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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.

The mean-variance model of asset choice has been proposed by Markowitz [

In the absence of a risk-free asset the investment opportunity set is characterized by the vector of expected returns on risky assets and the variance-covariance matrix. In the presence of a risk-free asset the investment opportunity set is characterized by the vector of excess returns on risky assets and the variance-covariance matrix. To demonstrate the classical characterization of the investment opportunity set one typically employs a two-dimensional standard deviation-expected return space. Next one illustrates the construction of the minimum-variance frontier of risky assets. In the absence of a risk-free asset the efficient part of the minimum-variance frontier of risky assets coincides with the investment opportunity set. In the presence of a risk-free asset one draws a straight line (whose intercept and slope are equal to the risk-free rate of return and the maximum Sharpe ratio, resp.) which is tangent to the efficient frontier of risky assets.

In this paper we show that in the presence of a risk-free 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. In other words, two apparently different sets comprised of the same number of risky assets (different in terms of the values of expected returns and standard deviations), that produce apparently different minimum-variance frontiers of risky assets, will generate exactly the same investment opportunity set if the risky assets in the two sets have the same Sharpe ratios and correlation matrix.

The alternative characterization of the investment opportunity set in the presence of a risk-free asset implies that the two-dimensional standard deviation-expected return space, although instructive in teaching, can be redundant and misleading in practice. A more straightforward depiction of the characteristics of individual risky assets and the investment opportunity set can be done in one-dimensional Sharpe ratio space. For the case of two risky assets the alternative characterization allows to visualize the maximum Sharpe ratio as a function of the correlation coefficient.

In addition, 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. In other words, if we change the values of the expected returns and standard deviations of individual risky assets, will it result in an improvement or worsening of the investment opportunity set? Using the classical characterization of the investment opportunity set, the answer is not clear. With the alternative characterization of the investment opportunity set by means of the Sharpe ratios of individual risky assets, the answer is rather trivial. We show that if the weight of a risky asset in the optimal risky portfolio is positive, then an increase in the Sharpe ratio of this asset results in an improvement of the investment opportunity set. Moreover, if the market is in equilibrium, then an increase in the Sharpe ratio of any risky asset results in an increase in the maximum Sharpe ratio.

Finally, the alternative characterization of the investment opportunity set has a clear appeal to the investment practice. In particular, it is well known that the risk-return characteristics of individual risky assets and the correlations among them are changing over time. Existing academic studies usually address only the changing nature of correlations and try to deduce how these changes affect the investment opportunity set. The alternative characterization of the investment opportunity set motivates the idea that one also needs to examine the changing nature of the Sharpe ratios, in addition to that of the correlation structure, in order to have a full picture of the evolution of the investment opportunity set.

The classical CAPM is, in fact, none other than the Markowitz portfolio theory in the presence of a risk-free asset combined with an equilibrium condition. Therefore, the Sharpe ratio thinking motivates reconsidering the CAPM relationship in terms of Sharpe ratios. We show that the classical expected return-beta relationship can alternatively be represented by a Sharpe ratio-rho relationship (rho denotes the correlation coefficient between the returns on a risky asset and the market portfolio). Moreover, the Sharpe ratio thinking motivates adjusting Jensen's alpha in order to properly measure abnormal portfolio performance. Instead of Jensen's alpha we propose to use Jensen's alpha divided by the total risk. In this manner the adjusted Jensen's alpha becomes a true reward-to-risk performance measure that cannot be manipulated by leverage.

The rest of the paper is organized as follows. In Section

The purpose of this section is to introduce the notation, the investor's optimal portfolio choice problem, and to derive the expression for the maximum Sharpe ratio that characterizes the investment opportunity set in the presence of a risk-free asset. We do not provide all details of the derivation because there are no new results in this section. Our exposition in this section is similar to that in Huang and Litzenberger [

We suppose that the investment universe consists of

We further suppose that the investor exhibits mean-variance preferences. In particular, this investor prefers portfolios that have minimum variance for various levels of expected rate of return. The classical Markowitz portfolio optimization procedure consists in finding a portfolio that has minimum variance for a given level of expected return,

First we introduce the vector of the Sharpe ratios of individual risky assets and the correlation matrix

The maximum Sharpe ratio can be characterized in terms of the vector of Sharpe ratios of individual risky assets and the correlation matrix. In particular,

Define the diagonal matrix of standard deviations of the returns on individual risky assets

In case the returns on the risky assets in the investment universe are not correlated, that is,

In this case

Now we turn to the presentation of some examples.

