# Aggregation of $\rho$ and $p$ for a vasicek model

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