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Value-at-risk and expected shortfall

1. Value-at-risk and expected shortfall

In this chapter, you are going to measure the risk of some simple financial portfolios. But what does it mean to measure the risk of a portfolio?

2. Value-at-risk (VaR)

Consider a fixed time period, like one day or one week, and consider the distribution of losses that would be incurred over that time period if no further trading or rebalancing of the portfolio took place. This is essentially the approach that banks take when they measure the risks of their trading book. They consider the distribution of losses they could incur over two trading weeks if the trading book positions are held fixed. Now a risk measure is really just a statistic that describes the loss distribution. You could take the mean of the distribution or the standard deviation, but what is most common in QRM is to take a statistic that describes the tail of the loss distribution and the potential for large losses. In particular, it is common to take a quantile of the distribution in the tail, such as the 95th, the 99th, or the 99 point 5th percentile. This is the value-at-risk concept. The alpha-VaR is the alpha quantile of the portfolio loss distribution for a specified time period; it is often called the VaR at the alpha confidence level. With a probability of alpha, the loss should be no more than the VaR over the time period. Of course, with a probability of one minus alpha, the loss could be larger and, depending on the heaviness of the tail, even quite a lot larger. The VaR risk measure is used as a guide to determine how much capital a bank or other institution should have so that it can remain solvent in the face of large trading losses.

3. 95% VaR illustrated

To understand the concept better, consider the picture on the slide. The density of a stylized loss distribution is shown, and the 95% VaR is marked with a red vertical line. The probability mass in the red shaded area is 5%. In this distribution, the mean loss is minus two point four; in other words, a gain of two point four and the 95% VaR is one point six. This means that there is a 95% chance the loss would be no greater than one point six. In practice, of course, the loss distribution has to be estimated from data in practice, and then the corresponding estimate of VaR has to be calculated. That will be the subject of later R exercises.

4. Expected shortfall (ES)

There is another risk measure that is commonly used in practice and which is becoming more important for determining capital requirements in banking regulation. This risk measure is variously known as tail VaR, conditional VaR, or expected shortfall. As noted, by definition, there is a probability of 1-alpha that the loss could exceed the alpha-VaR. The expected shortfall at level alpha is the expected size of the loss given that the alpha-VaR is exceeded. It can be thought of as the expectation of the tail of the distribution starting at the alpha quantile.

5. 95% of ES illustrated

In the picture, the 95% expected shortfall for the stylized loss distribution is shown with a blue vertical line. If the red shaded part of the distribution beyond the 95% VaR is considered as a distribution in its own right, the blue line would mark the mean of this distribution. The value, in this case, is three point three, which is more than double the 95% VaR. In this case, the loss distribution is a skewed and fairly heavy-tailed distribution.

6. Let's practice!

In the exercise that follows, you are going to use qnorm() to calculate the VaR of a normal distribution and the function ESnorm() in the QRM package to calculate the associated expected shortfall. So why don't you try that out?