A) 1.8% B) 1.9% C) 2.0% D) 2.1% E) 2.2%

Exam FM

Questions

1. Consider the following yield curve:

Year Spot Rate

1 5.5%

2 5.0%

3 5.0%

4 4.5%

5 4.0%

Find the four year forward rate.

A) 1.8% B) 1.9% C) 2.0% D) 2.1% E) 2.2%

2. Find the Macaulay duration of a 10-year 1000 par value bond with 8%

annual coupons and an effective annual interest rate of 6.5%.

A) 7.2 B) 7.4 C) 7.6 D) 7.8 E) 8.0

3. At an effective annual rate of interest i, a person can pay off a loan of K in two ways:

1) 475 now and 475 in 1 year, or 2) 570 in 2 years and 570 in 3 years.

Calculate K.

A) 893 B) 901 C) 909 D) 917 E) 925

4. A 10-year annuity-immediate pays 100 quarterly for the first year. In each subsequent year, each payment is increased by 5% over the payment for the previous year. There is a nominal annual interest of 8% convertible

quarterly. Find the present value of this annuity.

A) 2997 B) 3075 C) 3108 D) 3225 E) 3333

Practice Exam 1

5. The present value of a 10-year annuity-immediate with level annual payments and interest rate i is X. The present value of a 20-year annuity- immediate with the same payments and interest rate is 1.5X. Find i.

A) 7.2% B) 7.4% C) 7.6% D) 7.8% E) 8.0%

For Problems 6 and 7 use the following account summary:

Date Balance Before

Activity

Deposits Withdrawals January 1 100,000

March 1 105,000 10,000

September 1 112,000 30,000

December 31 95,000

6. Find the time-weighted yield for this account.

A) 17.2% B) 17.5% C) 17.9% D) 18.1% E) 18.5%

7. Find the dollar-weighted yield for this account.

A) 14.9% B) 15.3% C) 15.6% D) 16.1% E) 16.4%

8. An investor has 3000 worth of 5-year bonds with a modified duration of 4.615, 7000 worth of 10-year bonds with a modified duration of 9.323 and 10,000 worth of 20-year bonds with a modified duration of 19.085. What is the modified duration of this entire portfolio?

A) 13.5 B) 13.7 C) 13.9 D) 14.1 E) 14.3

9. A company has liabilities of 1000, 3000 and 5000 payable at the end of years 1, 2 and 3 respectively. The investments available to the company are the following zero-coupon bonds:

Maturity (years)

Effective Annual Yield

Par

1 7% 1000

2 8% 1000

3 9% 1000

Determine the cost for matching these liabilities exactly.

A) 6918 B) 7024 C) 7165 D) 7368 E) 7522

10. A man creates a retirement fund by depositing payments at the end of each month for 20 years. For the first 10 years the deposits are 100 per month and for the last 10 years the deposits are 200 per month. The fund earns interest at a nominal rate of 6% per year converted monthly. Upon

retirement he uses the proceeds to purchase a 30-year annuity-immediate with monthly payments. The annuity earns at a nominal rate of 8%

converted monthly. What are monthly payments from this annuity?

A) 408 B) 425 C) 437 D) 441 E) 459

11. An annuity pays annual payments at the beginning of each year for 20 years.

For the first 10 years the payments are 100. Starting with payment 11 each payment is increased by 6% over the previous payment. The annuity earns at an annual effective rate of 8%. Find the present value of this annuity.

A) 1177 B) 1190 C) 1202 D) 1213 E) 1225

12. A corporate bond is priced to yield 7.2% and has a price of 972.48. The Macaulay duration is D = 7.1245. Estimate the change in price if rates increase by 0.10%.

A) -6.463 B) -6.685 C) -6.814 D) -7.012 E) -7.163

13. A 40-year loan is paid with level annual payments at the end of each year.

The principal paid in the 20^th payment is 166.59 and the principal paid in the 25^th payment is 244.78. Find the interest rate for this loan.

