Abstract
In the presence of a risk-free asset the investment opportunity set obtained via the Markowitz portfolio optimization procedure is usually characterized in terms of the vector of excess returns on individual risky assets and the variance-covariance matrix. We show that the investment opportunity set can alternatively be characterized in terms of the vector of Sharpe ratios of individual risky assets and the correlation matrix. This implies that the changes in the characteristics of individual risky assets that preserve the Sharpe ratios and the correlation matrix do not change the investment opportunity set. The alternative characterization makes it simple to perform a comparative static analysis that provides an answer to the question of what happens with the investment opportunity set when we change the risk-return characteristics of individual risky assets. We demonstrate the advantages of using the alternative characterization of the investment opportunity set in the investment practice. The Sharpe ratio thinking also motivates reconsidering the CAPM relationship and adjusting Jensen's alpha in order to properly measure abnormal portfolio performance.
Highlights
The mean-variance model of asset choice has been proposed by Markowitz [1] and used extensively in finance principally due to a strong intuitive appeal and the existence of closed-form solutions to the optimal portfolio choice and equilibrium problems
We show that the investment opportunity set can alternatively be characterized in terms of the vector of Sharpe ratios of individual risky assets and the correlation matrix
What if we increase both the expected return and standard deviation? An alternative characterization of the investment opportunity set by means of the Sharpe ratios of individual risky assets makes it possible to provide the answer to this question
Summary
The mean-variance model of asset choice has been proposed by Markowitz [1] and used extensively in finance principally due to a strong intuitive appeal and the existence of closed-form solutions to the optimal portfolio choice and equilibrium problems. In this paper we show that in the presence of a riskfree asset the investment opportunity set can alternatively be characterized in terms of the vector of the Sharpe ratios of risky assets and the correlation matrix. This implies that the changes in the characteristics of individual risky assets that preserve the Sharpe ratios of risky assets and the correlation matrix do not change the investment opportunity set. The alternative characterization of the investment opportunity set provides a simple answer to the question of what happens with the investment opportunity set when we change the risk-return characteristics of individual risky assets.
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