Sharpe (Ratio) Thinking about the Investment Opportunity Set and CAPM Relationship

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.

1. Introduction

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. The papers by Merton [2] and Roll [3] are the two seminal papers where the authors provide a rigorous characterization of the investment opportunity set in the absence and in the presence of a risk-free asset.

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 2 we review the classical characterization of the investment opportunity set in the presence of a risk-free asset. In Section 3 we derive the alternative characterization of the investment opportunity set and perform a comparative static analysis. In this section we also present several theoretical examples and one real-world example that illustrate the derived relationships and demonstrate the advantages of the alternative characterization of the investment opportunity set. In Section 4 we reconsider the CAPM relationship and introduce an adjusted Jensen's alpha. Section 5 summarizes the paper.

2. The Maximum Sharpe Ratio

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 [4, Chapter  3].

We suppose that the investment universe consists of 𝑛+1 assets with returns 𝑥_𝑖,𝑖=1,…,𝑛. Asset 0 is a risk-free asset which provides a return of 𝑟. The market is frictionless and the assets can be bought and sold short without any limitations. We suppose that the investor allocates a proportion 𝑤_𝑖 of his wealth to asset 𝑖. That is, the return on the investor's complete portfolio is given by 𝑥_𝑝=𝑤₀𝑟+^𝑛_𝑖=1𝑤_𝑖𝑥_𝑖,s.t.^𝑛_𝑖=0𝑤_𝑖=1. (1) This allows us to write the return on the complete portfolio as 𝑥_𝑝=𝑟+^𝑛_𝑖=1𝑤_𝑖𝑥_𝑖−𝑟. (2) We denote the expected return on asset 𝑥_𝑖 by 𝐸[𝑥_𝑖] and the variance by Var[𝑥_𝑖]=𝜎²_𝑖. We also denote Cov(𝑥_𝑖,𝑥_𝑗)=𝜎_𝑖𝑗=𝜌_𝑖𝑗𝜎_𝑖𝜎_𝑗, where 𝜌_𝑖𝑗=𝜌_𝑗𝑖 is the correlation coefficient between the returns on assets 𝑖 and 𝑗. In matrix notation the vectors of expected returns on the risky assets, the portfolio weights of the risky assets, and the variance-covariance matrix are given by 𝐸𝑥𝐞=¹𝐸𝑥₂⋮𝐸𝑥_𝑛𝑤,𝐰=¹𝑤₂⋮𝑤_𝑛,𝜎𝐕=¹¹𝜎₁₂…𝜎_1𝑛𝜎₂₁𝜎₂₂…𝜎_2𝑛𝜎⋮⋮⋱⋮_𝑛1𝜎_𝑛2…𝜎_𝑛𝑛. (3) Consequently, in matrix notation 𝐸𝑥_𝑝=𝐰^′(𝐞−𝟏𝑟)+𝑟,Var𝑥_𝑝=𝐰^′𝐕𝐰, (4) where 𝟏 is a vector of ones. We suppose that the return on any risky asset cannot be expressed as a linear combination of the returns on other assets. Under this assumption, asset returns are linearly independent and the variance-covariance matrix is nonsingular (which also means that it is invertible). The variance-covariance matrix is symmetric because 𝜎_𝑖𝑗=𝜎_𝑗𝑖. Since the portfolio variance is positive, we conclude that the variance-covariance matrix is positive definite.