In this simplest case the investment universe consists of one risky asset and a risk-free asset. The application of (

In this case the investment universe consists of two risky assets and a risk-free asset. The computation of the maximum Sharpe ratio gives

Maximum Sharpe ratio of two risky assets versus their correlation coefficient. In this example

In this case the investment universe consists of three risky assets and a risk-free asset. After more tedious but still straightforward calculations we obtain

This example illustrates the classical and alternative characterizations of the investment opportunity set in the presence of a risk-free asset. We suppose that the investment universe consists of a risk-free asset which provides the return of 5% and three risky assets that have the following Sharpe ratios:

Figure

Classical and alternative characterizations of the investment opportunity set.

Classical

Alternative

A natural question to ask is what happens with the investment opportunity set when we change the values of the expected returns and standard deviations of individual risky assets. Will it result in an improvement or worsening of the investment opportunity set? Using the classical characterization of the investment opportunity set, the answer is not clear. A good guess would be the following: if we increase the expected return of a single risky asset while keeping the standard deviation of this asset at the same level, this should improve the investment opportunity set. In contrast, if we increase the standard deviation of a single risky asset while keeping the expected return at the same level, this should worsen the investment opportunity set. But 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.

To answer this question we need to find the first-order derivatives of the maximum Sharpe ratio with respect to the Sharpe ratios of single risky assets. The vector of partial derivatives of

The sign of the first-order derivative of the maximum Sharpe ratio with respect to the Sharpe ratio of risky asset

This theorem says that if the weight of risky asset

If the market is in equilibrium and all risky assets are in positive net supply, then the weights of all risky assets in the market portfolio are positive. Hence, if the market is in equilibrium, then an increase in the Sharpe ratio of any risky asset results in an increase in the maximum Sharpe ratio.

The vector of partial derivatives of

The signs of the weights of risky assets in the investor's optimal risky portfolio (as given by (

Consider the case where the investment universe consists of two risky assets and a risk-free asset. Suppose that

The computation of the first-order derivatives of the maximum Sharpe ratio with respect to individual Sharpe ratios gives

Considerable academic research documents the benefits of international diversification; see, for example, Grubel [

Our sample of 15 developed markets spans the period from January 1975 to December 2007. The 15 developed markets are Australia, Belgium, France, Germany, Hong Kong, Italy, Japan, the Netherlands, Norway, Singapore, Spain, Sweden, Switzerland, the UK, and the USA. The monthly value-weighted dollar returns for each country are obtained from the data library of Kenneth French (see

The maximum Sharpe ratio (computed using either (

At the end of this example we would like to emphasize the advantage (in the investment practice) of using Sharpe ratios and correlations to describe the investment opportunity set. In particular, this example convincingly demonstrates that it is not enough to examine only the changes in diversification opportunities in order to deduce the changes in the investment opportunity set. One also needs to study the changes in the Sharpe ratios in order to have a complete picture of the evolution of the investment opportunity set.

So far we have reconsidered some results of the Markowitz portfolio theory in the presence of a risk-free asset. The classical CAPM is, in fact, none other than the Markowitz portfolio theory in the presence of a risk-free asset paired with an equilibrium condition. Therefore, it is pretty straightforward to reconsider the CAPM relationship in terms of Sharpe ratios.

The standard CAPM (expected return-beta) relationship is as follows:

The difference between the actual and equilibrium expected rates of return on risky asset

To mitigate the leverage problem inherent in alpha one can employ the Treynor ratio (see Treynor [

Figure

Classical and alternative illustrations of the CAPM relationship.

Classical

Alternative

In this paper we showed that in the presence of a risk-free asset the investment opportunity set can be characterized in terms of the vector of the Sharpe ratios of risky assets and the correlation matrix. Consequently, this implies that the changes in the risk-return characteristics of individual risky assets that preserve the Sharpe ratios and the correlation matrix do not change the investment opportunity set. We performed the comparative static analysis which provides a simple answer to the question of what happens with the investment opportunity set when we change the characteristics of individual risky assets. We demonstrated the advantages of using the alternative characterization of the investment opportunity set in the investment practice. Using the Sharpe ratio thinking we reconsidered the CAPM relationship and proposed how to adjust the Jensen's alpha in order to properly measure abnormal portfolio performance.

The author is grateful to Steen Koekebakker and the anonymous referees for their comments.