What Practitioners Need To Know

. . . About Retum and Risk

14

Mark KritzmanWindham CapitalManagement

At first glance, return and riskmay seem to be straightforwardconcepts. Yet closer inspectionreveals nuances that can have im-portant consequences for deter-mining the appropriate methodfor evaluating financial results.This column reviews variousmeasures of return and risk withan emphasis on their suitabilityfor alternative uses.

RetumPerhaps the most straightforwardrate of return is the holding-period return. It equals the in-come generated by an investmentplus the investment's change inprice during the period the in-vestment is held, all divided bythe beginning price. For example,if we purchased a share of com-mon stock for $50.00, received a$2.00 dividend, and sold the stockfor S55.00, we would haveachieved a holding-period returnequal to 14.00%. In general, wecan use Equation (1) to computeholding-period returns.

Eq. 1

HPR = {I + E - B)/B

whereHPR = holding-period retum,

I = income,E = ending price andB = beginning price.

Holding-period returns are alsoreferred to as periodic retums.

Dollar-Weighted versusTime-Weighted Rates of

RetumNow let us consider rates of re-turn over multiple holding peri-

ods. Suppose that a mutual fundgenerated the following annualholding-period returns from 1988through 1992:

19881989199019911992

-5.00%-15.20%

3.10%30.75%17.65%

Suppose further that we had in-vested $75,000 in this fund bymaking contributions at the be-ginning of each year according tothe following schedule:

19881989199019911992

S5,000$10,000$15,000$20,000$25,000

By the end of 1992, our invest-ment would have grown in valueto $103,804.56. By discountingthe ending value of our invest-ment and the interim cash flowsback to our initial contribution,we can determine the invest-ment's dollar-weighted rate of re-turn, which is also referred to asthe internal rate of return:

5000= -10000 15000

(1 -H r) (1-H

20000 25000 1038057 "r P,

(1 -Hr)^ (1 +

DWR = 14.25%

We enter the interim contribu-tions as negative values, becausethey are analogous to negativedividend payments. Although wecannot solve directly for the dol-lar-weighted rate of return, mostfinancial calculators and spread-sheet software have iterative algo-

rithms that quickly converge to asolution. In our example, the so-lution equals 14.25%.

The dollar-weigh ted rate of re-tum measures the annual rate atwhich our cumulative contribu-tions grow over the measurementperiod. However, it is not a reli-able measure of the performanceof the mutual fund in which weinvested, because it depends onthe timing of the cash flows. Sup-pose, for example, we reversedthe order of the contributions.Given this sequence of contribu-tions, our investment would havegrown to a higher value—$103,893.76. The dollar-weightedrate of return, however, wouldhave been only 9.12%:

25000= -

10000

20000 15000

(1 + r) (1-H r)^

5000 103894

DWR = 9.12%

In order to measure the underly-ing performance of the mutualfund, we can calculate its time-weighted rate of retum. This mea-sure does not depend on thetiming of cash flows.

We compute the time-weightedrate of return by first adding oneto each year's holding-period re-turn to determine the retum'swealth relative. Then we multiplythe wealth relatives together,raise the product to the power 1divided by the number of years inthe measurement period, andsubtract 1. Equation (2) showsthis calculation:

Eq. 2

n= 1

+

1/n

- 1

where

TWK

n =

time-weighted rate ofreturn,holding-period returnfor year i andnumber of years inmeasurement period,

If we substitute the mutual fund'sholding-period returns into Equa-tion (2), we discover that thefund's time-weighted rate of re-turn equals 5.02%.

The time-weighted rate of returnis also called ihegeometric retumor the compound annual return.Although the geometric retumand the compound annual retumare often used interchangeably,technically the geometric returnpertains to a population whereasthe compound annual return per-tains to a sample. I use the termgeometric return to refer to both.It is the rate of return that, whencompounded annually, deter-mines the ending value of ourinitial investment assuming thereare no interim cash flows. Forexample, suppose we invest$10,000 in a strategy that pro-duces a holding-period rate ofretum of 50% in the first year and—50% in the second year, At theend of the second year, we willend up with $7500. The geomet-ric return over the rw^o-year mea-surement period equiils —13.40%:

- 1 = - 0.1340

If we multiply $10,000 times (1 -0.1340) and then multiply thisresult again by (1 — 0.1340), wearrive at the ending value of thisinvestment—$7500,00.

