Where the risk-free rate, portfolio returns, and standard deviations are supplied, the appropriate risk-adjusted comparison is the Sharpe ratio:

Sharpe ratio = (Portfolio return − Risk-free rate) ÷ Standard deviation

For Portfolio W:

(18% − 6%) ÷ 25% = 0.48

For Portfolio X:

(19% − 6%) ÷ 30% = 0.433

For Portfolio Y:

(21% − 6%) ÷ 20% = 0.75

For Portfolio Z:

(23% − 6%) ÷ 22% = 0.773

Portfolio Z therefore produces the highest excess return per unit of total portfolio risk.

Although Portfolio Y has the lowest standard deviation and an attractive return, its Sharpe ratio of 0.75 remains slightly below Portfolio Z's approximately 0.77. Portfolio Z's higher absolute return more than compensates for its slightly greater volatility.

This illustrates why portfolios should not be evaluated solely by absolute return. A portfolio generating a higher return may be inferior if it requires disproportionately greater risk. Risk-adjusted performance measures normalize returns for the risk undertaken, allowing portfolios with different volatility profiles to be compared on a consistent basis.

FPII reference/topic: Investment and Tax Planning — modern portfolio theory; standard deviation; risk-free return; Sharpe ratio; risk-adjusted performance.

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