Deriving the Expected Shortfall Formula from Value at Risk
Summary
This note derives a relationship between Conditional Value at Risk (CVaR), Value at Risk (VaR), and the expected excess of a random loss over its VaR threshold. It defines VaR through an upper-tail probability and CVaR as the average of VaR quantiles over tail probability levels. The derivation then expresses the expected positive excess as an integral over the distribution and changes variables to quantiles, linking that excess to the integrated VaR values.
Combining the integral identity with the CVaR definition yields CVaR as VaR plus the expected excess divided by the tail probability. The worked argument assumes a continuous cumulative distribution function, so the derivation as presented does not address atoms or other complications in quantile conventions. The note supplies a proof rather than empirical evidence, and the result depends on keeping the stated tail-probability parameterization consistent.
Key ideas
- VaR is defined here as a quantile corresponding to an upper-tail probability.
- CVaR is the average of VaR across tail probability levels up to the selected level.
- The expected positive excess above VaR can be rewritten as an integral of quantiles.
- Under the stated continuity assumption, CVaR equals VaR plus scaled expected excess beyond VaR.
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Full text
# How to prove the following relation of Conditional Value-at-Risk and Value-at-Risk?
# How to prove the following relation of Conditional Value-at-Risk and Value-at-Risk?
How to prove the following relation of Conditional Value-at-Risk $\text{CVaR}_{\alpha}(X)$ and Value-at-Risk $\text{VaR}_{\alpha}(X)$, \begin{equation} \text{CVaR}_{\alpha}(X) = \text{VaR}_{\alpha}(X)+\frac{1}{\alpha}E[(X-\text{VaR}_{\alpha}(X))^{+}]? \end{equation} Here are the definations of Value-at-Risk and Conditional Value-at-Risk.
Value-at-Risk
Suppose $X$ is a random variable, the value-at-risk (VaR) of $X$ at a confidence level $1-\alpha$ where $0<\alpha<1$ is defined as \begin{equation} \text{VaR}_{\alpha}(X) := \inf\left\{x :Pr\{X>x\}\leq\alpha\right\}. \end{equation}
Conditional Value-at-Risk
Based on the definition of Value-at-Risk, the Donditional Value-at-Risk (CVaR) of $X$ at a confidence level $1-\alpha$ (namely, significance level $\alpha$) is defined to be \begin{equation} \mathrm{CVaR}_{\alpha}(X) = \frac{1}{\alpha}\int_{0}^{\alpha}\mathrm{VaR}_{s}(X)ds. \end{equation}
## Answer by Magic is in the chain (score 2, accepted)
https://quant.stackexchange.com/a/41862
A slightly different take here:
## Answer by Gordon (score 3)
https://quant.stackexchange.com/a/41859
Let $F$ be the cumulative distribution function of $X$. We assume that $F$ is continuous. Then, for $x\ge 0$, \begin{align*} F^{-1}(x) = \inf\{s: F(s) \ge x \}. \end{align*} Moreover, \begin{align*} \text{VaR}_{\alpha}(X) &= \inf\left\{x :1-F(x) \le \alpha\right\}\\ &=F^{-1}(1-\alpha). \end{align*} Consequently \begin{align*} E\Big(\big(X-\text{VaR}_{\alpha}(X)\big)^+\Big) &= \int_{-\infty}^{\infty} \Big(x-\text{VaR}_{\alpha}(X)\Big)^+ dF(x)\\ &=\int_{\text{VaR}_{\alpha}(X)}^{\infty} \Big(x-\text{VaR}_{\alpha}(X)\Big) dF(x)\\ &=\int_{1-\alpha}^1 \Big(F^{-1}(y)-\text{VaR}_{\alpha}(X)\Big) dy\\ &=\int_{1-\alpha}^1 F^{-1}(y) dy - \alpha \text{VaR}_{\alpha}(X) \\ &=\int_{1-\alpha}^1 \text{VaR}_{1-y}(X) dy - \alpha \text{VaR}_{\alpha}(X) \\ &=\int_0^{\alpha} \text{VaR}_{s}(X) ds - \alpha \text{VaR}_{\alpha}(X). \end{align*} That is, \begin{align*} \text{VaR}_{\alpha}(X)+\frac{1}{\alpha}E\Big(\big(X-\text{VaR}_{\alpha}(X)\big)^+\Big) &= \frac{1}{\alpha}\int_{0}^{\alpha}\text{VaR}_{s}(X)ds. \end{align*}Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.