Question

Consider the two empirical models for excess returns (Ri-Rf) of stocks A and B. The risk free rate (Rf) over the period was 6%, and the market’s average return (Rm) was 14%. Stock A Stock B Estimated market models Ri-Rf= 1% + 1.2(Rm – Rf) Ri-Rf = 2% + 0.8(Rm – Rf) Standard deviation of excess returns 21.6% 24.9% Find the following for each stock: i) Alpha ii) Sharpe ratio iii) Treynor ratio

Answer 1

i) Alpha of stock A = Actual return - expected return

Alpha = Actual return - (Risk free rate + beta (market return - risk free rate)

Alpha = Actual return - Risk free rate - beta (market return - risk free rate)

Actual return - Risk free rate = Alpha + beta (market return - risk free rate)

Ri - Rf = Alpha + beta (Rm - Rf)

Seeing the above equation

Ri - Rf = 2% + beta ( Rm - Rf)

Alpha of Stock A = 1%

Alpha of Stock B = 2%

ii) Expected return of stock A = Risk free rate + beta (Market return - risk free rate)

= 6% + 1.2 (14% - 6%)

= 6% + 1.2 (8%)

= 6% + 9.6%

= 15.6%

Sharpe ratio of stock A= Expected return of Stock A - risk free rate / standard deviation of excess return

= 15.6% - 6% / 21.6%

= 9.6% / 21.6%

= 0.44

Expected return of Stock B= Rf + beta (Rm - Rf)

= 6% + 0.8 (14% - 6%)

= 6% + 0.8 (8%)

= 6% + 6.4%

= 12.4%

Sharpe ratio of Stock B = Expected return of Stock B - Risk free rate / standard deviation of excess return

= 12.4% - 6% / 24.9%

= 6.4% / 24.9%

= 0.26

iii) Treynor ratio of stock A = Expected return - risk free rate / beta

= 15.6% - 6% / 1.2

= 9.6% / 1.2

= 8

Treynor ratio of Stock B= Expected return - risk free rate / beta

= 12.4% - 6% / 0.8

= 6.4% / 0.8

= 8