From: Subject: The Sharpe Ratio Date: Sat, 14 Mar 2009 08:16:13 -0500 MIME-Version: 1.0 Content-Type: multipart/related; type="text/html"; boundary="----=_NextPart_000_004F_01C9A47D.238AB550" X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.5579 This is a multi-part message in MIME format. ------=_NextPart_000_004F_01C9A47D.238AB550 Content-Type: text/html; charset="iso-8859-1" Content-Transfer-Encoding: quoted-printable Content-Location: http://www.stanford.edu/~wfsharpe/art/sr/sr.htm The Sharpe Ratio

The Sharpe Ratio

William F. Sharpe
Stanford University

Reprinted fromThe Journal of Portfolio = Management,=20 Fall 1994

 This copyrighted material has been reprinted with = permission=20 from The Journal of Portfolio=20 Management.
Copyright =A9 Institutional = Investor, Inc.,=20 488 Madison Avenue, New York, N.Y. 10022,
a Capital Cities/ABC, Inc. = Company.=20 Phone (212) 224-3599.

 

. Over 25 years ago, in Sharpe = [1966], I introduced a measure for the performance of mutual funds = and=20 proposed the term reward-to-variability ratio to describe it = (the=20 measure is also described in Sharpe = [1975] ). While the measure has gained considerable popularity, the = name has=20 not. Other authors have termed the original version the Sharpe Index (Radclif= f=20 [1990, p. 286] and Haugen [1993,=20 p. 315]), the Sharpe Measure (Bodie, = Kane and=20 Marcus [1993, p. 804], Elton and = Gruber [1991, p. 652], and Reilly [1989,=20 p.803]), or the Sharpe Ratio (Morn= ingstar=20 [1993, p. 24]). Generalized versions have also appeared under various = names=20 (see. for example, BARRA = [1992, p.=20 21] and Capaul,=20 Rowley and Sharpe [1993, p. 33]).

Bowing to increasingly common usage, this article refers to both the = original=20 measure and more generalized versions as the Sharpe Ratio. My goal here = is to go=20 well beyond the discussion of the original measure in Sharpe = [1966] and Sharpe = [1975], providing more generality and covering a broader range of=20 applications.

THE RATIO

Most performance measures are computed using historic data = but=20 justified on the basis of predicted relationships. Practical=20 implementations use ex post results while theoretical discussions focus = on ex=20 ante values. Implicitly or explicitly, it is assumed that historic = results have=20 at least some predictive ability.

For some applications, it suffices for future values of a measure to = be=20 related monotonically to past values -- that is, if fund X had a higher = historic=20 measure than fund Y, it is assumed that it will have a higher future = measure.=20 For other applications the relationship must be proportional - - that = is, it is=20 assumed that the future measure will equal some constant (typically less = than=20 1.0) times the historic measure.

To avoid ambiguity, we define here both ex ante and ex post versions = of the=20 Sharpe Ratio, beginning with the former. With the exception of this = section,=20 however, we focus on the use of the ratio for making decisions, and = hence are=20 concerned with the ex ante version. The important issues associated with = the=20 relationships (if any) between historic Sharpe Ratios and unbiased = forecasts of=20 the ratio are left for other expositions.

Throughout, we build on Markowitz' mean-variance paradigm, which = assumes that=20 the mean and standard deviation of the distribution of one-period return = are=20 sufficient statistics for evaluating the prospects of an investment = portfolio.=20 Clearly, comparisons based on the first two moments of a distribution do = not=20 take into account possible differences among portfolios in other moments = or in=20 distributions of outcomes across states of nature that may be associated = with=20 different levels of investor utility.

When such considerations are especially important, return mean and = variance=20 may not suffice, requiring the use of additional or substitute measures. = Such=20 situations are, however, beyond the scope of this article. Our goal is = simply to=20 examine the situations in which two measures (mean and variance) can = usefully be=20 summarized with one (the Sharpe Ratio).

The Ex Ante Sharpe Ratio

Let Rf represent the return on fund F in the forthcoming = period=20 and RB the return on a benchmark portfolio or security. In = the=20 equations, the tildes over the variables indicate that the exact values = may not=20 be known in advance. Define d, the differential return, as: =

Let d-bar be the expected value of d and sigmad be the = predicted=20 standard deviation of d. The ex ante Sharpe Ratio (S) is:

In this version, the ratio indicates the expected differential return = per=20 unit of risk associated with the differential return.

