Capital Asset Pricing Model, Part One – Normal Distribution

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We left off last time showing how the Security Characteristic Line indicates the beta of an asset under Harry Markowitz’s Modern Portfolio Theory (MPT). We are now ready to discuss asset pricing models, and we’ll begin by documenting the Capital Asset Pricing Model (CAPM).   This model was developed in the 1960’s by several independent researchers, including Sharpe, Treynor, Lintner and Mossin, building on Markowitz’s previous work.

CAPM is an equation that indicates the required rate of return (ROR) one should demand for holding a risky asset as part of a diversified portfolio, based on the asset’s beta.   If CAPM indicates a rate of return that is different from that predicted using other criteria (such as P/E ratios or stock charts), then one should, in theory, buy or sell the asset depending on the relationship of the different estimates.  For instance, if stock charting indicates that the ROR on Asset A should be 13% but CAPM estimates only a 9% ROR, one should sell or short the asset, which cumulatively should drive the price of Asset A down.

Over the next several articles, we are going to critically examine the assumptions made by CAPM. Then we’ll look at the CAPM equations.  But today we’ll start the ball rolling by looking at CAPM’s assumption that returns are normally distributed. A normal distribution is bell-shaped, with a centered mean at its peak. The probabilities that a particular return lies within 1, 2, or 3 standard deviations of the mean in a normal distribution are 68%, 95% and 99.7% respectively.

Only two normally distributed variables, return and risk, are needed to understand the expected behavior of a given asset or portfolio. The rate of return is the discount rate of the asset’s future cash flows.  The higher the beta of an asset, the higher the required ROR. The risk is measured by an asset’s volatility of returns, expressed as a standard deviation.

If an asset’s returns are normally distributed, then the marginal utility of adding the asset to a diversified portfolio takes on the shape of a concave curve, as the above chart shows. That is, the slope of a utility curve decreases with increasing asset; this is termed a quadratic utility function. For a risk-adverse investor, the marginal utility of buying one more unit of an asset must be weighed against spending the money on a substitute asset, or not spending the money at all. Theory posits that the marginal utility of an asset diminishes as you own more of it.  For example, if you already have 3 gallons of milk in the refrigerator, you may not want to pay full price for the 4th gallon (unless you are an ice-cream maker on a hot summer day!).

The use of a normal distribution to assess an asset’s marginal utility is often unrealistic, in that an asset’s distribution of returns can be skewed and/or have an unexpected higher probability of being extreme (“fat tails”).  When skewed, the distribution’s mean is not centered symmetrically, and several hedge fund strategies demonstrate a negative skew.  Fat tails and negative skew are not favored by risk-adverse investors, as they indicate the probability of overpaying for an asset given its risk and return. One might consider the 2008 market meltdown as an unpleasant reminder that fat tails do exist – that is, the chances that an unlikely market event can occur are understated by a normal distribution.

Using variance (or equivalently standard deviation) as a measure of risk is also misleading in a normal distribution, due to the fact that investors equate risk only to loss.

An example of a hedge fund strategy not realistically modeled by a normal distribution is shorting naked far out-of-the-money options. This strategy is exposed to the occurrence of an extreme event.  These so-called disaster options yield attractive premium income in most situations, but can be ruinous to the seller during an extreme event. If return distributions are fat-tailed, then the premium collected for a disaster option is too low.  Another way to say this is that the profit contribution to fund performance of these options is overstated, since they are not properly adjusted for risk.  To compensate, some commentators suggest subtracting the prices of far out-of-the-money option from the fund’s reported return, but this practice is extremely rare.

We’ll pick up next time by evaluating CAPM’s assumption that there is only a single source of systematic risk – beta (market risk) – in a portfolio.

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Copyright 2011 Eric Bank, Freelance Writer

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