Conditional Value at Risk (CVaR) is given as: $$CVaR_\alpha(X)=\frac{1}{\alpha}\int_{0}^{\alpha}VaR_\beta(X)d\beta=-E(X|X\leq-VaR_\alpha(X))=-\frac{1}{\alpha}\int_{-\infty}^{-VaR_\alpha(X)}x \cdot f(x)\,dx$$
I am not sure if the last term is correct regarding multiplication with $1/\alpha$?
The average is already only up to $VaR_\alpha$.
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 define $\text{VaR}_\alpha (X) = - F_X^{-1}(\alpha)$ then you probably want to define $\text{CVaR}_\alpha(X) = - \mathbb{E}(X|X\leq -\text{VaR}_\alpha(X))$ you will get your result.
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 amount. The calculation is like a weighted average where every observation above VaR has a zero weight. You do the sum and then you divide by the weighted divisor. But since we know that the VaR is at a particular confidence, the divisor should equal that confidence every time.