The Ex Post Sharpe Ratio

Let RFt be the return on the fund in period t, = RBt the=20 return on the benchmark portfolio or security in period t, and = Dt the=20 differential return in period t:

Let D-bar be the average value of Dt over the historic = period from=20 t=3D1 through T:

and sigmaD be the standard deviation over the period 1= :=20

The ex post, or historic Sharpe Ratio (Sh) is:

In this version, the ratio indicates the historic average = differential return=20 per unit of historic variability of the differential return.

It is a simple matter to compute an ex post Sharpe Ratio using a = spreadsheet=20 program. The returns on a fund are listed in one column and those of the = desired=20 benchmark in the next column. The differences are computed in a third = column.=20 Standard functions are then utilized to compute the components of the = ratio. For=20 example, if the differential returns were in cells C1 through C60, a = formula=20 would provide the Sharpe Ratio using Microsoft's Excel spreadsheet = program:

AVERAGE(C1:C60)/STDEV(C1:C60)

The historic Sharpe Ratio is closely related to the t-statistic for = measuring=20 the statistical significance of the mean differential return. The = t-statistic=20 will equal the Sharpe Ratio times the square root of T (the number of = returns=20 used for the calculation). If historic Sharpe Ratios for a set of funds = are=20 computed using the same number of observations, the Sharpe Ratios will = thus be=20 proportional to the t-statistics of the means.

Time Dependence

The Sharpe Ratio is not independent of the time period over which it = is=20 measured. This is true for both ex ante and ex post measures.

Consider the simplest possible case. The one-period mean and standard = deviation of the differential return are, respectively, = d-bar1 and=20 sigmad1. Assume that the differential return over T periods = is=20 measured by simply summing the one-period differential returns and that = the=20 latter have zero serial correlation. Denote the mean and standard = deviation of=20 the resulting T-period return, respectively, d-barT and=20 sigmadT. Under the assumed conditions:

and:

Letting S1 and ST denote the Sharpe Ratios for = 1 and T=20 periods, respectively, it follows that:

In practice, the situation is likely to be more complex. Multiperiod = returns=20 are usually computed taking compounding into account, which makes the=20 relationship more complicated. Moreover, underlying differential returns = may be=20 serially correlated. Even if the underlying process does not involve = serial=20 correlation, a specific ex post sample may.

It is common practice to "annualize" data that apply to periods other = than=20 one year, using equations (7) and (8). Doing so before computing a = Sharpe Ratio=20 can provide at least reasonably meaningful comparisons among strategies, = even if=20 predictions are initially stated in terms of different measurement = periods.

To maximize information content, it is usually desirable to measure = risks and=20 returns using fairly short (e.g. monthly) periods. For purposes of=20 standardization it is then desirable to annualize the results.

To provide perspective, consider investment in a broad stock market = index,=20 financed by borrowing. Typical estimates of the annual excess return on = the=20 stock market in a developed country might include a mean of 6% per year = and a=20 standard deviation of 15%. The resulting excess return Sharpe Ratio of = "the=20 stock market", stated in annual terms would then be 0.40.

Correlations

The ex ante Sharpe Ratio takes into account both the expected = differential=20 return and the associated risk, while the ex post version takes into = account=20 both the average differential return and the associated variability. = Neither=20 incorporates information about the correlation of a fund or strategy = with other=20 assets, liabilities, or previous realizations of its own return. For = this=20 reason, the ratio may need to be supplemented in certain applications. = Such=20 considerations are discussed in later sections.

Related Measures

The literature surrounding the Sharpe Ratio has, unfortunately, led = to a=20 certain amount of confusion. To provide clarification, two related = measures are=20 described here. The first uses a different term to cover cases that = include the=20 construct that we call the Sharpe Ratio. The second uses the same term = to=20 describe a different but related construct.

Whether measured ex ante or ex post, it is essential that the Sharpe = Ratio be=20 computed using the mean and standard deviation of a differential = return=20 (or, more broadly, the return on what will be termed a zero investment=20 strategy). Otherwise it loses its raison d'etre. Clearly, the = Sharpe=20 Ratio can be considered a special case of the more general construct of = the=20 ratio of the mean of any distribution to its standard deviation.

