The two blogs to which this article refers (The comparison of distribution parts 0ne and two) show an alternate method of modelling investment performance to the standard capital asset pricing model’s alpha and beta. Criticisms of the capm have a very long history and a very broad scope, but this is not intended as yet another example of that genus. In general, the problem is the comparison of (returns) distributions.
The concerns motivating the proposed approach are twofold. The first is the uncertainty associated with the standard estimates of alpha and beta. This is a well-recognised issue in the academic literature, but is overlooked in almost all practitioner usage. The second is the degree to which the capm alpha and beta model faithfully represents the portfolio in question. In an illustrative case, the returns projected for the portfolio from the application of the alpha-beta model fail the most basic test, that these projected returns are drawing from the same distribution as the portfolio returns.
The model being proposed is simple; it is affine transformation of the portfolio returns such that they share a common range of support with the benchmark returns, and can be directly compared. Affine transformation is an elementary mathematical operation, translation and rescaling of the portfolio returns.
There are similarities between the two approaches in that alpha can be seen as a translation of returns, with beta being a scaling or leverage factor. However, there are also significant differences. Alpha in the capm case is based upon the mean of the distribution. In the affine case, it is the minimum return. In a sense, the capm alpha, though it is the projection to the risk-free level, includes and is biased by the risk contained within the portfolio distribution. The affine translation or alpha is entirely outside of the distribution. If viewed as measures of the location of the portfolio, the capm alpha is using the mean, which is clearly within the risky distribution, while the proposed method is entirely outside; in this sense, the capm alpha mixes both risky and risk-free elements, while the proposed affine translation does not.
This separation of location under the affine approach allows standard risk and return analysis in a gain-loss framework within the domain of the (risky) distribution. Keating and Shadwick’s Omega function methods are a gain-loss approach to risk and return within a distribution.
The scaling or beta under the capm is defined as: , where sigma denotes the standard deviation of the distributions and rho, the correlation between them. By contrast, the rescaling factor of the affine transformation is simply the ratio of the ranges of support of them: . The major difference here is that Beta depends upon the detail of the distributions of returns while the affine approach only their relative ranges.
After rescaling of returns, the portfolio and benchmark mean returns are sufficient for comparison. These returns are returns and may be added to or subtracted from the translation term; though standard deviation is measured in units of return, it differs from a return, and is not additive in this manner.
The affine approach proposed offers further new insights. For example, into the effects of the search for alpha on the risk and return profile of a portfolio, and into the trade-offs between leverage and value-added. One of its greatest attractions is that is framed in the most intuitive world of investment gain and loss.
The comparison of distributions part one
The comparison of distributions part two
