Now that major financial institutions and companies world-wide are sensitive to the macro factor risk they face, implementation of accurate valuation and stress testing with respect to these macro factors is very important. In March, 2009, U.S. bank regulators mandated the 19 largest banks to stress test their portfolios with respect to changes in home prices, real gross domestic product, and unemployment rates. This post talks about how to do this when the number of counterparties is realistic--that is, too large for a brute force approach. "Reduced reduced form models" are the answer and a key to what Kamakura does in both Kamakura Risk Manager and our Kamakura Risk Information Services KRIS-cdo credit portfolio management simulator.
Let's take the example of a hard working financial institution that has already built a suite of logistic regression models for its counterparties from retail borrowers to small businesses to major corporations and sovereigns. The discussion that follows applies to any class of counterparties, and we'll pick public corporations as the case in point. The December 2008 Journal of Finance article "In Search of Distress Risk" by John Y. Campbell (Harvard), Jens Hilscher (Kamakura Corporation and Brandeis University) and Jan Szilagyi (Duquesne Capital) includes financial ratios like these among the explanatory variables:
Market value/book value ratio
Cash/(liabilities + market capitalization) ratio
Net income/(liabilities + market capitalization) ratio
Other explanatory variables could include equity returns, equity volatilities, and macro economic factors like stock index returns. This kind of default model using logistic regression is called a "reduced form model" because it models default as a random process driven by these kinds of factors without modeling the capital structure of the firm and the process by which bankruptcy takes place directly. As pointed out in many other publications on this site, this is a much more accurate way of predicting bankruptcy than using a very highly simplified model of the capital structure like the Merton (1974) model of risky debt.
Let's say we have the model and we're able to use one common logistic regression to simulate the default probabilities of 1,000 public firm counterparties. Can we do that? Surprisingly, the answer is no. We have two problems. The first problem is that the three financial ratios described above are correlated with key macro factors like unemployment, real GDP and home prices but we don't know what the links are. The second problem is more serious. Even if we could use these 3 financial ratios directly in the simulation, we have 3,000 financial ratios to simulate--3 for each of our 1,000 counterparties. Not only that, we haven't even begun to count the other inputs to the corporate default models, let alone the retail and small business models. Why are 3,000 financial ratios a problem? To simulate them forward with the proper correlation with each other, it turns out that we need to be able to invert the 3,000 x 3,000 variance-covariance matrix of these "risk factors" in order to create random monte carlo scenarios of their values at multiple points of time for a large number of scenarios. This inversion, as every linear algebra student learns, is impossible if either the rows or the columns are linear combinations of each other. As the number of counterparties gets larger and the number of risk factors goes from 50 to 100 to 200 to 500 to 1000, it becomes more and more unlikely that this matrix can be inverted, due both to inevitable rounding error that is inescapable in computer science and to accidental or coincidental linearity of the rows or columns of this variance/covariance matrix.
So what do we do? It turns out that there is a simple solution that the staff at Kamakura calls "reduced reduced form" models.
Here is how it works. We start with the logistic regression model discussed above, and we use it to produce historical estimated default probability values for each of our corporate counterparties. Let's take General Motors as an example, using the default model that we have in the KRIS default probability service. We have the monthly history of General Motors default probabilities from January 1990 to the present for default probabilities with maturities of 1 month, 3 months, 6 months, 1 year, 2 years, 3 years and 5 years. We assume these default probabilities are "true" or correct. If our forward looking simulation will have monthly time steps (which is best practice), we want to "reduce" the "reduced form" default probabilities for General Motors to a "reduced reduced form" model that has only macro factors and one idiosyncratic risk factor as inputs to a function that best explains the evolution in the "true" one month default probability for GM. To derive this reduced reduced form model, we fit the logistic regression formula to the monthly history of GM 1 month default probabilities with the unemployment rate, real GDP, home prices and other macro factors as the only inputs. If we select a list of macro factors that is rich enough, any errors in fitting the "true" time series of GM default probabilities will not only be uncorrelated with any macro factors--they will also be uncorrelated with the errors of any other counterparty. This is an insight of Jarrow, Lando and Yu. For the major world economies, Kamakura has found that 20-40 macro economic factors are able to produce logistic equations for each counterparty that typically explain 50-80% of the 19 year history of their default probabilities. The remainder of the variation in their default probabilities is idiosyncratic and diversifiable. The size of this idiosyncratic risk for each counterparty is retained and becomes one of the risk factors that is simulated. Since this risk factor is uncorrelated with any others, it is not necessary to include them in the matrix that needs to be inverted. All we need to invert is a 40 x 40 matrix that is the variance-covariance matrix of the macro factors. This is a vast improvement on the 3,000 x 3,000 matrix we had to start with.
Isn't this a lot of work if we have a lot of counterparties? Not really. In the KRIS default probability service, we have already fitted a logistic function linking macro factors to default probability histories for each of the 22,000 public companies in KRIS. These functions can be invoked with a mouse click for simulations in the credit portfolio management tool KRIS-cdo or loaded into the Kamakura Risk Manager enterprise wide risk management system. The result is a very fast, efficient and highly accurate simulation and stress testing capability that provides a "credit risk CAT scan" that looks through each individual counterparty to the true macro factors driving their risk. This is the capability that Board members or CEOs at Merrill Lynch, UBS, and Citigroup said they needed in order to avoid the huge home price risk that caused tens of billions of dollars of losses.
Donald R. van Deventer
Kamakura Corporation
Honolulu, March 19, 2009