Example Exam Solution

7/31/2019 Example Exam Solution

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Question 1

Suppose an investor contacted a broker on Monday, January 8 and placed an order to take a short position in 5 March on gold futures contracts. Further suppose that at the time the order wasexecuted, the March gold futures price was 650 dollars per troy ounce. The size of each contract

is 100 troy ounces.

The broker required the investor to post an initial margin of $2,500 per contract. The broker alsoinformed the investor that the maintenance margin was $1,000 per contract.

Answer the following questions.

A. Assuming the investor closes out the position at the settlement price on Friday, January 12,fill out the following table.

Date

March Gold

FuturesSettlementPrices

Mark-to-Market

Add/Withdraw

To/FromMarginAccount

End-of-DayBalance

1/8 open 650 Nothing needs to be entered here

Nothing needs to be entered here

1/8 settle 645

1/9 settle 655

1/10 settle 660

1/11 settle 670

1/12 settle 615 0

B. Compute the gain/loss on the futures position. Ignore commission charges.


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Solution:A.

Date

March GoldFutures

SettlementPrices

Mark-

to-Market

Add/Withdraw To/From

MarginAccount End-of-DayBalance1/8 open 650 Nothing needs to

be entered here $12,500 Nothing needs to

be entered here 1/8 settle 645 $2,500 $15,000

1/9 settle 655 $5,000 $10,000

1/10 settle 660 $2,500 $7,500

1/11 settle 670 $5,000 $10,000 $12,500

1/12 settle 615 $27,500 $40,000 0

B. 40,000 12,500 10,000 = (5)(100)(650 615) = $17,500

Question 2

The price of gold is currently $500 per ounce. Forward contracts are available to buy or sell goldat $700 for delivery in one year. An arbitrageur can borrow money at 10% per annum. Whatshould the arbitrageur do? Assume that the cost of storing gold is zero and that gold provides noincome.

Solution:The arbitrageur should borrow money to buy a certain number of ounces of gold today and shortforward contracts on the same number of ounces of gold for delivery in one year. This meansthat gold is purchased for $500 per ounce and sold for $700 per ounce. As long as the cost of

borrowed funds is less than 40% per annum this generates a riskless profit.

700=500*(1+r) r = 40%


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Question 3

Suppose the spot price of corn (a pure consumption commodity) is $4.75 per bushel. Storagecosts are $0.60 per bushel per year payable semiannually in advance, i.e., $0.30 now for the first

six months and $0.30 in six months for the next six months. The interest rate is 6 percent per year with continuous compounding. Answer the following questions.

A. Suppose the actual quoted price for a six-month forward contract is $5.50 per bushel. Explainwhether or not there is an arbitrage opportunity. If one does exist, use an arbitrage table todemonstrate how you can make a riskless arbitrage profit. Note that the arbitrage table shouldhave the following column titles: Transaction, Payoff (now), and Payoff (6 months).

B. Suppose the actual quoted price for a six-month forward contract is $5.05 per bushel. Explainwhether or not there is an arbitrage opportunity. If one does exist, use an arbitrage table todemonstrate how you can make a riskless arbitrage profit. Note that the arbitrage table should

have the following column titles: Transaction, Payoff (now), and Payoff (6 months).

C. Using the actual quoted forward price of 5.05 per bushel in part B., compute the impliedconvenience yield, y, of corn.

Solution:

A. For a consumption commodity:

20.5e)30.075.4(e)US(F )50.0)(06.0(rTactual

Since the actual forward price is greater than 5.20, an arbitrage position is feasible. In this case,short the forward contract and buy corn. See table below.


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Transaction Payoff (now) Payoff (6 months)

Short Forward

Buy Spot

Borrow to Buy Spot

Pay Storage

Borrow 0.30 for 6Months

0

4.75

+ 4.75

0.30

+ 0.30

5.50 TS

+ TS

)50.0)(06.0(e75.4

)50.0)(06.0(e30.0

Net 0

bushelper30.0

e)30.075.4(50.5 )50.0)(06.0(

B. Since the actual quoted forward price, 5.05, is less than 5.20, there is not an arbitrageopportunity. For a pure consumption commodity, selling the commodity and taking a long

position in the forward contract is not feasible.

C. Using the actual quoted forward price of 5.05 per bushel in part B., compute the impliedconvenience yield, y, of corn.

)50.0)(06.0()50.0)(y( e)30.075.4(e)05.5(

)50.0)(06.0()50.0)(y( e)05.5(e)05.5(

)50.0)(06.0()50.0)(y( ee

y = 0.06 or 6%


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Question 4

The current spot price of silver is $12.50 per ounce, and the risk-free rate of interest is 5.13% per year with annual compounding. The standard cost to store large quantities of silver in a securestorage facility is $0.05 per ounce every four months payable in advance. Answer the following

questions.