In the investment arena, a number of authors associated with BARRA (a = major=20 supplier of analytic tools and databases) have used the term = information=20 ratio to describe such a general measure. In some publications , = the ratio=20 is defined to apply only to differential returns and is thus equivalent = to the=20 measure that we call the Sharpe Ratio (see, for example, Rudd and = Clasing=20 [1982, p. 513] and Grinold<= /A>=20 [1989, p. 31]). In others, it is also encompasses the ratio of the mean = to the=20 standard deviation of the distribution of the return on a single = investment,=20 such as a fund or a benchmark (see, for example, BARRA = [1993, p.=20 22]). While such a "return information ratio" may be useful as a = descriptive=20 statistic, it lacks a number of the key properties of what might be = termed a=20 "differential return information ratio" and may in some instances lead = to wrong=20 decisions.

For example, consider the choice of a strategy involving cash and one = of two=20 funds, X and Y. X has an expected return of 5% and a standard deviation = of 10%.=20 Y has an expected return of 8% and a standard deviation of 20%. The = riskless=20 rate of interest is 3%. According to the ratio of expected return to = standard=20 deviation, X (5/10, or 0.50) is superior to Y (8/20, or 0.40). According = to the=20 Sharpe Ratios using excess return, X (2/10, or 0.20) is inferior to Y = (5/20, or=20 0.25).

Now, consider an investor who wishes to attain a standard deviation = of 10%.=20 This can be achieved with fund X, which will provide an expected return = of 5.0%.=20 It can also be achieved with an investment of 50% of the investor's = funds in Y=20 and 50% in the riskless asset. The latter will provide an expected = return of=20 5.5% -- clearly the superior alternative.

Thus the Sharpe Ratio provides the correct answer (a strategy using Y = is=20 preferred to one using X), while the "return information ratio" provides = the=20 wrong one.

In their seminal work, Treynor = and=20 Black [1973], defined the term "Sharpe Ratio" as the square = of the=20 measure that we describe. Others, such as Rudd and = Clasing=20 [1982, p. 518] and Grinold<= /A>=20 [1989, p. 31], also use such a definition.

While interesting in certain contexts, this construct has the curious = property that all values are positive -- even those for which the mean=20 differential return is negative. It thus obscures important information=20 concerning performance. We prefer to follow more common practice and = thus refer=20 to the Treynor-Black measure as the Sharpe Ratio squared = (SR2). 2= :=20

We focus here on the Sharpe Ratio, which takes into account both risk = and=20 return without reference to a market index. [Sharpe<= /A> 1966,=20 1975] discusses both the Sharpe Ratio and measures based on market = indices, such=20 as Jensen's alpha and Treynor's average excess return to beta ratio. =

Scale Independence

Originally, the benchmark for the Sharpe Ratio was taken to be a = riskless=20 security. In such a case the differential return is equal to = the=20 excess return of the fund over a one-period riskless rate of = interest.=20 Many of the descriptions of the ratio in Sharpe<= /A>=20 [1966, 1975] focus on this case .

More recent applications have utilized benchmark portfolios designed = to have=20 a set of "factor loadings" or an "investment style" similar to that of = the fund=20 being evaluated. In such cases the differential return represents the = difference=20 between the return on the fund and the return that would have been = obtained from=20 a "similar" passive alternative. The difference between the two returns = may be=20 termed an "active return" or "selection return", depending on the = underlying=20 procedure utilized to select the benchmark.

Treynor = and=20 Black [1973] cover the case in which the benchmark portfolio is, in = effect,=20 a combination of riskless securities and the "market portfolio". Rudd and = Clasing=20 [1982] describe the use of benchmarks based on factor loadings from a=20 multifactor model. Sharpe<= /A>=20 [1992] uses a procedure termed style analysis to select a mix = of asset=20 class index funds that have a "style" similar to that of the fund. When = such a=20 mix is used as a benchmark, the differential return is termed the fund's = selection return. The Sharpe Ratio of the selection return can = then=20 serve as a measure of the fund's performance over and above that due to = its=20 investment style. 3= :=20

Central to the usefulness of the Sharpe Ratio is the fact that a = differential=20 return represents the result of a zero-investment strategy. = This can be=20 defined as any strategy that involves a zero outlay of money in the = present and=20 returns either a positive, negative or zero amount in the future, = depending on=20 circumstances. A differential return clearly falls in this class, since = it can=20 be obtained by taking a long position in one asset (the fund) and a = short=20 position in another (the benchmark), with the funds from the latter used = to=20 finance the purchase of the former.

