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}