May 23

Correlations in Basel II IRB (Learning Objective "in the news")

by David Harper, CFA, FRM, CIPM

FRM |

basel2PicIntro

Basel's internal ratings-based (IRB) approach is called ratings-based because an exposure's expected and unexpected loss (EL, UL) is a function of obligor-specific qualities:

  • Probability of default (PD)
  • Loss given default (LGD)
  • Exposure at default (EAD)
  • Maturity

Under advanced IRB, the bank can calibrate its own parameter inputs. But they must use Basel's IRB formula. And, as we move from individual obligors to credit portfolios, they must use the correlation parameters supplied by Basel. (A classic FRM question asks, can a bank use its own formula under advanced IRB? Answer: no.) To reiterate, under advanced IRB: bank can supply PD, LGD, and EAD but they use Basel's formula and Basel's correlations.

Under Basel II, an obligor's contribution to the portfolio's capital charge (the solved-for K below) is a function of asset correlation (given by Greek rho in the IRB function below); e.g., portfolios containing assets with higher correlations should exhibit greater volatility of unexpected losses.

The Basel approach, despite its mathematically elegance, takes theoretical shortcuts in pursuit of usability. It assumes the bank's portfolio is already diversified such that the obligor's idiosyncratic risk is already nullified and only its systematic risk matters (i.e., it's correlation with the portfolio. Or, as Basel says, the asset's and the portfolio's mutual dependence on the "state of the economy" but that is a bit of sleight of hand since economic state is not really in the formula.).

An analogy is to the one-factor market model (CAPM) where only systematic risk (beta) contributes to expected return and idiosyncratic risk vanishes into the portfolio's diversification (credit goes to jyothi on the forum for this nice insight).

So a key parameter is the correlation (rho) which varies by asset class:

irp_equation_4

One way to grasp this is to imagine a correlation of zero (i.e., an obligor with no systemic exposure and only idiosyncratic risk). The formula above, giving no weight to the idiosyncratic, reduces to zero (LGD*PD - LGD*PD). But zero cannot be the input, as in Basel, the asset correlations are bounded at 12% on the low and 24% on the high (for corporates, banks, and sovereigns).

In short, the asset's contribution to capital charge is largely a function of the correlation parameter. Because it is the proxy for its systematic contribution to the portfolio's risk. And even more specifically, the correlation (rho) is a function of:

  • Asset class; e.g., commercial mortgage, and
  • Probability of default (PD, at least in the case of corporate, banks, and sovereign). Here Basel's assumption is that asset correlations tend to decrease with increasing PD.

Fitch evaluates asset correlations

Fitch conducted an empirical study with the aim of discerning true asset correlations. They found the Basel's correlations to be appropriately higher (more conservative):

  Basel II Fitch Empirical
Credit cards 4.0% 1.3%
Residential Mortgage 15% 2-6.76%
Consumer lending 9% 1.3%
Comm'l Mortgage 18.65% 18.26%
Corporates 15.28% 5.15%

But, unlike assumptions built into Basel, the study does not find support for the link between PD and correlation: "Contrary to widely-held views, there does not appear to be a uniform statistical relationship between asset correlations and default probabilities"

Comments

  1. Be the first to leave a comment!

Leave a Comment