Hedge fund course phần 5 doc

Restating equation 7.1 slightly:7.2This simplifies to: 7.3 Sometimes, this nominal return is modified slightly to acknowledgethat the return shown in the numerator increases the investme

CHAPTER 7

Performance Measurement

Competitive investors seek attractive returns Beauty remains in the eye

of the beholder, though Clearly, higher returns are better than lowerreturns Investors would prefer to accept less risk to achieve a given return

It is important to understand performance measurement First thereader may be called upon to conduct a performance return Second, thereader should be able to review critically the performance measurementcalculated by others Finally, the hedge fund returns are not directly com-parable to the yields on alternative assets.1However, hedge fund returnscan 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 toknow 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 ponent of most of the other return measures To calculate nominal return,divide the gain in value by the starting value of the investment

com-(7.1)

Initial Investment Value

=

107

Restating equation (7.1) slightly:

(7.2)This simplifies to:

(7.3)

Sometimes, this nominal return is modified slightly to acknowledgethat the return shown in the numerator increases the investment base in thedenominator as in equation (7.4):

is not possible to compare returns Generally, nominal returns are adjusted

to a period equal to one year

(7.5)Incorporating equation (7.1):

Initial Investment Value Fraction of Year

Nominal Return Final Investment Value

(Initial Investment Value Final Investment Value)

=

Nominal Return Final Investment Value

Initial Investment Value

Nominal Return Final Investment Value Initial Investment Value

Initial Investment Value

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 partwaythrough the year are not available for reinvestment during the period Thisannualized 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 becausethis interest is then available for reinvestment Compound returns accountfor this potential to earn interest on interest Also, compound returns cal-culated from hedge fund returns that may not make periodic payments areimportant because this return can be compared directly with other invest-ment alternatives

Semiannual Compound Return Most bonds pay periodic interest ments during the life of the investment In the United States, most gov-ernment and corporate bonds pay half the annual income in twoinstallments per year Interest from first payments can be reinvested inthe second period, so the gain to the investor is greater than in statedcoupon rate

pay-Consider the following specific example A bond pays 10 percent 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

inter-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 couldreinvest that amount for the second half of the year The future value of thesemiannual bond will slightly exceed the future value of the annual bond

Suppose for simplicity that the coupon could be reinvested in an identicalbond 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 bondwith a 10.25 percent coupon This means that a 10 percent semiannualbond has an effective annual yield of 10.25 percent

Daily Compounding In the 1970s, savings institutions used this interest effect to pay a higher effective rate than the allowable ceiling If abank paid 10 percent interest compounded daily, the investor would have abalance (future value) of $110.5156 at the end of one year The formulafor daily compounding is shown in equation 7.13:

Using Excel:

= (1 + 10%/365)^365*100 produces $110.5156Therefore, a 10 percent interest rate paid daily is equivalent to an annualpayment of 10.5156 percent

Continuous Compounding The logical limit to compounding in an ing system is daily Most interest accrual systems don’t break down a yearany finer than daily However, mathematicians followed this progressionfrom annual to semiannual to daily to the mathematical extreme If interestcould be paid every infinitesimally small fraction of a second and that in-terest was available for immediate reinvestment, the formula for the futurevalue is given by equation (7.16):

where T is the time until repayment in years.

Future Value = 2.7182810%×1× $100 = $100.5171 (7.17)Using Excel:

= exp(10%)*100 produces $100.5171Notice 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 generallyreported monthly or quarterly The mathematics follows the same pattern

as already described See equations (7.18) to (7.21) for details:

In the previous examples, a 10 percent rate was converted to the annualequivalent The examples that follow find the rates required to attain thesame effective annual rate.

