As European IFRS reporters enter 2017, the first generation of Expected Credit Loss (ECL) models have generally been developed, and granular transitional impacts quantified.
The start of 2017 also marks the point in many firms’ IFRS 9 journeys where validation teams, Internal Audit functions and regulators have their first opportunity to lift the lid on detailed methodological assumptions, input parameters, and uncertainty in input parameters.
This article investigates how the mathematical concept of an Expectation has been decomposed at many firms, and highlights two particular sources of bias which may impact accounting values as well as business decisions such as pricing which are informed by IFRS 9 model outputs.
Expectation of Loss
The IFRS 9 standard requires firms to quantify expectations of lifetime default risk and credit losses for in-scope instruments. The standard further recognises that future losses are uncertain and asks firms to evaluate a range of possible outcomes. In other words, expected credit loss (ECL) is a random variable with some distribution; and the standard requires estimation of the Expectation of that distribution.
Mathematically, the Expectation of a random variable can be computed by integrating over all possible outcomes. Many market participants have chosen to approximate “all” with a finite set of scenarios. In this article we focus on estimating a single scenario-conditional expectation of loss (i.e. a single point on the loss distribution).
At firms which have used PIT calibrations of Basel models to quantify ECL, the most commonly observed approach is to define scenario-conditional loss over some observation period as the product of:
- PD, the probability of default event occurring during the observation period;
- SR, the survival rate (i.e. probability that the account has not prepaid at the start of the period);
- EAD, the exposure at default; and
- LGD, the loss given default.
Expected Loss is therefore the expectation of this product, i.e. E[Loss] = E[PD ∙ SR ∙ EAD ∙ LGD].
Many firms have supposed that the Expectation operator is multiplicative and that the expression for scenario-conditional ECL may be expressed as the product E[PD] ∙ E[SR] ∙ E[EAD] ∙ E[LGD]. However, for random variables x and y by definition E[XY] = E[X]E[Y] + Cov(X,Y). In other words, to suppose that the Expectation is multiplicative is only correct if the covariance is zero. Otherwise, the multiplicativity assumption is guaranteed to mis-state ECL.
Whilst PD and LGD are often independent (within the same scenario and time period), the same cannot be said of EAD and LGD: Any LGD model which uses EAD as an input will by-design introduce EAD-LGD correlation and hence violate the zero covariance assumption. To quantify the impact a numerical assessment is likely to be required.
Functional non-invariance assumption
The task of jointly estimating a set of input calibrations is often simplified by forming scenario-conditional expectations of drivers such as a credit cycle index, cure rates and forced sale haircuts. Typical approaches therefore replace expectations with transformations of expectations of inputs. That is to say, and without loss of generality:
- E[PD] = g1 (E[PD Inputs])
- E[SR] = g2 (E[SR Inputs])
- E[EAD] = g3 (E[EAD Inputs])
- E[LGD] = g4 (E[LGD Inputs])
where g(.) is some arbitrary mapping function such as PD models’ log-odds or workout LGD models’ (1/LTV). However, it can be shown that in general for random variable x then g(E[X]) ≠ E[g(X)]. Two further useful results are that:
- Where g(.) is a convex function then g(E[X]) ≤ E[g(X)] and an over-statement of ECL will result; and
- For the special case where g(.) is a linear function then g(E[X]) = E[g(X)] and no mis-statement will result.
Mapping functions can broadly be grouped into three classes:
- Functions which are, by design, linear and can be shown trivially to introduce no bias – i.e. the partial derivative is constant.
- Functions which are, by design, not linear but where the partial derivative of ECL with respect to the input can be assessed analytically and shown numerically to be approximately constant.
- Functions which cannot be differentiated, such as a given scenario’s forward collateral valuation’s relationship with respect to the time to sale input of a workout LGD model.
The impact of functions which fall into the third category can only be assessed numerically.
Although the likelihood of either a multiplicative or functional non-invariance assumption leading directly to a material mis-statement of an ECL accounting value seems remote, it is nevertheless incumbent on model developers to prove that this kind of approximation has had an immaterial impact on both ECL and lifetime PD estimates used for stage allocation.
On a final note, biases which are immaterial in absolute terms from an accounting perspective can be profound in relative terms for applications such as pricing, as firms who deploy their IFRS 9 models for other application without validation are likely to discover to their peril.