How bad can things get? A risk measure provides an answer.

Suppose you are presented with a wager costing 0.4 which returns 1 with probability 0.5 and otherwise zero. Such a wager has an expected return of 25%. You can make this wager once a day over the next 10 days. Suppose you do. How bad can things get? We know that there’s a small probability (\(0.5^{10}\)) that you lose 4 (i.e., you lose all 10 wagers). And there’s a larger probability, \(10 \cdot 0.5^{10}\), that you lose 3 (i.e., win one, lose 9). Although these are useful pieces of information, we might wish for a single-number summary of the riskiness of our planned series of bets. *Expected shortfall* is one such measure of risk. It describes how much you should expect to lose if things go bad. \(\newcommand{\VaR}{\text{VaR}} \newcommand{\E}{\text{E}}\)

Expected shortfall, also known as conditional value at risk, is typically defined in terms of *value at risk* (*VaR*). Let’s begin by defining value at risk. (VaR is also a risk measure, just not a particularly good one.) For a given set of wagers (“portfolio”) over a defined period of time and a specified confidence level α, the VaR is the (1-α)-quantile of the portfolio’s loss distribution. If the profit-and-loss distribution is X then \(\VaR_α(X)\) is the (1-α)-quantile of Y := -X. (Losses are negative, profits positive.) Suppose our portfolio X has a standard normal distribution. Then the VaR at level 0.05 of our portfolio is 1.645. 1.645 is the 0.95-quantile of -X. Suppose, alternatively, that our portfolio Z has a probability of 0.05 of losing 2 million or more. The VaR at level 0.05 of the portfolio is 2 million. 2 million is the 0.95-quantile of -Z.

Expected shortfall, like value at risk, is defined with respect to a period of time and a confidence level α. It is the expected loss in the worst α of cases. If the profit-and-loss distribution is X and X follows a continuous probability distribution then the expected shortfall of X at level α is the E S_{α} = E[- X | X ≤ - VaR_{α}(X)]. This is the left-tail conditional expectation below -\(\VaR_α(X)\). Suppose again that our portfolio X has a standard normal distribution. Then the expected shortfall at level 0.05 of our portfolio is 2.0627. Expected shortfall, like value at risk, is a single-number measurement of how much we should expect to lose if things go bad.

Expected shortfall is a better measure of downside risk than VaR because expected shortfall gives us the average loss when things go bad. VaR, by contrast, tells you a single example of a bad outcome. Consider the following example of a portfolio X which has an expected 2% return. The portfolio costs 1 to acquire and returns 1.6 with probability 0.95, 0 with probability 0.04, and -50 with probability 0.01. The VaR at level 0.05 of this portfolio is 0. The expected shortfall at level 0.05 is 10. 0 is an example of a bad outcome (VaR). 10 is a measure of the average badness, given a bad outcome. Expected shortfall provides a richer description of downside risk than VaR. For this reason people tend to prefer it whenever it is available.

In a prediction market setting, use expected shortfall to characterize the downside risk of a portfolio of wagers. Consider the example of a portfolio which consists of 10 independent wagers. Each wager costs 0.5 and pays out 1 with probability 0.51. (This portfoilio has an expected return of 2%.) To calculate the expected shortfall we calculate the left-tail conditional expectation below \(-\VaR_α(X) = 2\). Calculating this can be accomplished in at least two ways. Simulating the portfolio is likely the easiest way. We arrive at an expected shortfall of 3.21.

When considering different portfolios which we might purchase, it is natural to ask about the typical case: How much do we stand to gain (or lose) on average? It is also natural—or should be natural—to ask how bad things can get. Expected shortfall provides an answer. Expected shortfall describes how much you should expect to lose if things go bad.

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