Formula for previous months interest payment and balance - Reverse Amortization

Question

I am trying to resolve an issue I am having where a system calculated incorrect additional payments on a loan for a couple of months, resulting in the new current balance being incorrect. I want to fix the balance but I need to find a way of determining what the original starting balance was, since it wasn't logged anywhere.

I know the formulas for normal amortization but I need to be able to figure out the reverse with the following information:

• Current Balance
• Interest Rate
• Monthly Payment
• Additional Principal Payment if used.
• No number of payments or length of loan known.

So knowing that, and if you were to create an amortization spreadsheet, how would you get the previous couple months?

Current formula for regular amortization.

Interest Rate = (X.XX/100)/12

New Balance = Current Balance + ((Minimum Payment - (Current Balance * Interest Rate)) + Additional Payment)

I need the reverse of the above. I've tried subtracting the values from Current Balance and looping it a couple times - I get close to what the original balance may be, but it's off a couple decimals of interest.

Update: Below is an example set of numbers using the normal amortization schedule that does work, and the new set of numbers to determine the original balance. However as you can see I am off. No additional payment in example below.

Normal monthly direction:

Starting Balance: $5000

Interest Rate: 5%

Monthly Payment: $300

After 3 Auto Payments the total is: $4,159.01 (This number is correct and has been verified in spreadsheet).

$Int = (5.0/100)/12;
$Months = 1;
$NewBalance = 5000;
while($autoMonths <= 3){
    $NewBalance = $NewBalance - ((300 - ($NewBalance * $Int)));
    $Months++;
}

Now reverse:

Starting Balance: $4,159.01

Interest Rate: 5%

Monthly Payment: $300

3 Auto Payments BACK, the total is: $5,003.49

$prevBalance = 4159.01;
$prevMonths = 1;
while($prevMonths <= 3){
    $prevBalance = $prevBalance + ((300 - ($prevBalance * $Int)));
    $prevMonths++;
}

Answer 1

Given

s = principal
r = periodic rate
d = periodic payment

the balance b remaining in month x is

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

Applying your figures

s = 5000
r = 5.0/100/12
d = 300

x = 3
b = (d + (1 + r)^x (r s - d))/r = 4159.01

Starting value, three months back

x = 0
b = (d + (1 + r)^x (r s - d))/r = 5000

Or, in your code

$Int = (5.0/100)/12;
$Months = 1;
$NewBalance = 5000;
while($Months <= 3){
    $NewBalance = ($NewBalance * (1 + $Int)) - 300;
    $Months++;
}

$prevBalance = 4159.01;
$prevMonths = 1;
while($prevMonths <= 3){
    $prevBalance = ($prevBalance + 300) / (1 + $Int);
    $prevMonths++;
}