Explain how the Sharpe Index Model is different from Multi Index Models.

The Sharpe single index model imposes restrictions on how security returns can co-vary. In particular, it is assumed that all covariance arises through an index. This leads to a dramatic reduction in complexity. The multi index model assumes a return generating process that is a linear function of many factors. In this approach each factor is a source of systematic risk. Since investors cannot diversify systematic risk, they are assumed to be compensated for bearing this risk. As a result a security's sensitivity to each factor affects the assumed return generating process for the security. This sensitivity is captured by a beta factor. The last term is the source of idiosyncratic or diversifiable risk. Except for the factor that multiple factors make this a richer model than the single index model, the use of a multi factor model is similar to the single index model.