# The Best Ways to Determine the Par Value of Fixed Rate Loans

by Steve Williams Vol. 48 No. 1 2021
As common as sales of fixed rate loans are, there isn’t a uniform way to determine their par value when sales occur on a non-scheduled payment date. Steve Williams offers up two effective methods — the two-loan method and the present value method — as a means to make these calculations.

Steve Williams,
Managing Director,
CF Software Solutions

Fixed rate loans are sold every day, yet when a loan is sold on a non-scheduled payment date, the methods used to determine its book or par value can vary widely. This is not a pricing issue but an issue of allocating a loan’s purchase price into it its par value and premium fee components, which is important for accounting and documentation purposes. This article will examine two methods of determining the par value of a fixed rate loan on a non-scheduled payment date.

Review of Basic Concepts

When an existing fixed rate loan is sold, any adjustments to the economics of the loan must occur outside of the loan itself as fees paid either to the seller or to the buyer.

Let’s review the relationship between a loan and its amortization. A loan’s ending balance on any scheduled payment date is equal to the present value of the remaining scheduled payments as of that date. This means that on a scheduled payment date, the par value of the loan is the same for the seller and the buyer.

However, difficulty arises when a fixed rate loan is sold on a non-scheduled payment date. To get a better understanding of the problem, let’s look at an example.

Consider a \$10 million loan that amortizes quarterly over five years at an interest rate of 5% (Loan A). The chart in the upper portion of the next page shows the amortization of Loan A for the first four payments.

Determining the par value of a fixed-rate loan only becomes an issue when it is sold on a non-scheduled payment date. For example, if Loan A was sold on July 15, 2020, its outstanding balance on that date would be \$8,742,639.70.1 However, the present value of the remaining scheduled payments is equal to \$8,742,442.50.2 Furthermore, if a buyer purchased Loan A on that date and used the outstanding balance as the par value, then the amortization of that loan would be different than the original amortization of Loan A.

This can be seen by calculating the ending balance for the first scheduled payment date after the sale. We start with a beginning balance of \$8,742,639.70 and add \$19,428.09 of accrued interest3 and then subtract the Aug. 1, 2020, scheduled payment of \$568,203.90. The result is an ending balance of \$8,193,863.90, which is \$197.65 greater than Loan A’s ending balance for that date.4 So, the par value of the loan is different for the seller and the buyer on a non-scheduled payment date. A paradox emerges where the par value for the seller results in a change in the original loan amortization for the buyer, and a par value that maintains the original loan amortization for the buyer results in a loss for the seller.

Various methods have been used in the market to address this issue. The failure of the market to adopt a consistent approach is probably because the differences between buyer and seller par values are small. However, the administrative costs become significant when dozens of loans are purchased and the book values do not match the original documents. What is needed is a simple method of calculating par values on any non-scheduled payment date. We will look at two approaches now.

Two-Loan Method

One leading provider of lease/loan pricing software uses the two-loan method to resolve this paradox. When using this vendor’s software to “chop” a loan on a non-scheduled payment date, the program creates two loans. Why is that? Let’s look at an example assuming Loan A is sold on July 15, 2020.

Loan 1 is sized and structured to match the amortization of Loan A. Loan 1’s funding amount is then equal to the sum of Loan A’s scheduled principal reduction for the first payment date after the sale (\$460,032.66) and the ending balance for that date (\$8,193,666.24).

Loan 2 is sized to equal the difference between the outstanding balance of Loan A on the sale date (\$8,742,639.70) and the amount of Loan 1. The interest rate on Loan 2 is equal to 0%. The loan also funds on the sale date but matures on the first scheduled payment date after the sale. Any premium or discount is determined in relation to that amount.

The beauty of the two-loan approach is that when the first scheduled payment after the sale date is paid, Loan 2 is retired and you are left with just Loan 1, which perfectly matches the amortization of Loan A.

Present Value Method

Another method for determining a loan’s par value is the present value method. To explore this method, let’s take a closer look at what occurs when a fixed rate loan is sold on a non-scheduled payment date.