In the original applications of the ratio, where the benchmark is = taken to be=20 a one- period riskless asset, the differential return represents the = payoff from=20 a unit investment in the fund, financed by borrowing. 4= :=20 More generally, the differential return corresponds to the payoff = obtained from=20 a unit investment in the fund, financed by a short position in the = benchmark.=20 For example, a fund's selection return can be considered to be = the=20 payoff from a unit investment in the fund, financed by short positions = in a mix=20 of asset class index funds with the same style.

A differential return can be obtained explicitly by entering into an=20 agreement in which a party and a counterparty agree to swap the = return=20 on the benchmark for the return on the fund and vice-versa. A = forward=20 contract provides a similar result. Arbitrage will insure that the = return=20 on such a contract will be very close to the excess return on the = underlying=20 asset for the period ending on the delivery date. 5= : A=20 similar relationship holds approximately for traded contracts such as = stock=20 index futures , which clearly represent zero-investment strategies. 6= :=20

To compute the return for a zero-investment strategy the payoff is = divided by=20 a notional value. For example, the dollar payoff for a swap is = often=20 set to equal the difference between the dollar return on an investment = of $X in=20 one asset and that on an investment of $X in another. The net difference = can=20 then be expressed as a proportion of $X, which serves as the notional = value.=20 Returns on futures positions are often computed in a similar manner, = using the=20 initial value of the underlying asset as a base. In effect, the same = approach is=20 utilized when the difference between two returns is computed.

Since there is zero net investment in any such strategy, the = percent=20 return can be made as large or small as desired by simply changing the = notional=20 value used in such a computation. The scale of the return thus = depends=20 on the more- or-less arbitrary choice of the notional value utilized for = its=20 computation. 7= :=20

Changes in the notional value clearly affect the mean and the = standard=20 deviation of the distribution of return, but the changes are of the same = magnitude, leaving the Sharpe Ratio unaffected. The ratio is thus = scale=20 independent. 8= :=20

The Influence of a Zero Investment Strategy on Asset Risk and = Return

Scale independence is more than a mathematical artifact. It is key to = understanding why the Sharpe Ratio can provide an efficient summary = statistic=20 for a zero- investment strategy. To show this, we consider the case of = an=20 investor with a pre-existing portfolio who is considering the choice of = a zero=20 investment strategy to augment current investments.

The Relative Position in a Zero Investment Strategy

Assume that the investor has $A in assets and has placed this money = in an=20 investment portfolio with a return of RI. She is considering=20 investment in a zero-investment strategy that will provide a return of d = per=20 unit of notional value. Denote the notional value chosen as V (e.g. = investment=20 of V in a fund financed by a short position of V in a benchmark). Define = the=20 relative position, p, as the ratio of the notional value to the = investor's assets:

The end-of-period payoff will be:

Let RA denote the total return on the investor's initial = assets.=20 Then:

If R-barA denotes the expected return on assets and R-=20 barI the expected return on the investment:

Now, let sigmaA, sigmaI and sigmad = denote=20 the standard deviations of the returns on assets, the investment and the = zero-investment strategy, respectively, and rhoId the = correlation=20 between the return on the investment and the return on the = zero-investment=20 strategy. Then:

or, rewriting slightly:

The Risk Position in a Zero Investment Strategy

The parenthesized expression (p sigmad) is of particular = interest.=20 It indicates the risk of the position in the zero-investment strategy = relative=20 to the investor's overall assets. Let k denote this risk = position

For many purposes it is desirable to consider k as the relevant = decision=20 variable. Doing so states the magnitude of a zero-investment strategy in = terms=20 of its risk relative to the investor's overall assets. In effect, one = first=20 determines k, the level of risk of the zero- investment strategy. Having = answered this fundamental question, the relative (p) and absolute (V) = amounts of=20 notional value for the strategy can readily be determined, using = equations (17)=20 and (11). 9= :=20

