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ccc_mccrary_ch06_87-106.qxd 10/6/04 1:43 PM Page 106 CHAPTER 7 Performance Measurement C ompetitive investors seek attractive returns. Beauty remains in the eye of the beholder, though. Clearly, higher returns are better than lower returns. Investors would prefer to accept less risk to achieve a given return. It is important to understand performance measurement. First the reader may be called upon to conduct a performance return. Second, the reader should be able to review critically the performance measurement calculated by others. Finally, the hedge fund returns are not directly com- parable to the yields on alternative assets. 1 However, hedge fund returns can be readily adjusted to facilitate comparison to bond and money mar- ket returns. CALCULATING RETURNS Investors commit funds to a particular investment for a variety of reasons. The return on the investment is usually very important. Yet return is calcu- lated many different ways to serve different purposes. Investors need to know how return is calculated to properly understand the investment re- sults they receive. Nominal Return The nominal return is the simplest type of return calculation and is a com- ponent of most of the other return measures. To calculate nominal return, divide the gain in value by the starting value of the investment. (7.1) Nominal Return Gain Initial Investment Value = 107 ccc_mccrary_ch07_107-126.qxd 10/7/04 1:35 PM Page 107 Restating equation (7.1) slightly: (7.2) This simplifies to: (7.3) Sometimes, this nominal return is modified slightly to acknowledge that the return shown in the numerator increases the investment base in the denominator as in equation (7.4): (7.4) In equation (7.4), the return is divided by the average value of the invest- ment. Annualized Return It is difficult to compare the nominal returns of different assets. Clearly, higher returns are better than lower returns, but it also matters how long it takes to achieve a particular return. Without some adjustment for time, it is not possible to compare returns. Generally, nominal returns are adjusted to a period equal to one year. (7.5) Incorporating equation (7.1): (7.6) This method of converting a nominal return to an annual return presumes that you can repeat an investment over and over successively, until a year Annualized Return Nominal Return Fraction of Year = Annualized Return Gain Initial Investment Value Fraction of Year =× 1 Nominal Return Final Investment Value (Initial Investment Value Final Investment Value) = + − /2 1 Nominal Return Final Investment Value Initial Investment Value =−1 Nominal Return Final Investment Value Initial Investment Value Initial Investment Value = − 108 HEDGE FUND COURSE ccc_mccrary_ch07_107-126.qxd 10/7/04 1:35 PM Page 108 has passed. The return earned in a full year is the sum of the returns earned in all the subperiods of the year. In its simplest form, returns partway through the year are not available for reinvestment during the period. This annualized return can be compared with simple interest rates on invest- ment alternatives. Compound Returns Many investments pay interest regularly during the life of the investment. Investors prefer to receive frequent partial payments of interest because this interest is then available for reinvestment. Compound returns account for this potential to earn interest on interest. Also, compound returns cal- culated from hedge fund returns that may not make periodic payments are important because this return can be compared directly with other invest- ment alternatives. Semiannual Compound Return Most bonds pay periodic interest pay- ments during the life of the investment. In the United States, most gov- ernment and corporate bonds pay half the annual income in two installments per year. Interest from first payments can be reinvested in the second period, so the gain to the investor is greater than in stated coupon rate. Consider the following specific example. A bond pays 10 percent inter- est and repays principal at the end of one year. The repayment in one year (per $100 bond) is $100 principal plus $10 ($100 × 10 percent) or a total of $110. This value is sometimes called the future value. Equation (7.7) shows the future value of an annual-pay