How to calculate the monthly fee on a loan?

Given is:

- a: an amount to loan.
- b: the loan period (number of months).
- c: the interest rate p.a. (interests is calculated and added every month, 1/12 of the interest is added. So if the interest is on 12%, 1% interest is added every month).
- d: the amount of money owed after the end of the period.

This problem is a bit different than the usual since, the goal is not to have the loan payed after the loan period has ended, but to still owe an amount that is given. I have been able to find an algorithm so solve the problem if I wanted to pay the entire amount, but it will of course not work for this problem where the goal is to end up owing a given amount rather than not owing anything.

I managed to make a solution to this problem by starting with an guess and then keep on improving that guess until it was close enough. I wondered however, if there is a better way to simply calculate this, rather than just guessing.

**Edit: Here's how I'm doing it now.**

```
def find_payment(start, end, months, interest):
difference = start
guess = int(start / months * interest)
while True:
total = start
for month in range(1, months + 1):
ascribe = total * interest / 12
total = total + ascribe - guess
difference = total - end
# See if the guess was good enough.
if abs(difference) > start * 0.001:
if difference < 0:
if abs(difference) < guess:
print "payment is %s" % guess
return evolution(start, guess, interest, months)
else:
mod = int(abs(difference) / start * guess)
if mod == 0:
mod = 1
guess -= mod
else:
mod = int(difference / start * guess)
if mod == 0:
mod = 1
guess += mod
else:
print "payment is %s" % guess
return evolution(start, guess, interest, months)
```

evolution is just a function that displays how the loan would look like payment for payment and interest for interest, summing up total amount of interest paid etc.

An example would be if I wanted to find out the monthly payments for a loan starting with $100k and ending at $50k with an interest of 8% and a duration of 70 months, calling

```
>>> find_payment(100000, 50000, 70, 0.08)
payment is 1363
```

In the above case I would end up owing 49935, and I went through the loop 5 times. The amount of times needed to go through the loop depends on how close I want to get to the amount and it varies a bit.

This is a basically a mortgage repayment calculation.

Assuming that start is greater than end, and that interest is between 0 and 1 (i.e. 0.1 for 10% interest)

First consider the part of the payment you want to pay off.

```
Principal = start - end
```

The monthly payment is given by:

```
pay_a = (interest / 12) / (1 - (1+interest/12) ^ (-months))) * Principal
```

You then need to consider the extra interest. Which is just equal to the remaining principal times the monthly interest

```
pay_b = interest / 12 * end
```

So the total payment is

```
payment = (interest / 12) * (1 / (1 - (1+interest/12) ^ (-months))) * Principal + end)
```

On the example you gave of

```
Start: 100000
End: 50000
Months: 70
Interest: 8%
pay_a = 896.20
pay_b = 333.33
Payment = 1229.54
```

When I tested these values in Excel, after 70 payments the remaing loan was 50,000. This is assuming you pay the interest on the notional before the payment is made each month.

Perhaps the easiest way to think about this is to split the loan in two parts, one part which is to be repaid in full and another part where you don't pay off anything. You have already computed the monthly fee for the first part.

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