# Formula for calculating a reamortization

So my friend recently recasted/reamortized a loan. Initially, they had a principal loan balance of \$13,000 at an interest rate of 5.99% over 7 years (84 months). They verified the loan company's monthly payment math by using this formula for calculating monthly loan payments.

They reamortized it at the same interest rate when they had 77 payments left. The bank officer told them to figure out their monthly recasted payment, they just had to add total remaining interest to the total remaining principal and then divided by the number of payments left. Is that a correct method to do this? Is there a "formula" where you can plug some numbers in to arrive at the same result?

• Since the interest rate is the same, then yes that simplification works out. If the interest rate is different then it does not hold. But unless there are repayment penalties, one could always just pay more and effectively "re-amortize" the loan themselves - hopefully that didn't pay a fee to change the loan terms for something they could do for free. Jul 7, 2022 at 13:32
• @DStanley The simplification holds insofar as it calculates the payment amount. The problem is the payment amount is required to calculate the total remaining interest in the first place. Jul 7, 2022 at 14:13
• Did they pay extra? Recast only does anything when extra was paid. Otherwise it would be a no-operation. Jul 7, 2022 at 17:32
• @void_ptr, thank you, yes they paid \$750 per month extra in addition to the regular payment up until the time of recast. They made that payment 3 times.
– Nona
Jul 8, 2022 at 14:56

The loan formula is based on the principal `s` equalling the sum of the payments `d` discounted by the compounded periodic rate `r`

$s=\sum_{k=1}^{n}\frac{d}{(1+r)^k}=\frac{d-d(r+1)^{-n}}{r}$

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

so assuming a nominal annual rate of 5.99% compounded monthly

``````  s = 13000
r = 5.99/100/12
n = 84
∴ d = 189.85
``````

The balance `b` remaining in month `x` is given by

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

So the balance in month 7 (77 months remaining) is

``````  x = 7
∴ b = 12112.09
``````

For example, resetting `s = b` and calculating `d` over 77 months

``````  s = b
n = 77
∴ d = 189.85 (same as before, as expected)
``````

The total interest paid in this reset loan is

``````n d - s = 2506.27
``````

Incidentally, the total interest paid in the original loan is

``````84 d - 13000 = 2947.31
``````

Now trying the method: "add total remaining interest to the total remaining principal and then divide by the number of payments left"

``````(2506.27 + 12112.09)/77 = 189.85
``````

So the method finds the correct repayment amount.

Approaching the same from another direction, starting with the original loan again

``````  s = 13000
r = 5.99/100/12
n = 84
∴ d = 189.85
``````

and calculating forward 7 months . . .

``````x = 7
b = (d + (1 + r)^x (r s - d))/r = 12112.09

principal repaid = (d - r s) ((1 + r)^x - 1)/r = 887.91

b + principal repaid = 13000 (as expected)

interest paid = (d - d (1 + r)^x - r s + r (1 + r)^x s + d r x)/r = 441.03

interest remaining = (n d - s) - interest paid = 2506.27 (as previously)

(interest remaining + b)/77 = 189.85
``````

The recalculated payment amount is correct, (in this case equal to `d`).

However, what happens if something changes after the first 7 months, such as an additional windfall repayment of \$1000. No easy way to determine the interest remaining. Need to calculate the new repayment `d` in order to calculate the interest remaining. For example, reducing the reset loan principal by \$1000.

``````s = b - 1000 = 11112.09
n = 77
d = r s (1 + 1/((1 + r)^n - 1)) = 174.17

interest on reset loan = n d - s = 2299.35
``````

Now "add total remaining interest to the total remaining principal and then divide by the number of payments left"

``````(interest on reset loan + s)/77 = 174.17
``````

This is the correct repayment amount, but it was already known.

• Very well explained (+1) - In case of a windfall payment (no penalties charged) of any amount at any given time during that period, would the bank immediately recalculate the period or the prinicpal or do I need to ask them to do it (I couldn't see any statement regarding this matter in the fine-prints of my loan contract..)? Dec 5, 2022 at 14:39
• @iLuvLogix Thanks. I think you would need to contact your loan provider to find out how or if they handle overpayment. Dec 5, 2022 at 14:59