CVaR reformulation correct?
Conditional Value at Risk (CVaR) is given as:
[Math Processing Error]$$ CVaR_\alpha(X)=\frac{1}{\alpha}\int_{0}^{\alpha}VaR_\beta(X)d\beta=-E(X|X\leq-VaR_\alpha(X))=-\frac{1}{\alpha}\int_{-\infty}^{-VaR_\alpha(X)}x \cdot f(x),dx $$ 我不確定最後一項關於乘法是否正確[Math Processing Error] $ 1/\alpha $ ?
平均值已經只有 $ VaR_\alpha $ .
它是正確的！
你也可以這樣看：
[Math Processing Error]$$ \text{CVaR}\alpha(X)=\mathbb{E}(X|X\leq \text{VaR}\alpha(X)) = \frac{\int_{\mathbb{R}} x\cdot 1_{X\leq \text{VaR}\alpha(X)}dF(x)}{\int\mathbb{R}1_{X\leq \text{VaR}\alpha(X)}dF(x)} = \frac{1}{\alpha} \int{-\infty}^{\text{VaR}\alpha(X)}xdF(x) $$ 符號問題仍然存在（在兩個版本中）。如果你定義 $ \text{VaR}\alpha (X) = - F_X^{-1}(\alpha) $ 那麼你可能想要定義[Math Processing Error] $ \text{CVaR}\alpha(X) = - \mathbb{E}(X|X\leq -\text{VaR}\alpha(X)) $ you will get your result.