The purpose of this issue’s columnis to review the use of a widely usedrisk-adjusted portfolio performancemeasure—the Sharpe Ratio (SR). Thistopic may seem a bit humdrum andbasic, and for some readers this mightbe the case; on the other hand, webelieve that this review might be ofinterest to those who are still trying tomake sense of portfolio returns thatinclude significant losses resulting fromthe deep market losses experiencedduring the Great Recession.

As a reminder, Sharpe (Sharpe1966, 1994) originated the calculationfor what has since been termed the SR.The SR provides a means to compareportfolio performance on a risk-adjusted basis. The SR can be com-puted as follows:

SRi = (ui – rf ) /si

where,SRi = Sharpe Ratio of portfolioui = average return of portfoliorf = risk-free rate of returnsi = standard deviation of the excess

portfolio returnTwo aspects of the SR are appar-

ent. First, the measure of risk (volatil-ity) is standard deviation, whichimplies the evaluation of total, ratherthan systematic, risk. Second, the SRindicates the level of return per unit ofrisk taken. The SR of a portfolio isindicated by the slope of a line fromthe risk-free rate to the point of portfo-lio return (Scholz and Wilkens 2005).Holding other factors constant, thehigher the SR, the better the perform-ance. In effect, the SR standardizesreturns so that multiple portfolios canbe compared.

The use of SRs is widespreadwithin the financial service profession.Nearly every fund and asset manage-ment rating service reports the SR offunds and managed accounts. Financialadvisors use these figures to comparethe risk-adjusted performances of port-folio managers. Practically speaking,this is the primary use of such ratios—namely, to rank order investment port-folios. A secondary use of the SR is asan input into other measures of portfo-lio performance. For example, the M-squared measure, which was developedby Modigliani and Modigliani(Modigliani and Modigliani 1997),utilizes the SR as a formula input.

On the surface, the SR formulaseems straightforward. Nearly every

financial advisor already understandsthe basic calculation and how to usethe ratio in the portfolio develop-ment, implementation, and reviewprocess. Recently we ran into a per-plexing problem using the SR. Givenour situation—which is describedbelow—we wanted to take time inthis column to help others betterunderstand a particular nuance associ-ated with the ratio.

Not long ago, we were reviewingthe performance of two managedaccounts. It is important to put theanalysis into context. The returns beinganalyzed included data from the GreatRecession, which resulted in heavilyskewed negative returns. Here are theSR data inputs for the two accounts:Portfolio A• ui = -20.93 percent• si = 161.80 percent• rf = 3.50 percent average over

periodPortfolio B• ui = -24.24 percent• si = 182.69 percent• rf = 3.50 percent average over

periodIt should be obvious that Portfolio

A provided a superior return over theperiod. The nominal performance differ-ence was 3.31 percent. Additionally, thestandard deviation of returns was lowerfor Portfolio A. As is common practice inthe financial service field, we wanted toconfirm that Portfolio A also provided abetter return on a risk-adjusted basis.Turning to the SR, we calculated the fol-lowing for the two portfolios:

EIMEconomics & Investment Management

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Society of Financial Service Professionals. All rights reserved.

The Sharpe Ratio and Negative Excess Returns:The Problem and Solution

by John E. Grable, PhDSwarn Chatterjee, PhD

Abstract: The purpose of this issue’scolumn is to review the use of awidely used risk-adjusted portfolioperformance measure—the SharpeRatio. This is a fundamental topic,and for some readers this may be abasic review; on the other hand, webelieve that this review might be ofinterest to those who are still trying tomake sense of portfolio returns thatinclude significant losses resultingfrom the deep market losses experi-enced during the Great Recession.

ECONOMICS & INVESTMENT MANAGEMENT

Portfolio A: SRi = -0.151Portfolio B: SRi = -0.152Does this seem odd? The results

certainly caught us off guard.1 Wewere perplexed by this result until weremembered a crucial point about theSR. Specifically, the SR becomes veryunstable during times of marketdecline. Recall that the performanceof the two portfolios was measuredduring one of the worst bear marketsin history. It was not surprising thatboth accounts lost money; nor was itsurprising that the standard deviationof returns was so large. What wasunexpected was how close the SR out-comes were to each other. Conceptu-ally, it is not possible for the risk-adjusted performance of Portfolio Bto be within .001 of Portfolio A. Nosensible investor would even considerPortfolio B over Portfolio A, given thereturn and standard deviation infor-mation presented.

