Expected Shortfall (ES), also known as Conditional Value at Risk (CVaR), is a measure of risk that captures the expected loss in the tail of a returns distribution, given that the return is below a certain VaR level. It is considered a more comprehensive risk measure than Value at Risk (VaR) because it takes into account the tail risk and provides an estimate of the expected loss in extreme scenarios.
To calculate the Expected Shortfall, first, we need to determine the VaR level at a specific confidence level α, and then compute the expected value of the returns below that threshold. Let’s denote the returns distribution as r and its cumulative distribution function (CDF) as F(r). The Value at Risk (VaR) at level α is the value rα that satisfies:
F(rα) = α
Now, given the PDF (probability density function) f(r), the Expected Shortfall at confidence level α is calculated by taking the expected value of the losses below the VaR level:
$$ES_\alpha = \frac{1}{1-\alpha} \int_{-\infty}^{r_\alpha} r \cdot f(r) dr$$
To illustrate this with an example, let’s suppose the returns of a portfolio follow a normal distribution with mean μ and standard deviation σ:
r ∼ N(μ, σ2)
First, find the VaR level at a certain confidence level α. For the normal distribution, this corresponds to the quantile function rα = Q(α) = μ + σ ⋅ Φ − 1(α), where Φ − 1( ⋅ ) is the inverse of standard normal CDF.
Now we calculate the Expected Shortfall by integrating over the region below the VaR level:
$$ES_\alpha = \frac{1}{1-\alpha} \int_{-\infty}^{Q(\alpha)} r \cdot f(r) dr$$
For a normal distribution, we have:
$$f(r) = \frac{1}{\sigma\sqrt{2\pi}} e^{-\frac{(r-\mu)^2}{2\sigma^2}}$$
Thus, the Expected Shortfall becomes:
$$ES_\alpha = \frac{1}{1-\alpha} \int_{-\infty}^{Q(\alpha)} r \cdot \frac{1}{\sigma\sqrt{2\pi}} e^{-\frac{(r-\mu)^2}{2\sigma^2}} dr$$
Substituting the VaR rα = Q(α), we have:
$$ES_\alpha = \mu + \sigma \cdot \frac{\Phi(\Phi^{-1}(\alpha) - Z)}{1-\alpha}$$
Where, Z ∼ N(0, 1) is the standard normal random variable, and Φ( ⋅ ) is the standard normal CDF. This gives the Expected Shortfall for a portfolio with normal returns distribution.
Note that this example assumes normal distribution of the returns, but the Expected Shortfall calculation can be applied to any distribution given its PDF and CDF.