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, 𝜇. That is, 𝐰 is a solution to the following quadratic program: min_𝐰12𝐰^′𝐕𝐰,s.t.𝐰^′(𝐞−𝟏𝑟)+𝑟=𝜇. (5) Forming the Lagrangian, we know that 𝐰 is the solution to the following: min_𝐰,𝜆1𝐿=2𝐰^′𝐕𝐰+𝜆𝜇−𝐰^′(𝐞−𝟏𝑟)−𝑟. (6) Since matrix 𝐕 is positive definite, this insures that the first-order condition is necessary and sufficient for a global minimum. The first-order condition for the optimality of 𝐰 gives 𝐕𝐰−𝜆(𝐞−𝟏𝑟)=0. (7) Solving with respect to 𝐰, we obtain 𝐕𝐰=⁻¹(𝐞−𝟏𝑟)𝜇−𝑟(𝐞−𝟏𝑟)^′𝐕⁻¹(𝐞−𝟏𝑟). (8) The investment opportunity set is defined as all risk-return combinations available to the investor. The computation of the variance of the minimum-variance portfolio gives Var𝑥_𝑝=𝐰^′𝐸𝑥𝐕𝐰=^𝑝−𝑟²(𝐞−𝟏𝑟)^′𝐕⁻¹(𝐞−𝟏𝑟). (9) It is easy to observe that in the standard deviation—expected return plane the investment opportunity set represents a straight line:𝐸𝑥_𝑝=𝑟+𝑆Var𝑥_𝑝, (10) where 𝑆=(𝐞−𝟏𝑟)^′𝐕⁻¹(𝐞−𝟏𝑟) (11) denotes the slope of this straight line. Note that since 𝐸𝑥𝑆=^𝑝−𝑟Var𝑥_𝑝, (12)𝑆 can be interpreted as the Sharpe ratio of any portfolio in the investment opportunity set (excluding the case where 𝐸[𝑥_𝑝]=𝑟). Observe that the Sharpe ratio of any portfolio in the investment opportunity set has the maximum Sharpe ratio among all feasible portfolios 𝑥_𝑝 that satisfy the condition ∑^𝑛_𝑖=0𝑤_𝑖=1. Therefore, we will refer to 𝑆 as the maximum Sharpe ratio. Consequently, the investment opportunity set in the presence of a risk-free asset can be characterized by the maximum Sharpe ratio. Finally recall that the investment opportunity set is tangent to the efficient frontier of risky assets. The composition of the tangency portfolio is the same as the composition of any risky portfolio in the investment opportunity set and the weights of the tangency portfolio must sum up to 1. This gives us 𝟏^′𝐰=𝜇−𝑟(𝐞−𝟏𝑟)^′𝐕⁻¹()𝟏𝐞−𝟏𝑟^′𝐕⁻¹(𝐞−𝟏𝑟)=1. (13) Note that (this will be used later in the paper) 𝟏^′𝐕⁻¹(𝐞−𝟏𝑟)>0 (14) since for a portfolio in the investment opportunity set 𝜇−𝑟>0 and (𝐞−𝟏𝑟)^′𝐕⁻¹(𝐞−𝟏𝑟)=𝑆²>0. The combination of (13) and (8) gives us the following expression for the weights of risky assets in the tangency portfolio:𝐰_𝑡=𝐕⁻¹(𝐞−𝟏𝑟)𝟏^′𝐕⁻¹(𝐞−𝟏𝑟). (15)

3. Alternative Characterization of the Maximum Sharpe Ratio

First we introduce the vector of the Sharpe ratios of individual risky assets and the correlation matrix 𝑆𝑥𝐒=¹𝑆𝑥₂⋮𝑆𝑥_𝑛=𝐸𝑥₁−𝑟𝜎₁𝐸𝑥₂−𝑟𝜎₂⋮𝐸𝑥_𝑛−𝑟𝜎_𝑛,𝐑=1𝜌₁₂⋯𝜌_1𝑛𝜌₂₁1⋯𝜌_2𝑛𝜌⋮⋮⋱⋮_𝑛1𝜌_𝑛2.⋯1 (16) Note that since matrix 𝐕 is invertible and matrix 𝐑 is essentially a normalized version of matrix 𝐕, this means that matrix 𝐑 is also invertible. Now we are ready to state the main result.

Theorem 1. 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, 𝑆=𝐒^′𝐑⁻¹𝐒. (17)