bond: Future Value = Principal + (Principal × r) (7.7) which factors down to: Future value = Principal × (1 + r) (7.8) = $100 × (1.10) = $110 (7.9) If the bond paid half the coupon after six months, the investor could reinvest that amount for the second half of the year. The future value of the semiannual bond will slightly exceed the future value of the annual bond. Performance Measurement 109 ccc_mccrary_ch07_107-126.qxd 10/7/04 1:35 PM Page 109 Suppose for simplicity that the coupon could be reinvested in an identical bond. The future value is given by equation (7.9): (7.10) (7.11) (7.12) Using Excel: = 1.05^2*100 produces $110.25 The semiannual bond has the same future value as an annual-pay bond with a 10.25 percent coupon. This means that a 10 percent semiannual bond has an effective annual yield of 10.25 percent. Daily Compounding In the 1970s, savings institutions used this interest-on- interest effect to pay a higher effective rate than the allowable ceiling. If a bank paid 10 percent interest compounded daily, the investor would have a balance (future value) of $110.5156 at the end of one year. The formula for daily compounding is shown in equation 7.13: (7.13) (7.14) (7.15) Future Value =+       ×+       ×+       +       =+       =+       ×= 1 365 1 365 1 365 1 365 1 365 1 10 365 100 110 5156 365 365 rrrr r K % $$. Future Value Principal Principal =+       ×+       ×         =+       × =×= 1 2 1 2 1 2 1 05 100 110 25 2 2 rr r (. ) $ $ . 110 HEDGE FUND COURSE ccc_mccrary_ch07_107-126.qxd 10/7/04 1:35 PM Page 110 Using Excel: = (1 + 10%/365)^365*100 produces $110.5156 Therefore, a 10 percent interest rate paid daily is equivalent to an annual payment of 10.5156 percent. Continuous Compounding The logical limit to compounding in an account- ing system is daily. Most interest accrual systems don’t break down a year any finer than daily. However, mathematicians followed this progression from annual to semiannual to daily to the mathematical extreme. If interest could be paid every infinitesimally small fraction of a second and that in- terest was available for immediate reinvestment, the formula for the future value is given by equation (7.16): Future Value = e rT (7.16) where T is the time until repayment in years. Future Value = 2.71828 10%×1 × $100 = $100.5171 (7.17) Using Excel: = exp(10%)*100 produces $100.5171 Notice that nearly all the benefit of interest on interest has already been re- alized under daily compounding. Monthly and Quarterly Compounding Hedge fund performance is generally reported monthly or quarterly. The mathematics follows the same pattern as already described. See equations (7.18) to (7.21) for details: Quarterly: (7.18) (7.19) Future Value =+       =+       ×= 1 4 1 10 4 100 110 3813 4 4 r % $$. Performance Measurement 111 ccc_mccrary_ch07_107-126.qxd 10/7/04 1:35 PM Page 111 Monthly: (7.20) (7.21) Using Excel: = (1+ 10%/4)^4*100 produces $110.3813 = (1+ 101%/12)^12*100 produces $110.4713 Finding Equivalent Interest Rates for Different Compounding Frequencies It should be clear that a particular rate can imply different economic returns, depending on the frequency of compounding. For this reason, it is not pos- sible to compare rates of different compounding frequencies without fur- ther analysis. Fortunately, it is possible to convert a rate using one compounding frequency to the equivalent rate using any other frequency. In the previous examples, a 10 percent rate was converted to the annual equivalent. The examples that follow find the rates required to attain the same effective annual rate. For example, suppose that a hedge fund has been providing an annu- alized monthly return of 10 percent. To find a semiannual rate that is equivalent, find a rate that creates the same future value after one year. Equations (7.22) to (7.25) derive the equivalent rate relative to equal fu- ture values. Find the future value from the monthly return: (7.22) Find the semiannual rate giving the same future value: (7.23) 1 104713 1 2 2 . =+       r Semiannual Future Value =+       =1 10 12 1 104713 12 % . Future Value =+       =+       ×= 1 12 1 10 12 100 110 4713 12 12 r % $$. 