It turns out that the SR will fre-quently provide a biased result whennegative excess returns (ui – rf ) arepresent (Israelsen 2005). When facedwith this type of situation, financialadvisors need to make a modificationto the SR. The modified SR formula isshown below: ERi

SR'i = ERi / (si(absERi))

where,SRi = modified Sharpe RatioERi = excess return of portfolio,

were ERi = ui – rfabsERi = absolute value of excess

return of portfoliosi = standard deviation of the

excess portfolio returnRanking of portfolios using either

the SR or the modified SR will beexactly the same when expected returnsare positive; however, in situations

where the excess return is negative, therankings may be significantly differ-ent. The change in SR when the mod-ification is made results from the factthat the result of ERi will always

absERi

be -1.0, at least when negative excessreturns are persistent. As illustratedbelow, the modified SR is markedlydifferent from the traditional SR out-come for the two accounts.

Portfolio A: SR'i = -0.3952

Portfolio B: SR'i = -0.507While the absolute ranking of the

two portfolios did not change, themagnitude of the ranking shifted dra-matically after using the modified SRformula. In effect, making the adjust-ment to the exponent in the denomi-nator normalized the risk-adjustedreturn calculation to account for thepresence of negative excess returns.

The key takeaway is this: remem-ber that the SR may give a counterin-tuitive indication of risk-adjusted per-formance when the portfolios beingcompared have negative excess returns.In these situations it is always a goodidea to use the modification to the SRformula as described here. The out-come will be a more realistic estimateof the level of return per unit of risktaken. A second takeaway is that thoseadvisors who also use the TreynorRatio when comparing diversifiedportfolios and accounts should con-sider making the same type of adjust-ment when negative excess returns arepresent. The only difference betweenthe SR and the Treynor Ratio is theuse of beta (b) in the formula’s denom-inator. Hopefully, moving forward,financial advisors will not need to usea modified SR formula, but if the mar-kets ever do sustain another signifi-

cant bear trend, remembering this sim-ple adjustment may help clarify port-folio comparisons. �

Dr. John E. Grable holds an Athletic Associ-ation Endowed Professorship at the Uni-versity of Georgia, where he conductsresearch and teaches financial planning. Dr.Grable is best known for his work related tofinancial risk tolerance assessment and psy-chophysiological economics. He serves asthe Co-Director of the Financial PlanningPerformance Laboratory at UGA. He maybe reached at grable@uga.edu.

Dr. Swarn Chatterjee is an Associate Pro-fessor of Financial Planning at the Univer-sity of Georgia. He has published morethan 40 peer-reviewed papers and teachesclasses in investing, portfolio manage-ment, and behavioral finance. He servesas the Co-Director of the Financial Plan-ning Performance Laboratory at UGA. Hecan be reached at swarn@uga.edu.

(1) The problem can be seen with even moreclarity if the standard deviations are reducedto, say, 62 percent and 83 percent for PortfoliosA and B, respectively. The traditional SharpeRatio results in an output of -0.39 for PortfolioA and -0.33 for Portfolio B. These results leadto the incorrect conclusion that Portfolio B issuperior to Portfolio A. (2) This result was obtained in Excel as follows:(-.2093 - .035)/(1.618^-1).

ReferencesIsraelsen, C. (2005). A Refinement to the Sharpe

Ratio and Information Ratio. Journal ofAsset Management 5(6): 423-427.

Modigliani, F. and Modigliani, L. (1997). Risk-Adjusted Performance. The Journal ofPortfolio Management 23(2): 45-54.

Scholz, H. and Wilkens, M. (2005). A JigsawPuzzle of Basic Risk-Adjusted Perform-ance Measures. The Journal of Perform-ance Measurement Spring: 57-64.

Sharpe, W. F. (1966). Mutual Fund Perform-ance. Journal of Business 39(1): 119-138.

Sharpe, W. F. (1994). The Sharpe Ratio. Journalof Portfolio Management Fall: 49-58.

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