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$457,800.

New position value = 20 Ã— 42,000 Ã— $0.555 = $466,200, for a loss of

Day 1:

$8,400. Your margin account balance is now $15,600, so you face a

margin call. Put another $8,400 in your account to bring it up.

New position value = 20 Ã— 42,000 Ã— $0.560 = $470,400, for a loss of

Day 2:

$4,200. Your margin account balance is now $19,800.

B-58 CORRADO AND JORDAN

New position value = 20 Ã— 42,000 Ã— $0.540 = $453,600, for a profit of

Day 3:

$16,800. Your margin account balance is now $36,600

New position value = 20 Ã— 42,000 Ã— $0.520 = $436,800, for a profit of

Day 4:

$16,800. Your margin account balance is now $53,400.

Your total profit is thus $53,400 - $8,400 - $24,000 = $21,000

F = $90(1 + .06 - .04)1/4 = $90.45

14.

Parity implies that F = 1,800(1 + .06 - .03)1/2 = 1,826.80. If the parity relationship holds,

15.

the futures price should be 1,826.80. At 1,850, the futures are currently overpriced; thus,

you would want to buy the index and sell the futures.

The closing value of the Midcap 400 index futures is 366 Ã— $500 = $183,000, so the

16.

desired hedge is 1.15 Ã— $200M/$183,000 = 1,257 contracts. Assuming the mutual fund is

long stocks, the likely hedge would then be to sell 1,257 Midcap 400 futures.

17. In reality, two factors in particular make stock index arbitrage more difficult than it might

appear. First, the dividend yield on the index depends on the dividends that will be paid

over the life of the contract; this is not known with certainty and must, therefore, be

estimated. Second, buying or selling the entire index is feasible, but index staleness

(discussed in our first stock market chapter) is an issue; the current up-to-the-second price

of the index is not known because not all components will have just traded. Trading costs

have to be considered as well.

Thus, there is some risk in that the inputs used to determine the correct futures

price may be incorrect, and what appears to be a profitable trade really is not. Program

traders usually establish bounds, meaning that no trade is undertaken unless a deviation

from parity exceeds a preset amount. Setting the bounds is itself an issue. If they are set

too narrow, then the risks described above exist. If they are set too wide, other traders will

step in sooner and eliminate the profit opportunity.

18. The spot-futures parity condition is:

F = S(1 + r - d)T,

where S is the spot price, r is the risk-free rate, d is the dividend yield, F is the futures

price, and T is the time to expiration measured in years.

Plugging in the numbers we have, with 1/2 for the number of years (6 months out 12),

gets us:

1200 = 1194(1 + X)1/2

Solving for X, the difference between r and d, we get 1 percent.

FUNDAMENTALS OF INVESTMENTS B-59

19. The formula for the number of U.S. Treasury note futures contracts needed to hedge a

bond portfolio is:

DÃ—V

= P P

Number of contracts

D Ã—V

F F

where VP is the value of the bond portfolio,

DP is the duration of the bond portfolio,

DF is the duration of the futures contract,

VF is the value of a single futures contract.

The duration of the futures contract is the duration of the underlying instrument, plus the

time remaining until contract maturity, i.e.,

DF = DU + MF

where DF is the duration of the futures contract,

DU is the duration of the underlying instrument, and

MF is the time remaining until contract maturity.

In our case, the duration of the underlying U.S. Treasury note is 9 years and the futures

contract has 90 days to run, so DF = 9.25. The face value of the note contract is 102

percent of $100,000, or $102,000. Plugging in the numbers, we have:

6 Ã— $600,000,000

3,816 contracts =

9.25 Ã— $102,000

You therefore need to sell 3,816 contracts to hedge this $600 million portfolio.

B-60 CORRADO AND JORDAN

Chapter 17

Diversification and Asset Allocation

Answers to Questions and Problems

Core Questions

.2 Ã— (-.10) + .5 Ã— (.20) + .3 Ã— (.30) = 17%

1.

.2 Ã— (-.10 - .17)2 + .5 Ã— (.20 - .17)2 + .3 Ã— (.30 - .17)2 = .0201; taking the square root, Ïƒ

2.

= 14.1774%.

(1/3) Ã— (-.10) + (1/3) Ã— (.20) + (1/3) Ã— (.30) = 13.333 . . . %

3.

(1/3) Ã— (-.10 - .17)2 + (1/3) Ã— (.20 - .17)2 + (1/3) Ã— (.30 - .17)2 = .030233 ; taking the

square root, Ïƒ = 17.3877%.

4.

Calculating Expected Returns

Roll Ross

(1) (2) (3) (4) (5) (6)

State of Probability of Return if Product Return if Product

(2) Ã— (3) (2) Ã— (5)

Economy State of Economy State State

Occurs Occurs

Bust .30 -10% -.03 40% .12

Boom .70 50% .35 10% .07

E(R) = 32% E(R) = 19%

FUNDAMENTALS OF INVESTMENTS B-61

5.

(1) (2) (3) (4) (5)

State Probability of Return Deviation Squared Product

(2) Ã— (4)

of Economy State of from Expected Return

Economy Return Deviation

Roll

Bust .30 -.42 .1764 .05292

Boom .70 .18 .0324 .02268

Ïƒ2 = .0756

Ross

Bust .30 .21 .0441 .01323

Boom .70 -.09 .0081 .00567

Ïƒ2 = .0189

Taking square roots, the standard deviations are 27.4955% and 13.7477%.

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