Understanding the Mathematics of Personal Finance


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

Подпись: R У 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.

У V y)

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.


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

Подпись: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

Подпись: S = Pi Подпись: i -1 in -1

Solving the above for S,

Подпись: У image056

and recalling the definition of i,

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Understanding the Mathematics of Personal Finance


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