A. Compute the delivery price for a 12-month forward contract taken out today (i.e., the forward price that is consistent with a zero value for the forward contract).

B. Suppose the 12-month forward contract referred to in part A is to buy 100,000 ounces of silver, and suppose that we are now 8 months into the life of the contract (i.e., the contract has aremaining maturity of 4 months). Compute the market value of this previously-issued forwardcontract assuming the spot price of silver in 8 months is $15.55 per ounce.

Solution:

A. 05.0050027493.0)0513.1ln(r

1475.0e05.0e05.005.0U )12 / 8)(05.0()12 / 4)(05.0(

30.13e)6475.12(e)1475.050.12(e)US(FK )1)(05.0()1)(05.0(rT

B. 86.15e)05.055.15(F )12 / 4)(05.0(

69.768,251$e)30.1386.15)(000,100(f )12 / 4)(05.0(

or

97.982,251$)e30.1305.055.15)(000,100(f )12 / 4)(05.0(


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Question 5

In the following situations, indicate whether a long or short position is appropriate and computethe optimal number of contracts.

A. A jewelry manufacturer would like to hedge the purchase of 1,000 ounces of gold using goldfutures. One gold futures contract is for the delivery of 100 ounces of gold.

B. A wholesaler of pork bellies would like to hedge the sale of 280,000 pounds of pork belliesusing pork belly futures. One pork belly futures contract is for the delivery of 40,000

pounds.

C. An energy products distributor has a fixed-price contract to sell 36,000 barrels of oil at a price of $28 per barrel in 3 months. One oil futures contract is for the delivery of 1,000 barrels of oil.

D. Statistical analysis indicates that the secondary-market price of Pokeman cards is inverselycorrelated with the futures price of baseball cards. Indeed, a regression of the secondary-market price of Pokeman cards on the futures price of baseball cards has a statisticallysignificant (slope) coefficient of 2.60 and an R 2 of 75%. A dealer in Pokeman cards wouldlike to use this information to hedge variation in the value of his inventory of 10,000Pokeman cards. One baseball card futures contract is for the delivery of 500 baseball cards.

E. Platinum and gold prices tend to be highly correlated. However, platinum prices tend to beless volatile than gold prices. A regression of the spot price of platinum on the futures priceof gold has a statistically significant (slope) coefficient of 0.65 and an R 2 of 89%. A Russian

platinum mine would like to use this information to hedge the sale of 1,200 ounces of platinum. One gold futures contract is for the delivery of 100 ounces of gold.

F. Heritage Rare Coins (located in Dallas and the largest rare coin dealer in the world) hasagreed to sell its entire inventory of 2,000 $20 gold pieces (each minted in the 1900s andeach approximately one ounce of pure gold) for $1,750 per coin to another rare coin dealer inJune of this year. Heritage would like to hedge this transaction using gold futures. Aregression of $20 gold piece prices on the futures price of gold has a statistically significant(slope) coefficient of 5.80 and an R 2 of 55%. One gold futures contract is for the delivery of 100 ounces of gold.

Solution:A. LONG: (1,000)/(100) = 10 futures contractsB. SHORT: (280,000)/(40,000) = 7 futures contractsC. LONG: (36,000)/(1,000) = 36 futures contractsD. LONG: (2.6)[(10,000)/(500)] = 52 futures contractsE. SHORT: (0.65)[(1,200)/(100)] = 7.8 or 8 (rounded up) futures contractsF. LONG: (5.80)[(2,000)/(100)] = 116 futures contracts


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Question 6

A company would like to hedge the purchase of pink widgets (PW) with a long position infutures contracts on magenta widgets (MW). A regression of the spot price of pink widgets (

PWS ) on the futures price of magenta widgets ( MWF ) resulted in the following equation:

86.0RF50.075.0S 2MWPW

The company plans to purchase 100,000 pink widgets in two months. The current spot price of pink widgets is $7 per widget and the current futures price of magenta widgets for delivery in 3months is $12.50 per widget. Each futures contract is for delivery of 10,000 magenta widgets.Answer the following questions.

A. What is the minimum variance hedge ratio ( h ) and optimal number of futures contracts (N )?

B. The company projects two extreme outcomes in two months: (1) 75.9SPW and 18FMW

and (2) 75.3SPW and 6FMW . Compute the net cost to the company under each

outcome in order to show that your h in part A is the minimum variance hedge ratio.

