IFRS 9 rules for calculating the lifetime expected credit loss (ECL) continue to create confusion. Currently, there is a lot of talk about post-model adjustments that need to be made to IFRS 9 projections in response to shifting geopolitical risk.
But there is a much more impactful issue hiding in the shadows: calculating probability of default (PD) across the lifetime of a loan. Today, it’s difficult to know which PD approach (e.g., marginal or conditional) is the most logical to use for this projection, because little guidance is provided under IFRS 9. Moreover, even the definitions of these approaches lack clarity.
Marco Folpmers
Before we compare and contrast marginal and conditional PD, it’s important to understand why the choice between the two is important. Figuring out the best PD implementation is imperative, because, under IFRS 9, banks with loans that migrate to Stage 2 (increased level of credit risk) and 3 (loss impairment) must calculate the lifetime ECL. For these exposures, banks model the future default probabilities (along with the expected recoveries and outstandings), through the maturity of the loan.
The lifetime ECL, in short, is the discounted sum of the future yearly expected credit losses, which are based upon future values for PD, LGD and EAD. (The future values for PD will be the focus of this article.)
Unlike Basel III’s internal ratings-based (IRB) approach to credit risk, having only one PD value per exposure is not sufficient for IFRS 9. Instead, the financial accounting standard requires that a PD curve be modeled as a vector of PD values for a given exposure – annually, until the end of a loan’s contract.
The modeling of these fluctuating future PD values is a complex problem. Right now, though, let's focus on only one question: which PDs should be used to calculate lifetime ECL under IFRS 9? In our search for an answer, we must first consider the differences between marginal and conditional PDs.
To get a better grip on marginal and conditional PDs, let’s make sure that we have our language straight. A simple example is provided in the table below, which covers a loan portfolio of 100 loans across a nine-year timeframe.
Table 1: Year-by-Year Default Statistics
For a given cohort (a group of homogeneous loans with the same origination date), analysis of past defaults has taught us that, on average, two loans default during the first year. A year later, another two loans default, followed by one additional loan in year three. This information has been captured in the “Nr of Defaults” column in Table 1.
If we sum up the number of defaults of this column, we see that, on average, 20 loans default out of 100 across a nine-year period. This means that the “lifetime PD” equals 20%. Let’s now assume that this cohort of 100 loans is typical of the portfolio in scope.
If this is a representative cohort, we can estimate the cumulative PD by simply dividing the cumulative number of defaults by the size of the loan portfolio – 100, in this case. After nine years, the cumulative PD is 20%, as it should be.
Following the calculation of the cumulative PD, we can proceed to the “marginal PD.” This is the PD for each single year. It is calculated by subtracting last year’s cumulative PD from this year’s cumulative PD. For example, in year five (see Table 1), the marginal PD equals 2% – i.e., 10% minus 8%.
After allocating the defaults to single years and then adding the marginal PDs, we arrive again at the expected lifetime PD of 20% – the same number as the sum of the marginal PDs. As we consider which PD approach is better, it is crucial to remember that the sum of the marginal PDs reconciles exactly with the lifetime PD.
Conditional PD, typically calculated using Bayes’ theorem, is the other option for firms that need to calculate lifetime ECL. This, in short, is the probability of a default in a specific year, given its survival up to that same year.
In our example, this conditional probability is calculated as the marginal PD in year t, divided by the survival probability up to (and including) year t-1. This survival probability up to year t equals the complement of the cumulative default probability of year t-1. So, for example, in year five, the conditional PD is 2.17%. This is the probability that a loan defaults in year five, given its survival up to that year. In our example above, we divide the marginal PD of 2% for year five by 92%.
In general, for meaningful cumulative PD and survival rates (between 0 and 1), the conditional PDs are higher than the marginal PDs, due to a denominator effect. As is typically the case in cohort statistics, cumulative PD, marginal PD and conditional PD are the same during the first year. However, after that, in a meaningful setting with positive default probabilities, conditional PDs are higher than marginal PDs.
This also means that the sum of the conditional PDs does not reconcile with lifetime PD. In our example, the sum of conditional PDs is 21.98% – roughly 10% larger than lifetime PD.
The cumulative, marginal and conditional PD curves are depicted in the graph below. We see clearly the common start of all three, while the red conditional PD curve rises above the blue marginal PD curve in subsequent years.