Asset Risk and Expected Return

It is straightforward to determine the manner in which asset risk and = expected return are related to the risk position of the zero investment=20 strategy, its correlation with the investment, and its Sharpe Ratio. =

Substituting k in equation (16) gives the relationship between 1) = asset risk=20 and 2) the risk position and the correlation of the strategy with the=20 investment:

To see the relationship between asset expected return and the = characteristics=20 of the zero investment strategy, note that the Sharpe Ratio is the ratio = of=20 d-bar to sigmad. It follows that

Substituting equation (19) in equation (14) gives:

or:

which shows that the expected return on assets is related directly to = the=20 product of the risk position times the Sharpe Ratio of the strategy. =

By selecting an appropriate scale, any zero investment strategy can = be used=20 to achieve a desired level (k) of relative risk. This level, plus the = strategy's=20 Sharpe Ratio, will determine asset expected return, as shown by equation = (21).=20 Asset risk, however, will depend on both the relative risk (k) and the=20 correlation of the strategy with the other investment (rhoId = ). In=20 general, the Sharpe Ratio, which does not take that correlation into = account,=20 will not by itself provide sufficient information to determine a set of=20 decisions that will produce an optimal combination of asset risk and = return,=20 given an investor's tolerance of risk.

Adding a Zero-Investment Strategy to an Existing Portfolio

Fortunately, there are important special cases in which the Sharpe = Ratio will=20 provide sufficient information for decisions on the optimal risk/return=20 combination: one in which the pre-existing portfolio is riskless, the = other in=20 which it is risky.

Adding a Strategy to a Riskless Portfolio

Suppose first that an investor plans to allocate money between a = riskless=20 asset and a single risky fund (e.g. a "balanced" fund). This is, in = effect, the=20 case analyzed in Sharpe<= /A>=20 [1966,1975].

We assume that there is a pre-existing portfolio invested solely in a = riskless security, to which is to be added a zero investment strategy = involving=20 a long position in a fund, financed by a short position in a riskless = asset=20 (i.e., borrowing). Letting Rc denote the return on such a = "cash=20 equivalent", equations (1) and (13) can be written as:

and

Since the investment is riskless, its standard deviation of return is = zero,=20 so both the first and second terms on the right-hand side of equation = (18)=20 become zero, giving:

The investor's total risk will thus be equal to that of the position = taken in=20 the zero investment strategy, which will in turn equal the risk of the = position=20 in the fund.

Letting SF represent the Sharpe Ratio of fund F, equation = (21) can=20 be written:

It is clear from equations (24) and (25) that the investor should = choose the=20 desired level of risk (k), then obtain that level of risk by using the = fund (F)=20 with the greatest excess return Sharpe Ratio. Correlation does not play = a role=20 since the remaining holdings are riskless.

This is illustrated in the Exhibit. Points X and Y represent two = (mutually=20 exclusive) strategies. The desired level of risk is given by k. It can = be=20 obtained with strategy X using a relative position of px = (shown in=20 the figure at point PxX) or with strategy Y using a relative position of = pY (shown in the figure at point PyY). An = appropriately-scaled=20 version of strategy X clearly provides a higher mean return (shown at = point MRx)=20 than an appropriately-scaled version of strategy Y (shown at point MRy). = Strategy X is hence to be preferred.

The Exhibit shows that the mean return associated with any = desired=20 risk position will be greater if strategy X is adopted instead of = strategy Y.=20 But the slope of such a line is the Sharpe Ratio. Hence, as = long as=20 only the mean return and the risk position of the zero-investment = strategy are=20 relevant, the optimal solution involves maximization of the Sharpe Ratio = of the=20 zero-investment strategy.

Consider, for example, a choice between fund XX, with a risk of 10% = and an=20 excess return Sharpe Ratio of 0.20 and fund YY with a risk of 20% and an = excess=20 return Sharpe Ratio of 0.25. Assume the investor has $100 to invest and = desires=20 a level of risk (here, k) equal to 15%.

The optimal strategy involves investment of $100 in the riskless = asset plus a=20 zero-investment strategy based on fund YY. To make the risk of the = latter equal=20 to 15%, a relative position (p) of 0.75 should be taken. This, in turn, = requires=20 an investment of $75 in the fund, financed by $75 of borrowing (i.e. a = short=20 position in the riskless asset). The net position in the riskless asset = will=20 thus be $25 ($100 - $75), with $75 invested in Fund YY.

