## Expected Return Returns

In addition to the expected return, an investor now has to consider the following. First, the spread of the actual returns around the expected return is captured by the variance or standard deviation of the distribution; the greater the deviation of the actual returns from expected returns, the greater the variance. Second, the bias towards positive or negative returns is captured by the skewness of the distribution. The distribution above is positively skewed, since there is a greater likelihood of large positive returns than large negative returns. Third, the shape of the tails of the distribution is measured by the kurtosis of the distribution; fatter tails lead to higher kurtosis. In investment terms, this captures the tendency of the price of this investment to "jump" in either direction.

In the special case of the normal distribution, returns are symmetric and investors do not have to worry about skewness and kurtosis, since there is no skewness and a normal distribution is defined to have a kurtosis of zero. In that case, it can be argued that investments can be measured on only two dimensions - (1) the 'expected return' on the investment comprises the reward, and (2) the variance in anticipated returns comprises the risk on the investment. Figure 3.3 illustrates the return distributions on two investments with symmetric returns-

Figure 3.3: Return Distribution Comparisons

Figure 3.3: Return Distribution Comparisons

In this scenario, an investor faced with a choice between two investments with the same standard deviation but different expected returns, will always pick the one with the higher expected return.

In the more general case, where distributions are neither symmetric nor normal, it is still conceivable, though unlikely, that investors still choose between investments on the basis of only the expected return and the variance, if they possess utility functions1 that allow them to do so. It is far more likely, however, that they prefer positive skewed distributions to negatively skewed ones, and distributions with a lower likelihood of jumps (lower kurtosis) over those with a higher likelihood of jumps (higher kurtosis). In this world, investors will trade off the good (higher expected returns and more positive skewness) against the bad (higher variance and kurtosis) in making investments. Among

1 A utility function is a way of summarizing investor preferences into a generic term called 'utility' on the basis of some choice variables. In this case, for instance, investor utility or satisfaction is stated as a function of wealth. By doing so, we effectively can answer questions such as - Will an investor be twice as happy if he has twice as much wealth? Does each marginal increase in wealth lead to less additional utility than the prior marginal increase? In one specific form of this function, the quadratic utility function, the entire utility of an investor can be compressed into the expected wealth measure and the standard deviation in that wealth, which provides a justification for the use of a framework where only the expected return (mean) and its standard deviation (variance) matter.

the risk and return models that we will be examining, one (the capital asset pricing model or the CAPM) explicitly requires that choices be made only in terms of expected returns and variances. While it does ignore the skewness and kurtosis, it is not clear how much of a factor these additional moments of the distribution are in determining expected returns.

In closing, we should note that the return moments that we run into in practice are almost always estimated using past returns rather than future returns. The assumption we are making when we use historical variances is that past return distributions are good indicators of future return distributions. When this assumption is violated, as is the case when the asset's characteristics have changed significantly over time, the historical estimates may not be good measures of risk.

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