A) 7.7% B) 8.0% C) 8.2% D) 8.5% E) 8.8%

14. Linus deposits 100 into an account at the end of each year for 20 years. This account earns interest at an annual effective rate of 5%. Lucy deposits money into an account at the end of each year for 20 years. Her account also earns interest at an annual effective rate of 5%. Her deposits are:

P, 2P, …. , 20P. At the end of 20 years the accumulated amounts are the same. Find P.

A) 10.93 B) 11.05 C) 11.12 D) 11.23 E) 11.35

15. Schroeder borrows money to buy a new piano. He agrees to pay back the loan with level annual payments at the end of each year for 30 years. The annual interest rate is 7%. The interest in his 10^th payment is 366.74. What is the interest in his 20^th payment?

A) 221.86 B) 229.64 C) 244.18 D) 250.72 E) 253.80

16. A woman makes a deposit into an account. For the first 5 years the account accumulates with a force of interest of 0.05. For the next 10 years the fund accumulates with an annual nominal discount rate of 6% convertible quarterly. For the 15 year period, what is the annual nominal interest rate convertible monthly?

A) 5.59% B) 5.71% C) 5.83% D) 5.96% E) 6.04%

17. Violet purchases a 10-year 1000 par bond with 8% semiannual coupons. The bond is priced to yield 7.5% convertible semiannually. She reinvests the coupon payments in a fund that pays a nominal rate of 7% convertible semiannually. What is her nominal annual yield convertible semiannually?

A) 7.36% B) 7.41% C) 7.48% D) 7.56% E) 7.63%

18. You are given the following yield curve:

Year Spot Rate

1 4.0%

2 4.2%

3 4.6%

4 -- 5 5.1%

If i4,5 = 6.1%, find s4.

A) 4.81% B) 4.83 C) 4.85% D) 4.87% E) 4.89%

19. A 20-year annuity-immediate has annual payments. The first payment is 100 and subsequent payments are increase by 100 until they reach 1000. The remaining payments stay at 1000. The annual effective interest rate is 7.5%.

What is the cost of this annuity?

A) 6201 B) 6372 C) 6413 D) 6584 E) 6700

20. A woman buys a 1000 par 5-year zero coupon priced to yield 6%. At the same time she buys a 5-year 1000 par bond with 8% semiannual coupons which is priced to yield 7%. The coupon payments are reinvested at 6.5%

convertible semiannually. What is her annual effective yield for the combined investment?

A) 6.0% B) 6.2% C) 6.4% D) 6.6% E) 6.8%

21. The S&R index currently has a price of 1300. The price of a six month forward contract is 1320. What annual interest rate (compounded

continuously) is implied by this forward price? Note that the S&R has no dividend.

A) .02481 B) .02500 C) .0305 D) .0355 E) .0411

22. The S&R index currently has a price of 1300. The price of a three month 1320-strike put is 81.41 . The annual interest rate is 4% compounded

continuously. A buys this put, and B enters into a long forward contract. In three months A and B have the same profit. What is the price of the index in three months?

A) 1310 B) 1297 C) 1289 D) 1291 E) 1275

23. The current value of the a stock isS0 =25, and the continuously compounded risk free rate is r=.04. The price of a six month (T=.5) 26-strike call is 1.7152 and the price of a six month (T=.5) 26-strike put is 2.5726. Find the continuously compounded dividend yield δ .

A) 1% B) 2% C) 3% D) 4% E) 5%

24. Investor C buys the S&R index at time 0 for 1100 and buys an 1100-strike put with T=.25 for a price of 81.51.If the interest rate is r=.04, what is his minimum profit (loss)?

A) -93.38 B) -63.015 C) -57.64 D) -48.50 E) There is no minimum

25. The current (spot) rate for corn is 1.60 per bushel. The 6 month forward price is $1.50 per bushel. The continuously compounded annual rate is

.035

r= . Farmer Brown, has total fixed and variable costs of 1.44 per bushel, and plans to produce 100,000 bushels for $144,000.

A six month (T=.5) put with a strike price of 1.52 per bushel is available at a price of 0.12. What are the minimum and maximum profits for Farmer Brown in six months if he is hedged with a purchase of this put?

