X-Sigma-Rho and Market Efficiency

DOI: http://dx.doi.org/10.18533/jefs.v2i02.133 © 2014 The Authors. This is an open access article under the terms of the Creative Commons Attribution License 4.0, which allows use, distribution and reproduction in any medium, provided the original work is properly cited. 1.0 Introduction A fundamental assertion of financial economic theory is that individuals will accept risk only if compensated through return. Moreover, the return must be increasing in the measure of risk implying that individuals have an aversion to risk and will judge expected return relative to underlying risk. Menchero and Davis (2011) define XSigma-Rho as a risk metric that shows the marginal contribution to risk when a security is added to a portfolio of assets. In this analysis the “X-Sigma-Rho” result is extended by adding marginal contribution to return. If both marginal risk and marginal return are considered under the equilibrium condition that all assets have the same reward to risk ratio, say Sharpe’s ratio, the capital/security market line (CML/SML) easily obtains from the XSigma-Rho framework. Thus, the Sigma-Rho part of Menchero and Davis’ metric is not surprisingly a determinant of expected return using either total or systematic risk. When the CML is expressed in Sigma-Rho terms the impact of diversification on relevant risk is evident. If there is no diversification effect (the security’s correlation with the market portfolio is +1.0) the CML provides an estimate of expected return based on total risk. For all other cases where at least some diversification exists (the security is not perfectly positively correlated with the market portfolio) the expected return is from the security market line or X-sigma-rho and market efficiency Baigent, G.C., JEFS (2014), 02(02), 41–44 Journal of Economic and Financial Studies. Page 42 Page 42 capital asset pricing model [Sharpe, 1964] and is dependent upon systematic risk. In this way the CML is a special case of the SML when Rho is equal to +1.0. In this framework the expected rate of return is a function of a security’s standard deviation relative to the standard deviation of the market (risk factor), but this risk factor is scaled by the correlation, rho, and can become a maximum of  i M or a minimum of    i M . Regardless of the value of rho, a normal rate of return can be determined so that the equilibrium or risk-based rate of return can be compared to the observed return to identify mis-priced assets, a measure that is a form of Jensen’s Alpha. The Sigma-Rho approach is compelling because it highlights the impact of correlation on portfolio formation. Strategies such as creating a zero-beta portfolio, a market-neutral portfolio, or the use of precious metals and other commodities have at their roots the correlation of securities; portfolios formed on this basis should tend towards the efficient frontier. The end result is, for two assets with the same risk-return characteristics, the security with the lower correlation to the existing portfolio will contribute the greatest efficiency. This can only be achieved if both return and risk are considered. 2.0 X-sigma-rho revisited Menchero and Davis (2011) and Davis and Menchero (2012) study the attribution of risk in the context of standard deviation and correlation. For completeness, X-Sigma-Rho is reviewed in this section using the same notation. Define a portfolio’s return as R in equation (1) where m g and m x are the return and proportional weight (or exposure) for each security m .


Introduction
A fundamental assertion of financial economic theory is that individuals will accept risk only if compensated through return.Moreover, the return must be increasing in the measure of risk implying that individuals have an aversion to risk and will judge expected return relative to underlying risk.Menchero and Davis (2011) define X-Sigma-Rho as a risk metric that shows the marginal contribution to risk when a security is added to a portfolio of assets.In this analysis the "X-Sigma-Rho" result is extended by adding marginal contribution to return.
If both marginal risk and marginal return are considered under the equilibrium condition that all assets have the same reward to risk ratio, say Sharpe's ratio, the capital/security market line (CML/SML) easily obtains from the X-Sigma-Rho framework.Thus, the Sigma-Rho part of Menchero and Davis' metric is not surprisingly a determinant of expected return using either total or systematic risk.
When the CML is expressed in Sigma-Rho terms the impact of diversification on relevant risk is evident.If there is no diversification effect (the security's correlation with the market portfolio is +1.0) the CML provides an estimate of expected return based on total risk.For all other cases where at least some diversification exists (the security is not perfectly positively correlated with the market portfolio) the expected return is from the security market line or capital asset pricing model [Sharpe, 1964] and is dependent upon systematic risk.In this way the CML is a special case of the SML when Rho is equal to +1.0.
In this framework the expected rate of return is a function of a security's standard deviation relative to the standard deviation of the market (risk factor), but this risk factor is scaled by the correlation, rho, and can become a maximum of Regardless of the value of rho, a normal rate of return can be determined so that the equilibrium or risk-based rate of return can be compared to the observed return to identify mis-priced assets, a measure that is a form of Jensen's Alpha.
The Sigma-Rho approach is compelling because it highlights the impact of correlation on portfolio formation.
Strategies such as creating a zero-beta portfolio, a market-neutral portfolio, or the use of precious metals and other commodities have at their roots the correlation of securities; portfolios formed on this basis should tend towards the efficient frontier.The end result is, for two assets with the same risk-return characteristics, the security with the lower correlation to the existing portfolio will contribute the greatest efficiency.This can only be achieved if both return and risk are considered.

