Beyond Mean-variance: Risk and Performance Measures for Portfolios with Nonsymmetric Return Distributions

Most practitioners measure investment performance based on the CAPM, determining portfolio"alphas" or Sharpe Ratios. But the validity of this analysis rests on the validity of the CAPM, whichassumes either normally distributed (and therefore symmetric) returns, or mean-variance preferences.Both assumptions are suspect: even if asset returns were normally distributed, the returns of options ordynamic strategies would not be. And investors distinguish upside from downside risks, implyingskewness preference. This has led to the adoption of ad hoc criteria for measuring risk andperformance, such as "Value at Risk" and the "Sortino Ratio."We consider a world in which the market portfolio (but not necessarily individual securities) hasidentically and independently distributed (i.i.d.) returns. In this world the market portfolio will bemean-variance inefficient and the CAPM alpha will mismeasure the value added by investmentmanagers. The problem is particularly severe for portfolios using options or dynamic strategies.Strategies purchasing (writing) fairly-priced options will be falsely accorded inferior (superior)performance using the CAPM alpha measure.We show how a simple modification of the CAPM beta can lead to correct risk measurement forportfolios with arbitrary return distributions, and the resulting alphas of all fairly-priced options and/ordynamic strategies will be zero. We discuss extensions when the market portfolio is not assumed to bei.i.d. Figure 1 plots the security market line, the straight line joining the riskfree asset point(ß = 0, E(r) = 0.05) with the market portfolio point (ß = 1, E(r) = 0.12).The annual standard deviation of the market portfolio return is 15%.The dashing line above the security market line is the plot of rebalancing or value strategies foralternative strike prices of the call option sold. The points along this line range from strikeprice 90 (lower left) to strike price 140 (upper right).The large dashing line below the security market line is the plot of momentum or portfolioinsurance strategies for alternative strike prices of the put option bought. The points along thisline range from strike price 140 (lower left) to strike price 90 (upper right).Alpha is measured by the vertical distance between the plotted point and the securitymarket line. TABLE I:CAPM-based ß and a vs. Modified B, A Rebalancing or Value Strategies: Long the Market; Short 1 Call(i)(ii)(iii)(iv)(v)Strike PriceE(r)ßaBA 905.51% .0380.24% .07301006.76% .1630.62% .25101108.61% .3940.85% .515012010.27% .6500.72% .753013011.30% .8380.57% .900014011.77% .9390.20% .9670 Portfolio Insurance or Momentum Strategies: Long the Market; Long 1 Put(i)(ii)(iii)(iv)(v)Strike PriceE(r)ßaBA 9011.49% .962-0.24%.927010010.24% .837-0.62%.74901108.40% .606-0.84%.48501206.73% .351-0.72%.24701305.70% .163-0.44%.10101405.24% .062-0.19%.0340 Column (i) is computed assuming a lognormal market portfolio with annual mean = 12%, and std.dev. = 15%, and the distributions this implies for portfolios with options.Column (ii) computes equation (1), using the assumptions in column (i).Column (iii) computes equation (3), with E[rp | M] = column (i). rf = 5%.Column (iv) computes equation (7). Equation (8) implies b = 3.63.Column (v) computes equation (9). TABLE IIValues of Bp (ßp) for Lognormally Distributed Assets ?p,mkt.25.50.75______________________________________________________.15.256 (.248).508 ( .498).756 ( .748) sp .25.415 (.405).819 ( .813)1.213 (1.224) .35.561 (.551)1.103 (1.108) 1.625 (1.670)______________________________________________________ Table II assumes the portfolio p and the market portfolio returns are jointly lognormal.The market has annual mean = 12% and std. dev. = 15%. The annual riskfree rate is 5%.Portfolios p with differ with respect to their correlations with the market (columns), and differentvolatilities (rows).Table entries are the computed Bp and, in parentheses, CAPMßp.