Currently viewing the tag: "variance"

Ocean lIner sinking at severe angleWe have devoted a lot of blog space in the past examining the pros and cons of the Capital Asset Pricing Model (CAPM).  The model predicts the amount of excess return (return above the risk-free rate) of an arbitrary portfolio that can be ascribed to a relationship (called beta[1]) to the excess returns on the underlying market portfolio. It assumes that investors seek maximum utility from their portfolios, and that investors expect to be compensated for taking on additional risk. In CAPM, no portfolio can outperform a mix of the risk-free asset and the market portfolio on a risk-adjusted basis.

Risk and return are the only two variables of importance in CAPM.  Risk is measured by beta, which is rooted in the variance (or more properly, its square root, standard deviation) of returns, and is an indication of volatility.

Recall that CAPM makes a number of simplifying (and criticized) assumptions to work.  A couple of the assumptions that have been the subject to criticism are:

1)     Portfolio returns are distributed symmetrically around a mean.

2)     Portfolio returns are assumed to have no outliers (or “fat tails”).

Empirical evidence suggests otherwise. Researchers have thus sought alternatives to a variance-based beta that relaxes both of these assumptions, and one popular candidate is called semivariance – a measure of the dispersion of those values in a distribution that fall below the mean or target value of a data set. In short, semivariance-based modifications to CAPM concentrate on downside-risk only. Semivariance is a better statistic when dealing with asymmetric distributions, as it automatically incorporates the notion of skewness. Ignoring skewness, by assuming that variables are symmetrically distributed when they are not, will cause any model to understate the risk of variables with high skewness

One popular semivariant CAPM alternative is called D-CAPM (Downside-CAPM).  Regular old beta is replaced by downside-beta (βD). Different researchers have supplied different technical definitions for βD; here we will use the one provided by Javier Estrada:

βD = downside covariance between asset and market portfolio / downside variance of market portfolio.

There are several ways to calculate βD, but I will spare you the details. The important point is that empirical studies[2] indicate D-CAPM gives better predictions compared to CAPM, especially for emerging markets.  This may be due to the hypothesis that returns from emerging markets are less normal and more skewed than returns from developed markets.

D-CAPM is generally well-regarded because of its plausibility, supporting evidence, and the widespread use of D-CAPM.  For instance, a study by Mamoghli and Daboussi[3] concluded “It comes out from these results that the D-CAPM makes it possible to overcome the drawbacks of the traditional CAPM concerning the asymmetrical nature of returns and the risk perception…

However, D-CAPM is not without its critics.  Let me quote from one[4], in which downside correlation is taken to task: “This measure ignores the ability of upside returns of one asset to hedge the downside returns of another asset in a portfolio. Within the scope of D-CAPM the standard equation to calculate portfolio’s downside semideviation cannot be correct, since the Estrada’s downside correlation equates upside returns to zeros and does not represent the true downside correlations. Specifically, the downside correlation cannot be measured, because the portfolio’s semideviation depends on the weights of assets, their standard deviations and correlation between them, rather than on the semivariance. This formula is specified for normally distributed and symmetrical returns and there is no formula invented to calculate the portfolio’s semideviation yet”.

This criticism, if true, is especially relevant to hedge fund traders, who use long-short strategies as a matter of course.

Nevertheless, Professor Estrada responded[5] as follows: First, it’s true that the downside beta I suggest in my paper does not account for something that would be legitimately called risk, which is the market going up and the asset going down. The way I suggest to calculate downside beta only accounts for down-down states. But there is a reason for this: It’s the only way to achieve a symmetric semivariance-semicovariance matrix, necessary for a solution of the problem. For example, the only way to apply standard techniques to solve an optimization problem with downside risk is to have a symmetric matrix. Second, and importantly, my several papers on downside beta, D-CAPM, and downside risk in general have been published in several journals and therefore peer-reviewed. I believe Mr. Cheremushkin’s comment is an unpublished piece.”

