The Robustness Of Single Index Models In Crop Markets: A Multiple Index Model Test: Comment

A recent article by Steven C. Blank purports to evaluate the assumptions underlying the single index model (SIM) when it is applied in the farm planning context. Instead, it evaluates the extent to which the capital asset pricing model (CAPM) equilibrium conclusions arise in crop returns. While these issues have been dealt with previously (Hutchinson and McKillop; Collins), the important differences between the CAPM and the SIM apparently are not yet well understood by many. The SIM regresses each individual set of crop returns (CRi) on an index I, CRi = a, + ,I + Ei, and uses the estimates of the three parameters to approximate a full covariance matrix in a normative farm planning model as an alternative to the more complex Markowitz model. The CAPM, on the other hand, is a model of equilibrium returns to risky assets which states that if a long list of asset homogeneity and market efficiency assumptions are met, the equilibrium expected return on a risky asset (Ri) should be a linear function of its beta coefficient, Ri = Rf + f(RM - R), where Rf is the riskless rate of return and RM is the expected return to a market index of all risky assets. The SIM simply is an approximation technique. The CAPM is a model of equilibrium returns for risky assets. While the SIM may or may not be useful as an approximation to the Markowitz portfolio model, it is clear that there is no reason for crop returns in a particular region (on their excess returns over the rental rate) to be precisely related to their systematic risk. Therefore, Blank's analysis is fundamentally flawed because it does not test the SIM in any way, but instead tests and rejects CAPM hypotheses that should be rejected a priori. Sharpe (1963) proposed the SIM as a normative tool which could be used to approximate the Markowitz portfolio model. His primary objectives were conservation of computer time and dealing with large portfolio problems on small computers. However, an additional feature was that the beta coefficient obtained from regressing the individual activity returns on an index could be regarded as a measure of the risk that a particular activity would add to a well diversified portfolio. No such measure exists for the Markowitz model. For a model that considers n potential activities, one must consider the combined effects of an activity's variance along with the effects of n - 1 covariances. In this spirit, Collins and Barry proposed using the SIM for farm planning as an approximation of a full covariance model for farm planning. The SIM has several possible advantages for farm planning in the context of diversified farming. It allows large problems to be quickly solved on very small computers (or even hand calculators) and the betas provide a basis for comparing the risk contributions of alternative crops. In addition, only 3n parameters must be estimated to evaluate the risk of a SIM portfolio, while (n 2 + n)/2 parameters must be estimated for the full covariance model. This reduces the estimation required from limited data when n > 5. Most of all, they thought the SIM might have more intuitive appeal to farmers than the Markowitz model. But the important point is that the SIM is intended to have a normative purpose, i.e., provide a simple approximation of a Markowitz model for formulating approximately risk efficient farm plans. Tests of the robustness should evaluate this objective, i.e., the SIM is robust when it approximates