112 HEDGE FUND COURSE ccc_mccrary_ch07_107-126.qxd 10/7/04 1:35 PM Page 112 (7.24) 10.210% = r Semiannual (7.25) Using Excel: =(SQRT(1.104713) – 1) × 2 produces 10.210% It is also possible to convert the annualized monthly performance numbers to continuously compounded returns. See equations (7.26) to (7.27): 1.104713 = e rContinuousT (7.26) Take the natural logarithm of each side. Recall that T = 1, so it drops out: ln(1.104713) = r Continuous = 9.9586% (7.27) Using Excel: ln(1.10473) produces 9.9586% Effect of Taxes Suppose an individual investor paid a 40 percent income tax (federal plus state tax) on the return. Suppose that the investor made a $100 investment that provided a nominal return of $30 or 30 percent. The $30 return would create a $12 tax liability, reducing the after-tax return to $18 or 18 percent. The after-tax return is approximated by equation (7.28): r AfterTax = r BeforeTax × (1 – Tax Rate) (7.28) Notice that it is also possible to calculate the after-tax return directly, by reducing the future value in equation (7.18) by the amount of the taxes paid and resolving for the return consistent with this reduced future value. Equation (7.18) is only an approximation because the timing of the tax payment may affect the true return. Certain taxes like the capital gains tax can be postponed indefinitely. Other taxes are payable several months after the end of a tax year. For the investor who makes estimated tax pay- ments quarterly, the approximation may be accurate. 1 104713 1 2 . =+ r Semiannual Performance Measurement 113 ccc_mccrary_ch07_107-126.qxd 10/7/04 1:35 PM Page 113 AVERAGING RETURNS Hedge fund investors generally want to know how well a fund has per- formed over a period of time, so that the return is not overly influenced by short-term performance. This performance may be the basis of com- parison between two hedge funds of the same strategy or between two hedge fund strategies, or comparison of a hedge fund against a bench- mark return. Calculating the Arithmetic Average Return The simplest way to generate an average return is to add up a series of re- turns and divide by the number of periods in the sum. This method is called the arithmetic average return. Refer to the performance of a hypo- thetical hedge fund in Table 7.1. The arithmetic average is calculated in the way most familiar to read- ers. First, the 12 monthly numbers are totaled (22.15 percent). Next, this total is divided by 12, the number of data points in the table. This arith- metic average (1.85 percent) is also called the simple average or un- weighted average. 114 HEDGE FUND COURSE TABLE 7.1 Monthly Hedge Fund Performance Month Return 1 1.50% 2 –3.00% 3 3.75% 4 7.50% 5 7.20% 6 9.00% 7 –5.80% 8 1.80% 9 6.90% 10 –1.80% 11 0.20% 12 –5.10% Total 22.15% Arithmetic Average 1.85% ccc_mccrary_ch07_107-126.qxd 10/7/04 1:35 PM Page 114 Calculating the Geometric Mean Return Table 7.2 extends the monthly performance from Table 7.1. The “Wealth Relative” column represents a $1 investment in the fund with reinvestment. At the end of one year, $1 grows to $1.2282. Obviously, the fund has produced an annual return of 22.82 percent. This information is sufficient to determine the geometric average monthly return. Equations (7.29) to (7.32) show how the monthly average is calculated: (7.29) To simplify, take the 12th root of both sides: (7.30) 1.73% = r GeometricAverage (Monthly) (7.31) 20.73% = 1.37% × 12 (Annual) (7.32) 1 2282 1 12 12 . GeometricAverage −= r Future Value GeometricAverage =+       1 12 12 r Performance Measurement 115 TABLE 7.2 Monthly Performance with Wealth Relatives Month Return Wealth Relative January 1.50% 1.0150 February –3.00% 0.9846 March 3.75% 1.0215 April 7.50% 1.0981 May 7.20% 1.1771 June 9.00% 1.2831 July –5.80% 1.2087 August 1.80% 1.2304 September 6.90% 1.3153 October –1.80% 1.2916 November 0.20% 1.2942 December –5.10% 1.2282 ccc_mccrary_ch07_107-126.qxd 10/7/04 1:35 PM Page 115 [...]... August September October November December 1 .50 % –3.00% 3. 75% 7 .50 % 7.20% 9.00% 5. 80% 1.80% 6.90% –1.80% 0.20% 5. 10% Wealth Relative High-Water Mark 1.0 150 0.9846 1.02 15 1.0981 1.1771 1.2831 1.2087 1.2304 1.3 153 1.2916 1.2942 1.2282 1.0 150 1.0 150 1.02 15 1.0981 1.1771 1.2831 1.2831 1.2831 1.3 153 1.3 153 1.3 153 1.3 153 Drawdown 0.00% –3.00% 0.00% 0.00% 0.00% 0.00% 5. 