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.