Proof. Define the diagonal matrix of standard deviations of the returns on individual risky assets 𝚺=diag𝜎₁,…,𝜎_𝑛=𝜎₁0⋯00𝜎₂⋯0⋮⋮⋱⋮00⋯𝜎_𝑛. (18) Since the entries 𝜎₁,…,𝜎_𝑛 of the diagonal matrix are all nonzero, the diagonal matrix of standard deviations is invertible. The variance-covariance matrix can, thus, be written as (note that symmetry of 𝚺 implies 𝚺^′=𝚺) 𝐕=𝚺𝐑𝚺. (19) Since matrixes 𝚺 and 𝐑 are invertible, (𝚺𝐑𝚺)⁻¹=𝚺⁻¹𝐑⁻¹𝚺⁻¹. (20) Consequently, the maximum Sharpe ratio can be rewritten as (now note that, since 𝚺 is symmetric, 𝚺⁻¹ is also symmetric) 𝑆=(𝐞−𝟏𝑟)^′(𝚺𝐑𝚺)⁻¹()=𝐞−𝟏𝑟𝚺⁻¹(𝐞−𝟏𝑟)^′𝐑⁻¹𝚺⁻¹(𝐞−𝟏𝑟). (21) Let us find the expression for 𝚺⁻¹=adj(𝚺)(𝚺)=1det(𝚺)Σdet¹¹Σ₂₁⋯Σ_𝑛1Σ₁₂Σ₂₂⋯Σ_𝑛2Σ⋮⋮⋱⋮_1𝑛Σ_2𝑛⋯Σ_𝑛𝑛, (22) where det(𝚺) and adj(𝚺) are the determinant and the adjoint of 𝚺, respectively, and Σ_𝑖𝑗 is an element of the cofactor matrix of 𝚺. Since all off-diagonal elements of 𝚺 are zeros, we arrive at det(𝚺)=^𝑛_𝑘=1𝜎_𝑘,Σ_𝑗𝑖=0for𝑖≠𝑗,_{𝑛𝑘=1𝑘≠𝑖}𝜎_𝑘for𝑖=𝑗. (23) Note that, whereas det(𝚺) is a product of all 𝜎_𝑘 (𝑘∈[1,…,𝑛]), Σ_𝑖𝑖 is a product of all 𝜎_𝑘 except 𝜎_𝑖. Consequently, 𝚺⁻¹=1𝜎₁010⋯0𝜎₂1⋯0⋮⋮⋱⋮00⋯𝜎_𝑛. (24) Consider the vector 𝚺⁻¹(𝐞−𝟏𝑟). It equals 𝚺⁻¹()=1𝐞−𝟏𝑟𝜎₁010⋯0𝜎₂1⋯0⋮⋮⋱⋮00⋯𝜎_𝑛𝐸𝑥₁𝐸𝑥−𝑟₂⋮𝐸𝑥−𝑟_𝑛=𝐸𝑥−𝑟₁−𝑟𝜎₁𝐸𝑥₂−𝑟𝜎₂⋮𝐸𝑥_𝑛−𝑟𝜎_𝑛=𝐒. (25) Finally, (21) for the maximum Sharpe ratio can be rewritten as (17).

Corollary 1. In case the returns on the risky assets in the investment universe are not correlated, that is, 𝜌_𝑖𝑗=0 for 𝑖≠𝑗, then the maximum Sharpe ratio equals 𝑆=^𝑛_𝑖=1𝑆²𝑥_𝑖. (26)

Proof. In this case 𝐑=𝐈_𝑛 where 𝐈_𝑛 is the identity matrix of order 𝑛. For the identity matrix 𝐈_𝑛⁻¹=𝐈_𝑛. Consequently, 𝑆=𝐒^′𝐑⁻¹𝐒=𝐒^′𝐈_𝑛⁻¹𝐒=𝐒^′𝐈_𝑛𝐒=𝐒^′𝐒=^𝑛_𝑖=1𝑆²𝑥_𝑖. (27)

Now we turn to the presentation of some examples.

Example 1 (𝑛=1). In this simplest case the investment universe consists of one risky asset and a risk-free asset. The application of (17) gives 𝑆=𝑆²𝑥₁=𝑆𝑥₁, (28) which means that the maximum Sharpe ratio equals the Sharpe ratio of the single risky asset (supposing that 𝑆(𝑥₁)>0).

Example 2 (𝑛=2 with graphical illustration). In this case the investment universe consists of two risky assets and a risk-free asset. The computation of the maximum Sharpe ratio gives 𝑆=𝑆²𝑥₁+𝑆²𝑥₂−2𝜌₁₂𝑆𝑥₁𝑆𝑥₂1−𝜌²₁₂. (29) It can be easily checked that when 𝜌₁₂=0 then 𝑆=𝑆²𝑥₁+𝑆²𝑥₂. (30) The alternative characterization of the maximum Sharpe ratio by means of Sharpe ratios of individual risky assets and the correlation matrix allows, at least for the case of two risky assets, to visualize the dependence of the maximum Sharpe ratio on the correlation coefficient, see Figure 1.

Figure 1 

Maximum Sharpe ratio of two risky assets versus their correlation coefficient. In this example 𝑆(𝑥1)=0.4, 𝑆(𝑥2)=0.25, and we vary the correlation coefficient 𝜌12 in [−0.95,0.95].

