Ambiguity When Performance is Measured by the Securities Market Line 1055 INDICES AND SECURITIES MARKET LINES USED BY THE THREE JUDGES IN THE PORTFOLIO SELECTION CONTEST Return Slope Intercept Percentage of Index in Security) Index Securities Market linea l14 (44 31823721523.375 11.0 a Standard errors are in parentheses. These lines were fitted cross-sectionally by ordinary least quare applied to mean returns and betas of the fifteen portfolios in the contest b Judge three actually specified more precise weights. The weights reported here have been rounded. (For example, the proportion of his index in asset 4 was actually 23. 3214% )The exact weights were used in all calculations Coefficients of Correlation Between Indices 3 982 920 What about their assessments of"winners"and"losers"? Judge no. I ranks th 15 contestants from best (largest positive deviation from his securities market line) to worst as follows Rankings of Contestants by Judge no. 1 (above the line) (below the line) 613 Losers 9871241131←( Worst a graph of the contestants' positions relative to the judge's criterion is given by Figure 2(A) The second judge has a different set of assessments, as shown in the following list and depicted in Figure 2(B) Losers
1056 The Journal of Finance 391014 FIGURE 2. Securities Market Lines and Positions of Selected Portfolios as Perceived by the Three Judges of the Contest nes are ordinary least squares estimated securities market lines fitted through the fifteen portfolios. The dotted lines pass through portfolios 12-15, which were exactly efficient ex post Although some contestants were similarly rated by both judges, (e.g, contestant 15 was ranked second by both and contestant 3 was ranked 14th by judge no, I and 15th by judge no. 2), other contestants were rated quite differently; (e.g,the number one winner according to judge no. I was a loser and ranked 13th out of 15 by judge no. 2. )The rank correlation between the decisions of these two judges is only 0036 and the lack of agreement is clearly evident in the figure As for judge no. 3, after calculating his securities market line and plotting the selected portfolios, he observes Figure 2(). Every single contestant is exactly on the line. Of course, this judge is unable to construct a ranking and can draw no inference about the relative abilities of contestants c This example was not constructed to generate bizarre results. The same results an be obtained from every sample of asset returns. They were not caused by any of the example's parameters, by the numbers of assets and contestants, nor by the pattern of returns The results in the example and the ambiguity in the securities market line criterion can be attributed to the following fact: corresponding to every index, there is a beta for every individual asset(and thus for every portfolio); but these betas can be different for different indices and will be different for most. To consider the beta as an attribute of the individual asset alone is a significant mistake. For every asset, an index can be found to produce a beta of any desired magnitude, however large or small. Thus, for every asset(or portfolio) judicious hoice of the index can produce any desired measured"performance, (positive or negative), against the securities market line. IL. MATHEMATICS OF THE SECURITIES MARKET LINE CRITERION In this section, the mathematical causes of the preceding numerical results will be made more precise. Since there are general principles involved that would bring