Solution:

A. contracts5000,10000,100

)50.0(contractfuturesoneUnits

hedgedUnitshN50.0

dFdS

hMW

PW

B.Extreme outcome #1: 75.9SPW and 18FMW

Purchase at spot: (100,000)(9.75) = 975,000Gain on futures: (5)(10,000)(18 12.50) = 275,000

Net cost: 975,000 275,000 = 700,000

Extreme outcome #2: 75.3SPW and 6FMW

Purchase at spot: (100,000)(3.75) = 375,000Gain on futures: (5)(10,000)(6 12.50) = 325,000

Net cost: 375,000 + 325,000 = 700,000


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Question 7

Hector Lopez is a bond portfolio manager at PNY. His portfolio has a market value of $500million and a duration of 9.8 years. Answer the following questions.

A. Compute an estimate for the percentage change in the value of the portfolio if interest ratesdecrease by 50 basis points. Use the formula that assumes continuous compounding.Compute an estimate for the market value of the portfolio after the decrease in interest rates?

B. Mr. Lopez would like to use T-bond futures to hedge the risk of an increase in interest rates.He decides to use the September contract, which has a futures price quote of 96-22(96+22/32). The duration of the cheapest-to-deliver bond for the September contract isexpected to be 8.7 years, and he estimates that the average duration of his portfolio bySeptember will be 9.6 years. What position must he establish (long or short) and why? Howmany contracts must this position have to achieve a minimum variance hedge? Assume themarket value of the portfolio is $500 million.

C. Suppose that after setting up this hedged position interest rates increase by 75 basis points.Show that the decrease in the bond portfolio is offset by the gain on the T-bond futures

position. The September T-bond futures price decreases to 90-12 (96+12/32), and the bond portfolios average duration is 9.6 years. As in part A, use the continuous compoundingformula.

Solution:

A. 049.0)005.0)(8.9(yDBB

Portfolio value = (1.049)($500 million) = $524,500,000

B. Short: Both the portfolio value and T-bond futures price decrease as interest rates increase.He must establish a short position to make money on the futures position to offset thedecrease in the value of the portfolio.

26.706,5)7.8)(5.687,96(

)6.9)(000,000,500(DFDS

NF

S or 5,706 contracts

C. Change in Value of Portfolio:

000,000,36)000,000,500)(0075.0)(6.9()S)(y)(D(S S

Change in Futures Position: 125,019,36)375,905.687,96)(706,5(F


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Question 8

Michael Johnson made delivery on his short position in five December T-bond futures contractson Wednesday, December 13, 2006. Using the information below, compute the cash that

Michael received on the delivery day.

December T-bond futures settlement prices

Monday, December 11, 2006 110-07Tuesday, December 12, 2006 110-15Wednesday, December 13, 2006 110-23Thursday, December 14, 2006 111-01Friday, December 15, 2006 110-30

Available T-bonds to deliver

Bond Coupon (%) Quoted Price Maturity CF1 7.00 109-05 Nov. 15, 2022 1.11562 6.00 108-23 Nov. 15, 2023 1.00003 6.50 109-15 Nov. 15, 2025 1.06234 7.50 110-02 Nov. 15, 2026 1.1812

Day counts

Time Period Number of Days Nov. 15, 2006 to Dec. 11, 2006 26 Nov. 15, 2006 to Dec. 12, 2006 27 Nov. 15, 2006 to Dec. 13, 2006 28

Nov. 15, 2006 to Dec. 14, 2006 29 Nov. 15, 2006 to Dec. 15, 2006 30 Nov. 15, 2006 to May 15, 2007 181

Solution:

For one contract, the cash received by the short is

Cash = (Settlement contract price on position day)(CF) + Accrued interest

Position day = Monday, December 11

Settlement price on position day = 110-07 or 110.21875

Contract price = (100,000/100)(110.21875) = $110,218.75

Find the cheapest-to-deliver T-bond:

Quoted price (Quoted futures price on position day)(CF)

1 109.15625 (110.21875)(1.1156) = 13.8038


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2 108.71875 (110.21875)(1.0000) = 1.50003 109.46875 (110.21875)(1.0623) = 7.61664 110.06250 (110.21875)(1.1812) = 20.1279 Cheapest -to-deliver

Accrued interest for cheapest-to-deliver bond:

Nov. 15, 2006 to Dec. 13, 2006 = 28 days Nov. 15, 2006 to May 15, 2007 = 181 days

For one bond: (100,000)(0.075/2)(28/181) = $580.11

The cash received by Michael for 5 contracts is therefore equal to

Cash = 5 {110,218.75 1.1812 + 580.11} = $653,852.49


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Question 9Intelilogic Inc. enters into a $50 million notional principal swap in which it agrees to pay fixedand receive LIBOR. Payments will be made every six months with the floating payments based

on LIBOR six months prior. The swap lasts for two years. For simplicity, assume there areexactly 180 days in a six-month period and 360 days in a year. Answer the following questions.