Figure 1: PD Curves
For the IFRS 9 ECL formula, we need forward-looking PDs annually. Suppose that the PDs have been calibrated, based on cohort data, to the curves depicted in Figure 1. In this scenario, which PDs do we use annually? The marginal or the conditional PDs?
Based on the data in Table 1 and Figure 1, the only conclusion that can be drawn is that marginal PD is superior to conditional PD for ECL calculation. Remember, when using marginal PD, one ends up with a prospective total number of defaults for a homogeneous group of loans that aligns exactly with the projected lifetime PD.
Conditional PD, on the other hand, runs the risk of overestimating the ECL. This overestimation translates linearly to the ECL. In our example, using conditional PD, we would overestimate the ECL by 10%.
Interestingly, in my experience collaborating with clients, European banks often resort to conditional PD. David Harper, the founder of Bionic Turtle, an exam preparation provider, shares this perspective. “In practice, I am quite confident that everybody tends to use conditional PDs,” he recently said in response to a question about PD approaches in the Bionic Turtle forum.
Theoretically, Harper elaborated, unconditional (aka marginal) PD is often more appropriate. Why, then, is the industry often resorting to the conditional PD when calculating ECL for IFRS 9?
Firstly, the conditional PD seems to make sense. It is easy to reason that, for a loan to default in a future year t, it first needs to have survived up to that year. Based on this factor alone, one could make the argument that the conditional PD is better.
However, while this reasoning is valid in areas related to survival analysis, it does not hold true for IFRS 9’s ECL formula. Remember, under that formula, projected loan losses need to be precisely allocated to the years between now and the end of the lifetime of a loan. Marginal PDs do exactly that – while conditional PDs do not reconcile with the lifetime PD.
Under IFRS 9, there is also a lot of confusion about PD terms.
Marginal PD, for example, has been described both as an unconditional PD (as its interpreted in this article) and as a conditional PD. Consequently, some practitioners choose not to use the term “marginal PD.”
Moreover, computation of the conditional PD under IFRS 9 also suffers from a lack of clarity. Many interpret it as the PD in a specific future year, conditional upon survival up to that same year. (This calls for it to be computed using Bayes’ theorem – i.e., dividing PD by 1 + the survival rate.)
However, conditional PD has also been interpreted by others as the multiplication of an unconditional PD and the survival rate. (The latter approach would actually make conditional PDs lower than the incoming PDs, which is certainly not common practice!)
Complicating matters further, there is not much guidance on IFRS 9 ECL calculation and how to deal precisely with the PD forward curve. This is exactly the reason the experts are hesitant to offer specific recommendations about using marginal versus conditional PDs. After all, this choice depends not only on the definitions but also on the broader modeling approach.
Adding to the confusion, the European Banking Authority (EBA) goes off track by using unconditional PD for the through-the-cycle PD, and by subsequently contrasting these with point-in-time PDs (which are then, apparently, “conditional”).
Since specific rules for the IFRS 9 formulae are not prescribed in an RWA kind of expression, it remains unclear which PD approach is favored. If one hears, amid this confusion, that peers are mainly using “conditional PD” for IFRS 9 lifetime loss calculations, it may be tempting to follow the mainstream approach, under the false assumption that the “majority will have it right.”
Given all these muddled definitions and ideas, it’s vital for practitioners to define the PD concepts they use before embarking on IFRS 9 modeling and documentation.
Detailed documentation of the IFRS 9 ECL formulae, and especially of PD curves, is important and necessary for European banks.
If we use marginal PD in the fashion outlined in this article (see Table 1), it’s clearly the superior choice to conditional PD for calculating ECL under IFRS 9. Marginal PD, after all, reconciles exactly with lifetime ECL, whereas conditional PD can potentially overestimate it.
Going forward, in addition to weighing the benefits of marginal PD versus conditional PD, it would be wise for European banks to promote consistency of the language used for different PD approaches. When meeting with the EBA, moreover, banks should also ask the supervisor not to confuse through-the-cycle PD with unconditional PD or point-in-time PD with conditional PD.
The clearer the industry can make PD definitions, the better.
Dr. Marco Folpmers (FRM) is a partner for Financial Risk Management at Deloitte the Netherlands.