Again, let’s use Loan A and a sale date of July 15, 2020. If not for the sale, the \$88,940.80 of accrued interest wouldn’t be paid until the next scheduled payment date on Aug. 1, 2020. By purchasing Loan A on July 15, 2020, the buyer is prepaying that portion of the accrued interest in 16 days. The prepaid accrued interest is then capitalized to the buyer’s purchase price, which then accrues interest on itself. So, the problem of determining the par value on a non-scheduled payment is essentially a present value problem.

If we quantify the impact of prepaying the accrued interest and subtract that difference from the loan’s outstanding balance on the sale date, then we should arrive at the loan’s true par value on the sale date. Using this approach allows us to generalize that the par value of a fixed rate loan on any non-scheduled payment date is equal to the applicable principal balance plus accrued interest less the present value of prepaying the accrued interest.

Since the present value of prepaying the accrued interest is an adjustment to the loan’s economics, it must be handled in the form of a fee paid by the buyer to the seller (the PAI fee). Specifically, the PAI fee is calculated as follows:

Accrued Interest — (Accrued Interest /
(1 + Daily Interest Rate * No.of Days)

where the number of days is the day count from the sale date to the first scheduled payment date following the sale.

Now let’s calculate the PAI fee for Loan A and the July 15, 2020, purchase date that we used earlier.

PAI Fee = \$88,940.80 – (\$88,940.80/ (1 + 0.05/360 *16)) = \$197.21

Under the present value approach, the par value of Loan A on that date is equal to the following:

Par Value = Ending Principal Balance + Accrued Interest – PAI Fee = \$8,742,442.505 5

Now let’s check the results. If we use the calculated par value as the purchased loan’s starting balance, then its ending balance on the first scheduled payment date after the sale must be equal to Loan A’s ending balance on that date (Aug. 1, 2020). As the previous amortization table shows, the ending balances for that date do in fact match.6 Under the present value method, you would end up with one par value loan and two fees: a prepaid accrued interest fee and a premium fee.

In the electronic version of this article, a table shows the par values at various sales dates calculated by using the PAI fee and par value formulas shown in this article and demonstrates how the present value approach works for any sale date.

Comparing the Methods

Both approaches work well. I prefer the present value approach because it is easier to book two fees than two loans when purchasing one loan.

It is worth noting that both approaches are mathematically the same. Under the two-loan method, if Loan 2 were to accrue interest at Loan A’s interest rate of 5% instead of at 0%, then Loan 2 would accrue \$197.65 of interest.7 The present value of that amount on July 15, 2020, is \$197.21,8 which is equal to the PAI fee we calculated earlier. Effectively, a 0% interest loan of \$88,940.80 that matures on Aug. 1, 2020, is equal to a fee of \$197.21 paid on July 15, 2020.

1. The sum of the ending principal balance of \$8,653,698.90 plus accrued interest of \$88,940.80.

2. \$8,742,639.50 + 19,427.65 of interest – payment of \$568,203.90 = \$8,193,666.24 (8/1/20 ending balance)

3. 16 days at \$1,214.256 per day

4. \$8,193,863.90 – \$8,193,666.24 = \$197.65

5. \$8,742,639.71 – \$197.21

6. The accounting for interest is as follows: The original loan produces \$108,171.24 of interest for the period of 5/1/20 to 8/1/20. The seller accrues \$88,940.79 of interest for 5/1/20 to 7/15/20 and the par value of the loan produces \$19,427.65 of interest for the period of 7/15/20 through 8/1/20. The total interest between seller and buyer is then \$108,368.44. This amount exceeds the \$108,171.24 by the amount of the PAI fee.

7. \$88,940.79 of principal times a daily rate of 0.0138889% times 16 days

8. \$197.65/ (1 + 0.050/360 days per year * 16 days)

Steve Williams is the managing director for CF Software Solutions, which is a provider of desktop applications to commercial lending and leasing companies and is a registered Microsoft developer. He can be reached at [email protected].