In order to manipulate geometricreturns, we must first conventhem to wealth relatives and raise

the v/ealth relatives to a powerequal to the number of years inthe measurement period. Thenwe multiply or divide the cumu-lative wealth relatives, annualizethe result, and subtract one inorder to conven the result backinto a geometric return.

Suppose, for example, that fiveyears ago we invested $100,000 ina fund that we thought wouldearn a geometric return of 8.00%over a 20-year horizon so that, atthe end of the horizon, we wouldreceive $466,095.71. During thepast five years, however, thefund's geometric retum was only6.50%. What must its geometricreturn be for the next 15 years ifwe are to reach our original goalof $466,095.71? We start by raising1.08 to the 20th power, whichequals 4.6609571. This valueequals the cumulative wealth rel-ative of the anticipated geometricreturn. We then raise the wealthrelative of the geometric retumrealized thus far to the 5th power,which equals 1.3700867. We thendivide 4.6609571 by 1.3700867,raise this value to the power 1over 15, and subtract 1, to arriveat 8.50%.

Alternatively, we can convertwealth relatives to continuous re-turns by taking their natural loga-rithms and manipulating theselogarithms. For example, the nat-ural logarithm of 1.08 equals0.076961, and the natural loga-rithm of 1,065 equals 0.062975.We multiply 0.062975 times 5 andsubtract it from 0.076961 times20, which equals 1.2243468. Thenwe divide this value by 15 and usethe base of the natural logarithm2.718281 to reconvert it to anannual wealth relative, whichequals 1,085.^

Geometric Retum versusArithmetic Retum

It is easy to see why the geometricreturn is a better description ofpast performance than the arith-metic average. In the example inwhich we invested $10,000 at areturn of 50% followed by a re-turn of -50%, the arithmetic av-

Table

19831984198519861987

I Annualturns

22.63%5.86%

31.61%18,96%5,23%

S&P

19881989199019911992

500 Re-

16.58%31,63%-3.14%31,56%

7,33%

erage overstates the retum onour investment. It did not grow ata constant rate of 0%, but de-clined by 13.40% compoundedannually for two years. The arith-metic average will exceed thegeometric average except whenall the holding-period retums arethe same; the two retum mea-sures will be the same in thatcase. Furthermore, the differencebetween the two averages willincrease as the variability of theholding-period returns increases.

If we accept the past as prologue,which average should we use toestimate a future year's expectedreturn? The best estimate of afuture year's return based on arandom distribution of the prioryears' retums is the arithmeticaverage. Statistically, it is our bestguess for the holding-period re-tum in a given year. If we wish toestimate the ending value of aninvestment over a multiyear hori-zon conditioned on past experi-ence, however, we should use thegeometric return.

Suppose we plan to invest$100,000 in an S&P 500 indexfund, and we wish to estimate themost likely value of our invest-ment five years from now. Weassume there are no transactioncosts or fees, and we base ourestimates on yearly results endingDecember 31, 1992, which areshown in Table 1. The arithmeticaverage equals 16.83%, while thegeometric average equals 16.20%.Our best estimate for next year'sretum, or any single year's returnfor that matter, equals 16.8396,because there is a 1-in-lO chanceof experiencing each of the ob-served returns. However, the bestestimate for the terminal value of

16

our fiind is based on the geomet-ric average. It equals $211,866.94,which we derive by raising 1.1620to the 5th power and multiplyingthis value by $100,000.

Here is what we mean by "bestestimate." Consider an experi-ment in which we randomlychoose five retums from a sampleof 1000 returns and calculate theterminal value of $1 invested inthese five retums. We repeat thisexperiment 200 times. The bestestimate is the value that corre-sponds to the typical or mostlikely terminal wealth of theseexperiments. It equals 1 plus thegeometric average of the 1000returns raised to the 5th power.