2.0
X-sigma-rho revisited Menchero and Davis (2011) and Davis and Menchero (2012) study the attribution of risk in the context of standard deviation and correlation.For completeness, X-Sigma-Rho is reviewed in this section using the same notation.
Define a portfolio's return as R in equation ( 1) where m g and m x are the return and proportional weight (or exposure) for each security m .
Equation ( 1) is a weighted-average of the component returns constrained by 1), it follows that the variance or volatility of returns is The main thrust of Menchero and Davis' (2011) analysis is to derive a risk metric they refer to as X-Sigma-Rho.In


is the correlation of security m with the remainder of the portfolio, R .The volatility of the portfolio is, Menchero and Davis show that differentiating (2b) with respect to m x gives the marginal contribution to risk (4).
Equation ( 4) describes the marginal contribution to risk for a security as a function of its individual volatility scaled by its correlation to the remainder of the portfolio's securities.This is a very compelling framework because of the boundaries of  .That is, if the full risk of security m applies as there is no diversification effect since portfolio risk is identical to weighted-average risk.However, if , there is a diversification effect where portfolio risk is less than weighted-average risk and the difference between the two is dependent upon the value of  .Their analysis is even more intuitive with the observation that the marginal contribution to portfolio risk can be shown as monotone decreasing in  where the lower the correlation, the lower the standard deviation, ceteris paribus.

3.0
Sigma-rho-equilibrium Menchero and Davis' X-Sigma-Rho equation has a broader application when considered in the context of marginal return   MR .Differentiating (1) with respect to m x gives the marginal contribution to return as The marginal return of security m is the contribution to a portfolio's return but it should be examined in the context of adjustment for risk.To make this adjustment, the commonly used Sharpe measure is employed as a ratio of the marginal contribution of return to the marginal contribution to risk and comparing it to a "market portfolio", M , or some other efficient benchmark.A long-only portfolio manager should want to add (remove) security m if its marginal risk-adjusted contribution to return is positive (negative).It might also be the case that a portfolio manager does not include a security with a positive Sharpe ratio because it does not contribute enough return to compensate for its risk.This issue was addressed by Elton, Gruber, and Padberg (1978).[In this analysis the typical and sometimes controversial assumptions for the market portfolio apply.]The Sharpe ratio for security In equilibrium the expectation is that all assets will have the same reward to risk ratio so that (7a) obtains, where Equation ( 7b) is Sharpe's (1964)


Equation ( 8) defines "alpha" as a function of the risk free rate of return, individual standard deviations, the efficiency of the benchmark return  g M  , and the correlation of the security with the benchmark.Ceteris paribus, as   decreases, alpha increases because the risk premium decreases.This level of efficiency should be the objective of the analyst.
Using the methodology described here, there should be a tendency towards efficient markets with correctly priced assets as (8) describes arbitrage opportunities.In well functioning liquid markets, mis-priced assets should be restored to equilibrium.However, this process will not take place without the presence of return so that it is necessary to extend Mencharo and Davis' (2011) analysis.

Conclusion and managerial implication
The risk analysis conducted by Menchero and Davis adds to the literature as X-Sigma-Rho but additionally provides a framework for deriving the capital market line and the capital asset pricing model under equilibrium conditions.If we require that all assets have the same marginal contribution to return after adjusting for marginal contribution to risk, the CAPM easily obtains.Analysts can improve the efficiency of portfolios by jointly considering the marginal rate of return and the marginal contribution of risk.Given two securities with the same risk-return characteristics, the security with lower correlation to the portfolio will provide the higher improvement in efficiency.
deviation or volatility of security m and   R g m ,

For
capital asset pricing model (CAPM) or the security market line, and when is equivalent to the capital market line, the difference being the consideration of total risk versus systematic risk.(Equation 7b can also be easily derived under no-arbitrage conditions as shown in the Appendix.)