The Term Structure of Default Probabilities

The term structure of the interest rate market is comparatively well-known to most participants in the international financial markets. What is less clear, however, is that there is also a term structure to credit risk. In particular, there is a term structure to default probabilities.

Given the term structure of risk-free rates, and the term structure of borrowing by a potentially defaulting borrower, and making certain assumptions about the proportion of a loan lost in the event of default, we may derive a term structure to the implied probability of default of that borrower. Having derived this term structure, it is important to understand it, and in particular how it changes with time.

Note that there are two different ways of deriving the term structure of default probabilities: the absolute probability of default in each time period can be determined, or the conditional probability ( that is, the probability of default in a given time-bucket given no earlier default ) can be used. Each approach has its own advantages and disadvantages, and both are covered below, as similar statistical techniques can be applied to both. When only one approach is being discussed, this should be clear from the context.

For many companies, the conditional probability of default can be expected to increase with time. This is a fairly natural expectation: in general, the future is unpredictable, and the probability that one or more factors damaging to a company exceeds a critical value increases with time. ( We might call this a "positively-sloping", or, perhaps, "normal" term structure of default probabilities. ) There will be some cases, however, where the probability of default in a given time-bucket is actually lower than that of a preceding time-bucket. This can be interpreted as meaning that the company faces some shorter-term risks which, if successfully overcome, will then be followed by a more stable set of trading circumstances. ( This corresponds to an "inverted" term structure. ) For some borrowers, the probability of default may decline with time to some minimum, and thereafter increase again. This "downwardly flexed" curve corresponds to the case where it is known that the company faces some short-term challenges, which may be overcome, but may then face further challenges in the more distant future. ( This might occur with a company in a regulated industry facing immediate management difficulties, and facing a more deregulated future from some already-known date, for example. ) An "upwardly flexed" curve can reasonably be expected to be comparatively rare, as the interpretation of this curve - that default probabilities increase with time to some point, beyond which they decline with time - corresponds to an extremely rare market situation.

In general, the term structure of default probabilities can be expected to be relatively smooth. Exceptions to this can be expected largely where some potentially adverse structural change in the markets to which the borrower is exposed is already known. These kinds of structural changes are, typically, rare.

Once the immediately observable term structure of default probabilities for some borrower has been determined, the next problem is to understand how this structure might change with time.

Given adequate information, it is possible to determine the term structure of default probabilities for a particular borrower for a whole range of previous days.

The natural way to analyse this data is by considering changes in the shape of this term structure from day to day. That is, it is natural to seek to build a correlation matrix, displaying the correlations of changes at each grid-point ( that is, the probability of default in different time periods ) with each other. This correlation matrix is then susceptible to analysis in much the same way as any other. The natural way to begin the process of understanding this correlation matrix is by performing a principal component decomposition of the matrix. We seek to decompose the matrix like this:

where P is the correlation matrix, and E is the matrix of the product of eigenvectors and the square root of eigenvalues of P.

We would, at first sight, expect to find that this decomposition will show a first component which corresponds to a "parallel" shift in default probabilities, a second component which corresponds to a "steepening" shift, and so on. In practice this expected result may not occur; the eigenvector with the corresponding eigenvalue may actually look more like a "steepening" shift, as the conditional nature of the default probabilities being examined may actually result in changes in shorter-term default probabilities having a lower impact further out along the term structure. ( Think of it like this: if the probability of default in the first year increases, it can only do so at the expense of lowering the probability of default elsewhere in the term structure, as the total of all the default probabilities should be 1. However, we are dealing with probabilities which are, for the most part, comparatively low, we are likely to only be examining the "near" part of the term structure of default probabilities, and we are examining changes rather than outright default probabilities, so this effect should not be terribly marked, though it should be detectable. )

Having performed a principal component analysis of the correlation matrix, the next natural step is to perform a factor analysis. We seek to decompose the matrix like this:

where L is the matrix of factors proposed, and E is a diagonal matrix of error terms.

Performing this analysis will allow us to "impose" our own expectations of the behaviour of the term structure of default probabilities on the correlation matrix, though these expectations will, of course, be significantly coloured by the results of the earlier principal component analysis. In particular, if the principal component analysis has delivered a relatively unambiguous first component, we might begin our factor analysis simply by using this component ( or a slightly smoothed version of this ) as the only factor in a "single factor" analysis.

Factor analysis allows us to impose our own ideas of the behaviour of the term structure of default probabilities on the actual historically observed correlation matrix, but what might these ideas actually be ?

The most important observation to make about the term structure of absolute default probabilities is that it is constrained in a way that other term structures - notably the term structure of interest rates - is not. The constraint is that the total of all the probabilities in the term structure must be a constant: 1. This is quite different from a term structure in which there is no such constraint, and it is this constraint that gives the behaviour of default probabilities its flavour.

Consideration of the fact that the sum of all absolute default probabilities in the term structure must be a constant would suggest that a parallel shift in this term structure is unlikely to appear in the principal component analysis. This is in principle true, but in practice a further point should be borne in mind: although the sum of all the default probabilities must be 1, the default probability for each individual time-bucket is typically very small. Further, when we consider simply the term structure of default probabilities out to ( say ) ten years, or fifteen years, forward, the sum of all of these default probabilities is still very small. Thus there is ample opportunity for one of the components of a principal component analysis to appear, and in practice be, a parallel shift. This shift simply reflects the possibility that default probabilities over the time-horizon being examined may all rise together, with the corresponding fall in default probabilities dictated by the "constant sum" condition occurring beyond this horizon.

