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Buy, Sell or Hold Tight?

Written by Andrew Clark

 –  September 01, 2011

The vast majority of asset returns, 80-90 percent of cases, follow the smooth, bell-curve shape of the normal distribution. But when markets become volatile, investors naturally focus on the 5 percent or 1 percent of events where the assumptions of the normal distribution do not apply. And, in particular, they wish to know how much they might lose.

One measure of risk, called “expected shortfall” (ES), can be used to quantify this. ES has an intuitive appeal, because it answers the question: “How much might I lose, on average, in the worst five of 100 cases?” So ES is not a measure of frequency—in other words, how often will a certain event occur—but rather tells us how much we might lose, on average, if that event occurs.

It also has good mathematical appeal because “tail” events and the skewness of the distribution of returns are part of the ES calculation. And its ability to account for kurtosis (the length and thickness of the tail) and skewness means we will gain significant insight into expected losses.

To illustrate the difference between ES and the normal distribution, it’s sufficient to point out that under ES, we have calculated the loss on the S&P 500 index for the worst 1 percent of daily events as 7.5 percent. The equivalent calculation using a normal distribution is an expected loss of 2.6 percent.

For monthly S&P returns we compute losses of 14.1 percent (under ES) and 10.6 percent (under the normal distribution). For quarterly losses ES gives 26.6 percent and the normal distribution 18.8 percent.

We can see that using the normal distribution to estimate expected losses in volatile times will more than likely lead to a severe underestimation of risk.

Below, we look at the expected shortfall on three commonly tracked indices, two following equities and one commodities: the S&P 500, the Euro Stoxx 50 and the TRJ/CRB indices. We’ve based our study on daily index return data from January 1994 (when the current version of the TRJ/CRB index started) to August 2011.

Expected 1 Day Loss on 10,000 Euros

Worst Cases
S&P 500
Euro Stoxx 50
5 in 100
1 in 100





If we look at one day returns, it is clear that stock indices suffer significantly larger losses as we change from looking at the worst 5 percent of scenarios to the worst 1 percent.

For the commodity index, most of the damage occurs at the 5 percent level, with little change at the 1 percent level. This is because the daily return “tails” of the TRJ/CRB distribution are longer but thinner than under a normal distribution. This means the amplification of losses that you see with stocks (whose daily return distribution has much “thicker” tails) is lacking in the TRJ/CRB. However, we will find that this lack of amplification is not the case for the TRJ/CRB index’s monthly returns.

Expected Monthly Loss on 10,000 Euros

Worst Cases
S&P 500
Euro Stoxx 50
5 in 100
1 in 100





When it comes to the worst expected shortfall over one month, as opposed to one day, the absolute levels of losses are much higher. Since the losses are so high, 10 percent or more in five in 100 cases for all the indices, it begs the question of what your loss aversion is. In other words, how much money are you willing to lose on your investment?

If we look at the worst cases, the one in 100 events, the expected loss measure over one month is even higher: 14 percent for the S&P 500, 21 percent for the Euro Stoxx 50 and 49 percent for the commodity index.


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