Example 3 (𝑛=3). 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 𝑆=𝐶1−𝜌²₁₂−𝜌²₁₃−𝜌²₂₃+2𝜌₁₂𝜌₁₃𝜌₂₃, (31) where 𝐶=𝑆²𝑥₁1−𝜌²₂₃+𝑆²𝑥₂1−𝜌²₁₃+𝑆²𝑥₃1−𝜌²₁₂−2𝑆𝑥₁𝑆𝑥₂𝜌₁₂−𝜌₁₃𝜌₂₃−2𝑆𝑥₂𝑆𝑥₃𝜌₂₃−𝜌₁₂𝜌₁₃−2𝑆𝑥₁𝑆𝑥₃𝜌₁₃−𝜌₁₂𝜌₂₃. (32) Again, it can be easily checked that, if 𝜌₁₂=𝜌₁₃=𝜌₂₃=0, then 𝑆=𝑆²𝑥₁+𝑆²𝑥₂+𝑆²𝑥₃. (33)

Example 4 (illustration of classical versus alternative characterizations for 𝑛=3). 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: 𝑆(𝑥₁)=0.50, 𝑆(𝑥₂)=0.40, and 𝑆(𝑥₃)=0.25. Given a particular value of a Sharpe ratio, there are infinite combinations of the expected return and standard deviation that produce this Sharpe ratio. For the sake of simplicity, we consider only two sets of risky assets, where each of the risky assets has the same Sharpe ratio as given above. In particular, the first set of parameters for the risky assets is 𝐸[𝑥₁]=20%, 𝜎₁=30%, 𝐸[𝑥₂]=20%, 𝜎₂=37.5%, 𝐸[𝑥₃]=10%, and 𝜎₃=20%. The second set of parameters for the risky assets is 𝐸[𝑥₁]=27.5%, 𝜎₁=45%, 𝐸[𝑥₂]=15%, 𝜎₂=25%, 𝐸[𝑥₃]=15%, and 𝜎₃=40%. The correlation coefficients between the returns of the risky assets are the same in both cases: 𝜌₁₂=0.5 and 𝜌₂₃=𝜌₃₁=−0.2.
Figure 2(a) illustrates the classical characterization of the investment opportunity set. This is a two-dimensional standard deviation-expected return space. Observe that in this space the risk-return characteristics of the two sets of risky assets are seemingly different with different minimum-variance frontiers of risky assets and different compositions of the tangency portfolios. However, in both cases the investment opportunity set is the same straight line. Figure 2(b) presents an illustration of the alternative characterization of the investment opportunity set. This is a one-dimensional Sharpe ratio space. In contrast to the classical illustration, the two sets of risky assets fully coincide. The investment opportunity set is represented by the maximum Sharpe ratio.

(a) Classical

(b) Alternative

Figure 2 

Classical and alternative characterizations of the investment opportunity set.

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 𝑆 with respect to 𝑆(𝑥_𝑖) is given by 𝜕𝑆=𝜕𝜕𝐒𝑆𝜕𝑆𝑥₁𝜕𝑆𝜕𝑆𝑥₂⋮𝜕𝑆𝜕𝑆𝑥_𝑛=𝜕𝐒^′𝐑⁻¹𝐒=𝐑𝜕𝐒⁻¹𝐒𝐒^′𝐑⁻¹𝐒=𝐑⁻¹𝐒𝑆. (34) Intuitively we expect the maximum Sharpe ratio to increase when we increase the Sharpe ratio of a single risky asset in the investment universe. However, this intuition is not always correct. The subsequent theorem characterizes the condition at which this intuition is perfectly correct.

Theorem 2. The sign of the first-order derivative of the maximum Sharpe ratio with respect to the Sharpe ratio of risky asset 𝑖 coincides with the sign of the weight of risky asset 𝑖 in the optimal risky portfolio. In particular, 𝜕sgn𝑆𝜕𝑆𝑥_𝑖=sgn𝑤_𝑖, (35) where sgn(⋅) is the sign function.

Remark 1. This theorem says that if the weight of risky asset 𝑖 in the optimal risky portfolio is positive, then an increase in the Sharpe ratio of asset 𝑖 results in an increase in the maximum Sharpe ratio. Conversely, if the weight of risky asset 𝑖 in the optimal risky portfolio is negative, then an increase in the Sharpe ratio of asset 𝑖 results in a decrease in the maximum Sharpe ratio.

