Chapter 26

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1. Award: 10.00 points Problems? Adjust credit for all students. A hedge fund with $1 billion of assets charges a management fee of 2% and an incentive fee of 20% of returns over a money market rate, which currently is 5%. Required: Calculate total fees, both in dollars and as a percent of assets under management, for the portfolio returns in the table below. Note: Enter your answers for Total Fee ($ millions) in millions not dollars. $ $ $ $ Portfolio Rate of Return (%) Total Fee ($ million) Total Fee (%) a. -5 20 2 b. 0 20 2 c. 5 20 2 d. 10 30 3 Explanation: Management fee = 0.02 × $1 billion = $20 million For a single period (assuming no highwater mark), the incentive fee is $0 or $10,000,000, depending on the returns (above or below the hurdle rate): Incentive Fee = r Incentive × [ V 0 × (1 + r Assets ) − V 0 × (1 + r Hurdle )] = [ V 0 × ( r Assets r Hurdle )] For r Assets = 0.10(part d.) Incentive Fee = 0.20 × [$1,000,000,000 × (0.10 − 0.05)] = $10,000,000 Portfolio Rate of Return (%) Management Fee ($ million) Incentive Fee ($ million) Total Fee ($ million) Total Fee (%) a. −5 20 0 20 2 b. 0 20 0 20 2 c. 5 20 0 20 2 d. 10 20 10 30 3 Worksheet Difficulty: 2 Intermediate Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 26: Alternative Assets > Chapter 26 Problems - Algorithmic & Static References
2. Award: 10.00 points Problems? Adjust credit for all students. A hedge fund with net asset value of $62 per share currently has a high-water mark of $66. Required: Is the value of its incentive fee more or less than it would be if the high-water mark were $67? Incentive fee Less Explanation: The incentive fee is typically equal to 20 percent of the hedge fund’s profits beyond a particular benchmark rate of return. However, if a fund has experienced losses in the past, then the fund may not be able to charge the incentive fee unless the fund exceeds its previous high-water mark. The incentive fee is less valuable if the high-water mark is $67, rather than $66. With a high-water mark of $67, the net asset value of the fund must reach $67 before the hedge fund can assess the incentive fee. The high-water mark for a hedge fund is equivalent to the exercise price for a call option on an asset with a current market value equal to the net asset value of the fund. Worksheet Difficulty: 2 Intermediate Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 26: Alternative Assets > Chapter 26 Problems - Algorithmic & Static References
3. Award: 10.00 points Problems? Adjust credit for all students. A hedge fund with net asset value of $62 per share currently has a high-water mark of $66. Suppose it is January 1, the standard deviation of the fund’s annual returns is 50%, and the risk-free rate is 4%. The fund has an incentive fee of 20% of annual returns, but its current high-water mark is $66, and net asset value is $62. Required: a. What is the value of the annual incentive fee according to the Black-Scholes formula? (Treat the risk-free rate as a continuously compounded value to maintain consistency with the Black-Scholes formula.) Note: Do not round intermediate calculations. Round your answer to 2 decimal places. 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? Note: Do not round intermediate calculations. Round your answer to 2 decimal places. 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? Note: Do not round intermediate calculations. Round your answer to 2 decimal places. d. Recalculate the incentive fee value for part ( b ) if an increase in fund leverage increases volatility to 60%. Note: Do not round intermediate calculations. Round your answer to 2 decimal places. $ $ $ $ a. Annual incentive fee 2.34 per share b. Annual incentive fee 2.65 per share c. Annual incentive fee 2.45 per share d. Annual incentive fee 3.12 per share Explanation: a. First, compute the Black Scholes value of a call option with the following parameters: S 0 = $62 X = $66 r = 0.04 σ = 0.50 T = 1 year = 0.2050 = 0.2950 N ( d 1 ) = N (0.2050) = 0.5812 N ( d 2 ) = N (−0.2950) = 0.3840 C 0 = S 0 N ( d 1 ) − Xe rT N ( d 2 ) = $62 × 0.5812 − $66 × e -0.04 × 1 × 0.3840 = $36.0344 − $24.3489 = $11.6853 $11.69 The value of the incentive fee is 0.20 × $11.6853 $2.34 b. Here we use the same parameters used in the Black-Scholes model in part (a) with the exception that X = $62. = 0.33 = −0.17 N ( d 1 ) = N (0.33) = 0.6293 N ( d 2 ) = N (−0.17) = 0.4325 C 0 = S 0 N ( d 1 ) − Xe rT N ( d 2 ) = $62 × 0.6293 − $62 × e −0.04 × 1 × 0.4325 = $13.253 ≈ $13.25 The value of the annual incentive fee is 0.20 × C 0 = 0.20 × $13.25 ≈ $2.65 c. Here we use the same parameters used in the Black-Scholes model in part (a) with the exception that: X = S 0 × e 0.04 = 62 × e 0.04 = 64.5303 = 0.25 = −0.25 N ( d 1 ) = N (0.25) = 0.5987 N ( d 2 ) = N (−0.25) = 0.4013
C 0 = S 0 N ( d 1 ) − Xe rT N ( d 2 ) = $62 × 0.5987 − $64.5303 × e −0.04 × 1 × 0.4013 ≈ $12.24 The value of the annual incentive fee is 0.20 × C 0 = 0.20 × $12.24 ≈ $2.45 d. Here we use the same parameters used in the Black-Scholes model in part (a) with the exception that X = 62 and σ = 0.60 = 0.3667 = −0.2333 N ( d 1 ) = N (0.3667) = 0.6431 N ( d 2 ) = N (−0.2333) = 0.4078 C 0 = S 0 N ( d 1 ) − Xe rT N ( d 2 ) = $62 × 0.6431 − $62 × e −0.04 × 1 × 0.4078 = $15.58 The value of the annual incentive fee is 0.20 × C 0 = 0.20 × $15.58 = $3.12 Worksheet Difficulty: 2 Intermediate Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 26: Alternative Assets > Chapter 26 Problems - Algorithmic & Static References
