Calculating Loan Payments

Before learning how the payment for a loan is calculated, it is necessary to understand compound interest. For example, someone invests $1000 at t0 and keeps it in for n periods of time where interest is paid into the account at the end of each period.

Let's assume 10% interest every period. After the first period the account will have $1100. When interest is credited at the end of the second period, it will be not on just the original $1000 but also on the interest already earned. Instead of having a value of $1200, it will be worth $1100 * 1.1 or $1210. Likewise, it's value after three periods is $1000 * (1.10) * (1.10) * (1.10) = $1331. The number 1 + the interest rate per period (expressed as a decimal) would be multiplied by itself n times, then by the original investment P to arrive at a future value.

If we use P to represent the present value, F to represent the future value, and r for the interest rate (as decimal), the relationship between the two can be expressed with the formula:
F = P * (1 + r)n

t0-----------t1-----------t2-----------t3-----------. . . . .-----------tn-3-----------tn-2-----------tn-1----------- tn

Assume a first payment is made of amount A after one period (t1) and every period after that until tn. The future value will be the sum of each payment times (1 + r) multiplied by itself (i.e., raised to a power) however many periods remain until tn. At the other end, the last payment has no chance to earn interest. The Future value is the sum of the series:

1) F       = A * (1 + r)n - 1+A * (1 + r)n - 2 + . . . . +A * (1 + r)2+A * (1 + r)1+A
This can be calculated by multiplying both sides by (1 + r) then subtracting the first formula from it.
Only the end of the first formula and beginning of the new one will remain.
2) F * (1+r) = A * (1 + r)n+A * (1 + r)n - 1+A * (1 + r)n - 2 + . . . . +A * (1 + r)2+A * (1 + r)1
1) F       =   A * (1 + r)n - 1+A * (1 + r)n - 2 + . . . . +A * (1 + r)2+A * (1 + r)1+A
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F * r    = A * (1 + r)n - A
Combining the A's on the right side:
F * r    = A * [(1 + r)n - 1]
Solving for A:
A    =      F * r      
[(1 + r)n - 1]
Since F = P * (1 + r)n, we can substitute it and A is terms of P, the value of the loan,
r, the interest rate, and n, number of periods:
A    =P * r * (1 + r)n
   (1 + r)n - 1
The top and bottom of that can be divided by (1 + r)n to simplify it further:
A    =      P * r       
1 - 1 / (1 + r)n

What about the first payment is 45 days after the loan date? Assume the loan was made at t-0.5 and multiply the principal and monthly interest rate by (45-30)/30. That gives a new t0 fifteen days after the loan was made and that amount is used in the equation.

Interest is usually expressed as an annual rate. When the periods are a month long, the interest rate used is 1/12th the annual rate.
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