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Investors commit funds to a particular investment for a variety of reasons
Competitive 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 comparable to the yields on alternative assets.1 However, hedge fund returns can be readily adjusted to facilitate comparison to bond and money market 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 calculated many different ways to serve different purposes. Investors need to know how return is calculated to properly understand the investment results they receive.
The nominal return is the simplest type of return calculation and is a component of most of the other return measures. To calculate nominal return, divide the gain in value by the starting value of the investment.
Initial Investment Value
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):
Final Investment Value
(Initial Investment Value + Final Investment Value) / 2
In equation (7.4), the return is divided by the average value of the investment.
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.
Initial Investment Value Fraction of Year Incorporating equation.
Fraction of Year
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 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 investment alternatives.
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 calculated from hedge fund returns that may not make periodic payments are important because this return can be compared directly with other investment alternatives.
Semiannual Compound Return Most bonds pay periodic interest payments during the life of the investment. In the United States, most government 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 interest 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: 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.
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-oninterest 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. 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 accounting system is daily. Most interest accrual systems dont 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 interest was available for immediate reinvestment, the formula for the future value is given by equation
Notice that nearly all the benefit of interest on interest has already been realized 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:
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 possible to compare rates of different compounding frequencies without further 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 annualized 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.
It is also possible to convert the annualized monthly performance numbers to continuously compounded returns.
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
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 payments quarterly, the approximation may be accurate.
Hedge fund investors generally want to know how well a fund has performed over a period of time, so that the return is not overly influenced by short-term performance. This performance may be the basis of comparison between two hedge funds of the same strategy or between two hedge fund strategies, or comparison of a hedge fund against a benchmark return.
Calculating the Arithmetic Average Return
The simplest way to generate an average return is to add up a series of returns and divide by the number of periods in the sum. This method is called the arithmetic average return. Refer to the performance of a hypothetical hedge fund.
The arithmetic average is calculated in the way most familiar to readers. 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 arithmetic average (1.85 percent) is also called the simple average or unweighted average.
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:
Notice that the geometric average is lower than the arithmetic average. The geometric average will generally be below the arithmetic average when the returns differ from month to month. Consider an example that may be familiar. Suppose a hedge fund made 50 percent in one month and lost 50 percent in the second month. The arithmetic average return is zero because (50% - 50%)/2 = 0. The geometric return is negative. A $1 investment would grow to $1.50 at the end of the first month then decline to $.75 after the second month.2
Time -Weighted Returns
Investors often hear about time-weighted returns. Portfolio managers like to publish the time-weighted returns because the results are not influenced by whether investors made additional investments just before a good or a bad month. Instead, the time-weighted returns reflect a constant investment in the fund, changing only by the amount reinvested each period.
In fact, the time-weighted return is nearly the same as the geometric average return. For hedge fund returns averaging evenly spaced time intervals (months or quarters), they are identical.
Dollar -Weighted Returns
Investors may prefer to see the performance they have experienced with a particular hedge fund. The investor may have not made a single investment. Instead, the investor may have made additional investments over time and have greater sensitivity to recent performance. The dollarweighted return reflects the economics of a particular investor and specifically considers the impact of the timing of the investments.
Suppose an investor contributed $1 million to a hedge fund that experienced the returns in Table 7.1. After six months, the fund had experienced monthly returns of over 4 percent (by both the arithmetic and geometric means). In fact, based on Table 7.2, the $1 million investment would have grown to $1,228,200. Suppose that the investor put in an additional $1 million on June 30. At the end of the year, the combined investment was worth $2,185,466 (less than the value on June 30) because the fund lost around 0.75 percent per month in the final six months of the year.
This return is considerably below the 22.82 percent reported in Table 7.2 because more money was invested during the later, losing period than the earlier months that produced gains.
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