For example, suppose that a hedge fund has been providing an alized monthly return of 10 percent To find a semiannual rate that isequivalent, 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

annu-Find the future value from the monthly return:

Take the natural logarithm of each side Recall that T = 1, so it

drops out:

ln(1.104713) = rContinuous= 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 plusstate tax) on the return Suppose that the investor made a $100 investmentthat provided a nominal return of $30 or 30 percent The $30 returnwould create a $12 tax liability, reducing the after-tax return to $18 or 18percent The after-tax return is approximated by equation (7.28):

rAfterTax= rBeforeTax× (1 – Tax Rate) (7.28)Notice that it is also possible to calculate the after-tax return directly, byreducing the future value in equation (7.18) by the amount of the taxespaid and resolving for the return consistent with this reduced future value.Equation (7.18) is only an approximation because the timing of thetax payment may affect the true return Certain taxes like the capital gainstax can be postponed indefinitely Other taxes are payable several monthsafter 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 = +rSemiannual

com-Calculating the Arithmetic Average Return

The simplest way to generate an average return is to add up a series of turns and divide by the number of periods in the sum This method iscalled the arithmetic average return Refer to the performance of a hypo-thetical hedge fund in Table 7.1

re-The arithmetic average is calculated in the way most familiar to ers First, the 12 monthly numbers are totaled (22.15 percent) Next, thistotal 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

read-TABLE 7.1 Monthly Hedge Fund

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 withreinvestment

At the end of one year, $1 grows to $1.2282 Obviously, the fund hasproduced 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:

Using Excel:

= 1.2282^(1/12) – 1 (Monthly) produces 1.73%Notice that the geometric average is lower than the arithmetic average.The geometric average will generally be below the arithmetic average whenthe returns differ from month to month Consider an example that may befamiliar Suppose a hedge fund made 50 percent in one month and lost 50percent in the second month The arithmetic average return is zero because(50% – 50%)/2 = 0 The geometric return is negative A $1 investmentwould grow to $1.50 at the end of the first month then decline to $.75 af-ter 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 abad month Instead, the time-weighted returns reflect a constant invest-ment in the fund, changing only by the amount reinvested each period

In fact, the time-weighted return is nearly the same as the geometricaverage return For hedge fund returns averaging evenly spaced time inter-vals (months or quarters), they are identical

Dollar-Weighted Returns

Investors may prefer to see the performance they have experienced with aparticular hedge fund The investor may have not made a single invest-ment Instead, the investor may have made additional investments overtime and have greater sensitivity to recent performance The dollar-weighted return reflects the economics of a particular investor and specifi-cally considers the impact of the timing of the investments

Suppose an investor contributed $1 million to a hedge fund that perienced the returns in Table 7.1 After six months, the fund had experi-enced monthly returns of over 4 percent (by both the arithmetic andgeometric means) In fact, based on Table 7.2, the $1 million investmentwould have grown to $1,228,200 Suppose that the investor put in an ad-ditional $1 million on June 30 At the end of the year, the combined in-vestment was worth $2,185,466 (less than the value on June 30) becausethe fund lost around 0.75 percent per month in the final six months ofthe year

ex-The dollar-weighted return is the rate that makes equation (7.33) true:

(7.33)

Equation (7.33) is true at a return of 11.82 percent compoundedmonthly or 12.49 percent compounded annually.3This return is consid-erably below the 22.82 percent reported in Table 7.2 because moremoney was invested during the later, losing period than the earliermonths that produced gains

MEASURES OF INVESTMENT RISK

Hedge funds measure risk in a variety of ways In this chapter, risk sures will be derived from the reported performance values Other mea-sures of risk can be calculated from the characteristics of the positions Foradditional reading, review material discussing value at risk (VaR), Risk-Metrics, CreditMetrics, bond duration, and option “Greeks.”

mea-Standard Deviation as a Measure of Performance Risk

The most common measure of portfolio risk used by both practitionersand academics is the standard deviation of return The measure appliesthe textbook definition of this summary statistic; see equations (7.34)and (7.35):

N

i i

N

2 1

Using Excel:

= sstdev(a1 : a12) or = stdeva(a1 : a12) produces sample

standard deviationThe standard deviation of returns from Table 7.1 or Table 7.2 was 5.11percent Typically, this monthly standard deviation would be annualized

