The connection between Basel ASRF formula and the Merton model: an intuitive understanding

The BCBS paper "An Explanatory Note on the Basel II IRB Risk Weight Functions" explains the Basel II risk weight formulas in a non-technical way by describing the economic foundations as well as the underlying mathematical model and its input parameters. However, those who do not have an understanding of the Merton model of firm default will find it difficult to grasp the logic behind the ‘normal’ and ‘inverse normal’ functions used in the Basel ASRF formula to adjust the probability of default (PD), as it was the Merton model that was extended by Vasicek to an ASRF credit portfolio model. I have attempted to explain the similarity between the Merton model and the ASRF model, while leaving out any discussion on the other aspects of the Basel ASRF formula like asset correlation and maturity adjustment.

What is a normal distribution?

Ask a non-technical person what a normal distribution is, and he will tell you that it is a bell shaped curve. But it is like ‘defining’ a cat as a cute animal. Being cute is not a definitive characteristic of a cat, as there are other animals too that are cute. Just because an animal is cute, we cannot conclude that the animal in question is a cat. Similarly all symmetrical bell shaped curves are not normal. A distribution is normal, if and only if, the probability density at any given value x is equal to:

One can input any value for x in the above equation and can find out the probability density corresponding to the said value of x.

Integrating the probability density function (pdf) will yield the cumulative density function (cdf) for any given value of x. We need not bother ourselves with the technicalities of how to do this, as there are more than enough softwares available today that can do this task for us. Using the cumulative density function, one can see that for a value of x which is 1.96 standard deviations away from the mean, 95% of observations will fall in the range between mu – x and mu + x. Similarly, for a value of x which is 2.58 standard deviations away from the mean, 99% of observations will fall in the range between mu – x and mu + x.

1.96 and 2.58 mentioned above are just two examples. We can do the computation for any value of x.

Now that we know what proportion of values (which in other words is the 'probability' as per the frequentist interpretation of probability) falls between mu – x and mu + x for any given value of x, it is possible to compute the value of x for a given probability, by employing a reverse process. For example, if someone tells us that 95% of the values lie between mu – x and mu + x, we can employ the reverse process and find out the value of x.

The two computations mentioned above can be carried out using the NORMS.DIST and NORMS.INV functions respectively of Microsoft Excel software.

A standard normal distribution is a special case of normal distribution in which the mean (mu) equals 0 and standard deviation (sigma) equals 1. So, in a standard normal distribution 95% of values fall between -1.96 and +1.96 (since mean is zero and standard deviation is 1)

Merton model of default and the Basel ASRF formula

In the Merton model of firm default, the firm defaults when the value of its assets go below a threshold point. The threshold point in this context is the value of the firm’s liabilities. Thus, if the value of a firm’s assets go below the value of the firm’s liabilities, the firm can be deemed to have been defaulted, since it cannot repay its liabilities. Merton assumes that the liabilities can be represented by a single zero coupon bond that matures at the end of the time horizon, say after 1 year.

He also assumes that the asset value follows a normal distribution. This makes it very easy to estimate the probability of the asset value taking any specified value, if we know the mean and standard deviation from historic asset values. The asset value which we are interested in, is the threshold value at which the firm defaults (it defaults because beyond this point liabilities will exceed assets). Since asset value follows a normal distribution, we can find out from the normal table (or using an application like MS Excel), the probability that assets take a value less than the threshold value. Incidentally, this is also the Probability of Default (PD), since the firm will default if the asset value becomes less than or equal to the threshold value! This point is a very important aspect of the Basel formula, and we cannot comprehend the logic behind the Basel formula without understanding it. Fortunately the logic is very intuitive, and I shall make it clear using a simple example:

“When asset value goes below the threshold, firm defaults. If we find that there is a 2% probability of this (asset value going below the threshold) happening, then PD is also 2%. Conversely if we know (somehow) that the PD is 2%, then the probability of asset value going below the threshold value is also 2%. This last point enables us to estimate the ‘threshold value’ if the PD is known, by using an inverse of the normal distribution.”

When a portfolio consists of a large number of relatively small exposures, idiosyncratic risks associated with individual exposures tend to cancel out one-another and only systematic risks that affect many exposures have a material effect on portfolio losses. In the ASRF model used for the Basel formula, all systematic (or system-wide) risks, that affect all borrowers to a certain degree, like industry or regional risks, are modelled with only one (the “single”) systematic risk factor. It is also assumed that the Single Risk Factor follows a normal distribution (just like the asset value in the Merton model). When the value of the Single Risk Factor crosses a threshold, default takes place. If we know the PD (from long term history) we can estimate (using Inverse Normal function) the threshold value of the Single Risk Factor. The ultimate objective is to get a conservative estimate of PD, for which this is the first step. Basel Committee’s guidance to Banks about LGD is that Banks have to estimate a ‘downturn’ LGD, which will be conservative than the LGD during normal times. But as far as PD is concerned, Banks’ estimate does not correspond to a ‘downturn PD’. It is just a long run average value, averaged over a complete business cycle. We know that ‘Capital’ is meant to cover ‘Unexpected’ losses. If so, it is a conservative PD that should go into the formula, and not an average PD. It is the Basel Committee formula that converts the Bank-estimated PD into a Conservative PD. As mentioned earlier the first step is to estimate the SRF value corresponding to the Bank-estimated PD using an inverse normal function (highlighted in the equation below).

Capital requirement (K) = [LGD * N [(1 - R) ^ -0.5 * G (PD) + (R / (1 - R)) ^ 0.5 * G (0.999)] - PD * LGD] * (1 - 1.5 x b (PD)) ^ -1 × (1 + (M - 2.5) * b (PD)

In the next step, we estimate the value of the Single Risk Factor corresponding to a probability of 99.9%. This value will of course be an extremely conservative value.

Capital requirement (K) = [LGD * N [(1 - R) ^ -0.5 * G (PD) + (R / (1 - R)) ^ 0.5 * G (0.999)] - PD * LGD] * (1 - 1.5 x b (PD)) ^ -1 × (1 + (M - 2.5) * b (PD)

A weighted average (correlation-weighted) of these two SRF values, ie. a weighted average of G(PD) and G(99.9%) will yield an ‘appropriately conservative’ value of the Single Risk Factor.

Capital requirement (K) = [LGD * N [(1 - R) ^ -0.5 * G (PD) + (R / (1 - R)) ^ 0.5 * G (0.999)] - PD * LGD] * (1 - 1.5 x b (PD)) ^ -1 × (1 + (M - 2.5) * b (PD)

Applying the normal function to this ‘appropriately conservative’ value of the Single Risk Factor will yield an ‘appropriately conservative’ value of the PD. 

Capital requirement (K) = [LGD * N [(1 - R) ^ -0.5 * G (PD) + (R / (1 - R)) ^ 0.5 * G (0.999)] - PD * LGD] * (1 - 1.5 x b (PD)) ^ -1 × (1 + (M - 2.5) * b (PD)

The connection with the Merton model ends here. The rest of the formula pertains to multiplying the ‘appropriately conservative’ value of the PD estimated above by the LGD to arrive at the unexpected loss, and then subtracting PD * LGD (the expected loss) from the unexpected loss, and finally doing a maturity adjustment to adjust for different maturities of different exposures.  

(Views expressed are personal)