Question

Consider the two (excess return) index-model regression results for stocks A and B. The risk-free rate over the period was 7%, and the market’s average return was 15%. Performance is measured using an index model regression on excess returns.

	Stock A	Stock B
Index model regression estimates	1% + 1.2(rM − rf)	2% + 0.8(rM − rf)
R-square	0.641	0.469
Residual standard deviation, σ(e)	11.4%	20.2%
Standard deviation of excess returns	22.7%	27.1%

a. Calculate the following statistics for each stock: (Round your answers to 4 decimal places.)

	Stock A Stock B i. Alpha % % ii. Information ratio iii. Sharpe ratio iv. Treynor measure

Answer 1

Solution a) Aplha is calculated as follows:

b) Information Ratio is calculated as:

For Stock A, Information Ratio = 1%/11.4% = 0.087719 = 0.0877

For Stock A, Information Ratio = 2%/20.2% = 0.09901 = 0.0990

c) Sharpe Ratio is calculated as follows:

Thus, Sharpe Ratio for Stock A: [1% + 1.2(r_M − r_f)]/22.7%

r_M =15%

r_f = 7%

Putting the values, we get,

Sharpe Ratio for Stock A = [1% + 1.2*(15% - 7%)]/22.7% = [1% + 1.2*8%]/22.7% = [1% + 9.6%]/22.7% = 10.6%/22.7% = 0.46696

= 0.4670

Thus, Sharpe Ratio for Stock B: [2% + 0.8(r_M − r_f)]/27.1%

r_M =15%

r_f = 7%

Putting the values, we get,

Sharpe Ratio for Stock B = [2% + 0.8*(15% - 7%)]/27.1% = [1% + 0.8*8%]/27.1% = [1% + 6.4%]/27.1% = 7.4%/27.1% = 0.276063

= 0.2761

d) Treynor Ratio is calculated as follows:

For Stock A, Treynor Ratio = [1% + 1.2(r_M − r_f)]/1.2

r_M =15%

r_f = 7%

Putting the values, we get,

Treynor Ratio for Stock A = [1% + 1.2*(15% - 7%)]/1.2 = [1% + 1.2*8%]/1.2 = [1% + 9.6%]/1.2 = 10.6%/1.2 = 0.088333

= 0.0883

Treynor Ratio for Stock B: [2% + 0.8(r_M − r_f)]/0.8

r_M =15%

r_f = 7%

Putting the values, we get,

Treynor Ratio for Stock B = [2% + 0.8*(15% - 7%)]/0.8 = [1% + 0.8*8%]/0.8 = [1% + 6.4%]/0.8 = 7.4%/0.8 =

= 0.0925

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