In this case the investor's tasks include the selection of the fund = with the=20 greatest Sharpe Ratio and the allocation of wealth between this fund and = borrowing or lending, as required to obtain the desired level of asset = risk.=20

Adding a Strategy to a Risky Portfolio

Consider now the case in which a single fund is to be selected to = complement=20 a pre-existing group of risky investments. For example, an investor = might have=20 $100, with $80 already committed (e.g. to a group of bond and stock = funds). The=20 goal is to allocate the remaining $20 between a riskless asset ("cash") = and a=20 single risky fund (e.g. a "growth stock fund"), accepting the = possibility that=20 the amount allocated to cash might be positive, zero or negative, = depending on=20 the desired risk and the risk of the chosen fund.

In this case the investment should be taken as the pre-existing = investment=20 plus a riskless asset (in the example, $80 in the initial investments = plus $20=20 in cash equivalents). The return on this total portfolio will be = RI.=20 The zero- investment strategy will again involve a long position in a = risky fund=20 and a short position in the riskless asset.

As stated earlier, in such a case it will not necessarily be optimal = to=20 select the fund with the largest possible Sharpe Ratio. While the ratio = takes=20 into account two key attributes of the predicted performance of a=20 zero-investment strategy (its expected return and its risk), it does not = include=20 information about the correlation of its return with that of the = investor's=20 other holdings (rhoId). It is entirely possible that a fund = with a=20 smaller Sharpe Ratio could have a sufficiently smaller correlation with = the=20 investor's other assets that it would provide a higher expected return = on assets=20 for any given level of overall asset risk.

However, if the alternative funds being analyzed have similar = correlations=20 with the investor's other assets, it will still be optimal to select the = fund=20 with the greatest Sharpe Ratio. To see this, note that with = rhoId=20 taken as given, equation (18) shows that there is a one-to-one = correspondence=20 between sigmaA and k. Thus, for any desired level of asset = risk, the=20 investor chooses the corresponding risk position k given by equation = (18),=20 regardless of the fund to be employed.

But, as before, the expected return on assets will be:

which can be maximized by selecting the fund with the largest Sharpe = Ratio.=20

The practical implication is clear. When choosing one from among a = group of=20 funds of a particular type for inclusion in a larger set of holdings, = the one=20 with the largest predicted excess return Sharpe Ratio may reasonably be = chosen,=20 if it can be assumed that all the funds in the set have similar = correlations=20 with the other holdings. If this condition is not met, some account = should be=20 taken of the differential levels of such correlations.

The Choice of a Set of Uncorrelated Strategies

Suppose finally that an investor has a pre-existing set of = investments and is=20 considering taking positions in one or more zero-investment strategies, = each of=20 which is uncorrelated both with the existing investments and with each = of the=20 other such strategies. Such lack of correlation is generally assumed for = residual returns from an assumed factor model and hence applies to = strategies in=20 which long and short positions are combined to obtain zero exposures to = all=20 underlying factors in such a model.

In particular, this is assumed to hold for the "non-market returns" = which are=20 the residual returns in one-factor "market models" of the type employed = in Treynor-= Black=20 [1973]. It is also assumed to hold for the "active returns" that = constitute the=20 residual returns in a model of the type used by BARRA (described, for = example,=20 in Grinold<= /A>=20 [1989]).

Most germane, perhaps, for selecting funds, this is assumed to hold = for the=20 "selection returns" that constitute the residuals from the asset class = factor=20 model used in the style analysis procedure described in Sharpe [1992]. Note, however that the key results apply to all three = cases. 10:

Under the assumed conditions, the counterpart to equation (13) is: =

where pi represents the relative position taken in = strategy i and=20 di represents its return.

