OUTLINE OF THE BOOK
2.1 THE MEAN–VARIANCE FRAMEWORK FOR MEASURING FINANCIAL RISK
2.1.2 Limitations of the Normality Assumption
Nonetheless, the assumption of normality also has its limitations. Ironically, the key ones stem from the last point — that the normal distribution requires only two parameters. Generally speaking, any statistical distribution can be described in terms of its moments. The first moment is the mean, and
0.4 0.5
0.35 0.45
0.3
0.15 0.25 0.2
0.1 0.05 0
Probability
4 Quantile
Symmetric distribution Skewed distribution
1 2
0 3
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Figure 2.3 A skewed distribution.
the second moment corresponds to the variance. However, there are also higher moments, and the third and fourth moments can be of great importance.
The third moment gives an indication of the asymmetry or skewness of the distribution. This leads to the skewness:
Skew=E(x−µ)3/σ3 (2.3)
The skewness coefficient will be zero for a symmetric distribution, and non-zero for an asymmetric one. The sign of the coefficient indicates the direction of the skew: a positive skew indicates a short tail on the left and a long tail on the right, and a negative skew indicates the opposite.
An example of a positively skewed distribution is shown in Figure 2.3, along with the earlier symmetric normal distribution for comparison. The skew alters the whole distribution, and tends to pull one tail in whilst pushing the other tail out. If a distribution is skewed, we must therefore take account of its skewness if we are to be able to estimate its probabilities and quantiles correctly.
The fourth moment, the kurtosis, gives an indication of the flatness of the distribution. In risk measurement practice, this is usually taken to be an indication of the fatness of the tails of the distribution. The kurtosis parameter is:
Kurtosis=E(x−µ)4/σ4 (2.4)
If we ignore any skewness for convenience, there are three cases to consider:
r
If the kurtosis parameter is 3, the tails of our P/L distribution are the same as those we would get under normality.r
If the kurtosis parameter is greater than 3, our tail is fatter than under normality. Such fat tails are common in financial returns, and indicate that extreme events are more likely, and more likely to be large, than under normality.0.4 0.35 0.3
0.15 0.25 0.2
0.1 0.05 0
Probability
4 Quantile
Normal pdf
Fat-tailed pdf
1 2
0 3
−2 −1
−4 −3
Figure 2.4 A fat-tailed distribution.
r
If the kurtosis parameter is less than 3, our tail is thinner than under normality. Thin tails indicate that extreme events are less likely, and less likely to be large, than under normality.The effect of kurtosis is illustrated in Figure 2.4, which shows how a symmetric fat-tailed distribution — in this case, a Studentt-distribution with five degrees of freedom — compares to a nor- mal one. Because the area under the pdf curve must always be 1, the distribution with the fatter tails also has less probability mass in the centre. Tail-fatness — kurtosis in excess of 3 — means that we are more likely to gain a lot or lose a lot, and the gains or losses will tend to be larger, relative to normality.
The moral of the story is that the normality assumption is only strictly appropriate if we are dealing with a symmetric (i.e., zero-skew) distribution with a kurtosis of 3. If these conditions are not met — if our distribution is skewed, or (in particular) has fat tails — then the normality assumption is inappropriate and can lead to major errors in risk analysis.
Box 2.1 Other Risk Measures
The most widely used measure of risk (or dispersion) is the standard deviation (or its square, the variance), but the standard deviation has been criticised for the arbitrary way in which deviations from the mean are squared and for giving equal treatment to upside and downside outcomes. If we are concerned about these, we can use the mean absolute deviation or the downside semi- variance instead: the former replaces the squared deviations in the standard deviation formula with absolute deviations and gets rid of the square root operation; the latter can be obtained from the variance formula by replacing upside values (i.e., observations above the mean) with zeros.
We can also replace the standard deviation with other simple dispersion measures such as the entropy measure or the Gini coefficient (see, e.g., Kroll and Kaplanski (2001, pp. 13–14)).
A more general approach to dispersion is provided by Fishburnα−t measures, defined as t
−∞(t−x)αf(x)dx(Fishburn (1977)). This measure is defined on two parameters:α, which describes our attitude to risk, and t, which specifies the cut-off between the downside that we worry about and other outcomes that we don’t worry about. Many risk measures are special cases of the Fishburn measure or are closely related to it. These includes the downside semi- variance, which is very closely related to the Fishburn measure with α=2 andt equal to the mean; Roy’s safety-first criterion, where α→0; and the expected tail loss (ETL), which is closely related to the Fishburn measure withα=1. In addition, the Fishburn measure encom- passes the stochastic dominance rules that are sometimes used for ranking risky alternatives:2 the Fishburn measure with α=n+1 is proportional to the nth-order distribution function, so ranking risks by this Fishburn measure is equivalent to ranking by nth-order stochastic dominance.3
2.1.3 Traditional Approaches to Financial Risk Measurement