Aggregating Vasicek Credit Loss Models with Different PDs and Correlations
Summary
The document asks how to approximate a portfolio’s loss distribution with a single-factor Vasicek model when individual loans have different default probabilities and correlations to the risk factor. The stated goal is to match the aggregate expected loss and a high loss quantile while accounting for the absence of diversification in a sum of standalone credits.
The response suggests first choosing an average probability of default, either equally weighted or exposure weighted, and using probability buckets when borrower PDs vary widely. It then proposes solving for an aggregate correlation, analogous to inferring volatility from an option price. Base correlation in collateralized debt obligation tranches is offered as a related concept. This is practical calibration guidance rather than a derivation: it does not specify a unique averaging rule, solve the stated quantile-matching problem, or discuss how losses and loss given default affect the calibration.
Key ideas
- A single-factor Vasicek model can summarize loss risk through default probabilities and factor correlation.
- Possible aggregate PD choices include simple averaging and exposure-weighted averaging.
- Bucketing loans by PD may be useful when default probabilities differ substantially.
- After selecting aggregate PD and loss given default, correlation can be calibrated to the loss distribution.
- The proposed calibration is an outline and does not derive a unique solution to matching a target quantile.
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# Aggregation of $\rho$ and $p$ for a vasicek model
# 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
## Answer by Magic is in the chain (score 3, accepted)
https://quant.stackexchange.com/a/46907
You can first compute the average PD - few choices would be:
- Simple average of the individual PDs
- Exposure weighted average of the PDs
- 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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.