# How to model fixed-rate loans or mortgages with act/365 but constant payment

## My question

I have a question on how to model the cashflows of fixed-rate loans or mortgages.

Let's say the payments are monthly, and the rate remains constant throughout the life of the product; each month the borrower(s) will pay the same amount, easily calculated with the pmt() formula available in most spreadsheets or finance libraries (Excel, numpy_financial, Matlab, etc); as time goes by, the amount paid each month remains the same, but the interest portion of that payment goes down, and the principal portion goes up.

There are loads of examples everywhere, like in the Excel link above, but these examples are all based on periods of equal length, effectively assuming a 30/360 daycount convention. In reality, (maybe not in all but in many countries at least) most consumer loans/mortgages calculate interest daily based on act/365. How do you model this? How do you handle the fact that, this way, it might take one more period to fully amortise the loan (see example below)?

## Practical example

Let's consider a 100,000 loan (I am intentionally ignoring the currency as it's irrelevant) for a 5-year period at 10% per annum. If we assume months of equal length, so that the interest in each month will be = balance * rate /12, then, well, of course by the end of month 60 the balance is zero.

If, however, we consider act/365, keep the total payment constant, and simply calculate principal amortisation = total payment - interest payment, then, by the end of month 60, the balance won't be zero, but 5.96 - see screenshot below:

## Options

I can only think of the following options:

1. ignore act/365 and calculate the interest assuming every period has the same length (i.e. 30/360), even though that's not what happens
2. Assume one extra period (in the screenshot above, there is a small final payment in month 61)
3. Don't assume an extra period, but assume that the final payment may be slightly larger than the other (5.96 more at month 60 in the example)
4. Recalculate the total payment each month

Of these, 1. and 3. seem the least worst, so to speak. Probably 3 is better than 1, especially if this modelling is the basis of some kind of ABS/RMBS model, because it would mean the assets (the loans/mortgages) pay interest on the same basis as the liabilities (the bonds backed by those assets) - assuming of course the bonds are act/365.

Thoughts?

First and foremost, your premise that "most consumer loans/mortgages calculate interest daily based on act/365" is incorrect. At least in the United States, most single-family residential mortgages accrue interest on a 30/360 basis.

Using the US mortgage market as the context of this discussion, an important point to remember, and one which you allude to, is that most loans originated in the United States will be securitized into Fannie/Freddie guaranteed MBS securities so the characteristics of the originated mortgage will by design, follow the characteristics of the security it's delivered into.

If a loan does accrue interest on an actual 360 basis with a fixed rate, I think your last solution is actually the most correct one. Solution 2 is a non-starter, because again, most mortgages in the US are securitized and adding an extra payment at the end of the amortization schedule would make collateral indelible for Agency MBS. A 30 year loan, under the Solution 2 approach, would have 361 payments, which per Fannie Mae securitization guide would make that collateral ineligible for securitization:

Part B, Subpart B2 B2-1.5

Fannie Mae purchases or securitizes loans that have original terms up to 30 years. The term of a first mortgage may not extend more than 30 years beyond the date that is one month prior to the date of the first payment

I guess there is nothing preventing a Mortgage Originator from using that approach if they intend to keep a mortgage on their balance sheet. I just know from experience that's not what happens. Even if mortgages aren't sold to an Agency, mortgages are generally designated as "Agency eligible" or "Agency ineligible" and an Agency ineligible mortgage will price at a discount in the Secondary mortgage market, regardless of whether or not it's ever securitized.

Solution 3 is also a non-starter for securitization purposes, per the Fannie Mae securitization guide:

Part A, Subpart A2 A2-2

the mortgage loan will be fully amortized during a specified original term with no subsequent adjustments to the amount payable

Your last suggestion is how we would approach a loan that accrues on Act/360 or Act/365 basis save for the fact that we aren't necessarily "recalculating" the loan payment each month. The Amortization schedule is set at mortgage origination just like a level-pay loan, but the payments are adjusted to mirror the Act/360, Act/365 interest accrual. In an Excel calculator, you'd use an adjustment like YEARFRAC(), nested within the first argument of the PMT function. I worked at a mortgage originator and the number one question we received was from borrowers wanting to know why their payment had changed month over month when they were certain they had agreed to a fixed payment. As I mentioned, 95% of our mortgages were originated 30/360 but for those that weren't, most peopled don't realize a "fixed rate" mortgage doesn't necessarily mean and "fixed payment" mortgage if 30/360 convention is not followed.

Your last conclusion is pretty spot on, although most mortgage securitizations follow 30/360 accrual (again, because that's what most mortgages follow). Mortgage Originators and Servicers are best served by aligning the interest accrual convention of the loans they originate and/or service with the coupons on the bonds those loans have been securitized into. To my knowledge though, the Agencies make no requirement that originators do so. Rather, the Agencies make Originators and Servicers represent that they will follow an Scheduled/Scheduled Remittance schedule as it pertains to fulfilling bond coupons:

The Scheduled/Scheduled (S/S) remittance type is available on a negotiated basis for whole loans, including ARMs, and fully amortizing fixed-rate conventional monthly payment first mortgages. It is required for MBS mortgages. Lenders remit scheduled interest (net of servicing fees) and scheduled principal due, whether or not payment is collected from the borrower.

A misalignment of underlying mortgage interest accrual and securitized bond coupon accrual is a risk, but scheduled/scheduled remittance representation ensures that risk is placed squarely on the Originator/Servicer. Bond holders are due coupons regardless of whether or not the mortgages backing those bonds have properly aligned accrual schedules.

• I did write "not in all but in many countries"! :) Your answer is very useful in the context of the US market, but I was after a more generic, not US-centric type of answer. E.g. the terms and conditions of most banks in the UK specify act/365 (easy to google), but, for fixed-rate mortgages, the total instalment remains the same every month for the period the rate is set (2 to 4 years in most cases). Mar 5 at 12:15
• I see what you're saying...I could definitely hypothesize how non-standard interest accrual schedules are treated in other countries but I'd be guessing. Mar 8 at 16:09