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

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.