A) minimum = -4,212, maximum = 19,678 B) minimum =-6222, maximum = 19,678 C) minimum= -4,212, no maximum D) minimum = -6,242, no maximum E) none of the above

26. Company XYZ makes an aircraft which costs 80,000,000 to manufacture. It will be completed in six months. At that time it will sell either for 90,000,000 with probability .5 or 74,000,000 with probability .5. The company decides to enter into a forward contract to sell the unit for 85,000,000 in six months The company has a 40% tax rate, and has no tax benefit for losses. What is the company’s expected profit after tax?

A) -1,000,000 B) 0 C) 1,000,000 D) 2,000,000 E) 3,000,000

27. A stock has current price.S0 =25 The annual continuous interest rate is .03

r= . If the expiration time for a forward contract is T =.25 and the forward price is 25.15, what is the continuous dividend yield δ? A) 0.003 B) 0.006 C) 0.010 D) 0.015 E) 0.018

28. The S&R index has a spot price ofS0 =1100. The continuous interest rate is .03

r= and the continuous dividend yield is δ = The one year forward price 0 is 1133.50. Which of the following positions results in a synthetic long

forward contract?

A) Sell the index short for 1100 and lend the proceeds at r=.03 B) Sell the index short for 1100 and borrow 1000 at r=.03 C) Borrow 1000 at r=.03 and buy the index.

D) Borrow 1000 at r=.03 and sell the index short E) None of these.

In Problems 29-30, use the following table of quarterly oil forward prices and zero-coupon bond prices.

Quarter 1 2 3 4

Oil Forward Price 20.9 21.2 20.8 20.7 Zero-coupon bond price .984 .969 .953 .935 29. Find the price of a four quarter oil swap.

A) 21.18 B) 21.62 C) 20.90 D) 20.83 E) 20.78

30. Suppose you enter a three quarter interest rate swap. What net interest payment will be made to you in the second quarter if the spot interest rate for the second quarter is .018?

A) .0010 B) .0012 C) .0016 D) .0018 E) .002

Solutions

1. The four year forward rate i4,5 is given by 1 + i4,5 = (1 + s5)⁵/(1 + s4)⁴ = 1.04⁵/1.045⁴ = 1.020

i4,5 = .02 Answer C

2.

( )

( ) ( )

( )

10 10

10 10

10

10 10

10

10

80 10(1000)

Bond Price 10

1.065 0.532726

7.6561 (be sure calculator is in BGN mode) 35.8284

Bond price 1,107.83

Ia v

D

a v

Ia i

v a

Ia

−

⎡ + ⎤

⎣ ⎦

=

= −

= =

=

=

=

&&

&&

(Reset calculator to END mode. N = 10, PMT = 80, I/Y =6.5, FV = 1000.

CPT PV = -1.107.83)

D = [80(35.8284) + 5,327.26]/1,107.83 = 7.396 Answer B

3. K = 475 + 475v = 570v² + 570³

v² = [475(1 + v)]/[570(1 + v)] = 0.8333 ⇒ v = 0.91287 K = 475(1.91287) = 908.61

Answer C

4. The accumulated amount at the end of year one is 412.16.

(N = 4, I/Y = 2, PMT = 100, PV =0. CPT FV = - 412.16)

We can view the annuity as a 10-year annuity-immediate with annual payments, the first being 412.16 and subsequent payments are increase by 5% each year. The effective annual rate is i = (1.02)⁴ – 1 = 0.08243.