Letting sigmadi represent the risk of position i, asset = risk is=20 given by:

and expected asset return by:

Adding subscriptions to equations (21) and (18), and substituting the = results=20 gives:

and

Now, assume that the investor's goal is to maximize a standard risk- = adjusted=20 expected return of the form:

where tau represents risk tolerance (the marginal rate of = substitution of=20 variance for expected return). Substituting equations (30) and (31) in = (32)=20 gives:

Since the terms involving the initial investment will be unaffected = by the=20 decisions (ki's) concerning the zero investment strategies, = it=20 suffices to maximize:

To do so, the partial derivative with respect to each decision = variable=20 (ki) should be set to zero:

The optimal risk position in strategy i is thus:

Hence the risk levels of the strategies should be proportional to = their=20 Sharpe Ratios. Strategies with zero predicted Sharpe Ratios should be = ignored.=20 Those with positive ratios should be "held long", and those with = negative ratios=20 "held short". If strategy X has a positive Sharpe Ratio that is twice as = large=20 as that of strategy Y, twice as much risk should be taken with X as with = Y. The=20 overall scale of all the positions should, in turn, be proportional to = the=20 investor's risk tolerance.

An interesting application occurs when long and short positions can = be taken=20 (e.g. via financial futures) in the asset classes that underlie a style = analysis=20 model of the type described in Sharpe<= /A>=20 [1992]. In principle, funds should be selected based only on their = selection=20 returns, with the respective amounts of selection risk set in proportion = to the=20 funds' selection return Sharpe Ratios. The net exposures to asset = classes=20 required to implement this mixture of zero investment strategies can = then be=20 compared with the investor's desired passive asset mix to determine = needed net=20 positions.

Summary

The Sharpe Ratio is designed to measure the expected return per unit = of risk=20 for a zero investment strategy. The difference between the = returns on=20 two investment assets represents the results of such a strategy. The = Sharpe=20 Ratio does not cover cases in which only one investment return is = involved.

Clearly, any measure that attempts to summarize even an unbiased = prediction=20 of performance with a single number requires a substantial set of = assumptions=20 for justification. In practice, such assumptions are, at best, likely to = hold=20 only approximately. Certainly, the use of unadjusted historic (ex post) = Sharpe=20 Ratios as surrogates for unbiased predictions of ex ante ratios is = subject to=20 serious question. Despite such caveats, there is much to recommend a = measure=20 that at least takes into account both risk and expected return over any=20 alternative that focuses only on the latter.

For a number of investment decisions, ex ante Sharpe Ratios can = provide=20 important inputs. When choosing one from among a set of funds to provide = representation in a particular market sector, it makes sense to favor = the one=20 with the greatest predicted Sharpe Ratio, as long as the correlations of = the=20 funds with other relevant asset classes are reasonably similar. When = allocating=20 funds among several such funds, it makes sense to allocate funds such = that the=20 selection (residual) risk levels are proportional to the predicted = Sharpe Ratios=20 for the selection (residual) returns. If some of the implied net = positions are=20 infeasible or involve excessive transactions costs, of course, the = decision=20 rules must be modified. Nonetheless, Sharpe Ratios may still provide = useful=20 guidance.

Whatever the application, it is essential to remember that the Sharpe = Ratio=20 does not take correlations into account. When a choice may affect = important=20 correlations with other assets in an investor's portfolio, such = information=20 should be used to supplement comparisons based on Sharpe Ratios.

All the same, the ratio of expected added return per unit of added = risk=20 provides a convenient summary of two important aspects of any strategy = involving=20 the difference between the return of a fund and that of a relevant = benchmark.=20 The Sharpe Ratio is designed to provide such a measure. Properly used, = it can=20 improve the process of managing investments.

Endnotes

1. We use the formula for the standard deviation of = a=20 population, taking the observations as a sample. For applications in = which the=20 value of T is the same for all the funds being measured, the standard = deviation=20 of the historic data (in which the denominator is T rather than T-1) can = generally be used instead, since the relative magnitudes of the = resulting=20 measures would be the same.

2. Treynor and Black showed that if resources are = allocated=20 optimally, the SR2 of a portfolio will equal the sum of the=20 SR2 values for its components. This follows from the fact = that the=20 optimal holding of a component will be proportional to the ratio of its = mean=20 differential return to the square of the standard deviation of its = differential=20 return. Thus, for example, components with negative means should be held = in=20 negative amounts. In this context, the product of the mean return and = the=20 optimal holding will always be positive. For completeness, it should be = noted=20 that Treynor and Black used the term appraisal ratio to refer = to what=20 we term here the SR2 of a component and the term Sharpe=20 Ratio to refer to the SR2 of the portfolio, although = other=20 authors have used the latter term for both the portfolio and its = components.=20

3. This type of application is described in BARRA = [1992, p.=20 21].

4. In this context, maximization of the Sharpe = Ratio is the=20 normative equivalent to the separation theorem first put forth in Tobin = [1958]=20 in a positive context.