This controversy will no doubt be resolved over the fullness of time. We welcome your views.


[1] Technically, beta is equal to the covariance of an asset with the market portfolio divided by the variance of the market portfolio.

[2] Estrada, Javier (2003). “Mean-Semivariance Behavior (II): The D-CAPM.”  University of Navarra.

[3] Mamoghli, Chokri and Daboussi, Sami (2008). Valuation of Hedge Funds Portfolios in a Downside Risk Framework.

[4] Cheremushkin, Sergei V. (2009). “Why D-CAPM is a big mistake? The incorrectness of the cosemivariance statistics.”  Mordovian State University.  Also private communication 9-Aug-10.

[5] Private communication 9-Aug-10.

Non-diversifiable risk

We are reviewing the underlying assumptions made by the Capital Asset Pricing Model (CAPM).  Recall from last time the assumption that returns are distributed normally (i.e. a bell-shaped curve) and how this fails to account for skew and fat tails. Today we’ll look at CAPM’s assumption that there is but a single source of priced systematic risk: market beta.

Beta, of course, refers to the correlation between the returns on an asset and the returns on the broad market. Specifically, it is defined as:

βi  = Cov(Rm, Ri) / Var(Rm)  for asset i

that is, the covariance of the asset’s returns with the those of the market divided by the variance of the market returns.  Another way to think about beta is as a discount factor: when divided into an asset’s expected return, it adjusts the expected return so that it equals the market’s expected return.

By making this assumption, CAPM is able to simplify the universe of different risks undertaken by a portfolio, such as a hedge fund, to one global number. That’s good for ease of calculation but bad for reflecting reality. It is also assumed that all non-systematic risk (alpha) has been diversified away by having a sufficiently large portfolio, so that the expected return on the portfolio is a function of its beta alone.

Many academics challenge the use of variance in the formula for beta. They point out that variance is a symmetrical measure of risk, whereas investors are only concerned about loss (just the left side of the curve). Instead of variance, they point to other measures of risk, such as coherent risks, that may better reflect investor preferences.  Coherent risks have four attributes:

1)      When comparing two portfolios, the one with better values should have lower risk

2)      The joint risk of two portfolios cannot exceed the sum of their individual risks

3)      Increasing the size of positions within a portfolio increases its risk

4)      Picture two identical portfolios except one contains additional cash. The one with the extra cash is less risky, to the exact extent of the extra cash.

Hedge funds clearly take on risks that are not reflected in beta. These risks are the basis for the returns of many strategies, such as long/short (liquidity risk), fixed income (credit risk), and sector trading (sector risk) to name a few.  To the extent that a hedge fund takes on risks that are not reflected by beta, it is modifying its long-term expected return for better or worse.  For instance, a portfolio may add an asset with a negative expected return in order to hedge another asset that has an expected positive return. This has the effect of lowering the expected return of the portfolio as it lowers the portfolio’s risk.

Recall that the expected market return above the risk-free rate is called the market’s risk premium.  Many hedge funds seek returns independent of the market risk premium. Rather, hedge funds supposedly choose their risk exposures, and hence their risk premia. Ideally, their market beta would be zero, so that hedge fund returns would reflect only risks other than market risk. Note that these non-market risks are often hard to diversify away.  For instance, it is difficult to diversify away the risks involved in “risk arbitrage” – a misnomer for the strategy of taking positions on mergers and acquisitions. By assuming this “event risk” (i.e. by betting on whether a proposed merger will succeed), fund managers hope to capture the associated risk premium, independent of market beta.

There are a large number of hedge fund strategies, and most are exposed to their own systematic risks.  Later on, we will review different strategies in detail, and look at the associated risk premia at that time. The important point here is that systematic risk, whether captured in market beta or due to strategy-specific risk(s), should not be conflated with alpha – the returns due to superior skill and/or timing. Ultimately we will be exploring the case for and against alpha when we examine hedge fund replication strategies.

Next time, we’ll finish discussing the assumptions underlying CAPM.