Remark 2. 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.

Proof. The vector of partial derivatives of 𝑆 with respect to 𝑆(𝑥_𝑖) is given by (34). Observe that 𝑆 is a positive scalar and 𝐑⁻¹𝐒 is a vector, where each element 𝑖 is given by ∑^𝑛_𝑗=1𝑆𝑥_𝑗𝑅_𝑗𝑖det(𝐑), (36) where 𝑅_𝑗𝑖 is an element of the cofactor matrix of 𝐑. Consequently, the vector of partial derivatives of 𝑆 with respect to 𝑆(𝑥_𝑖) is given by 𝜕𝑆𝜕𝑆𝑥₁𝜕𝑆𝜕𝑆𝑥₂⋮𝜕𝑆𝜕𝑆𝑥_𝑛=1det(𝐑)𝑆^𝑛𝑗=1𝑆𝑥_𝑗𝑅_{𝑛𝑗1𝑗=1}𝑆𝑥_𝑗𝑅_{𝑗2𝑛𝑗=1}⋮𝑆𝑥_𝑗𝑅_𝑗𝑛. (37)
The signs of the weights of risky assets in the investor's optimal risky portfolio (as given by (8)) coincide with the signs of the weights of the risky assets in the tangency portfolio (as given by (15)), that is, sgn(𝑤_𝑖)=sgn(𝑤_𝑡𝑖). Let us write the weights of the risky assets in the tangency portfolio as 𝐰_𝑡=𝑏𝐕⁻¹(𝐞−𝟏𝑟), (38) where 𝑏=1/𝟏^′𝐕⁻¹(𝐞−𝟏𝑟) is a positive scalar; see (14). Then in the proof we will employ the results derived in the proof of Theorem 1. We establish the following: 𝐰_𝑡=𝑏𝐕⁻¹(𝐞−𝟏𝑟)=𝑏(𝚺𝐑𝚺)⁻¹(𝐞−𝟏𝑟)=𝑏𝚺⁻¹𝐑⁻¹𝚺⁻¹(𝐞−𝟏𝑟)=𝑏𝚺⁻¹𝐑⁻¹𝐒. (39) We already know how the matrix 𝚺⁻¹ looks like and the elements of the vector 𝐑⁻¹𝐒. This knowledge gives us 𝑤_𝑡1𝑤_𝑡2⋮𝑤_𝑡𝑛=𝑏(𝐑)1det𝜎₁010⋯0𝜎₂1⋯0⋮⋮⋱⋮00⋯𝜎_𝑛^𝑛𝑗=1𝑆𝑥_𝑗𝑅_{𝑛𝑗1𝑗=1}𝑆𝑥_𝑗𝑅_{𝑗2𝑛𝑗=1}⋮𝑆𝑥_𝑗𝑅_𝑗𝑛=𝑏(𝐑)1det𝜎₁^𝑛_𝑗=1𝑆𝑥_𝑗𝑅_𝑗11𝜎₂^𝑛_𝑗=1𝑆𝑥_𝑗𝑅_𝑗2⋮1𝜎_𝑛^𝑛_𝑗=1𝑆𝑥_𝑗𝑅_𝑗𝑛. (40) Note that 𝑆>0, det(𝐑)>0, and 𝜎_𝑖>0 for all 𝑖. Finally observe from (37) that sgn(𝜕𝑆/𝜕𝑆(𝑥_𝑖))=sgn(∑^𝑛_𝑗=1𝑆(𝑥_𝑗)𝑅_𝑗𝑖) and from (40) that sgn(𝑤_𝑡𝑖)=sgn(∑^𝑛_𝑗=1𝑆(𝑥_𝑗)𝑅_𝑗𝑖). Thus, we conclude that sgn(𝜕𝑆/𝜕𝑆(𝑥_𝑖))=sgn(𝑤_𝑖).

