ambiguity when Performance is Measured by the securities Market Line TORIo Richard roll The Journal of Finance, Vol 33, No 4.(Sep, 1978), pp. 1051-1069 Stable url: http://inks.jstor.org/sici?sici=0022-1082%028197809%2933%3a4%3c105193aawpimb%3e2.0.c0%03b2-4 The Journal of finance is currently published by American Finance Association Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http://www.jstor.org/about/terms.htmlJstOr'sTermsandConditionsofUseprovidesinpartthatunlessyou have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at http://wwwjstor.org/journals/afina.html Each copy of any part of a jSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission jStOR is an independent not-for-profit organization dedicated to creating and preserving a digital archive of scholarly journals. For more information regarding JSTOR, please contact support@jstor. org http://www」]stor.or Thu Apr607:37:312006
JOURNAL OF FINANCE. VOL. XXXIII, NO. 4.SEPTEMBER 1978 The fournal of FINANCE VOL. XXXIII SEPTEMBER 1978 AMBIGUITY WHEN PERFORMANCE IS MEASURED BY THE SECURITIES MARKET LINE RICHARD ROLLS INTRODUCTION IMAGINE AN IDEALIZED ANALoG to the activities of professional money managers, a contest whose rules are as follows (a)Each contestant selects a portfolio from a specified set of individual assets (c) After each period of return observation, the portfolios are re-balanced to the nitial selections (d)After an interval consisting of several periods, winners""and"losers"are declared for that interval (e) Contestants choose a new portfolio, or keep the old one, and the process(b) through()is repeated (f After several intervals, consistent winners are declared to be superior port folio managers and consistent losers are declared inferior. In the absence of any consistency, everyone is declared non-superior The sponsors of the contest face only a single problem of intellectual interest. They must develop criteria to partition contestants at step(d)into winners and losers Of course,the criteria must be acceptable to participants and to disinterested obser vers. There should be a correspondence between"consistency in winning "and an tuitive notion of ability in portfolio selection We might think of many desirable qualities to be possessed by such criteria. For example, they should be robust to stochastic changes in the return sequence; true ability should be detectable over many intervals regardless of the sequence. If the criteria are employed by different sponsors, the same judgements about ability should be obtained. It should not be possible to reverse judgements by making hanges in the computation of the criteria, if such changes are deemed insignificant by all observers. In other words, the criteria must provide decisions about ability that are unambiguous to rational judges. criterion that is widely employed in the financial community for assessing portfolio performance is the"securities market line, the (linear) relation betwe Graduate-School of Management U C LA. Comments and suggestions by Alan Kraus, David Mayers, Stephen Ross and Eduardo Schwartz are gratefully acknowledged
1052 The Journal of finance mean returns on assets or portfolios and the betas of these assets or portfolios calculated against a market index. Judging from the academic literature, this criterion is even more widely-accepted by scholars as a tool for assessing the ex ante or ex post qualities of securities, portfolios and investment projects. There eems little doubt that it is currently the most widely-accepted criterion for inferences about the quality of risky assets It is quite simple to employ, particularly in a situation such as the contest mentioned above. There are only two steps to accomplish. First, an index must be osen by the judges; second, the betas must be computed against this index for each asset (and portfolio). The second step can be accomplished in many ways, including purely subjective, but the most common method is to employ historical data over the same interval as the contest itsel Given the computed betas and the returns on assets or portfolios, the criterion an be illustrated as in Figure 1. A line, R=Yo+y B is fit to the observations and assets are declared"winners "if they are above the line(such as a and b)or losers elow(C or D) My purpose here is to expose the ambiguity in this criterion. It is not robust, is likely to yield different judgements when employed by different judges, and can completely reverse its judgements after seemingly innocuous changes in its compu- curities Market Line FIGURE'I. The Securities Market Line Employed as a Criterion for Assessing Asset Quality l. The"beta "is the covariance of the asset and the market index returns divided by the variance of the market index'return The method of fit is not crucial for this ion. It is often done by regression, sometimes with sophisticated econometric corrections of the betas. It could also be done by choosing Yo and y, based on theories of market equilibrium. For example, the Sharpe [1964], Lintner [1965] theory requires that Ro the riskless rate of interest, and that YI=E(Rm)-Res the difference between the expecte eturn on the market index and the riskless rate of interest. Thus, estimates of E(Rm ) and of RF can be sed to fix the line. Alternatively, the Black[1972] theory would require y,=E(Rm-R)and Yo=E(R2) where z is a portfolio with minimal variance uncorrelated with m
