Fixed rate loans are sold every day, but when a loan is sold on an unscheduled payment date, the methods used to determine its book or face value can vary significantly. This is not a question of pricing but a question of allocating the purchase price of a loan to its face value and premium cost components, which is important for purposes of accounting and documentation. This article will look at two methods of determining the face value of a fixed rate loan on an unscheduled payment date.
Review of basic concepts
When an existing fixed rate loan is sold, any adjustment to the loan economy must occur outside of the loan itself in the form of fees paid to either the seller or the buyer.
Let’s review the relationship between a loan and its amortization. The ending balance of a loan on any scheduled payment date is equal to the present value of the remaining scheduled payments on that date. This means that on a scheduled payment date, the face value of the loan is the same for the seller and the buyer.
However, a difficulty arises when a fixed rate loan is sold on an unscheduled payment date. To better understand the problem, let’s take an example.
Consider a $10 million loan that is amortized quarterly over five years at an interest rate of 5% (Loan A). The table at the top of the next page shows the amortization of Loan A for the first four installments.
Determining the face value of a fixed rate loan only becomes a problem when it is sold at an unscheduled maturity. For example, if Loan A were 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} Also, if a buyer purchased loan A on that date and used the outstanding balance as face value, the amortization of that loan would be different from 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 an opening balance of $8,742,639.70 and add $19,428.09 in accrued interest^{3} then subtract the scheduled August 1, 2020 payment from $568,203.90. The result is a closing balance of $8,193,863.90, which is $197.65 more than the closing balance of Loan A on that date.^{4} Thus, the nominal value of the loan is different for the seller and the buyer on an unscheduled payment date. A paradox arises when the face value for the seller results in a change in the original amortization of the loan for the buyer, and a face value that maintains the original amortization of the loan for the buyer results in a loss for the seller.
Various methods have been used in the market to solve this problem. If the market hasn’t taken a consistent approach, it’s probably because the differences between the face values of the bid and ask are small. However, 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 face values at any unscheduled payment date. We will now consider two approaches.
Two loans method
A leading lease/loan pricing software vendor uses the twoloan method to resolve this paradox. When you use this vendor’s software to “hash” a loan on an unscheduled payment date, the program creates two loans. Why is that? Let’s take an example assuming Loan A is sold on July 15, 2020.
Loan 1 is sized and structured to match the amortization of Loan A. The financing amount of Loan 1 is then equal to the sum of the scheduled principal reduction of Loan A 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 be equal to the difference between the outstanding balance of Loan A on the date of sale ($8,742,639.70) and the amount of Loan 1. The interest rate on Loan 2 is equal to 0%. The loan is also funded on the date of sale, but matures on the first scheduled payment date after the sale. Any premium or discount is determined in relation to this amount.
The beauty of the twoloan approach is that when the first scheduled payment after the date of sale is paid, loan 2 is taken out and you are left with only loan 1, which perfectly matches the amortization of loan A .
Present value method
Another method for determining the face value of a loan is the present value method. To explore this method, let’s take a closer look at what happens when a fixed rate loan is sold on an unscheduled payment date.
Again, let’s use Loan A and a sale date of July 15, 2020. If it weren’t for the sale, the $88,940.80 of accrued interest would not be paid until the next scheduled payment date on August 1, 2020. By purchasing Loan A on July 15, 2020, the buyer prepays that portion of accrued interest in 16 days. The prepaid accrued interest is then capitalized into the buyer’s purchase price, which then generates interest on itself. Thus, the problem of determining the face value of an unscheduled payment is essentially a present value problem.
If we quantify the impact of prepayment of accrued interest and subtract this difference from the outstanding balance of the loan at the date of sale, we should then arrive at the true face value of the loan at the date of sale. Using this approach allows us to generalize that the face value of a fixed rate loan on any unscheduled payment date is equal to the applicable principal balance plus accrued interest minus the present value of the prepayment of accrued interest.
Since the present value of the prepayment of accrued interest is an adjustment to the economics of the loan, it should be treated as 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 * number of days)
where the number of days is the number of days between the date of sale and the first scheduled payment date following the sale.
Now let’s calculate the PAI fee for Loan A and the purchase date of July 15, 2020 that we used earlier.
DAP fee = $88,940.80 – ($88,940.80/(1+0.05/360*16)) = $197.21
Under the present value approach, the face value of loan A on that date is equal to the following:
Face Value = Ending Principal Balance + Accrued Interest – PAI Fee = $8,742,442.505 ^{5}
Now let’s check the results. If we use the calculated face value as the opening balance of the purchased loan, its ending balance on the first scheduled payment date after the sale must equal the ending balance of loan A on that date (August 1, 2020). As the previous amortization table shows, the closing balances for this date actually match.^{6} Under the present value method, you would end up with a face value loan and two fees: a prepaid accrued interest fee and a premium fee.
In the electronic version of this article, a table shows face values at various sale dates calculated using the PAI fee and face value formulas presented in this article and shows how the present value approach works for n any sale date.
Comparison of methods
Both approaches work well. I prefer the present value approach because it’s easier to account for two fees than two loans when buying a loan.
It should be noted that the two approaches are mathematically the same. Under the twoloan method, if Loan 2 were to accrue interest at Loan A’s interest rate of 5% instead of 0%, Loan 2 would accrue $197.65 in interest.^{7} The present value of this amount on July 15, 2020 is $197.21,^{8} which is equal to the PAI fee we calculated earlier. Indeed, a 0% interest loan of $88,940.80 that matures on August 1, 2020 is equivalent to a fee of $197.21 paid on July 15, 2020.

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

$8,742,639.50 + 19,427.65 interest – payment of $568,203.90 = $8,193,666.24 (closing balance 08/01/20)

16 days at $1,214.256 per day

$8,193,863.90 – $8,193,666.24 = $197.65

$8,742,639.71 – $197.21

The accrual of interest is as follows: the original loan yields $108,171.24 in interest for the period from 05/01/20 to 08/01/20. The seller accrues $88,940.79 in interest from 05/01/20 to 07/15/20 and the face value of the loan accrues $19,427.65 in interest for the period from 07/15/20 to 01/ 08/20. The total interest between the seller and the buyer is then $108,368.44. This amount exceeds the $108,171.24 IAP charge amount.

$88,940.79 principal multiplied by a daily rate of 0.0138889% multiplied by 16 days

$197.65/ (1 + 0.050/360 days per year * 16 days)
Steve Williams is the Managing Director of CF Software Solutions, which is a desktop application provider for commercial loan and rental companies and is a registered Microsoft developer. He can be contacted at [email protected].