So here's the problem:
"In finance the level of a risk of a portfolio is often measured by the value-at-risk, i.e. , the loss that is exceeded with probability $\alpha$, or by the expected shortfall, which is the expected loss, conditional on the loss exceeding the value at risk. If the loss distribution is exponential with mean $\mu$, find the value-at-risk and the expected shortfall."
Here's what I've done so far.
Let $x_\alpha$ denote the value at risk . Since X (which is the random variable that represents the loss) follows an exponential distribution with parameter $\lambda = \frac{1}{\mu} $, and since we know that $P(X>x_\alpha) = \alpha $, we have that: $$ 1 - e^{-x_\alpha/\mu} = 1-\alpha$$ From there we can isolate $x_\alpha$, and we will know the value risk in terms of $\mu$ and $\alpha$, which gives us: $$x_\alpha = -\mu \ \text{log}(\alpha)$$
Now the problem is to compute $E(X \ | \ X>x_\alpha )$. I tried to apply the definition of the expected value, which gives me $$\int_{x_\alpha}^{+\infty}x f(x)dx$$ where $f$ is the density of the exponential distribution. So we have: $$\frac{1}{\mu}\int_{x_\alpha}^{+\infty}x \ e^{-\frac{x}{\mu}}dx$$ which we can solve by integration by parts.
But this result does not give me a finite value, so I guess something in my reasoning is wrong.
If you could help , that would be nice.