Although from a customer’s point of view the monthly repayments on a loan remain the same over time, the rate at which the principal loan balance decreases follows the shape of an annuity curve: slow at first but increases steadily over the life of the loan.

This shape can be attributed to the nature of compound interest. In the early months, the majority of a payment goes towards paying off interest and so the principal debt remains almost the same. Because the balance is at its highest in the early days, so too is the interest charged. However, with each payment received the balance decreases slightly and along with it the interest charge. Because the monthly payment has remained constant, when the interest charge decreases more and more of that payment goes towards reducing the principal and so the principal balance begins to decrease ever faster.

The mathematical equation that describes this curve is shown below:

PV = x [ (1 – (1 + i) ^{-t })/ i ]

Where:

PV = the value of the loan at a point in time

x = the instalment amount

i = the interest rate

t = the number of terms remaining

This annuity curve has two important uses for the credit risk manager: calculating debt delinquency and assigning initial loan sizes.

**Aging Delinquent Debt**

In cases where data is scarce and the existing databases do not store an accurate measure of how far a particular loan is in arrears, the annuity curve can be used.

Perhaps counter-intuitively, this is done working backward from the end of the loan to today rather than forward from the start of the loan until today. In other words, the method relies on the term remaining and the current product parameters rather than any historical payment patterns. Mathematically, this is done in two steps.

The first step is to solve for PV in the annuity formula shown above. This gives us the balance which would exist if all payments had been met, given the current interest rate, monthly repayments and remaining term of the loan. This is the de facto target and so if the actual balance outstanding is higher than this (above it on the curve) the loan must be in arrears.

To convert the arrears balance into a more useful, time-based measure is a simple process. If the balance in arrears is equal to the difference between the actual balance and the expected balance then time in arrears can be estimated by dividing that difference by the monthly instalment.

The reason this approach works better in environments where data is scarce is that it doesn’t rely on a knowledge of any previous transactions, changes in interest rates, partial payments made, etc. All that it needs to work is for the current data to accurately reflect the status quo. Even is the interest rate has changed or the term has been extended, etc. the properties of the curve mean that these can be catered for without losing too much accuracy. For example, an up-to-date account will remain so even if the interest rate changes. In this case the monthly repayment will increase and the slope of the curve will become shallower again initially and steeper towards the loans termination.

**Assigning Initial Loan Sizes**

Because the annuity formula provides a link between monthly instalments and loan balance for any given product parameters, it can be used to assign affordable loan sizes.

The first step is to estimate a customer’s monthly free cash flow, which will in turn be used to estimate an affordable monthly repayment. For example, if a customer has €5 000 a month of free cash flow after all current expenses have been paid, a lender can allow a percentage of that to be used to cover a new monthly instalment. Typically the percentage allowed will vary between 30% and 60% but this depends on many factors which will not be discussed here. Let’s assume that in this example the lender has decided to allow 50% of free cash flows to be used as a monthly loan repayment.

Once an affordable monthly instalment has been established – in this example €2 500 – it can be entered into the annuity formula which is then solved for PV: the initial loan size. Assuming the lender wishes to make the loan over a 5 year period at 10%, they should lend the customer in question no more than € 25 000. Calculated as below:

PV = €2 500 [ (1 – (1 + 10%) ^{-60 })/ 10% ]

PV = €2 500 [ 0.9967 / 0.1 ]

PV = €24 918

In most cases, when used in this manner the result of the annuity formula would be passed through a second set of filters to ensure that the loan offered did not contravene product parameters: maximum or minimum loan amounts aren’t exceeded, etc. So, although the customer qualifies for €25 000 in this case, if the lender has already decided to never make unsecured loans for more than €20 000 to new customers, the offer would be decreased at this stage to €20 000.

In more sophisticated operations the annuity model method can also be used to accommodate risk-based loan assignment. This is most easily done by making the affordability ratio and maximum loan sizes risk based so that, for example, a low risk customer may be allowed to use 60% of their free cash flow to pay a monthly loan instalment while a high risk customer may only be able to use 35% of their free cash flow for the same purpose.