# For IFRS9, losses should be discounted with the EIR, why is that sensible?

Within the IFRS9 framework it is stated that one needs to determine the expected losses and discount these with the effective interest rate (EIR), i.e. the contractual rate at initiation. However, I would like to understand, both from an economical and a mathematical point of view, why this is logical.

Now, the reason I have difficulty in understanding this is due to the following. I believe that given a product there are two ways to determine the fair value, suppose we are looking at a mortgage:

1. If we assume that the contractual rate is the sum of the risk free rate and a spread for credit risk (neglecting other types of risk). Then taking the contractual cashflows and discounting these with the contractual rate corresponds to the discounted cash flow approach, where all the counterparty risk and time value of money is within the discount rate. To this end this results in a fair value of the product. To recap in formula's:

$$V_0=\sum_{i}\frac{C_i}{(1+r+\lambda)^i},$$

where $r$ is the risk-free rate, $\lambda$ the spread for credit risk, $r+\lambda$ the contractual rate, and $C_i$ the contractual cashflow at time $i$. $V_0$ is what the bank would write for it's bookvalue. And as one can understand, due to the discounting with the contractual rate it already takes into account expected credit losses. So why even model losses seperately?

1. Instead of looking at the contractual cashflows we will create a model that models the credit risk, that is, risk-neutral pricing, which would look as follows:

$$V_0=\mathbb{E}\left[\sum_{i}\frac{C_i\cdot1_{\tau>i}+R\cdot1_{i-1<\tau\leq i}}{(1+r)^i}\right],$$

where $\tau$ the default time and $R$ the recovery in case of a default. And, note that discounting is only with respect to the risk-free rate (which might be the funding rate for a bank, if that is more sensible). Now, in this case it would be sensible to model the losses, but if we do so, they should be discounted with the risk-free rate, and so should be the contractual cashflows.

How exactly do we mediate between method 1 where we apply the contractual rate, and method 2 where we model the losses, as for IFRS9 we have to do both things?

1. $$V_0$$ is what the bank would write for it's book value

This is only the case for items held on the balance sheet at Fair Value. Most banks will hold many assets/loans at amortized cost (principal outstanding). The book value will therefore be irrespective of any changes in interest rates and credit spreads. The ABA has a good summary of why this is the case here.

1. So why even model losses separately?

If a bank is holding an asset at amortized cost then it is important to model losses to hold provisions for any credit losses. One only needs to model losses if the assets are not held at FV trough the P&L in which case your approach described in 1. makes sense.

2. How exactly do we mediate between method 1 where we apply the contractual rate, and method 2 where we model the losses, as for IFRS 9 we have to do both things?

As stated above one does not need to do both for IFRS 9, as the EIR under IFRS 9 is with respect to the amortised cost holding value and not the Fair Value holding value.

This all aside IFRS 9 requires loan loss provisions to be held in addition to the fact that a credit spread is included in the amortized cost holding value. Note that if there is no change in credit risk of the asset then only 1 year of expected losses need to held. So the idea is that only lifetime expected losses should be held if the asset has become riskier (i.e. client missed a monthly installment).