Hedge Fund Investors course phần 5 ppsx

What are the limits on leverage for this pany?com-6.8 A hedge fund has positions long and short in stocks, index tures, commodity futures, and currencies.. 6.9 How can a hedge fund excee

$5 million in capital What are the limits on leverage for this pany?

com-6.8 A hedge fund has positions (long and short) in stocks, index tures, commodity futures, and currencies The fund’s prime brokerfinances all the cash positions and carries all the futures positions

fu-To what extent can SPAN margin rules reduce the margin requiredfor the positions?

6.9 How can a hedge fund exceed the limit of 50 percent margin required

on cash stock positions?

6.10 Your hedge fund has long positions in U.S Treasury securities ing $50 million and short positions worth $35 million You pay anaverage repo rate of 4.5 percent to finance your longs and receive anaverage reverse repo rate of 4 percent on your cash collateral cover-ing your short positions You post haircuts averaging 0.25 percent onthe long positions and 0.5 percent on the short positions Whatamount of capital is tied up in haircuts?

total-6.11 How levered is the Treasury part of the portfolio in question 6.10?

6.12 A hedge fund has $100 million in capital levered approximately 20:1

It maintains a position of long notes and bonds roughly equal to itsposition in short bonds The financing rate on the long positions av-erages about 0.5 percent (50 basis points) higher than the rate on theshort positions How much does the fund need to make trading (an-nually) to break even after financing costs?

6.13 You borrow 25,000 shares of XYZ common and deliver the shares tosatisfy a short sale While you are carrying the short, the stock pays adividend of $1 per share Who receives the dividend?

6.14 You borrow 25,000 shares of XYZ common and deliver the shares tosatisfy a short sale While you are carrying the short, the stock splits2:1 How does this affect the stock loan transaction?

6.15 You borrow 25,000 shares of XYZ common and deliver the shares tosatisfy a short sale While you are carrying the short, a proxy fightdevelops over control of the company How do you restore the vote

to the lender of the shares?

6.16 What is the tax treatment of a Treasury coupon or stock dividend ceived as a replacement payment from a securities borrower?

re-6.17 Why might it be reasonable to allow a hedge fund greater leveragefor risky positions held as outright futures (long or short) than forlevered positions in the underlying cash instruments?

6.18 Why would the U.S Federal Reserve Bank want to limit the amount

of leverage possible on securities loans?

6.19 A hedge fund has $10 million in marginal positions and has loans taling $8 million The fund is subject to maintenance margin of atleast 30 percent How much new cash would the hedge fund need todeposit to satisfy a margin call?

to-6.20 If the hedge fund in question 6.19 chose to liquidate positions ratherthan deposit additional margin, how much would need to be liqui-dated?

6.21 A hedge fund buys a six-month call on a common stock at a price of

$5 The strike of the option is $102 The current price of the stock is

$100 The short-term rate of interest is 5 percent The current value

of the call is $5.25 How much margin must the hedge fund put up tocarry the call?

6.22 What margin would the hedge fund need to post if it wrote (sold) anoption like that described in question 6.21? Assume the sale is anaked short sale (i.e., you have no position in the stock)

6.23 What is the standard deviation of the return on a hedge fund folio that is 100 percent invested in stock A and also carries anequal amount of stock B on leverage Stock A has a standard devia-tion of return equal to 22 percent Stock B has a standard deviation

port-of return equal to 25 percent The two stocks have a correlation port-of

75 percent

6.24 What is the expected return on the leveraged portfolio in Question6.23 if the expected return on each stock is 20 percent (unleveraged)and the risk-free rate is 5 percent?

6.25 What is the probability of loss for the portfolio in questions 6.23 and6.24?

6.26 What is the probability of losing 25 percent or more for the portfolio

in questions 6.23 and 6.24?

NOTES

1 “Hedge Funds, Leverage, and the Lessons of Long-Term Capital Management: Report of the President’s Working Group on Financial Markets, April 1999,” www.ustreas.gov/press/releases/reports/hedgfund.pdf.

2 For additional reading on options, see John Hull, Options, Futures, and Other Derivatives, 5th Edition, Pearson Education, 2002.

3 The cumulative probability is the area under the curve This value can be lated using the Excel function NORMDIST(Threshold Return, Expected Re-

calcu-turn, Standard Deviation of Recalcu-turn, TRUE or 1) {=NORMDIST(0%, 15%, 18%, 1)} returns 20.2 percent.

4 (=NORMDIST(0%, 25%, 36%, 1)} returns 24.4 percent.

5 (=NORMDIST(0%, 25%, 30.12%, 1)} returns 20.3 percent.

6 The cash generated by the short sale is also reinvested at the short-term rate of return equal to 5 percent.

7 (=NORMDIST(0%, 6%, 8.05%, 1)} returns 22.8 percent.

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:

1.104713 = e rContinuousT (7.26)

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