.

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

Determine X.

(A) 924 (B) 987 (C) 1024 (D) 1386 (E) 1442


6


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 rage of prices at expiration does Bill make a profit?

(A) 0 to 92.58 (B) 92.58 to



(C) 0 to 107.42 (D) 107.42 to



(E) 0 to 107.60


7


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

Determine X.

(A) 408 (B) 458 (C) 471 (D) 512 (E) 603


8


9. John buys a perpetuity-due with annual payments that are adjusted each year for inflation. The first payment is 100. Inflation is 3% for years 1-5 and 2% thereafter.

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

(A) 2750 (B) 2760 (C) 2770 (D) 2780 (E) 2790


9


10. Given the following information about the treasury market: Term 1 2 3

Coupon 0% 0% 0%

Price 96.62 X 88.9

It is known that the 2-year forward rate is 4.5%.

Calculate X.

(A) 87.65 (B) 89.70 (C) 92.90 (D) 93.45 (E) 95.50


10


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

Determine R.

(A) 2 (B) 3 (C) 4 (D) 5 (E) 6


11


12. Which of the following is not true?

(A) An asset insured with a floor is equivalent to investing in a zero-coupon bond and buying a call option on the asset. (B) A short position insured with a cap is equivalent to writing a zero-coupon bond and buying a put option on the asset. (C) A covered written call is equivalent to investing in a zero-coupon bond and writing a put option on the asset. (D) A covered written put is equivalent to writing a zero-coupon bond and writing a call option on the asset. (E) All of the above are true.


12


13. A fairly smart actuary (also know as an FSA) is offered the following rates on a loan: I. X% nominal annual rate of interest compounded monthly II. X% nominal annual rate of discount compounded monthly III. X% annual effective rate of interest IV. X% annual effective rate of discount V. X% constant force of interest

Which rate does the FSA take?

(A) I (B) II (C) III (D) IV (E) V


13


14. An annuity pays 1 at the beginning of each year for n years. Using an annual effective interest rate of i, the present value of the annuity at time 0 is 8.55948. It is also known that 1  i 

n



3.172169 .

Find the accumulated value of the annuity immediately after the last payment.

(A) 27.152 (B) 28.456 (C) 29.513 (D) 30.765 (E) 31.973


14


15. Deposits are made at the beginning of every month into a fund earning a nominal annual rate of 6% convertible monthly. The first deposit is 100 and 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


15


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 fund balance is $110. At year-end the balance is $95.

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

(A) June (B) July (C) August (D) September (E) October


16


17. A common stock pays annual dividends at the end of each year. The earnings per 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 dividend will be 0% for the next 5 years and 50% thereafter.

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

(A)

(B)

(C)

(D)

(E)

5 J

1.1

4

5 J 5

1.1 5 J

1.1

6

10 J

1.1

5

10 J

1.1


6

17


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

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

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

(A) 777 (B) 888 (C) 999 (D) 1111 (E) 2222


18


19. You are given the following prices for $100 zero-coupon bonds: Term 1 2 3

Price 95.24 89.00 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%


19


20. A fund earns simple interest of i. Which of the following are true? I. The effective rate of interest is constant for each period. II. The amount of interest is constant for each period. III. The equivalent force of interest is ln 1  i  .

(A) I only (B) II only (C) III only (D) II, III only (E) The correct answer is not given by (A), (B), (C) or (D).


20


21. If

 t 

t

1  t 2

, t  0 and

 Ia 

n



13.03567.

Determine n.

(A) 10 (B) 11 (C) 12 (D) 13 (E) 14


21


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

Effective Rate Interest Payments per Year

A

j%

paid directly to investor

B

5%

paid directly to investor

C

4%

automatically reinvested

An investor deposits $10,000 into fund A. He reinvests fund A’s interest payments 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


22


23. Which of the following are true? I. Modified duration is greater for bonds with higher par values. II. Modified duration is greater for yields compound more frequently. III. The modified duration of a single cash flow is the time remaining until the cash flow.

(A) I only (B) II only (C) III only (D) II, III only (E) The correct answer is not given by (A), (B), (C) or (D).


23


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 annuity?

(A) 10a5 a5 (B) 10a5 a5 (C) 10a5 a4 (D) 10a5 a4 (E) 10  Ia 5




10  Da 4

24


25. You are given the following term structure of interest rates. Term (in y ears ) 1 2 3 4

Spot Rat e 5.00% 5.75% 6.25% 6.50%

Calculate 1000  Da 4 rounded to the nearest 100.

(A) 8700 (B) 8800 (C) 8900 (D) 9000 (E) 9100


25


26. Suppose you desire to short-sale 200 shares of ABC stock, which has a bid price of $24.82 and an ask price of $25.01. You cover the short position three months 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 a 0.3% commission to close the position. He also requires a 50% haircut on the net proceeds of the short-sale. The market rate of interest is 6% convertible continuously and the short-rebate on the haircut is 4% convertible continuously. 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


26


27. Which of the following are true? I. The payoff on a short forward is the spot price at expiration minus forward price. II. Zero coupon bonds can be used to shift profit diagrams. III. For forward contacts, cash settlements often reduce transaction costs. (A) I only (B) II only (C) III only (D) II, III only (E) The correct answer is not given by (A), (B), (C) or (D).


27


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

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

Calculate the outstanding balance just after the 17th payment.

(A) 1185 (B) 1485 (C) 1885 (D) 1985 (E) 2185


28


29. A 10-year 1000 par value bond pays 4% coupons semiannually and is redeemable 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 up (B) 24.17 write down (C) 24.17 write up (E) 35.73 write down (E) 35.73 write up


29


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 annual effective rate of interest is 10%.

Determine X.

(A) 70 (B) 80 (C) 90 (D) 100 (E) 110


30


31. Which of the following expression does not represent a definition for

m|

an ?

(A) v m an (B) am

n

a

m

(C) v m n sn 

(D) v m  v m

1



 

v

m n

(E) All of the above represent a definition for


31

m|

an .


FM Sample Exam #1 Solutions

Last updated December 7, 2009

32.

(1 + i)x = 5 2(1 + i)y = 8 (1 + i)y = 4 3(1 + i)z = 20 (1 + i)z =

12(1 + i)3x

y +z

−2

x 3

20 3

12(5)3 20 12[(1 + i) ] (1 + i) 3 = = = 625 2 2 (4) [(1 + i)y ] z

 

33. After the 3 years of interest only payments the balance is still 100,000 so we

solve 100,000 = 1500an 0.01 Using the calculator we get 111 payments plus the 3 years of interest only payments equals 111 + 3(12) = 147 payments. 34. The sinking fund payment is Rs25 0.04 = 10,000 R = 240.12

The interest payment is 10,000 × 0.05 = 500 The sinking fund balance at time 11 is 240.12s11 0.04 = 3238.34 c 2008 The Inﬁnite Actuary, LLC

FM Sample Exam #1 Solutions

FM1 Solutions

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