Under a single factor model three stocks have alphas of 2%, 1% and 3% with idiosyncratic volatilities of 20%, 10% and 30%. What weight does the maximum Sharpe active portfolio put on the second stock, and what is the combined appraisal ratio?

Under a single factor model three stocks have alphas of 2%, 1% and 3% with idiosyncratic volatilities of 20%, 10% and 30%. What weight does the maximum Sharpe active portfolio put on the second stock, and what is the combined appraisal ratio?

Approach: Under a factor model the residuals are independent, so the active weights follow alpha over residual variance. Then combine the individual alpha to risk ratios.

6/11. The active weights are proportional to alpha_i divided by idiosyncratic variance: 0.02/0.04 = 0.5, 0.01/0.01 = 1.0 and 0.03/0.09 = 1/3. They sum to 11/6, so the second stock takes 1/(11/6) = 6/11 = 54.5% and the others take 3/11 and 2/11. The smallest alpha carries the largest weight because its residual risk is a quarter of the first stock's. Each name has the same appraisal ratio alpha_i/sigma_i = 0.1, and with independent residuals the combined appraisal ratio is sqrt(0.1^2 + 0.1^2 + 0.1^2) = 0.173. That adds in quadrature to the factor Sharpe, so a factor Sharpe of 0.5 gives a total of sqrt(0.25 + 0.03) = 0.529.

Follow-up: What happens to those weights if the residuals have a common correlation of 0.2 rather than being independent?

Key concepts: appraisal ratio, idiosyncratic variance, active weights, alpha.