Jack Xu, founder of Modtris Financial Modelling, outlines two ways of conceptualising probability of default and the challenges associated with both

Probability has been used to define, measure and price risks in the financial market for a long time. For CLO managers and investors in particular, default risk is firmly equated to the probability of default. The meaning of probability may be self-evident from the example of counting faces and tails in coin tosses, but is the probability of a company’s default just as trivial and straightforward? This article reviews two ways of conceptualising probability in the case of default – one attributes the probability to a large group of companies, and the other attributes the probability to a single company without reference to a group.

Probability and repeatable observations
Probability is always associated with a method of repetition because it is observable only when a sufficiently large number of trials are repeated. “The probability that a coin is tossed to face is one half” means that, if we toss a coin sufficiently many times, the percentage of getting a face becomes very close to half of the total tosses made. Of course, the repetition works only if the coin returns to the same initial state after each toss.

Conceptually, if the coin is such that it self-destructs whenever it is tossed to face, then the repetition becomes undoable with a single coin. But we can get around this problem if there are a large number of coins of the exact same kind. The probability of tossing a face can then be observed by tossing many coins at once and counting the coins that implode into smoke.

Probability of default of a group of companies
A company can be considered to be like the combustible coin. If a company defaults, it is gone and cannot be again used to observe another default. So, the probability of default of a company is normally associated with a large group of companies, which - for the purpose of observing default probability - are considered all identical.

However, unlike coins, companies are intrinsically individualistic. Hence a company must first be de-individualised, based on various contraptions of rules, before it is assigned to a group.

Within each group, one can observe the percentage of default over any chosen time period and apply that to any similar group of companies. This is normally how CLO managers view default probability within a portfolio of credits.

There are a few challenges with this approach that must be addressed by portfolio managers.

1. Group size. The group used to observe a default percentage normally contains a much larger number of companies than an actual portfolio does. It is essential then to observe how strongly the default percentage of subgroups in similar size as the portfolio deviates from that of the entire group.
2. Time-dependence. The default percentage observed within each group should ideally be time-independent or follow a predictable time series. If neither is true, portfolio managers have to contend with how much the percentages of default observed in the past will hold in the future.
3. Interconnectedness. Portfolio managers need to compare how interconnected the companies are in the groupings and in the actual portfolio. This may come up as “default correlation”, but I refrain from using the term here before discussing the concept in a follow-up article.

These challenges rise when a company is de-individualised into a group. The grouping is a method of approximation which transforms a large portfolio of company obligors into a much fewer number of groups. The simplification makes it numerically feasible to quantify default risk, but the drawback is that it may miss out the details essential to the default risk - both of individual companies and, in aggregate, of the entire portfolio.

It is worth imagining if the risk of a portfolio can be derived truly from the ‘first principles’ – by aggregating the default risk of each individual company.  This requires first a meaning to the default probability of an individual company, without any reference to a group.

As discussed, probability requires repetitive observations, but default is not an event that a single company can repeat. However, if a good understanding of the causes of a company’s default can be achieved, it is possible to link a company’s default dynamically to other events of the company that do repeat over time.

Probability of default of a single company
For the sake of argument, let’s assume that company XYZ’s default is determined completely by and only by its sales performance; specifically, the ratio of sales to new inventory which is observed quarterly. Suppose that the future path (or paths) of the sales ratios that will lead XYZ to default has been determined exactly.

Since the event of sale is repeated quarterly, the probability of a given value of sales ratio can be defined from the distribution of the observed values. Consequently, the probability of any given future path that the sales ratio follows can also be defined. Finally, the probability of default of XYZ is the probability of that path (or those paths) leading to default.

Default probability defined this way does not rely on assigning XYZ to any group; rather, it depends only on repeatable conditions of XYZ itself. Additionally, default risk defined this way depends not on just one quarterly result, but on specific trails of the quarterly result - meaning that not only the levels of performance, but also the specific timing of them impacts the default outcome.

It is helpful to put the idea into perspective with two other long-standing approaches – fundamental credit analysis and structural models. For fundamental credit analysts, using a company’s individual financial data to assess its default risk is what they do. The difference is that the fundamental analysis does not link causally the outcome of default to a process or a path leading to the default, so that both the risk and the timing of default can be quantified and aggregated across a portfolio.

Tying the default probability of a single company to a process is not a new idea either. Structural models - which began with Merton’s work in 1974 - define the default of a company as the occurrence that a company’s total asset value, in mark-to-market terms, falls below the company’s debt level which is assumed to be fixed.

In this way, the default probability of a single company is tied to the probability that the total asset value would follow a path that crosses the fixed debt level at some point in the future. The problem is that for non-financial companies, total asset value is seldomly ever marked to market, let alone repetitively, and is not at all the reason a company defaults.

This leads us to the challenge in the ‘first-principles’ approach of modelling the default probability of a single company. To make an analogy to animation software, the structural models animate the human motions but do so by simplifying the human figure into a ball shape, while the fundamental credit analysis draws the full anatomy of a human body but cannot animate how the different body parts move together. The challenge is how to combine the two approaches and realistically model the dynamics of a company’s multi-dimensional financial state.

Summary
There are two ways of defining the probability of the non-repeatable event of default of a company – grouping many companies together, or causally linking default to other repeatable events. The first approach simplifies the complexity of a large credit portfolio and allows quantitative risk assessment.

However, it faces the challenges highlighted in this article. As the economy changes more and more rapidly and companies become more and more inter-connected, these challenges can become serious problems for making truly forward-looking forecasts.

The second approach attempts to understand the causes of default of a company from its intrinsic financial conditions and aggregate the risk of a portfolio from ground up. It is still a work-in-progress, as the fundamental credit analysis has not been widely incorporated into dynamic models.

In my view, the failure is in part due to the collective abhorrence of the quantitative community towards understanding financial accounting. However, major progress is being made.

Before diving into the details, I will discuss in a follow-up article another concept that is as important as it is confusing – correlation.