At a fundamental level, the IMA DRC is trying to determine the likely loss to the bank in the event of a default by an obligor which may not be a direct counterparty of the bank. For example, the bank may hold XYZ Pty Ltd equities with no direct bank relationship with that company. If the company defaults, the equities will become worthless and so the bank will make a loss.
The size of this loss needs to be covered by a capital charge. The Basel Committee has said that the desired level of capital is that which would cover losses at a 99.9% confidence interval over a horizon of 1 year.
To calculate the required capital needs data inputs for each obligor to enable the probability of default (PD) of the obligor in the time horizon plus the likely loss in the event of default (loss given default LGD).
In addition, the committee requires some degree of correlation between defaults to be included in the model to incorporate any concentration risks held in the bank’s portfolio.
The description in the Basel specification makes the following points:
Default risk must be measured using a VaR model.
Banks must use a default simulation model with two types of systematic risk factors.
Default correlations must be based on credit spreads or on listed equity prices.
Banks have the discretion to apply a minimum liquidity horizon of 60 days to the determination of default risk charges for equity sub-portfolios.
The VaR calculation must be done weekly and be based on a one-year time horizon at a one-tail, 99.9 percentile confidence level.
Methodology
Following the Basel Committee FAQ release in late January 2017, we moved to looking at a Merton-style model. Merton developed a company valuation model in the 1980s using option-pricing techniques. Companies are deemed to have equity equal to their assets minus their debt. If equity falls to zero then the company is in default. This model wasn’t extended to multiple simultaneous valuations as is required in the IMA DRC.
The inclusion of country and industry factors also requires the incorporation of equity beta methodologies into the Merton model.
We designed a model based on country factors against which obligors would have beta coefficients (in the same way equity stocks have betas to represent how they behave relative to the stock market as a whole). Similarly industry factors were introduced. Following Merton, we thus used those betas plus the individual obligor’s excess returns in the market to generate the overall obligor return over a one year horizon. Simulation of these country, industry and obligor excess returns would yield a distribution of overall obligor returns and from that distribution we would be able to choose a critical value in line with the PD for that obligor. This approach seemed to required simulation.
However, some further thought and consideration of the probability distributions and their behaviour led us to be able to analytically determine a Normal Distribution representation of the obligor returns. Thus we would be able to quickly calculate a mean and variance for each obligor return distribution and provided we had suitable default correlations we would be able to model the overall loss much more easily. This seemed to be an important result, useful both in further exploring a completely analytical solution, or as a starting point for a simulation approach if that was necessary.
On-going further analysis has failed to generate a fully analytical solution. This is primarily because of the binary nature of the pay-off (default or not) combined with the need for default correlations to be included in the final calculation. Numerous attempts such as using principle components analysis failed to make this tractable. We believe at this time that it is impossible to avoid simulation for the final step. However, the nature of the final step means it is able to be conveniently multi-threaded, done cumulatively rather than requiring a full 10 million path run to be run before deriving the results and thus dramatically reducing the memory requirements for the calculation. This is very important in making the calculation meet reasonable performance requirements.