Sharpe's Single-Index Model

Sharpe's single-index model (1963) offers a structural shortcut. If every stock's return can be decomposed into a market-wide component and a stock-specific residual, and if those residuals are uncorrelated with each other, then the entire $N \times N$ covariance matrix collapses into $3N + 2$ parameters: an alpha, a beta, and a residual variance per stock, plus the mean and variance of the market. This reduces the estimation burden from $O(N^2)$ to $O(N)$ while preserving the key property that makes portfolio analysis tractable: the model tells you, analytically, how any portfolio's risk splits into a diversifiable part that vanishes as $N$ grows and a non-diversifiable part that does not.

• write down the single-index return-generating process and state its defining assumption precisely;
• derive the three consequences of the model — the formula for expected return, variance, and pairwise covariance of any stock;
• compute portfolio expected return and variance under the single-index structure;
• explain why diversification eliminates idiosyncratic risk but not systematic (beta) risk;
• estimate $\alpha_i$, $\beta_i$, and $\sigma^2_{e_i}$ from a time-series regression and interpret each parameter.

1. The input explosion in full covariance models
2. The single-index model
3. Three key results
4. Portfolio risk under the single-index model
5. Diversification and beta: a key insight
6. Estimating beta