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Measuring portfolio performance using a modified measure of risk Authors: Chris Adcock Source: Journal of Asset Management, vol. 7, n°6 Date: March 2007 |
This article considers two measures of portfolio risk: the beta from the CAPM and the measure of portfolio risk introduced by Rubinstein (1976). The appropriate measure of portfolio risk is important, as it will determine the value of alpha in portfolio performance measurement. The traditional CAPM assumes that asset or portfolio returns are distributed according to an elliptical symmetry. But this is not so for all assets or portfolios. In particular, assets tend to exhibit skewness. To deal with the problems associated with estimates of alpha using CAPM beta, Leland (1999) proposes a measure of risk referring to Rubinstein’s (1976) asset pricing model, which takes into account the possible existence of asset return skewness, making it potentially attractive for portfolios including options. Adcock refers to this measure as the Rubinstein-Leland (RL) beta.
In the present paper, the author describes the theory behind this risk measure and proposes a practical regression method of estimating it. Then he proposes statistical tests of the difference between RL-beta and traditional beta.
To compare traditional beta and RL-beta, an empirical study was done on 380 non-financial US stocks, using daily data over the period from 4 November 1993 to 3 November 2003. All the assets considered were part of the FTSE 350 index during the period from January 1990 to December 2002. The author reports results both for individual stocks and for 210 simulated portfolios made up of 5 to 300 stocks, and including portfolios of all sizes of stocks, as well as small-cap and large-cap portfolios. The author looks at both equal-weighted and cap-weighted portfolios. Dividing the period into five sub-periods, he notices first that nearly all stocks exhibit kurtosis, while about 42% exhibit skewness during the most recent sub-period (2001-2003). Similar results are obtained for the other sub-periods.
For each of the five sub-periods, Adcock computes both the standard market model and the RL-beta using the regression method and compares the two results using Hotelling’s test. The results show that, in the absence of skewness, beta is to be preferred to RL-beta, as a measure of risk for the period 2001-2003. In addition, even in the presence of skewness, there is no evidence to suggest the superiority of the RL-beta during this period. For other periods, i.e., between 1993 and 2001, there is little evidence indicating any statistically significant difference between the two models of risk, whether there is skewness in the returns or not. This investigation does not support the use of the RL-beta as a measure of risk, as far as stocks are concerned, but the author plans further research using portfolios including stock options, which tend to present more extreme skewness and for which this risk measure may well be of greater interest.
References
Leland, H. E., “Beyond Mean-Variance: Performance Measurement in a Non-symmetrical World”; Financial Analysts Journal, vol. 55, 1999, p. 27-36.
Rubinstein, M., “The Valuation of Uncertain Income Streams and the Pricing of Options”, Bell Journal of Economics, vol. 7, 1976, p. 407-425.
