# 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

## 1 Answer

You can first compute the average PD - few choices would be:

1. Simple average of the individual PDs
2. Exposure weighted average of the PDs
3. If the PDs range is too large, then you might want to bucket them and apply the Vasicek formula to each bucket - this is how Basel approaches it.

Once you have the PD and LGD, then you can solve for the correlation. This is very much like finding the Black scholes implied vol - correlation serves the same purpose in the Vasicek. And if you want to research further then please look up base correlation: In the CDO tranches etc, the base correlation, is like the correlation in the Vasicek distribution, and pretty much carries the same meaning as the Black Scholes IV.

• Thank you! But how would you aggregate these buckets? – Nasser Bin Jul 31 '20 at 13:39
• Probably the easiest to justify approach would be to bucket them by rating agencies grade. If more granularity can be supported then one can segment the rating buckets by sector and country/geography, and borrower type. – Magic is in the chain Jul 31 '20 at 16:09
• Thank you. I meant: after bucketing, you would obtain a VaR or a Basel Capital per bucket by simply applying PD and Rho of each rating class in Vasicek/Basel formula. how would you aggregate this VaR/Capital for the overall portfolio? What about LGD? – Nasser Bin Aug 1 '20 at 10:22
• thanks, in the context of this question, the aggregate capital is just a simple sum of the individual buckets' capital. You can model the correlation/dependence between buckets, but that is outside the scope of Basel. It does feature in FRTB standardised but the framework there is much simpler (variance-covariance type approach), which makes it easy to incorporate correlation between buckets. – Magic is in the chain Aug 1 '20 at 19:02