The present value of this annuity is

(412.16)[1 + (1.05/1.08243) + … + (1.05/1.08243)⁹]

= 412.16[1 – (1.05/1.08243)¹⁰]/(1.08243 – 1.05)

= 3333.30 Answer E

5. X = (1 – v¹⁰)/i, 1.5X = (1 – v²⁰)/i

Hence 1 + v¹⁰ = 1.5, v¹⁰ = 0.5, i = 0.072 Answer A

6. For the time-weighted yield

1 + j = (105,000/100,000)(112,000/115,000)(95,000/82,000) = 1.185 j = 0.185

Answer E

7. For the dollar-weighted yield,

I = 95,000 – 100,000 – (10,000 – 30,000) = 15,000

i = 15,000/[100,000 + (1 – 1/6)(10,000) – (1 – 2/3)(30,000)]

= 0.153 Answer B

8. The weights are 3/20, 7/20 and 1/2 respectively for the 5-year, 10-year and the 20-year bonds. The modified duration is

DM = (3/20)(4.615) + (7/20)(9.323) + (1/2)(19.085) = 13.498 Answer A

9. The company must invest the present values of 1000 in one year at 7%, 3000 in 2 years at 8% and 5000 in 3 years at 9%. The cost is

1000/1.07 + 3000/1.08² + 5000/1.09³ = 7367.51 Answer D

10. The deposits can be viewed as payments of 100 into a 20-year annuity- immediate and 100 into a 10-year deferred 10-year annuity-immediate.

The accumulated amount in the first annuity is 46,204.09.

(N = 240, I/Y = 0.5, PMT = -100, PV = 0. CPT FV = 46,204.09)

The accumulated amount in the second annuity is 16,387.94.

(Reset N = 120. CPT FV = 16,387.94) Total accumulation is 62,592.02.

The monthly payments from the 30-year annuity are 459.30.

(N = 360, I/Y =0.6667, PV = - 62,592.02, FV = 0. CPT PMT = 459.30) Answer E

11. The present value of this annuity is

10 9

100a&&10 + (106 / 1.08 )[1+ (1.06 / 1.08) + K + (1.06 / 1.08) ]

To get the value of the first term set the BA II Plus to BGN mode.

Set N = 10, I/Y = 8, PMT = -100, and FV = 0. CPT PV = 724.69.

The value of the second expression is

(106/1.08¹⁰)[1 – (1.06/1.08)¹⁰]/[1- (1.06/1.08)] = 452.03 Present value is 724.69 + 452.03 = 1,176.72

Answer A

12. The change is

ΔP = − (D)P(i)Δi/(1 + i)

= − (7.1245)(972.48)(0.001)/(1.072)

= − 6.463 Answer A

13. PRink is the amount of principal repaid in the k^th period.

Prink+n = (1 + i)ⁿ PRink . Let k = 20 and n = 5.

244.78 = (1 + i)⁵ (166.59).

i = (244.78/166.59)^1/5 – 1 ⇒ i = .08 Answer B

14. The accumulation in Linus’s account is 100s₂₀ =3, 306.60. (N = 20, I/Y = 5, PV = 0, PMT = -100. CPT FV = 3,306.60) The accumulation is Lucy’s account is ^{P Is}

( )

( ) (

)

20

20 294.385 0.05

3, 306.60

11.23 294.385

Is s

P

= − =

= =

&&

Answer D

15. Let P be the annual payment. The interest paid in the 10^th payment is P(1 – v^30−10+1) = P(1 – v²¹) = P(1 – 0.24151) = 366.74

P = 483.51

For the 20^th payment the interest is 483.51(1 – v¹¹) = 483.51(0.52491)= 253.80 Answer E

16. If Y is the amount deposited, then the accumulation is A = Ye^0.05(5)(1 − 0.015)⁻⁴⁰ = 2.3503Y.

There are 180 months in the 15 year period. If j is the monthly interest then j = 2.3503^1/180 – 1 = 0.00476

i = 12(0.00476) = 0.0571 Answer B

17. The price of the bond is 1034.74.

(N = 20, I/Y = 0.375, PMT = 40 and FV = 1000. CPT PMT = - 1034.74)

The accumulated amount of reinvested coupon payments is 40s₂₀ =1131.19. The total accumulation is 2131.19.

The semiannual yield on the investment is j = (2131.19/1034.74)^1/20 – 1 = .0368.

The annual yield is 2(.0368) = 0.0736.

