Absolute and Conditional Default Probabilities

This document introduces the most important concept in pricing credit risk: default probability. As participants in the international financial markets have focussed more on assessing and calculating credit risk, the need for a clear method for pricing credit risk has become greater, but there is still no unanimity on the best approach to this problem. This document takes the first steps towards one possible approach to credit risk.

We begin by noticing that credit risk can be thought of as developing from two factors: the possibility that a borrower may default, and the amount of money that may be lost on a given liability of that borrower in the event of default. We think of the first of these two factors as the default probability, and the second as the severity of the default. Another way of thinking about severity is by considering not the money that may be lost in the event of default, but the money that may in the event of default be returned to the investor in the borrowers liability; we then talk about recovery rate. The recovery rate and the severity are both expressed as proportions of the capital invested in a given liability. Writing S for the severity, and Rfor the recovery rate, we can in general write S=1-R

When considering default probabilities, the first thing to notice is that, ultimately, every borrower will default. Over a sufficiently long time horizon, the probability of default for every borrower tends to 1. This seems, at first sight, slightly implausible: a major corporation or institution, or a major country, seems certain not to default. This is not the case; unless an organisation has no outstanding obligations of any kind, it has a minuscule probability of default.

Eventually, though it may take time, business conditions, poor management, or just plain bad luck will lead to the default of any company. Is the same true of countries ? At first sight, it seems that this is not the case: unlike corporations, countries run their own currencies, so when there is a problem, a country can just crank up the presses, and print off some more money. But consider the case of Japan: Tokyo sits in an area known for earthquakes, and sooner or later the big one will hit. When it does, there is a chance that Japan will default on its debts: the printing presses at the mint will be damaged, or the records saying how much is due, to whom, and when, will be destroyed, or the communication system will fail. And if Japan gets through this earthquake, well, the ring of fire round the Pacific will get it eventually maybe not this earthquake, or the next one, but one day In the same way, sooner or later, there will be some kind of national disaster which will lead every country in the world to default. It may take many centuries, but eventually everyone who ever borrows defaults.

Given this rather depressing consideration, how can we discuss default probabilities sensibly ? The answer is, of course, that some borrowers are probably going to default sooner than others. A company with huge debts struggling for survival in a poor market is more likely to default in the immediate future than one which has borrowed only negligibly, has a healthy bank account, and excellent trading conditions. The key is to consider time. Different borrowers have different probabilities of default over finite time periods. Pemex, the Mexican oil company, has a different probability of defaulting over the next year from Exxon or Shell. In the long run, theyre all going down, but over the next one year, or two years, or any finite number of years forward, they have different probabilities of default.

In other words, the way to think about default probabilities is as probability of default in a given time period. For example, we could consider the probability that a company might default in the one year period that starts today, or the two year period starting today, or the three year period starting today, and so on. Alternatively, we could consider the probability that a company might default in the one year period starting today, or the one year period starting one year from today, or year after that, and so on; with this alternative, though, we have two choices: we could consider the outright probability that the company might default in any particular year, or the probability that the company might default in that year, given that it has not yet defaulted.

We call the first of these ways of thinking about the probability of default the cumulative probability of default.

Notice how the cumulative probability of default never decreases with time. Its easy to see that eventually, the cumulative probability of default ( after many years ) will tend towards 1.

The second way of thinking of default probabilities is as the outright chance of default in any particular time period. We call this the absolute probability of default.

Notice that the absolute probability of default in this example seems to be decreasing with time. This is a very common occurrence, but isnt sure to happen. Sometimes the absolute probability of default may increase for a few years. It can be proven, though, that the absolute probability of default tends to decline towards 0. To see why this must be so, consider the relationship between the absolute probability of default and the cumulative probability of default: the cumulative probability of default for some particular time period is just the sum of the absolute probabilities of default at all periods up to and including that period. For example, in the graphs shown above, we have the following relationships:

So the cumulative probability of default in year 5, say, is just the sum of the absolute probabilities of default in years 1, 2, 3, 4 and 5.