Ambiguity When Performance is Measured by the Securities Market Line 1053 ation. Reasons for these deficiencies will be explained in detail. By implication, the concept of the beta as an unambiguous measure of risk will be disputed A Numerical Ey To illustrate the ambiguity of the securities market line criterion, let us consider specific numerical example-the idealized contest conducted with 15 hypothetical contestants and a hypothetical four asset universe. Table 1 begins the example with portfolios selected by the fifteen contestants Nothing is unusual about the selections. The first four contestants plunged into the individual assets and contestants 14 and 15 sold short some securities, but there seems little reason to exclude such possibilities. The idealization of the contest is evident only because trading costs and restrictions on short sales have not even been mentioned PORTFOLIOS SELECTED BY FIFTEEN CONTESTANTS Asset Percentage Invested in Individual Asset 0. l00 20 10 7.69 13 3l9 13.4 14 42 29.0 33.3 496 24 53.l Note: The last four portfolios' weights may not sum to 100% because of nding. The exact weights were used in all calculations After the portfolios were selected, a sample period was observed with the results for individual assets reported in Table 2. Again, there is nothing abnormal nor pathological about these numbers. The mean returns were different on different ssets and the covariance matrix was non-singular, certainly the most usual and desirable feature of real asset return samples. Indeed, all of the qualitative results to be reported hereafter could be obtained from any other numbers with these same general characteristics. For the purposes of illustration, there is not even any need to be concerned with the statistical properties of the sample. The ambiguities are
1054 The Journal of finance not related in any way to the sampling error in the estimates. The same problems nen the true population means, va and used in computing the criterion TABLE 2 SAMPLE STATISTICS FOR INDIVIDUAL ASSETS WHICH CONSTITUTED THE UNIVERSE OF SECURITIES FOR THE PORTFOLIO SELECTION CONTEST Asset asset Mean return Sample varian ance 6. 5. The observed means and variances of the 15 portfolios are easy to compute by applying the compositions of Table I to the observed individual asset returns, variances and covariances of Table 2 Because of its wide acceptance, the securities market line criterion will be used y the three hypothetical sponsors of the contest in order to distinguish winners from losers. Let us suppose, however, that a dispute arises about the best index to use in calculating the"betas. "Sponsor/judge no. 1 admits total ignorance about this question. He concludes that an index composed of equal weights in the individual assets would be the most sensible portrayal of this ignorance and the fairest to all contestants. Sponsor/judge no. 2, however, has studied asset pricing theory and argues that the appropriate index should have weights proportional to the aggregate market values of individual assets. He finds that the aggregate values of assets 2 and 3 are roughly four times larger than those of assets I and 4, so he suggests the index(10%, 40%, 40%, 10%) Sponsor/judge no. 3, also a theorist, thinks that a good index should be mean- variance efficient in the sense of Markowitz(1959), so he makes some calculations and obtains an index with the same mean return as the indices of the other judges but with a different composi- tion. His index turns out to be (18.2%, 37.0%, 21.5%, 23.3%), the proportions being rounded to three significant digits The compositions of the indices, their observed mean returns and variances, and the securities market lines computed against them are reported in Table 3. Notice that the three indices have equal means, similar variances and closely adjacent securities market lines. ( The securities market lines would have been identical they had been fit using an asset pricing theory rather than by regression. See Note 3. I. e, a portfolio which has the smallest sample variance of return for a given level of sample mean