5. To see this, note that by borrowing money to = purchase the=20 underlying asset, one can obtain precisely the same asset at the = delivery date.=20 The ending value of such a strategy will be perfectly correlated with = the value=20 of the forward contract and neither will require any outlay. If the = payoffs at=20 the end of the period differ, one could take a long position in one = combination=20 (e.g. the forward contract or the asset/borrowing combination) and a = short=20 position in the other and obtain a guaranteed payment at the end of the = period=20 with no outlay at any other time. This is unlikely to be the case in a = market=20 populated by astute investors. In practice, transactions costs will = limit the=20 precision of the relationship.

6. Futures contracts are often not protected = against changes=20 in value due to (for example) dividend payments. They also generally = require=20 daily marking to market. For these reasons they differ from forward = contracts=20 with dividend protection, for which the arbitrage relationship will hold = within=20 the bounds of transactions costs. Futures contracts generally require = that=20 margin be posted. However, this is not an investment in the underlying = asset.=20

7. Despite this drawback, once a notional value has = been=20 selected, the actual rate of return can be used for comparison purposes. =

8. Indeed, a Sharpe Ratio can be computed without = regard to=20 notional value by simply using the mean and standard deviation of the=20 distribution of the final payoff.

9. To see the advantages of concentration on the = risk=20 position of a strategy, consider two funds. One (X) invests directly, = the other=20 (Y) borrows money at the riskless rate and invests in X, with a leverage = ratio=20 of 2 to 1. Let kx be the optimal position in fund X. Clearly = the=20 optimal position in fund Y will be half as large. However, the standard=20 deviation of return on fund Y will be twice that of fund X. Thus the = optimal=20 risk position in Y will be the same as that in X.

10. In fact, the basic relationship on which this = section=20 builds was first obtained by Treynor and Black [1973].

References

BARRA Newsletter, September/October 1992, May/June 1993, = BARRA,=20 Berkeley, Ca.

Bodie, Zvi, Alex Kane and Alan J. Marcus. Investments, 2d = edition.=20 Homewood, IL: Richard D. Irwin, 1993.

Capaul, Carlo, Ian Rowley, and William F. Sharpe. "International = Value and=20 Growth Stock Returns," Financial Analysts Journal, = January/February=20 1993, pp. 27-36.

Elton, Edwin J., and Martin J. Gruber. Modern Portfolio Theory = and=20 Investment Analysis, 4th edition. New York: John Wiley & Sons, = 1991.=20

Grinold, Richard C. "The Fundamental Law of Active Management," = Journal=20 of Portfolio Management, Spring 1989, pp. 30-37.

Haugen, Robert A. Modern Investment Theory, 3d edition. = Englewood=20 Cliffs, NJ: Prentice-Hall, 1993.

"Morningstar Mutual Funds User's Guide." Chicago: Morningstar Inc., = 1993.=20

Radcliff, Robert C. Investment Concepts, Analysis, Strategy, = 3d=20 edition. New York: HarperCollins, 1990.

Reilly, Frank K. Investment Analysis and Portfolio = Management, 3d=20 edition. Chicago: The Dryden Press, 1989.

Rudd, Andrew, and Henry K. Clasing. Modern Portfolio Theory, The=20 Principles of Investment Management. Homewood, IL: Dow-Jones Irwin, = 1982.=20

Sharpe, William F. "Mutual Fund Performance." Journal of = Business,=20 January 1966, pp. 119-138.

-----. "Adjusting for Risk in Portfolio Performance Measurement." = Journal=20 of Portfolio Management, Winter 1975, pp. 29-34.

-----. "Asset allocation: Management Style and Performance = Measurement,"=20 Journal of Portfolio Management, Winter 1992, pp. 7-19.

Tobin, James. "Liquidity Preference as Behavior Toward Risk." = Review of=20 Economic Studies, February 1958, pp. 65-86.

Treynor, Jack L., and Fischer Black. "How to Use Security Analysis to = Improve=20 Portfolio Selection." Journal of Business, January 1973, pp. = 66-85.=20

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