80% –4.10% 0.00% –1.80% –1.60% –6.62%... curve is, in fact, a map of probabilities of such outcomes.4 4 .50 0 4.000 Portfolio 1 µ = 10% σ = 10% 3 .50 0 Probability 3.000 2 .50 0 2.000 Portfolio 1 Probability of Loss = 15. 9% 1 .50 0 Portfolio 2 µ = 15% σ = 20% 1.000 0 .50 0 Portfolio 2 Probability of Loss = 22.7% 0.000 –100% – 75% 50 % – 25% 0% Return FIGURE 7.1 Normal Distributions 25% 50 % 75% 100% 119 Performance Measurement Other Statistical Models of... exempt hedge fund are only protected from fraud, and managers are charged with few duties beyond honesty and fairness Not surprisingly, hedge fund managers avoid being classified as managers of plan assets If plan assets comprise 25 percent or more of a hedge fund, the hedge fund is deemed to be plan assets and the manager and the fund are controlled by ERISA Hedge funds monitor the portions of their funds... (i.e., one hedge fund) Some managers sponsor more than one hedge fund In particular, a 130 HEDGE FUND COURSE manager may have mirror funds in the United States and offshore that carry nearly identical positions Some managers may also run separate accounts for certain investors with positions that closely resemble the hedge fund managed in parallel These relationships could require a hedge fund manager... still provide useful information about the risk of a hedge fund portfolio Hedge fund practitioners have developed several alternative measures of hedge fund portfolio risk 120 HEDGE FUND COURSE Largest Losing Month and Drawdown This measure calculates the largest cumulative loss on a hedge fund portfolio In the example in Table 7.1, the largest loss was 5. 8 percent Usually, these loss statistics are calculated... CHAPTER 8 Hedge Fund Legislation and Regulation he laws and regulations affecting a hedge fund represent one of the most complex topics addressed in this course book One law firm that sets up hedge funds and provides legal services to hedge funds created a handbook1 for its clients totaling 56 3 pages, with promises to produce at least a second volume to complete the tutorial In contrast, this course book... invests in a hedge fund as a limited partner The hedge fund admits the investor as a qualified purchaser on the basis of the investor’s net worth Later, the investor loses his hedge fund investment because of losses in the position of the hedge fund Can he sue for restitution based on the argument that due to his inexperience as an investor he was not a qualified purchaser? 8 .5 An offshore hedge fund has... are the arithmetic and geometric averages of the following monthly returns? January February March April May June July August September October November December 2. 25% –1. 75% 8. 15% –4.40% –3.30% 3 .55 % 9. 15% 1. 85% 3. 65% 2. 25% –3.60% 2. 95% 7.19 What is the standard deviation of return for the return series in question 7.18? 7.20 What is the semiannually compounded equivalent to the geometric average... Act will define a hedge fund as a commodity pool if a hedge fund trades futures or commodities Many hedge funds therefore register as commodity pools and their managers register as commodity pool operators There is no exemption used by hedge funds to escape regulation under the Commodity Exchange Act, except not trading futures and commodities (which really isn’t an exemption) Hedge Fund Legislation... additional investors into the fund? 8.6 A domestic hedge fund has nearly 499 investors What can the manager do to permit it to accept additional investors into the fund? 8.7 Under what circumstances does a tax-exempt investor need to worry that an investment in a hedge fund would require the hedge fund to pay unrelated business income tax? 8.8 What are some of the problems for hedge funds complying with the . of such outcomes. 4 118 HEDGE FUND COURSE FIGURE 7.1 Normal Distributions 4 .50 0 4.000 3 .50 0 3.000 2 .50 0 2.000 1 .50 0 1.000 0 .50 0 0.000 –100% – 75% 50 % – 25% 0% 25% 50 % 75% 100% Probability Return Portfolio. =+       ×+       ×+       +       =+       =+       ×= 1 3 65 1 3 65 1 3 65 1 3 65 1 3 65 1 10 3 65 100 110 51 56 3 65 3 65 rrrr r K % $$. Future Value Principal Principal =+       ×+       ×         =+       × =×= 1 2 1 2 1 2 1 05. average (1. 85 percent) is also called the simple average or un- weighted average. 114 HEDGE FUND COURSE TABLE 7.1 Monthly Hedge Fund Performance Month Return 1 1 .50 % 2 –3.00% 3 3. 75% 4 7 .50 % 5 7.20% 6