Now, we also know that the ultimate cumulative probability of default, if we go far enough forward, is 1. So the sum of the absolute probabilities of default must always be no more than 1. Now, suppose that the absolute probabilities of default in any period do not gradually tend towards 0as we look at more and more distant periods; say they tend to some other number, X; then, if we summed up this number over enough periods, wed get a cumulative probability of default of greater than 1, which is impossible, as probabilities must always be less than or equal to 1. So our initial suggestion, that absolute probabilities of default do not tend to 0as we look at more and more distant periods, must be false; that is, absolute probabilities of default really do tend to 0for more and more distant periods. ( To see how this might work with the example above, suppose that the absolute probability of default for each yearly period after year 10 is at least 0.01. Then by year 21 the cumulative probability of default would be greater than 1, which is impossible. Whatever number we choose, if the absolute probability of default does not tend to 0, we get an impossible result, so the absolute probability of default must tend to 0.)

We can see that the sum of all the absolute probabilities of default for a particular credit over all time periods must equal 1. And we can also see that these absolute probabilities of default must tend to get closer to 0as we look at periods of time further and further into the future. Somehow, though, this seems slightly strange. For most organisations it seems somehow unlikely that the probability of default declines with time the future is unpredictable, so a lower chance of default in a given time period seems quite unexpected. What is going on ?

The answer is that the absolute default probabilities that we have considered so far are numbers which answer the question: what is the outright probability of default in this or that named time period. But this figure doesnt address what must happen before that event can occur. A more useful question to ask is often: if the borrower has survived up to the beginning of this period, what is the probability that it will default in this period. That is, we consider a conditional probability of default. There is a simple way of relating the conditional and the absolute probabilities of default in any period: the absolute probability of default in a particular period is just the conditional probability of default in that period, multiplied by the probability that the borrower has not defaulted in any earlier period. This probability of no default in any earlier period is, in turn, just 1 - cumulativeprobabilityofdefault.

We can formalise the relationships between the various ways of thinking of default probabilities. Say that time is divided into intervals which are identified by the points in time at which they begin and end, and say that probabilities are defined with respect to these intervals. Then we can write the relationship between cumulative and absolute probabilities of default like this:

where is the cumulative probability of default between times 0and n,

and is the absolute probability of default between times iand i+1.

We can write the relationship between the absolute and conditional probabilities of default like this:

where is the absolute probability of default between times iand i+1.

and is the conditional probability of default between times iand i+1.

( To be completely clear, we should also note that ; that is, we start our consideration of the possibility of default at time 0. )

Looking at the example discussed earlier, we can now look fill in another column:

To help understand how conditional probabilities of default vary from cumulative and absolute probabilities of default, another example may help:

Notice how, as remarked earlier, the cumulative probability of default increases towards 1. Also, as discussed earlier, the absolute probability of default gradually moves towards 0.

When we look at conditional probabilities of default, a couple of observations can be made immediately. The first thing to notice is that, unlike absolute and cumulative probabilities of default, conditional probabilities of default dont, at first sight, seem to have to converge to some particular value as we look at further time horizons. To be quite clear about this, conditional probabilities of default may converge to some particular value for time periods in the future, but ( although some models of default probabilities may assume this ) there is no reason why they should ( and, indeed, some models of the conditional probabilities of default for some loans quite deliberately assume that they do not converge to a single value ). The second point to notice is that if a particular period has a non-zero conditional probability of default, and if the conditional probabilities of default for all periods after that period are 0, the conditional probability of default for that time period must be 1.

In practice, different practitioners will adopt different conventions when discussing probabilities of default. The tendency, though, seems to be to consider the conditional probability of default. The reason for this is quite straightforward: conditional default probabilities make it easy to compare the probabilities of default in different time periods. Looking at the examples presented above, it is much easier to understand the relationships between the likelihoods of defaults in different time periods by examining the conditional probabilities of default than by looking at the cumulative and absolute probabilities of default.