Understanding the Mathematics of Personal Finance

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,