RiskAlthough the time-weighted rateof return measures the constantannual rate of growth that deter-mines terminal wealth, it is none-theless limited as a measure ofperformance because it fails toaccount for risk. We can adjustretums for risk in several ways.One approach is to compute aportfolio's retum in excess of theriskless return and to divide thisexcess return by the portfolio'sstandard deviation. This risk-adjusted return, called the Sharpemeasure, is given by Equation(3).'

Eq. 3

C *—

where

S = the Sharpe measure,Rp = portfolio return,Rp = riskless retum and(Tp = the standard deviation of

portfolio returns.

Because it adjusts return based ontotal portfolio risk, the implicitassumption of the Sharpe mea-sure is that the portfolio will notbe combined with other riskyportfolios. Thus the Sharpe mea-

sure is relevant for performanceevaluation when we wish to eval-uate several mutually exclusiveponfolios.

The Capital Asset Pricing Model(CAPM) assumes that risk consistsof a systematic component and aspecific component. Risk that isspecific to individual securitiescan be diversified away, hence aninvestor should not expect com-pensation for bearing this type ofrisk. Therefore, when a portfoliois evaluated in combination withother portfolios, its excess returnshould be adjusted by its system-atic risk rather than its total risk.-'

The Treynor measure adjusts ex-cess retum for systematic risk.'* Itis computed by dividing a portfo-lio's excess retum, not by its stan-dard deviation, but by its beta, asshown in Equation (4).

Eq. 4

T =(Rp -

where

T = the Treynor measure,Rp = portfolio return,Rp = riskless retum andy3p = portfolio beta.

We can estimate beta by regress-ing a portfolio s excess returns onan appropriate benchmark's ex-cess returns. Beta is the coeffi-cient from such a regression. TheTreynor measure is a valid perfor-mance criterion when we wish toevaluate a portfolio in combina-tion with the benchmark portfo-lio and other actively managedportfolios.

The intercept from a regressionof the portfolio's excess returnson the benchmark's excess re-turns is called alpha. Alpha mea-sures the value-added ofthe port-folio, given its level of systematicrisk. Alpha is referred to as theJensen measure, and is given byEquation (5)-^

Eq. 5

a = (Rp - RF) - ^P(RB - RF).

where

a = the Jensen measure (al-pha),

Rp = ponfolio retum,Rp = riskless return,j8p = portfolio beta andRij = benchmark return.

The Jensen measure is also suit-able for evaluating a portfolio'sperformance in combination withother portfolios, because it isbased on systematic risk ratherthan total risk.

If we wish to determine whetheror not an observed alpha is due toskill or chance, we can computean appraisal ratio by dividing al-pha by the standard error of theregression:

Eq. 6

a

where

A = the appraisal ratio,a = alpha and

cTg = the Standard error of theregression (nonsystem-atic risk).

The appraisal ratio compares al-pha, the average nonsystematicdeviation from the benchmark,with the nonsystematic risk in-curred to generate this perfor-mance. In order to estimate thelikelihood that an observed alphais not due to chance, we can testthe null hypothesis that the meanalpha does not differ significantlyfrom 0%. If we reject the nullhypothesis, the alpha is not due tochance.

Suppose, for example, that a port-folio's alpha equals 3% and thatits standard error equals 4%, sothat the appraisal ratio equals1.33. If we look up this number in

a t distribution table, we discoverthat, given the amount of nonsys-tematic risk, there is a 10%chance of observing an alpha ofthis magnitude by random pro-cess. Hence, we would fail toreject the null hypothesis that al-pha does not differ significantlyfrom 0%.

Downside RiskIn the previous example, webased the probability estimate onthe assumption that alpha is nor-mally distributed, This assump-tion is reasonable for evaluatingreturns over short measurementperiods. Over multiple-year mea-surement periods, however, it isthe logarithms of the wealth rela-tives that are normally distrib-uted. The geometric returnsthemselves are lognormally dis-tributed, which means that theyare positively skewed. Therefore,in order to estimate the likeli-hood of experiencing particularoutcomes over multiple-year ho-rizons, we should calculate thenormal deviate based on themean and standard deviation ofthe logarithms of the wealth rela-tives.

