# Calculate interest rate of mortgage loan that has deferred (suspended) payments

Simple example to illustrate the question:

Suppose I have a 30-year \$100,000 mortgage loan at a fixed-rate of 5% interest that I took out exactly 10 years ago. I have been making timely payments each month. Now, because of Covid and the CARES act, I am able to enter loan forbearance and stop making payments for 15 months. Those payments will be due when I sell the house in 20 years. They are essentially a no-interest loan for 20 years and I will pay back the missed payments using 2041 dollars. But put all that aside: whatever my gain on the house when I sell minus those deferred payments will be my profit or loss.

My question: I took out a 5% interest loan. Over the 30 years of the loan, I am now paying less because of the deferred payments. How do I calculate the true interest rate, taking into account the 15-month hole in the payment schedule? This has relevance in case I want to refinance at a lower rate (and I understand the deferred payments would immediately come due in such a case). If anyone could provide an equation or a set of Excel functions, I would be indebted.

Glancing at the CARES Act Mortgage Forbearance page, it looks like you are just deferring payments, so I don't see that a "15-month hole in the payment schedule" translates to "a no-interest loan for 20 years". Below is my expectation of how it would play out, but I have added a final note in case the bank actually waives interest charges during the payment suspension, although that seems sadly unlikely.

With the following variables defined

``````s = principal
r = periodic rate
d = periodic payment
n = number of periods

s = 100000
r = 5/100/12
n = 30*12 = 360
``````

Arranging the standard loan equation for `d`, the monthly payment is

``````d = r (1 + 1/((1 + r)^n - 1)) s = 536.82
``````

The balance `b` after 10 years with normal payment is

``````x = 10*12 = 120

b = (d + (1 + r)^x (r s - d))/r = 81342.06
``````

And if you continued to pay normally for the next 20 years the standard loan equation confirms `b`, i.e.

``````n2 = 20*12 = 240

(d - d (1 + r)^-n2)/r = b
``````

And the final balance `fb` is zero, as expected.

``````fb = d + (1 + r)^n2 (r b - d))/r = 0
``````

However, you are not paying for the next 15 months so the balance accumulates interest

``````b2 = b (1 + r)^15 = 86576.93
``````

After that the previous payment amounts resume for 225 months

``````n3 = 20*12 - 15 = 225
``````

at the end of which the balance for payment is

``````b3 = (d + (1 + r)^n3 (r b2 - d))/r = 21131.83
``````

To compare to a normal 30 year loan, under normal circumstances the total payment is

``````ti1 = d (30*12) = 193255.78
``````

But with the deferred payments the total payment is

``````ti2 = d (30*12 - 15) + b3 = 206335.29
``````

If that was the total payment for a normal 30 mortgage the regular payment would be

``````d2 = ti2/(30*12) = 573.15
``````

Solving the standard load equation for `r` to find the implied rate

``````Solve  s = (d2 - d2 (1 + r)^-n)/r   for  r

r = 0.00465443
``````

giving a nominal APR of `12 r = 5.58532 %`

Deferring the payments has added over half a percent to the interest rate.

However, this does not help with refinancing. At the time the payment suspension ends you have a 5% balloon loan with \$86576.93 to repay in 225 months, implying the final (additional) balloon payment of \$21131.83 as calculated. You could take out a second loan to pay the balloon, or perhaps increase the regular payments to reduce it over time (which would decrease the total payment). Increasing payments to \$593.68 for 225 months would eliminate the balloon payment altogether.

``````d3 = r (1 + 1/((1 + r)^n3 - 1)) b2 = 593.68

total payment = d (10*12) + d3*225 = 204818.99
``````

In the doubtful event the bank waives interest charges during the payment suspension and payments are \$536.82 throughout

``````balloon = (d + (1 + r)^n3 (r b - d))/r = 7790.13

total payment = d (30*12 - 15) + balloon = 192993.59
``````

That is a lower total payment than the original loan because the loan would have effectively been converted to a 345 month 5% balloon loan, avoiding the final 15 months' interest charges.

• Chris, Thank you for the effort and especially the time you spent constructing this detailed answer. Much appreciated! I think I am correct about the interest accumulation, though. Here is how Freddie Mac describes it.
– Yof
Mar 2, 2021 at 15:02
• CMHC describes it as per my answer: "How do I repay the deferred amounts? ... The interest on your mortgage that hasn’t been paid during the deferral period continues to be added to the outstanding principal of your mortgage. When your payments start again, your mortgage payment might be based off the total amount you then owe to pay off your mortgage in accordance with the original payment schedule." Mar 2, 2021 at 17:26
• One bank in this link (Canandaigua) states: "Up to 90 days payment relief on Principal and Interest, Principal only, or Interest Only (determined by client need and credit facility)", so policy can clearly vary, but this just relates to how much of the payment is deferred; nothing about what happens to interest accumulating on the principal. Mar 2, 2021 at 17:58
• Your Freddie Mac link seems vague: "If your monthly mortgage payment is \$1,000, you will owe about (?) \$3,000 in missed payments at the end of a three-month forbearance period." Is that \$3,000 plus interest? And only 3 months? Mar 2, 2021 at 17:59