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 and F to represent the future value, the relationship between the two can be expressed with the formula:
F = P * (1 + i)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 + i) 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 + i)n - 1+A * (1 + i)n - 2 + . . . . +A * (1 + i)2+A * (1 + i)1+A
This can be calculated by multiplying both sides by (1 + i) 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+i) = A * (1 + i)n+A * (1 + i)n - 1+A * (1 + i)n - 2 + . . . . +A * (1 + i)2+A * (1 + i)1
1) F       =   A * (1 + i)n - 1+A * (1 + i)n - 2 + . . . . +A * (1 + i)2+A * (1 + i)1+A
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F * i    = A * (1 + i)n - A
Combining the A's on the right side:
F * i    = A * [(1 + i)n - 1]
Solving for A:
A    =      F * i      
[(1 + i)n - 1]
Since F = P * (1 + i)n, we can substitute it and A is terms of P, the value of the loan,
the interest rate, and n, number of periods:
A    =P * i * (1 + i)n
   (1 + i)n - 1
The top and bottom of that can be divided by (1 + i)n to simplify it further:
A    =      P * i       
1 - 1 / (1 + i)n

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.

What about a first payment 45 days after the loan is made? How can the amount be calculated when the payment is due at 1.5, 2.5, 3.5, etc.? It is much easier to assume the loan was made at t = -0.5 and let the value of the loan grow by 1/2 the interest on one month: P-.5 * (1 + i * 0.5) = P0. This new value of P0 works in the above formula without any further changes.

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