Example 5 (𝑛=2). Consider the case where the investment universe consists of two risky assets and a risk-free asset. Suppose that 𝑟=5%, 𝐸[𝑥₁]=9%, 𝜎₁=40%, 𝐸[𝑥₂]=20%, 𝜎₂=40%, and 𝜌₁₂=0.5. The Sharpe ratios of the assets are 𝑆(𝑥₁)=0.1 and 𝑆(𝑥₂)=0.375. The maximum Sharpe ratio in this case is given by (29) and equals 𝑆=0.3884. The weights of the risky assets in the tangency portfolio are 𝑤_𝑡1=−0.3684 and 𝑤_𝑡2=1.3684. Note that asset 1 is inferior to asset 2 (since asset 1 has a much lower Sharpe ratio as compared to asset 2), the correlation coefficient between the returns on these two assets is rather large, and it is optimal to sell short asset 1.
The computation of the first-order derivatives of the maximum Sharpe ratio with respect to individual Sharpe ratios gives 𝜕𝑆𝜕𝑆𝑥₁=𝑆𝑥₁−𝜌₁₂𝑆𝑥₂𝑆𝜕=−0.2253,𝑆𝜕𝑆𝑥₂=𝑆𝑥₂−𝜌₁₂𝑆𝑥₁𝑆=0.8368. (41) Observe that the signs of the first-order derivatives coincide with the signs of the weights. The sign of the first-order derivative of the maximum Sharpe ratio with respect to the Sharpe ratio of the first risky asset is negative which means that an increase in the Sharpe ratio of the first asset results in a decrease in the maximum Sharpe ratio. For the sake of illustration, suppose that the Sharpe ratio of the first asset increases to 𝑆(𝑥₁)=0.2. In this case the maximum Sharpe ratio decreases to 𝑆=0.3753. In contrast, the sign of the first-order derivative of the maximum Sharpe ratio with respect to the Sharpe ratio of the second risky asset is positive which means that an increase in the Sharpe ratio of the second asset results in an increase in the maximum Sharpe ratio. If, for instance, the Sharpe ratio of the second asset increases to 𝑆(𝑥₂)=0.475, the maximum Sharpe ratio increases to 𝑆=0.5008.

Example 6 (real-world example with 𝑛=15). Considerable academic research documents the benefits of international diversification; see, for example, Grubel [5], Levy and Sarnat [6], and Solnik [7]. The gains from international diversification supposedly stem from relatively low correlations among international stock markets. However, academic research also documents that the average correlation among international stock markets has been increasing over time; see King et al. [8], Longin and Solnik [9], and Goetzmann et al. [10]. Due to the deterioration in diversification opportunities, it is tempting to deduce that the efficiency of international investment has diminished (see, e.g., Eun and Lee [11]). In this example we consider the evolution of the international investment opportunity set through time from the point of view of an investor in the USA. In particular, our goal is to find out whether the efficiency of international investment has really deteriorated.
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 http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html). Monthly risk-free rates of return, proxied by the monthly US T-Bill rates, are obtained from the Ibbotson SBBI 2009 Classic Yearbook. To examine the evolution of the investment opportunity set through time, we divide our overall sample period 1975–2007 into two subperiods: 1975–1991 and 1992–2007.
The maximum Sharpe ratio (computed using either (11) or (17)) of the optimal international portfolio amounts to 0.268 for the first subperiod and to 0.290 for the second subperiod. Yet, consistent with previous findings, the average correlation among international stock markets has increased from 0.466 to 0.581. That is, despite a significant increase in average correlation among stock markets, the investment opportunity set has improved through time, at least for an American investor. Without the knowledge of the alternative characterization of the maximum Sharpe ratio, all we can say is that the improvement of the investment opportunity set has been caused by some favorable changes in the excess returns and standard deviations of returns on international stock markets. The knowledge of the alternative characterization of the maximum Sharpe ratio motivates us to investigate the changes in the average Sharpe ratio. We find that the average Sharpe ratio of international stock markets has increased from 0.126 to 0.148. Thus, whereas the increased correlations have exerted a negative impact, the increased Sharpe ratios have exerted a positive impact on the investment opportunity set. The combined effect of both the increased correlations and increased Sharpe ratios has exerted a positive impact on the efficiency of international investment.
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.

