I'm currently facing the problem of how properly (analytically) adjust the parameters of an aggregated Vasicek (2002) loss distribution so that it has the same expected loss and 99% quantile as the sum of the standalone credits (i.e. no diversification effects).
Remember the formula of the asymptotic one-factor cumulative loss distribution, where $$P(L\le x) = F(x;p;\rho)=N\biggl(\frac{\sqrt{1-\rho} N^{-1}(x)-N^{-1}(p)}{\sqrt{\rho}}\biggr)$$
Also remember that $L$ is the fraction loss of the portfolio, i.e. $L \in [0;1]$, $\rho$ is the correlation of each loan with the risk factor (equicorrelation) and and $p$ is the probability of default. $\rho$ and $p$ is the same for all loans.
Now the problem is, that I have a loan portfolio of $n$ loans which have no equal correlation with the risk factor (because they belong to different sectors) and no equal probability of default (which is more realistic).
I now want to calibrate analytically the distribution above (find an aggregated $\rho$ and $p$) such that it holds the following:
$$\sum_{i=1}^n F_i(x;p_i;\rho_i ) = F(x;p;\rho)$$
Kind regards