Introduction

In the realm of risk assessment, understanding measures beyond Value at Risk (VaR) is crucial for a comprehensive evaluation of potential losses. In this chapter, we delve into the concept of Expected Shortfall (ES), a risk measure that goes beyond VaR in capturing the magnitude of losses beyond a certain threshold. We will explore the intricacies of calculating ES, highlight its advantages over VaR, and provide insights into their similarities and differences.

Expected Shortfall, also known as Conditional Value at Risk (CVaR), is a risk measure that goes beyond VaR by considering the average of all losses that exceed a specific VaR level. Unlike VaR, which focuses solely on the probability of exceeding a certain loss threshold, ES takes into account the magnitude of losses beyond that threshold. This makes ES a more comprehensive measure for evaluating tail risk, as it considers the severity of extreme losses.

Calculating Expected Shortfall (ES)

ES is calculated as the average of all losses that exceed the VaR level. Mathematically, it can be expressed as:

$$E S_\alpha=\frac{1}{1-\alpha} \int_\alpha^1 V a R_\beta d \beta$$

Here, $\alpha$ represents the significance level (probability threshold), and $VaR_\beta$ is the VaR at the $\beta$ quantile.

Example: Let’s consider a portfolio of stocks with a 95% VaR of 100,000 USD. The ES at a 95% significance level can be calculated as follows:

$$E S_{0.95}=\frac{1}{1-0.95} \int_{0.95}^1 \operatorname{VaR}_\beta d \beta$$

Substituting the values:

$$\begin{aligned} & E S_{0.95}=\frac{1}{0.05} \int_{0.95}^1 100,000 d \beta \\ & E S_{0.95}=\frac{1}{0.05}(100,000 \times 0.05)=100,000 \end{aligned}$$

Advantages of Expected Shortfall (ES) over VaR

• Capturing Tail Risk: ES considers the magnitude of losses beyond the VaR threshold, providing a more accurate representation of extreme risks that VaR might overlook.

• Monotonicity: ES satisfies the property of monotonicity, ensuring that increasing the amount of investment in a risky asset will lead to an increased ES.

Comparison and Contrast between VaR and ES

• Focus on Magnitude: VaR focuses on the probability of loss exceeding a threshold, whereas ES emphasizes the magnitude of losses beyond that threshold.

• Sensitivity to Tail Risk: ES is more sensitive to tail risk, making it a valuable measure for evaluating extreme scenarios.

• Risk Management: ES offers a more comprehensive view of risk and can guide risk management strategies more effectively.

• Mathematical Complexity: Calculating ES is more complex than VaR due to its integration-based formula.