A. What is the swap rate that sets the value of the swap equal to zero at the onset? The currentLIBOR term structure is as follows.

LIBOR Rates with Semiannual CompoundingMaturity LIBOR (%)

6 month 6.01 year 6.5

18 month 7.02 year 7.5

B. Suppose that Intelilogic is exactly one year into the life of the swap. On the last paymentdate (today), LIBOR is 8.00% per year with semiannual compounding. Currently, the 1-year LIBOR rate is 8.5% per year with semiannual compounding. Compute the value of the swapfrom Inte lilogics perspective.

Solution:

A.4321

2075.0

1

000,000,50Coupon

207.0

1

Coupon

2065.0

1

Coupon

206.0

1

Coupon000,000,50

Coupon = 1,863,495.482

Swap Rate = 074539819.0000,000,50

482.495,863,12

B. Fixed payment = 1,863,495.482

Floating payment = 50,000,000(0.08/2) = 2,000,000

85.835,512,49

2085.01

482.495,863,51

208.01

482.495,863,1B

21fixed

000,000,50

208.0

1

000,000,50

208.0

1

000,000,2B

11float

(as expected!)


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15.164,487$BBV fixedfloatswap

Question 10

Consider the lower bound for a European put on a non-dividend-paying stock:

0),t(S)T,t(XBMax)T,t,X,S(p

Answer the following questions.

A. Prove that there will be an arbitrage opportunity if the lower bound is violated. Use thefollowing arbitrage table in your proof.

Transaction Payoff at t Payoff at T

B. Prove that 0))T(SX(0),T(SXMax .

C. Why doesnt the lower bound in A. also hold for American puts?


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Solution:

A.

Transaction Payoff at t Payoff at T

Buy put

Buy stock

Borrow XB

p

S

+XB

Max[X S(T), 0]

S(T)

X

Sum XB S p > 0 Max[X S(T), 0] (X S(T)) 0

Arbitrage: 0pSXBSXBpbuybuyborrow

B. If S(T) < X: X S(T) (X S(T)) = 0

If S(T) > X: 0 (X S(T)) = S(T) X > 0

Therefore 0))T(SX(0),T(SXMax .


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Question 11

You decide to use a collar option strategy on Cisco Systems, Inc. You plan to purchase 1,000shares of Cisco and simultaneously establish a long put option position in 10 contracts and a

short call option position in 10 contracts. The current price of Cisco is $65 per share. The putoptions have an exercise price of $55, a maturity of 3 months, and are priced at $1.50 per option.The call options have an exercise price of $80, a maturity of 3 months, and are priced at $1.00

per option. Answer the following questions.

A. Complete the following table. For the purpose of filling in the table, assume one share of stock, one put option, and one call option.

Position Now (t)

Expiration (T)

Net

B. Graph the profit of the position in A. as a function of S(T).

C. What other option strategy has a profit graph similar to that in B?

D. Compute the maximum gain, maximum loss and breakeven price of the strategy. Themaximum gain and loss should be based on 1,000 shares, 10 put contracts, and 10 callcontracts.

E. Compute the standstill return of the strategy. The standstill return should be based on 1,000shares, 10 put contracts, and 10 call contracts.


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Solution:

A.

Position Now (t)

Expiration (T)

S(T) < 55 55 S(T) < 80 S(T) 80

Long Stock

Long Put

Short Call

65

1.50

+ 1.00

S(T)

55 S(T)

0

S(T)

0

0

S(T)

0

(S(T) 80)

Net 65.50 55 S(T) 80

B. Your profit graph should look like that for a bull call (put) spread.

C. Bull call (put) spread.

D. Maximum gain: (1,000)(80 65.50) = $14,500

Maximum loss: (1,000)(55 65.50) = $10,500

Breakeven price: (1,000)(S(T) 65.50) S(T) = $65.50

E. S(T) = S = 65(1000)[S + Max(XP S, 0) Max(S XC, 0) (S + P C)]

(1000)[65 + 0 0 65 1.50 + 1.00]

(1000)[ 0.50]

$500


16/21


17/21

Question 13

The Dallas/Fort Worth International Airport would like to buy the option to purchase a large parcel of land on the edge of the city of Grapevine from a real estate investor. The option would

give the airport the right to buy the land in one year for $10 million. Real estate experts estimatethat the land will be worth either $12 million or $8 million in one year. The current value of theland is $9.5 million. If the risk-free rate of interest is 5% per year with continuous compounding,what is the fair market value of the option?

