CALCULATING THE PAYMENT AMOUNT
Spreadsheets and online calculators handle this calculation very nicely. This section, deriving the formula, can be skipped if you wish.
If R is the annual interest rate and y is the number of payments per year, which we’re assuming is the same as the number of compounding intervals per year, then
is the interest per payment period. As a notational convenience, let
I — 1 + .
The balance at the time of taking the loan, that is, after 0 payment periods, is just the principal. If we let Bn be the balance after the nth payment, then if P is the principal:
Bo — P.
The balance after the first payment period is just the principal plus the interest accrued during this period minus the payment (S):
Bi — P + PR- S — P|1 + R |-S — Pi - S.
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This relationship is recursive. In other words, to get B2, we use the same expression as above except that we replace B0 with B1,
B2 — B1i - s — [Pi - S]i - s — Pi2 - Si - S,
B3 — B2i - S — Pi3 - Si2 - Si - S.
The general expression is, therefore,
Bn — Pin - S£ ik.
This summation is actually a geometric series. If we let f be the summation,
f — £ ik — 1 + i + i2 +... + in-1,
if — i + i2 + i2 +... + in
if - f—f (i -1)—(i+i2+...+in-1+in )-(1+i+...+in-1)—in -1 and finally
in -1 i -1
Putting this result back into the expression for Bn,
Bn = Pin - S—.
If we want the loan to be paid off at the end of n payment periods, then we want
Bn = 0:
0 = Pin - Sl------- .
Solving the above for S,
and recalling the definition of i,