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You don't really have a multivariate case: we can only define VaR (in its usual sense) for a one-dimensional output. Recall that $$ \operatorname{VaR}_\alpha(X) = \inf\{v:F_X(v)\geq \alpha\} $$ and since in your case $X = X_1+X_2$ you just need to compute $F_X$ in terms of $X_1$ and $X_2$. For the notation of partial derivatives, I denote the generic ...


0

When you backtest VaR, you are essentially backtesting your model. You are essentially saying: "If I had this model back in 2007, what would be its calculated VaR?" The model is tested given all available information up to that time. A good model will adapt given new data and the VaR will change after an extreme event. Even one data point should make a ...


1

I also have been puzzled by the intuition of this formula in the past. What made sense to me was converting the integral to a summation. You can then do the calculation quite simply in an excel document with some simulated data and see where everything is coming from. CVaR is really just calculating the average return given that it is less than a certain ...


2

It is correct! You can also see it this way: $$ \text{CVaR}_\alpha(X)=\mathbb{E}(X|X\leq \text{VaR}_\alpha(X)) = \frac{\int_{\mathbb{R}} x\cdot 1_{X\leq \text{VaR}_\alpha(X)}dF(x)}{\int_\mathbb{R}1_{X\leq \text{VaR}_\alpha(X)}dF(x)} = \frac{1}{\alpha} \int_{-\infty}^{\text{VaR}_\alpha(X)}xdF(x) $$ The sign problem still remains (in both versions). If you ...



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