Math for Long-Term Financial Management

Instructor: Michael Eckert

Michael has a Bachelor's in Environmental Chemistry and Integrative Science. He has extensive experience in working with college academic support services as an instructor of mathematics, physics, chemistry and biology.

Math is integral in predicting financial outcomes and avoiding financial risk. While math-based finance can be a complex study involving statistics and calculus, there are some math concepts (implementing algebra), which serve a valuable purpose in this capacity. One such focus is on compounding interest.

Compounding Interest

When there exists a financial advantage --or positive interest-- of investing money (or principle) into something or somewhere, one can find out how much can be made from the investment by using compounded interest equations. Two such equations serve to represent this return:

  1. Annually compounded interest
  2. Continually compounded interest

Note that the general equation for annually compounded interest can be divided into further subgroups of equations where interest might be compounded in increments throughout a year. For instance, interest might be compounded onto the initial investment / principle semi-annually (or twice a year), quarterly (four times a year), monthly (twelve times a year) or daily (three-hundred sixty-five times a year).

Continually compounded interest is that which is most often seen in banking. Unlike annually compounded interest, this means of compounding is continual.

Interest Compounded Annually

The general equation for compounding interest on a yearly basis is as follows: A = P(1 + r)^t, where A = the unit amount of money after time (e.g. dollars), Principle = the amount of the initial investment (e.g. $), r = the rate of return on the initial investment (given as a decimal, ergo 10% would equal .10) and t = time (in years). To illustrate, say we invested a principle of $1000 with a positive rate of return of 10% (.10) over one year. If we used the general equation above: A = 1000 (1 + .10)^1, we would find that an initial investment of $1000 with a positive rate of return of 10% (r = .10) compounded annually over the course of one year t = 1 would yield A = $1100:

A = 1000(1 + .10)^1

For those instances where interest is accrued multiple times over one year: A = P(1 + r / n)^nt, where n is merely the number of increments by which interest is accrued:

Compounded Semi-annually:

Given an initial investment of P = $1000, with a rate of 10%, compounded semi-annually (two times in one year) for t = 1, A = 1000 (1 + .10 / 2)^2 (1)

Compounded Quarterly:

A = 1000(1 + .10 / 4)^4(1)

Compounded Monthly:

A = 1000(1 + .10 / 12)^12(1)

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