# 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

R

I — 1 + .

y

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,

and then

B3 — B2i – S — Pi3 – Si2 – Si – S.

The general expression is, therefore,

Bn — Pin – S£ ik.

k—0

This summation is actually a geometric series. If we let f be the summation,

f — £ ik — 1 + i + i2 +… + in-1,

k—0

then

if — i + i2 + i2 +… + in

and

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,

in -1

Bn = Pin – S—.

i -1

If we want the loan to be paid off at the end of n payment periods, then we want

Bn = 0:

in -1

0 = Pin – Sl——- .

i -1

Solving the above for S,

and recalling the definition of i,