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948
P A R T V I I
Applied Portfolio Management
10. A hedge fund with net asset value of $62 per share currently has a high water mark of $66. Is the
value of its incentive fee more or less than it would be if the high water mark were $67?
11. Reconsider the hedge fund in the previous problem. Suppose it is January 1, the standard devia-
tion of the fund’s annual returns is 50%, and the risk-free rate is 4%. The fund has an incentive
fee of 20%, but its current high water mark is $66, and net asset value is $62.
a. What is the value of the annual incentive fee according to the Black-Scholes formula?
b. What would the annual incentive fee be worth if the fund had no high water mark and it
earned its incentive fee on its total return?
c. What would the annual incentive fee be worth if the fund had no high water mark and it
earned its incentive fee on its return in excess of the risk-free rate? (Treat the risk-free
rate as a continuously compounded value to maintain consistency with the Black-Scholes
formula.)
d. Recalculate the incentive fee value for part ( b ) now assuming that an increase in fund lever-
age increases volatility to 60%.
12. Go to the Online Learning Center at www.mhhe.com/bkm , link to Chapter 26, and find there a
spreadsheet containing monthly values of the S&P 500 index. Suppose that in each month you
had written an out-of-the-money put option on one unit of the index with an exercise price 5%
lower than the current value of the index.
a. What would have been the average value of your gross monthly payouts on the puts over the
10-year period October 1977–September 1987? The standard deviation?
b. Now extend your sample by 1 month to include October 1987, and recalculate the average
payout and standard deviation of the put-writing strategy. What do you conclude about tail
risk in naked put writing?
13. Suppose a hedge fund follows the following strategy. Each month it holds $100 million of an
S&P 500 index fund and writes out-of-the-money put options on $100 million of the index with
exercise price 5% lower than the current value of the index. Suppose the premium it receives for
writing each put is $.25 million, roughly in line with the actual value of the puts.
a. Calculate the Sharpe ratio the fund would have realized in the period October 1982–September
1987. Compare its Sharpe ratio to that of the S&P 500. Use the data from the previous prob-
lem, available at the Online Learning Center, and assume the monthly risk-free interest rate
over this period was .7%.
b. Now calculate the Sharpe ratio the fund would have realized if we extend the sample period
by 1 month to include October 1987. What do you conclude about performance evaluation
and tail risk for funds pursuing option-like strategies?
14. The following is part of the computer output from a regression of monthly returns on Water-
works stock against the S&P 500 index. A hedge fund manager believes that Waterworks is
underpriced, with an alpha of 2% over the coming month.
Beta
R-square
Standard Deviation
of Residuals
.75
.65
.06 (i.e., 6% monthly)
a. If he holds a $2 million portfolio of Waterworks stock, and wishes to hedge market exposure
for the next month using 1-month maturity S&P 500 futures contracts, how many contracts
should he enter? Should he buy or sell contracts? The S&P 500 currently is at 1,000 and the
contract multiplier is $250.
b. What is the standard deviation of the monthly return of the hedged portfolio?
c. Assuming that monthly returns are approximately normally distributed, what is the prob-
ability that this market-neutral strategy will lose money over the next month? Assume the
risk-free rate is .5% per month.
15. Return to the previous problem.
a. Suppose you hold an equally weighted portfolio of 100 stocks with the same alpha, beta,
and residual standard deviation as Waterworks. Assume the residual returns (the e terms in
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