Loan Pricing

Effective Interest (Yield) Loan Discount Points Application

Many types of loans–mortgages in particular–allow a borrower to pay discount points at loan origination as a way to reduce the interest rate of a loan. For financial accounting and reporting purposes these discount points are treated as a prepaid interest asset that is expensed over the life of the loan, but the expense is negative--it is income to the loan. For purposes of setting prices, it is critical that up-front discount points be applied to income in a way that allows managers to compare product yields in a realistic way. The article that follows describes how to do this calculation. The procedure is analogous to the procedure for up-front fee expenses described in Effective Interest (Yield) Loan Fee Amortization, but the sign for the pre-paid interest asset is negative instead of positive; this causes the the asset to be amortized to income rather than expense. This approach it makes it possible to use the same fee amortization code needed for Effective Interest (Yield) Loan Fee Amortization without modification; this makes it much easier to maintain reporting and analysis applications.

The article is divided into the following sections:

Example Mortgage Loan

Although this type of up-front discount points can occur on many loan and lease types, the most common are mortgage loans. For our example, a mortgage loan will be used:

Principal $100,000
Interest Rate 3.5%
Term 360 months
Payment -$449.04

Note that the normal loan payment has a negative sign for cash flow and the discount points payment has a positive sign for the cash flow--in this case the loan payment reduces the principal balance and the amortization payment increases the principal balance. The “discount points” payment can be thought of as a regular prepayment to principal, but since it is contractually used to buy down the interest rate, we need to amortize this principal reduction over the life of the loan. To do that, we will treat the discount points payment as a type of loan and then calculate a payment to get the principal and income that must be amortized:

Discount points paid -$2,000
Discount points Interest Rate 3.5%
Term 360 months
Discount points amortization payment $8.98

Calculation for Amortization of Pre-paid Interest to Income

With this basic information, it is now time to calculate the discount points amortization to income 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 -$157.38, -$157.84 and -$158.30 respectively.

Similarly, the cyan columns show the calculation of the monthly discount points "principal" amortization to principal, with discount points amortization of $3.15, $3.16 and $3.17 respectively. The discount points are subtracted from the principal balance in period 0, and then the “principal” portion of the “amortization payment” is amortized back into the loan principal as the loan pays down. The signs for discount points amortization are the opposite of the signs for fee amortization; the fee amortization is a positive expense while the discount points amortization is a negative expense– income for all intents and purposes.

The light grey columns show the calculation of the level yield as the interest ($291.67 for period 0) divided by the level yield asset ($98,000) multiplied by 12 periods to annualize the result, which gives 3.57%. 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 ($98,000) and multiply by 12 to annualize it, we get 0.07%--the difference between the contracted 3.5% interest rate and the effective yield after discount points amortization.

It may seem counter-intuitive to reduce the principal by the amount of the discount points, but it may help to think of it in terms of net cash flows: the bank’s net disbursement is $98,000 but the bank gets paid interest on $100,000, just as in the up-front fee case the bank would have a net disbursement of $102,000 but would get interest on only $100,000.

Table 1: Pricing Approach Example Loan Principal and Discount points Amortization for Three Periods
Monthly Principal and Interest Level Yield Discount Points Amortization Detailed Calculation Level Yield Level Yield Simplified Calculation
Period Principal Payment Interest Applied Principal Discount points Balance Discount points Pseudo Payment Discount points Pseudo Interest Applied Discount points Principal Amortization Expense Level Yield Asset Yield After Discount points Amortization Amortization Income Increase to Contract Yield Simplified Calculation of Amortization Income Increase to Contract Yield Simplified Calculation of Amortization Expense
0$100,000.00-$449.04$291.67-$157.38-$2,000.00-$8.98$5.83-$3.15$98,000.003.570.070.07-$3.15
1$98,842-$449.04$291.21-$157.84-$1996.85-$8.98$5.82-$3.16$97,845.773.570.070.07-$3.16
2$98,684-$449.04$291.21-$158.30-$1993.70-$8.98$5.81-$3.17$97,691.093.570.070.07-$3.17

In this amortization case, the discount points asset is subtracted from the loan principal rather than added to the loan principal for purposes of yield calculation, which seems backward.

Calculating Discount points Amortization to Income for Prepayments

This approach to calculating the discount points 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 discount points amortization for a partial prepayment?

To do this, first we should look for an easier way to calculate the monthly discount points amortization amount in a way that doesn't require calculating both the pseudo payment and pseudo interest for the fee. Notice that the discount points pseudo payment is proportional to the fee balance divided by the principal balance--$2,000/$100,000 or 0.02 in this case. Similarly, the applied principal, $157.38 is proportional to the discount points amortization, $3.15.

From this we can calculate the monthly discount points amortization as

\[ \begin{aligned} \text{discount points amortization}&=&\text{principal reduction}*\frac{\text{fee balance}}{\text{principal balance}} \\ &=&157.38*\frac{2000}{100000} \\ &=&3.15 \end{aligned} \]

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 discount points amortization:

\[ \begin{aligned} \text{discount points amortization}&=&\text{principal reduction}*\frac{\text{fee balance}}{\text{principal balance}} \\ &=&(157.38+449.04)*\frac{2000}{100000} \\ &=&12.13 \end{aligned} \]

Calculation Methods for Loan Portfolios

The example above 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 discount points 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 discount points 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. The spreadsheet also contains a tab with an alternate calculation approach that may be appropriate for uses other than pricing use, but it would not reuse the code from fee amortization.

Software Implementation

Writing a program to calculate the level yield discount points amortization schedule is very simple, but writing the code to extract the necessary information from loan systems--and put the calculated values back into the loan system--can be quite involved. If you can implement this as an extract-calculate-report capability through a data warehouse or data mart, implementing this is fairly straightforward. If you need to put values back into your loan system, you should first work with your loan system vendor to find out if they can implement the capability as a new feature or as an add-on. If that is not possible, plan to spend a significant amount of time with your Information Technology staff working out how the calculated values will be put back into the loan system.

In either case, you will need to work out a way to handle loan modifications. If the modified loan is handled systematically as a new loan, you will need to figure out a way to calculate the remaining discount point balance and transfer that to the modified loan record.

Notes

The formula displays in this example are formatted using MathJax. If the formulas do not display in a recognizable way, you should check your browser to make sure that JavaScript is enabled; MathJax requires JavaScript to render the equations. If you want to copy the math displays, right click on the equation and you will get a menu of options. MathML can be imported into many versions of Microsoft Word by copying the MathML to the clipboard and pasting it into Word using the "Keep Text Only" paste option. It can also by copied and pasted in LaTeX format.

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