Having made this observation, a second observation follows naturally: the term structure of default probabilities for a higher quality credit is more likely to exhibit a quasi-parallel shift than a poorer quality one. The reasoning is simple: with a poorer quality credit, the individual probabilities of default in the different time-buckets are higher, and therefore the sum of the default probabilities beyond the time-horizon being examined is lower. The unobserved default probabilities must, in the event of an observed quasi-parallel shift, sum to a lower number than before the parallel shift has occurred. Since this number is lower for a poorer quality credit than for a higher quality one, it is more difficult for this number to "absorb" the impact of the parallel shift. That is, changes in the term structure of default probabilities are less likely to occur as (quasi-)parallel shifts for lower-grade credits.

In building a model of the term structure of default probabilities, it is useful to consider the underlying financial markets on which these probabilities depend. In particular, it is useful to consider the way the information enters into the credit markets, and is processed within those markets.

In general, lending activity occurs only when the lender is confident that the risks that he assumes in making a loan are equal to, or outweighed by, the potential returns available to him from the loan. This lending decision is made on the basis of data that enters the market from time to time, but cannot be made on the basis of data that has not entered the market. That is, the lending decision is based on assumptions about future data that will enter the market later. In particular, a lender makes the assumption that company data entering the market in the future will not support a different credit assessment of the company. If a lender had a perfectly-functioning crystal ball, he would not need to make any risk-return decision in deciding whether to lend money to a particular company; he would simply look at his oracle and, in the event of a positive outcome, lend as much as possible. In effect, the lender makes his lending decision on the basis of the expected loss that he might incur, and his objective is to minimise this expected loss over his entire portfolio.

The behaviour of companies and individuals, however, seldom conforms to that expected of them. This is because it is extraordinarily difficult to form useful expectations. When a lender decides to lend to a particular company, he will typically do so on the basis of comparatively little data. For a five-year loan, for example, the lender may look at five years' worth of historical data on the company's performance, but he may look at far less, and is unlikely to look at much more. From this data he will seek to extrapolate the company's future performance, and base his lending decision on this extrapolation. This is highly risky: the data that will affect the company's performance in the future is presently inaccessible, and it is easy to see that the further an extrapolation of this kind is pushed into the future, the less reliable it will become.

Lenders cope with this difficulty in predicting the future in the most human way: they ignore it. Decisions about the longer-term creditworthiness of a particular firm are dealt with by a simple assumption: what holds true for the moderately foreseeable future will also hold true for the distant future. That is, once a certain time-horizon has been breached, a lender will assume that thereafter creditworthiness, and therefore the probabilities of default in further-out time buckets, remains constant.

In terms of potential changes in the term structure of default probabilities, this means that an important fraction of the variability of default probabilities will be concentrated at the front of the curve. We would expect to find a factor which changes only the nearer default probabilities in the term structure, with the influence of this factor tailing off at about the point on the term structure beyond which the market is unable to make useful judgements about the future riskiness of lending to that company. That is, the difficulty in establishing a fair assessment of the future potential creditworthiness of the company make it likely that the term structure of default probabilities is more subject to statistically significant driving factors in the earlier time-buckets.

Related to this observation of the tendency of more statistically significant driving factors to occur towards the front of the curve, we may expect that the volatility of default probabilities is higher in the earlier time-buckets. That is, the term structure of default probabilities can be expected to be more volatile at the front of the curve; this precisely matches the reasonable intuitions that information entering the markets is more likely to have significance for the immediate financial viability of a borrower, and that the longer-term implications of data entering the market cannot be well-established by market participants, who therefore tend to leave longer-term default probabilities unscathed by seemingly less-significant news.

Having established some expectations of the factors likely to be driving changes in the term structure of default probabilities, these expectations can be included in a factor model of the term structure. This model can then be applied in the analysis of the structure, and in relating the model to information actually entering the market from time to time. In particular, a factor model can be applied in the assessment of value at risk for credit derivatives portfolios. Further, a suitable factor model of the term structure of default probabilities should have usefulness as a tool for the pricing of credit derivatives, particularly options.

Finally, some mention should be made of models of the term structure in which the last time-bucket in the model includes not just the probability of default for one time interval, but for all subsequent time intervals. When this approach to the term structure of absolute default probabilities is used, the constraint that the total of the probabilities in each bucket is fixed at 1 becomes obvious, and imposes the constraint that the sum of all the factors creating each day by day change in the term structure must be such as to preserve this total. Note, though, that it does not impose the constraint that the net change in total implied by any individual factor is zero. It is quite possible to imagine a factor whose effect, when applied in isolation, would be to take the sum of all the default probabilities in the term structure to some other number than 1; the constraint on the system would simply be that this factor could only occur at the same time as one or more compensating factors. Note further that because the compensation could be caused by more than one factor, nothing obvious can be said about the relationship between the factors in this case.