Loan Pricing

# Effective Interest (Yield) Loan Fee Amortization

Many indirect loan products require that fees be paid to the firm that originates the loan--an auto dealer for example. Many institutions amortize these fees using a straight-line method over a period of months approximately equal to the estimated life of the loan. For optimizing the performance of a loan portfolio, it is important to understand how the costs associated with these loans and how the up-front fees affect the effective yield on a loan; the straight-line method isn't very good at this.

In some cases, institutions must amortize these fees in a way that satisfies SFAS 91, Accounting for Nonrefundable Fees and Costs Associated with Originating or Acquiring Loans and Initial Direct Costs of Leases. The discussion that follows is designed to provide a way to calculate fee amortizations that will work for loan portfolio optimization calculations. It may (or may not) be helpful in calculating amortizations for SFAS 91; you should discuss this with your accountant.

Using this method to calculate fee amortizations for loan portfolio pricing does not require that an institution use this method for financial accounting.

## Example Calculation

The calculation is easiest to describe with an example loan:

 Principal $10,000 Fee paid$1,000 Interest Rate 7% Term 60 months Payment $198 Since level-yield calculations treat the unamortized fee as part of the loan balance, let's treat the fee amortization just like another payment with the same term and interest rate:  Fee Interest Rate 7% Term 60 months Fee pseudo payment$19.80

With this basic information, it is now time to calculate the amortization for a few periods, as shown in Table 1. The columns in coral show the calculation of the monthly principal portion of the monthly payment, with monthly principal of $139.89,$140.49 and $141.31. Similarly, the cyan columns show the calculation of the monthly fee "principal" amortization, with fee amortization of$13.97, $14.05 and$14.13 respectively.

The light grey columns show the calculation of the level yield as the interest ($58.33 for period 0) divided by the level yield asset ($11,000) multiplied by 12 periods to annualize the result, which gives 6.36%. Repeating this for the other periods confirms that the yield on the combined asset is the same for each period.

What happens to the $5.83 "pseudo interest" in the amortization calculation. If we divide this by the the level yield asset balance ($11,000) and multiply by 12 to annualize it, we get 0.64%--the difference between the contracted 7% interest rate and the effective yield after fee amortization.

Table 1: Example Loan Principal and Fee Amortization for Three Periods
Level Yield Amortization Detailed Calculation Level Yield Level Yield Simplified Calculation
Period Principal Payment Interest Applied Principal Fee Balance Fee Pseudo Payment Fee Pseudo Interest Applied Fee Principal Amortization Expense Level Yield Asset Yield After Fee Amortization Amortization Expense Reduction to Contract Yield Simplified Calculation of Amortization Expense Reduction to Contract Yield Simplified Calculation of Amortization Expense
0$10,000.00-$198.01$58.33-$139.68$1,000.00-$19.80$5.83-$13.97$11,000.006.360.640.64-$13.97
1$9,860.32-$198.01$57.52-$140.49$986.03-$19.80$5.75-$14.05$10,846.356.360.640.64-$14.05
2$9,719.83-$198.01$56.70-$141.31$971.98-$19.80$5.67-$14.13$10,691.816.360.640.64-$14.13

## Calculating Fee Amortization for Prepayments

This approach to calculating the fee amortization works fine until an asset prepays. For full prepayment, this is easy--the entire remaining balance is amortized all at once--but how do you calculate the fee amortization for a partial pre-payment?

To do this, first we should look for an easier way to calculate the monthly fee amortization amount in a way that doesn't require calculating both the pseudo payment and pseudo interest for the fee. Notice that the fee pseudo payment is proportional to the fee balance divided by the principal balance--$1,000/$10,000 or 0.1 in this case. Similarly, the applied principal, $139.68 is proportional to the fee amortization,$13.97.

From this we can calculate the monthly fee amortization as

\begin{aligned} \text{fee amortization}&=&\text{principal reduction}*\frac{\text{fee balance}}{\text{principal balance}} \\ &=&139.68*\frac{1000}{10000} \\ &=&13.97 \end{aligned}

To calculate the prepayment of an unusual amount--perhaps a double payment in month 0--we would just take the principal applied, and use the formula above to calculate the fee amortization:

\begin{aligned} \text{fee amortization}&=&\text{principal reduction}*\frac{\text{fee balance}}{\text{principal balance}} \\ &=&(139.68+198.01)*\frac{1000}{10000} \\ &=&33.77 \end{aligned}

## Calculation Methods for Loan Portfolios

The example also shows how this would be implemented in practice. For loan pricing optimization, the effective yield is needed for each loan type, term and credit grade--including prepayments. Calculating a pseudo payment for each loan and determining the fee amortization by month would be programmatically painful and inefficient. Since the principal portion of the payment would be present in most accounting systems at the loan level, this becomes an easy way to retroactively calculate the fee amortization for effective yield. This is also a calculation that is necessary if the institution decides to convert from one amortization method to a level yield method.

A complete example of this approach in an Excel spreadsheet can be downloaded here.

### Notes

The formula displays in this example are formatted using MathJax.

{calltoaction}