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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:
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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
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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
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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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