4. The CAPM Relationship Reconsidered

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:𝐸𝑥_𝑖=𝑟+𝛽_𝑖𝐸𝑥_𝑀−𝑟, (42) where 𝑥_𝑀 is the return on the market portfolio and 𝛽_𝑖 is given by 𝛽_𝑖=Cov𝑥_𝑖,𝑥_𝑀𝜎²_𝑀=𝜌_𝑖𝜎_𝑖𝜎_𝑀, (43) where 𝜌_𝑖 is the correlation coefficient between the returns on asset 𝑖 and the market portfolio and 𝜎_𝑀 is the standard deviation of returns on the market portfolio. The CAPM relationship can be rewritten as 𝐸𝑥_𝑖−𝑟𝜎_𝑖=𝜌_𝑖𝐸𝑥_𝑀−𝑟𝜎_𝑀, (44) that is, as the Sharpe ratio-rho relationship 𝑆𝑥_𝑖=𝜌_𝑖𝑆𝑥_𝑀. (45) This says that in equilibrium the Sharpe ratio of risky asset 𝑖 equals the Sharpe ratio of the market portfolio times the correlation coefficient between the returns on asset 𝑖 and the market portfolio. The Sharpe ratio is often alternatively called the price of risk. Thus, yet another formulation of the CAPM relationship could be the following: in equilibrium the price of risk of asset 𝑖 equals the market price of risk times the correlation coefficient between the returns on asset 𝑖 and the market. If 𝜌_𝑖=0, then 𝑆(𝑥_𝑖)=0, which means that there is no risk to be priced. The higher the correlation coefficient 𝜌_𝑖, the higher the Sharpe ratio 𝑆(𝑥_𝑖). Since 𝜌_𝑖≤1, we conclude that 𝑆(𝑥_𝑖)≤𝑆(𝑥_𝑀). That is, the Sharpe ratio of the market portfolio is the maximum Sharpe ratio.

The difference between the actual and equilibrium expected rates of return on risky asset 𝑖 is denoted by (Jensen's) alpha:𝛼𝑥_𝑖=𝐸𝑥_𝑖−𝑟−𝛽_𝑖𝐸𝑥_𝑀−𝑟. (46) Jensen's alpha is one of the most popular portfolio performance measures used in practice. However, it is widely known now that Jensen's alpha has one serious drawback. Namely, if 𝛼(𝑥_𝑖)>0, then the value of alpha can be manipulated by leverage (see, e.g., Ingersoll et al. [12]). In particular, by borrowing money at the risk-free rate one can create an asset with return 𝑦_𝑖=𝑎(𝑥_𝑖−𝑟)+𝑟, 𝑎>1. It is easy to check that 𝛼(𝑦_𝑖)=𝑎𝛼(𝑥_𝑖)>𝛼(𝑥_𝑖). That is, alpha can in theory be made as large as desirable by leverage. The problem with Jensen's alpha stems from the fact that alpha is the abnormal return unadjusted for risk.

To mitigate the leverage problem inherent in alpha one can employ the Treynor ratio (see Treynor [13]). However, there is another way around the leverage problem. Divide the left- and right-hand sides of (46) by 𝜎_𝑖. Then we obtain 𝛼^′𝑥_𝑖=𝛼𝑥_𝑖𝜎_𝑖=𝑆𝑥_𝑖−𝜌_𝑖𝑆𝑥_𝑀. (47) We will refer to 𝛼^′(𝑥_𝑖)=𝛼(𝑥_𝑖)/𝜎_𝑖 as the adjusted Jensen's alpha. The adjusted Jensen's alpha is easy to interpret: it is the abnormal return adjusted for risk, or the abnormal Sharpe ratio. The form of the adjusted Jensen's alpha resembles the form of the information ratio (also known as the appraisal ratio or the Treynor-Black ratio; see Treynor and Black [14]). However, whereas in the information ratio the Jensen's alpha is divided by the nonsystematic risk, in the adjusted version the Jensen's alpha is divided by the total risk. One can easily check that, for any 𝑎>0, 𝛼^′(𝑦_𝑖)=𝛼^′(𝑥_𝑖), so the adjusted Jensen's alpha cannot be manipulated by leverage.

Figure 3(a) presents the classical illustration of the CAPM relationship. This is a beta-expected return space. SML denotes the security market line. It is a straight line with intercept 𝑟 and slope 𝐸[𝑥_𝑀]−𝑟. If the market is in equilibrium, all assets should lie along the security market line. Figure 3(b) presents an alternative illustration of the CAPM relationship. This is a rho-Sharpe ratio space. SMLS denotes the security market line segment. It is a part of a straight line with zero intercept and slope 𝑆(𝑥_𝑀). If the market is in equilibrium, all assets should lie along the security market line segment. Figure 3 also demonstrates the values of the Jensen's alpha and the adjusted Jensen's alpha for asset 𝑖 which provides abnormal return.

(a) Classical

(b) Alternative

Figure 3 

Classical and alternative illustrations of the CAPM relationship.

5. Summary

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.

Acknowledgment

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