Answer A

18. 1 + i4,5 = (1 + s5)⁵/(1 + s4)⁴

(1 + s4)⁴ = (1 + s5)⁵/(1 + i4,5) = (1.051)⁵/(1.061) = 1.20864 1 + s4 = 1.0485, s4 = 0.0485

Answer C

19. This can be viewed as a 10-year increasing annuity and a 10-year deferred 10-year annuity.

The present value of the 10-year deferred annuity is

( )( )

10

1000v a10 =1000 0.48519 6.8641 =3, 330.39.

The present value of the increasing annuity is ^{100 Ia}

( )

( ) (

)

10

10 7.3789 4.8519

33.6933 0.075

a v

Ia i

− −

= && = =

Total cost is 3,330.39 + 3,369.33 = 6,699.72 Answer E

20. The price of the zero-coupon bond is 1000/1.06⁵ = 747.26.

To find the price of the second bond with the BA II Plus set N = 10, I/Y = 3.5, PMT = - 40 and FV = -1000. CPT PV = 1041.58.

The accumulation of the reinvested coupon payments is 463.87.

(N = 10, I/Y =3.25, PMT = -40 and PV =0. CPT FV = 463.87) Total investment is 747.26 + 1041.58 = 1788.84.

Total accumulation is 1000 + 1000 + 463.87 = 2463.87.

Annual effective yield is (2463.87/1788.84) – 1 = 0.066.

Answer D

21. F0,T =S e0 ^rT →1320=1300e^.5r → =r .0305 Answer C

22. The forward price is F0,T =S e0 ^rT =1300e^{.25 .04( )} =1313.07. The long forward profit is ST −F0,T =ST −1313.07.

The put profit is ^{max 0,1320}

(

)

(

)

Assume that ST <1320. Then the equality of prices implies that 1313.07 1320 82.23 1275.42

T T T

S − = −S − →S =

Answer E

23. By put-call parity

0 T rT

C− =P S e^−δ −Ke⁻

( )

.04 .5

1.7152 2.5726− =25e⁻.5^δ −26e⁻ → =r .03 Answer C

24. Buying the index and buying a put with strike 1100 creates a floor. The floor has the same profit function as a long call with strike 1100.

The minimum profit on the floor is the (negative) loss of the future value of the call premium when the call expires unexercised.

By parity the value of the call premium is 92.4554. The minimum profit is 92.4554e.01 93.38

− = −

Answer A

25. The profit from the put option is

( )

.035 .5

100, 000 max(0,1.52⎣⎡ −x) .12− e ⎦⎤=100, 000 max(0,1.52−x) 12, 211.85− .

The total profit for the hedged position is

( )

100, 000 144, 000 100, 000 max(0,1.52 ) 12, 211.85 4, 211.85, 1.52

100, 000 156, 211.85, 1.5

x x

x

x x

− + − −

− <

= ⎨⎧⎩ − ≥

Answer C

26. The calculations are in the table below. Values are given in millions.

With Short Forward at 85

Price Price

90 74

Pre-tax op income 10 -6 Income from Forward -5 11

Taxable Income 5 5

Tax @ 40% 2 2

After Tax Income 3 3

Answer E

27.

( ) (^.03 )^.25

0, 0 25.15 25

25.15

ln .0075 .25

25 .006

r T

F T S e ^δ e ^δ

δ δ

− −

= → =

⎛ ⎞ = −

⎜ ⎟

⎝ ⎠

=

Answer B

28. STOCK - ZERO COUPON BOND = LONG FORWARD

Thus you buy the index for 1000 and sell a zero coupon bond for 1000 (borrow the money to buy the stock.).

Answer C

29. We will use the general formula

( ) ( ) ( )

( ) ( ) ( ) ( )

0 1

1

0, 20.9 .984 21.2 .969 20.8 .953 20.7 .935

20.90 .984 .969 .953 .935

0,

n

i i

i n

i i

P t f t P

P t

=

=

+ + +

= = =

+ + +

∑

∑

Answer C

30. The guaranteed interest rate is the three year par coupon bond rate.

( )

( )

( )

( )

P P P

− −

= = =

+ + + +

The net rate paid to you will be .018- .0162 = .0018 Answer D