4. Award: 10.00 points Problems? Adjust credit for all students. A market-neutral fund ($150,000,000) strives for very low market risk (β = 0.15). It believes it can generate α = 0.04 per quarter. The β of the underlying portfolio is 1.35. The risk-free rate is 0.5% per quarter and the S&P 500 is currently priced at 4,000 (E-mini S&P 500 multiplier = $50). Required: If the S&P 500 is 3,800 at the end of quarter, what is the expected return on the portfolio? Note: Do not round intermediate calculations. Round your answer to 2 decimal places. Expected return 3.68 % Explanation: First compute the hedge ratio necessary to keep β low (but not zero). Hedge Ratio = (Portfolio Value ÷ (Index Price × Contract Multiplier)) × ( β Portfolio β Target ) = ($150,000,000 ÷ (4,000 × $50)) × (1.35 − 0.15) = 900 Contracts (Short) Next, calculate the portfolio value, the value of the futures proceeds, and the market return: Value Portfolio = Value 0 × (1 + r Portfolio ) = Value 0 × [1 + r f + β Portfolio × ( r m r f ) + α Portfolio + e ] = $150,000,000 × [1 + 0.005 + 1.35 × ( r m − 0.005) + 0.04 + e ] = $155,737,500 + $202,500,000 × r m + $150,000,000 × e Proceeds Futures = Contracts × Multiplier × ( F 0 F 1 ) = 900 × $50 × [ S 0 (1 + r f ) − F 1 ] = 900 × $50 × [4000 × 1.005 − 3,800] = $9,900,000 r m = (3,800 − 4,000) ÷ 4,000 = −0.05, or −5.00% Combining these values generates the hedged proceeds and the expected rate of return: Value Hedged = Value Portfolio + Proceeds Futures = $155,737,500 + $202,500,000 × (−0.05) + $150,000,000 e + $9,900,000 = $155,512,500 + $150,000,000 × e E( r Hedged ) = ( E (Value Hedged ) − Value 0 ) ÷ Value 0 = ($155,512,500 − $150,000,000) ÷ $150,000,000 = 0.0368, or 3.68% The expected return on the hedge portfolio is 3.68%. Note: Here is realized returns if the investor did not hedge: E( r Unhedged ) = ([1 + r f + α + β × ( r m r f )] × Value 0 − Value 0 ) ÷ Value 0 = ($145,612,500 − $150,000,000) ÷ $150,000,000 = −2.93% Worksheet Difficulty: 2 Intermediate Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 26: Alternative Assets > Chapter 26 Problems - Algorithmic & Static References
5. Award: 10.00 points Problems? Adjust credit for all students. The following is part of the computer output from a regression of monthly returns on Waterworks stock against the S&P 500 index. A hedge fund believes that Waterworks is underpriced, with an alpha of 2% over the coming month. Beta R -square Standard Deviation of Residuals 0.75 0.65 0.06 (i.e., 6% monthly) Required: a. If the fund holds a $4 million position in 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 it enter? Should it buy or sell contracts? The S&P 500 currently is at 4,000 and the contract multiplier is $50. 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 probability that this market-neutral strategy will lose money over the next month? Assume the risk-free rate is 0.5% per month. d. 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 on each of these stocks are independent of each other. What is the residual standard deviation of the portfolio? e. Calculate the probability of a loss on a market-neutral strategy involving equally weighted, market-hedged positions in the 100 stocks over the next month. Assume the risk-free rate is 0.5% per month. Required A Required B Complete this question by entering your answers in the tabs below. If the fund holds a $4 million position in 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 it enter? Should it buy or sell contracts? The S&P 500 currently is at 4,000 and the contract multiplier is $50. Required A Required B Required C Required D Required E Number of contracts 15 Should it buy or sell contracts? Sell Explanation: a. Since the hedge fund manager has a long position in the Waterworks stock, he should sell 15 contracts, computed as follows: ($4,000,000 × 0.75) ÷ ($50 × 4,000) = 15 contracts b. The standard deviation of the monthly return of the hedged portfolio is equal to the standard deviation of the residuals, which is 6 percent. The standard deviation of the residuals for the stock is the volatility that cannot be hedged away. For a market-neutral (zero-beta) position, this is also the total standard deviation. c. The expected rate of return of the market-neutral position is equal to the risk-free rate plus the alpha: 0.5% + 2.0% = 2.5% We assume that monthly returns are approximately normally distributed. The z -value for a rate of return of zero is −2.5% ÷ 6.0% = −0.4167, or 0.4167 standard deviations below the mean. Therefore, the probability of a negative return is N (−0.4167) = 0.3385 or 33.85% d. The residual standard deviation of the portfolio is smaller than each stock’s standard deviation by a factor of = 10 or, equivalently, the residual variance of the portfolio is smaller by a factor of 100. So, instead of a residual standard deviation of 6 percent, residual standard deviation is now 0.6 percent. e. The expected return of the market-neutral position is still equal to the risk-free rate plus the alpha: 0.5% + 2.0% = 2.5% Now the z-value for a rate of return of zero is: −2.5% ÷ 0.6% = −4.1667 Therefore, the probability of a negative return is N (−4.1667) = 0.0000155 or 0.00155% A negative return is very unlikely. This is because both sources of risk have been eliminated: market risk has been hedged using the futures, and idiosyncratic risk has been dramatically reduced through diversification. Worksheet Difficulty: 2 Intermediate Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 26: Alternative Assets > Chapter 26 Problems - Algorithmic & Static References
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