Question

You are given the following information concerning three portfolios, the market portfolio, and the risk-free asset:

Portfolio	RP		σP		βP
X	12.5	%	34	%	1.50
Y	11.5		29		1.20
Z	7.1		19		0.80
Market	10.5		24		1.00
Risk-free	6.2		0		0

What are the Sharpe ratio, Treynor ratio, and Jensen’s alpha for each portfolio?

Answer 1

Sharpe ratio of portfolio X=(Rp-Risk free)/(Standard deviation of portfolio X)=(12.5%-6.2%)/(34%)=0.185294
Sharpe ratio of portfolio Y=(Rp-Risk free)/(Standard deviation of portfolio y)=(11.5%-6.2%)/(29%)=0.1827586
Sharpe ratio of portfolio Z=(Rp-Risk free)/(Standard deviation of portfolio Z)=(7.1%-6.2%)/(19%)=0.0473684

Teynor ratio of portfolio X=(Rp-Risk free)/(Portfolio beta)=(12.5%-6.2%)/(1.5)=0.042
Teynor ratio of portfolio Y=(Rp-Risk free)/(Portfolio beta)=(11.5%-6.2%)/(1.2)=0.044167
Teynor ratio of portfolio Z=(Rp-Risk free)/(Portfolio beta)=(7.1%-6.2%)/(0.8)=0.01125

Jensen's alpha = Portfolio return - [Risk Free Rate + (Portfolio Beta)*(Market Return - Risk Free Rate)]

Jensen's alpha of portfolio X=12.5% - [6.2% + (1.5)*(10.5% - 6.2%)]=12.5% - (0.1265)=-0.0015 or -0.15%
Jensen's alpha of portfolio Y=11.5% - [6.2% + (1.2)*(10.5% - 6.2%)]=11.5% - (0.1136)=0.0014 or 0.14%
Jensen's alpha of portfolio Z=7.1% - [6.2% + (0.8)*(10.5% - 6.2%)]=7.1% - (0.0964)=-0.0254 or -2.54%