This question comes from the Merton-Vasicek model for credit risk. The distribution of the total losses $L$ a bank can suffer in a credit risk event has the following cumulative distribution function $$F_L(x)=P(L\le x)=\Phi\left(\frac{\sqrt{1-a^2}}{a}\Phi^{-1}(x)-\frac{1}{a}\Phi^{-1}(p)\right)$$ for $0<x<1$, where $|a|\le 1$ is a correlation parameter (of the bank's clients with the overall state of the economy) and $0\le p\le 1$ is the probability that a client is in default, i.e unable to meet its obligations with the bank (all clients are assumed to have the same $p$).
I'm trying to compute the mean $\mathbb{E}[L]$ of this distribution, which I know is equal to $p$, but I have been unable to prove it. My attempt: There is an explicit formula for $f_L=F_L'$ in terms of $\Phi'$ and $\Phi^{-1}$ and with the change of variables $y=\Phi^{-1}(x)$ I can get $$\mathbb{E}[L]=\int_{-\infty}^{\infty}xf_L(x)\,dx=\int_{0}^{1}xf_L(x)\,dx=$$ $$=\int_{-\infty}^{\infty} A\Phi(y)\Phi'(Ay + B)\,dy$$ where $$A=\frac{\sqrt{1-a^2}}{a}$$ $$B=\frac{-\Phi^{-1}(p)}{a}$$ Now, integration by parts gives $$\mathbb{E}=\left[\Phi(y)\Phi(Ay+B)\right]_{-\infty}^{\infty}-\int_{-\infty}^{\infty} A\Phi(Ay+B)\Phi'(y)\,dy$$ but integration by parts doesn't work. A change of variables doesn't work either. Also, trying to express $$\Phi(y)=\int_{-\infty}^y\Phi'(t)dt$$ and trying to change the order of integration doesn't seem to solve the integral.
Any ideas?
Thanks!
