This blog brought home many basic truths. However, I would like to emphasise a few points.
The question of volatility being inadequate as a measure of risk is the first. If we are to succeed in understanding financial risk we need to distinguish between symmetric and asymmetric risk. Symmetric risk – or volatility when returns are normally distributed – drives a wedge between the arithmetic average return and the compounded geometric return. It does so exponentially – for normally distributed returns, the geometric return lies 0.5% below the arithmetic when volatility is 10%, but 2% below when return volatility is 20%. The tendency is for symmetric returns to regress to the geometric return quite quickly. By contrast asymmetric returns are thoroughly pernicious – they return to the geometric return only very slowly, if at all – the obvious example which never returns is the total loss of the investment. This is why we worry about long tails on the downside and should love them on the upside.
The second concerns Omega functions as risk measures. In 2002, Bill Shadwick and I introduced a new performance metric known as an Omega function. Since then we have published widely on its statistical and mathematical properties. It is a simple pay-off function. The decision metric we advocated was that we should prefer high to low Omega but we cautioned heavily against its use at any single return level. In this blog response, we shall illustrate the nature of the problem using two elementary distributions, denoted A and B – the elementary statistics for these are shown below as table 1.
| | B | A |
| Returns | ||
| | -4 | 0 |
| | 6 | 0 |
| | 6 | 0 |
| | 6 | 11 |
| | ||
| Average | 3.5 | 2.75 |
| Standard Deviation | 5 | 5.5 |
| Skewness | -2 | 2 |
Table 1: Two returns distributions and their elementary descriptive statistics.
By elementary mean-variance performance metrics, we should prefer B to A as it has both a higher return and a lower variance – the Sharpe ratios are respectively 0.7 and 0.5. Note, however, there is a pronounced difference in skewness, which may lead us to prefer A over B. The (log) Omega functions for these distributions, A and B, are shown as Chart 1.
Chart 1: Log Omega functions for distributions A and B.
Using the decision rule that we prefer the higher log-Omega function we see that we prefer A to B over a considerable proportion of the functions but there is a range in which B is preferred to A.
By translation and rescaling (T&R) of distribution B we may compare these two distribution function and arrive at Chart 2, which shows the excess return of B over A. The translation and rescaling was equivalent to adding 4.75% to B everywhere – the situation where B dominates A everywhere. However it is evident that (T&R) B only dominates A by a little above 3% on average and that this outperformance of (T&R) B over A is insufficient to offset the translation and rescaling effects applied to achieve complete dominance.
Chart 2: Return differences.
It is also sometimes helpful to consider what happens under a large negative shock. To illustrate this point we introduce a shock of -10% into the previous distributions A & B. The two distributions now have identical volatility. Chart 3 shows the log Omega functions for these augmented distributions.
Chart 3: Log Omega functions for A & B, with a -10% return shock.
Now we see that we prefer B to A everywhere below returns of + 4% and as B has a higher mean return than A we should prefer B globally.
Is Omega the answer to everything? No. It is only as good as the representativeness of its data with respect to the future. However, contrary to the regulatory mantra, past performance may be partially indicative of future returns performance.
Con Keating, January 2012.