Solution:

Riskless Hedge Portfolio Approach:

t = 0: Form the portfolio: C)5.9)((

t = 1: Set the up and down values of the portfolio equal and solve for :

50.00)8)(()1012()12)((

Equate t = 0 value of portfolio to certain discounted value and solve for C:

94508230.0$Ce4C)5.9)(50.0( )1)(05.0( million or $945,082.30

Risk-Neutral Valuation Approach:

)1)(05.0(

e)8)(p1()12)(p(5.9

p = 0.496768853 and (1 p) = 0.503231146

)1)(05.0(e)0)(503231146.0()1012)(496768853.0(C

C = $0.94508230 million or $945,082.30


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Question 14

Sally Shoomaker is a market maker in options on ADP. She currently has a large inventory of three month call options on ADP and would like to create a delta and gamma neutral positionusing the one-month call option on ADP and the six-month put option on ADP. The delta and

gamma parameters for the one-month call option ( 1C ), the three-month call option ( 3C ), and thesix-month put option ( 6P ) are as follows.

Price Delta Gamma232.4C1 537.01C 039.01C 568.7C3 563.03C 023.03C

541.8P6 411.06P 016.06P

For each three month call option held long, find the corresponding positions in the one-monthcall option and the six-month put option so that the overall position is delta and gamma neutral.Then show that your position is both delta neutral and gamma neutral for small changes in thestock price of ADP.

Solution:

6P1C3 PxCxC

0xxS 6PP1CC3C

(1)

0xxS

6PP1CC3C2

2(2)

From (1) we may determine that1C

6PP

1C

3CC xx . Substituting this expression into (2) and

solving for Px gives

1C6P6P1C

3C1C1C3CPx

Substituting this expression for Px into1C

6PP

1C

3CC xx gives

1C6P6P1C

3C6P6P3CCx


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From these expressions for Px and Cx we may determine that

39.0390155.0024621.0009606.0

)039.0)(411.0()016.0)(537.0()023.0)(537.0()039.0)(563.0(

xP

75.0749807.0024621.0018461.0

)039.0)(411.0()016.0)(537.0()023.0)(411.0()016.0)(563.0(

xC

613 P)39.0(C)75.0(C

We now check whether the overall position is delta neutral and gamma neutral:

Delta Neutral 0)411.0)(39.0()537.0)(75.0(563.0S

Gamma Neutral 0)016.0)(39.0()039.0)(75.0(023.0S22

Question 15

Consider an option on a non-dividend-paying stock when the stock price is $56, the exercise price is $50, the risk-free interest rate is 5 percent per year with continuous compounding, thevolatility is 30 percent per year, and the time to maturity is three months. Answer the followingquestions. Note: To compute standard normal probabilities (e.g., )d(N 1 and )d(N 2 ), studentswere given the standard normal probability tables or they have TI-83.

A. What is the price of the option if it is a European call?

B. What is the price of the option if it is an American call?

C. What is the price of the option if it is a European put?

D. Verify that put-call parity holds.

E. Now assume that the stock pays dividends and that it is expected to go ex-dividend in 1.5months. The dividend is expected to be $0.50 per share. What is the price of the option if itis a European put?


20/21

Solution:A.

9139.025.0)30.0(

)25.0)(2 / )30.0(05.0()50 / 56ln(

T

T)2 / r()X / Sln(d

22

1

7639.025.0)30.0(9139.0Tdd 12

)d(N 1 = N(0.9139) = N(0.91) + (0.39)[N(0.92) N(0.91)]

= 0.8186 + (0.39)[0.8212 0.8186]

= 0.81961

)d(N 2 = N(0.7639) = N(0.76) + (0.39)[N(0.77) N(0.76)]

= 0.7764 + (0.39)[0.7794 0.7764]

= 0.77757

50.7)77757.0(e)50()81961.0)(56()d(NXe)d(SNC 25.005.02rT

1

B. C = 7.50

C.

)d(N 1 = )d(N1 1

= 1 0.81961

= 0.18039

)d(N 2 = )d(N1 2

= 1 0.77757

= 0.22243

88.0)18039.0)(56()22243.0(e)50()d(SN)d(NXeP 25.005.012rT

D. rTXeSPC

25.005.0e505688.050.7


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