Choice of a “reference portfolio” such as the Standard & Poor’s 500 — commonly termed a market index — plays a central role in “alpha-beta” based performance measurement procedures. The market risk, or beta, of the portfolio to be measured is computed in relation to the reference portfolio. The portfolio’s alpha is then computed by subtracting from its actual return the return consistent with that level of beta.

Richard Roll has pointed out that, because the alphas that result from this procedure depend on the choice of reference portfolio used to define betas, virtually any set of desired alphas can be obtained by choosing the appropriate reference portfolio. In short, because the choice of reference portfolio is arbitrary, the rankings that result from performance measurement are also arbitrary. Evidently any measurement procedure based on alpha is going to be subject to this problem.

Portfolio performance measurement cannot be separated from investment strategy. Any conventional measurement procedure implies that the investor should own the portfolio ranked highest, either alone or in conjunction with a reference portfolio (as in active-passive management). But a rational investor will allocate his assets among all available portfolios, taking into account not only the risk and return attributes of the individual portfolios, but how they combine. Portfolio performance rankings are irrelevant to the rational investor.