Some investment strategies pro-duce distributions that areskewed differently from a lognor-mal distribution. Dynamic tradingstrategies such as portfolio insur-ance or strategies that involve theuse of options typically generateskewed distributions.

A distribution that is positively(right) skewed has a long tailabove the mean. Although mostof the outcomes are below themean, they are of smaller magni-tude than the fewer outcomesthat are above the mean. A distri-bution that is negatively (left)skewed has a long tail below themean. It has more outcomesabove the mean, but they aresmaller in magnitude than thosebelow the mean.

Skewness is calculated as the sumof the probability-weightedcubed deviations around themean. Risk-averse investors pre-fer positive skewness, because

there is less chance of large neg-ative deviations.

When we are evaluating strategiesthat have skewed distributionsother than lognormal, we cannotrely on only return and standarddeviation to estimate the likeli-hood of achieving a particularresult. Nor can we cx^mpare port-folios or strategies that have dif-ferent degrees of skewness usingonly these two characteristics. In-stead, we have to specify a targetreturn and base our evaluation onthe dispersion of returns beloivthis target, rather than the disper-sion of returns around the mean.*^An alternative method for dealingwith skewness is to define utilityas a function of mean, varianceand skewness and then to evalu-ate portfolios and strategies basedon their expected utilities.

I have attempted to shed somelight on the subtleties that distin-guish various measures of returnand risk. When we rely on thesesummary statistics to evaluate pastresults or to predict future conse-quences, it is important that weunderstand their precise mean-ing/

Footnotes7. For a review of logarithms and con-

tinuous returns, see M. Kriizman,"What Practitioners Need to KnowAbout Lognormaiityi," Financial Ana-lysts Journal, July/August 1992.

2. See Vt̂ Sharpe, "Mutual Fund Perfor-mance. " Journal of Busine.S5, Janu-ar^' 1966.

3- For a discussion of the Capital AssetPricing Model, see M. Kritzman,"What Practitioners Need to KnowAbout the Nobel Prize," FinancialAnalysts Journal, January/February1991.

4. See]. Treynor, "How to Rate Man-agement of Investment Funds," Har-vard Business Review, January-Feb-ruary 1965

5. See M.Jensen, "The Performance ofMutual Funds in the Period 1945-1964." Joumal of Finance, May1968.

6. See W. V. Harlow, "Asset Allocationin a Downside-Risk Framework,"Financial Analysts Journal September/October 1991

7. I thank Robert Ferguson for his help-ful comments.

footnotes concluded from page60.

of the S&P 500 and the MerrillLynch High Yield Master Index.

6. See Altman. Eherhart and Zekavat,"Priority Provisions." op. ciL

7. The arithmetic unweighted indexdid considerably better in 1991.with a retum of over 100%. Thisreflected some exceptionally highreturns on a number of small is-sues.

8. The Merrill Lynch Master Index ofHigh Yield Debt increased by 34.6%in 1991—a/so reversing relativelypoor years in 1989 and 1990-

9. This rate does not include priordefaults in the population base ofhigh-yield bonds' it only includesthose bonds that could have de-faulted during the year.

10. Ward and Griepentrog, "Risk andRetum in Defaulted Bonds," op, ciL

11. E. Altman. Corporate Financial Dis-tress and Bankruptcy, 2nd ed. (NewYork: John Wiley & Sons. 1993)-

12. For an analysis of this market, seeAltman, The Market for DistressedSecurities and Bank Loans, op, ciLand Carlson and Fabozzi, TheTrading and Securitization of Se-nior Bank Loans, op, cit-

13- I appreciate the support of the Foot-hill Croup (Los Angeles) and MerrillLynch & Co., as well as the assis-tance of Suzanne Crymes and Vel-lore Kishore of the NYU SalomonCenter.

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