To annualize, multiply this monthly standard deviation by the square root

of the number of time periods per year: 5.11% × sqrt(12) = 17.70%.The standard deviation is a powerful, concise measure of risk if the returns of the portfolio are normally distributed and if the standard devia-tion is constant or predictable Refer to Figure 7.1, which shows the distri-bution of returns for two portfolios One portfolio has an expected return

of 10 percent and a standard deviation of return of 10 percent The secondportfolio has a higher expected return of 15 percent and a standard devia-tion of return of 20 percent Assuming the returns of the two portfolios areindeed normally distributed, the standard deviation contains detailed in-formation about the chance of virtually any outcome For example, thechance of losing money on the first portfolio is 15.9 percent in the nextyear, while the second portfolio should lose money 22.7 percent of thetime This kind of information is available because the familiar bell curve

is, in fact, a map of probabilities of such outcomes.4

FIGURE 7.1 Normal Distributions

Portfolio 2 Probability of Loss = 22.7%

Portfolio 2

µ = 15%

σ = 20%

Other Statistical Models of Risk

Standard deviation has historical precedence as a risk measurement, butmost investors worry less about very good performance than very bad per-formance Realizing that investors are more concerned about outcomes onthe left sections of the bell curves in Figure 7.1, researchers have con-structed a number of measures to focus on the bad performance

The measure called “downside deviation” is essentially the formula forstandard deviation with the favorable deviations omitted This concept isstated more formally in equation (7.36):

(7.36)

Note that a variable r* substitutes for the mean return in equation (7.36) This threshold return is the break point When r i is below r*, the deviation is included; and when r i is above r*, the deviation is ignored When r* equals zero, the downside deviation includes only the losing re-

turns for each month or quarter The data from Tables 7.1 and 7.2 have

a downside deviation of 2.56 percent (not annualized) using a thresholdreturn of 0 percent

A related measure called downside semivariance corresponds to thevariance of returns Just as the standard deviation is the square root ofvariance, downside deviation is the square root of downside semivariance.See equation (7.37):

(7.37)

The data in Tables 7.1 and 7.2 have a downside semivariance of 0.07 cent (not annualized) using a threshold return of 0 percent

per-Other Measures of Portfolio Risk

Not all distributions are normal, but the statistical methods just describedmay still provide useful information about the risk of a hedge fund portfo-lio Hedge fund practitioners have developed several alternative measures

of hedge fund portfolio risk

N

01

2 1

N

012 1

Largest Losing Month and Drawdown This measure calculates the largestcumulative loss on a hedge fund portfolio In the example in Table 7.1, thelargest loss was 5.8 percent Usually, these loss statistics are calculatedfrom inception, so if additional performance data was available, it wouldalso be analyzed to find a larger loss.

In addition, if an additional loss occurs before an earlier loss is madeback, the cumulative effect of the losses is measured as a drawdown Table7.3 extends the performance calculations from Table 7.1 and Table 7.2 anddemonstrates one way that the drawdown can be calculated

The monthly returns and the wealth relative columns in Table 7.3 arethe same as they appeared in the previous tables The new column labeled

“High-Water Mark” preserves the previous wealth relative after a loss (Ifthe net asset value of the hedge fund is known, that value may be substi-tuted for the wealth relative values because these wealth relatives are justnormalized net asset values.) The column labeled “Drawdown” measuresthe cumulative loss from the previous high-water mark

Months to Earn Back Loss Another measure of risk is to track the number

of months before a loss is made up by later gains In our sample data, theloss in February lasted only one month because the gain in March restoredthe value of the fund to levels greater than the value at the beginning ofFebruary The loss in July lasted two months The loss that began in Octo-ber has stretched to three months and continues to be a loss period.Investors look at a variety of statistics related to these measures oftime How long did it take to recover the largest monthly loss? What was

TABLE 7.3 Drawdown Calculations

Month Return Wealth Relative High-Water Mark Drawdown