Sanderson M. Smith

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INVERSE FUNCTIONS AND FINANCE

Here is a rule (function) P that tells you how to accumulate a deposit of $1,000 for 20 years an an interest rate of x% per annum compounded monthly:

Take the rate, divide it by 12, add 1, raise what you have to the power 20(12) = 240, then multiply by 1,000.

If we let y represent the accumulated value of this investment, we can write

y = P(x) = 1000(1 + x/12)²⁴⁰

If we evaluate P(.08), we will find the accumulation of $1000 at 8% compounded quarterly. Let's do it:

P(.08) = 1000(1 + .08/12)²⁴⁰ = 4,926.80 (dollars).

OK, now we want to come up with the inverse function, P^-1. In other words, we want a rule that will provide us with the annual rate compounded quarterly when a value for y (the accumulation of the investment of $1,000 after 20 years) is provided. To do this, we need to find y in terms of x. Let's do it:

y = 1000(1 + x/12)²⁴⁰

==> (1 + x/12)²⁴⁰ = y/1000

==> 1 + x/12)= (y/1000)^1/240

==> x/12 = (y/1000)^1/240 - 1

==> x = 12[(y/1000)^1/240 - 1]

Hence, the inverse rule says:

Take the accumulated amount, divide it by 1000, take the 240-th root of what you have, subtract 1, then multiply by 12.

So, if x now represents the accumulated amount, we can write

y = P^-1(x) = 12[(x/1000)^1/240 - 1]

where y now represents the annual interest rate compounded quarterly.

If x = $7,000, for instance, then

y = P^-1(7000) = 12[(7000/1000)^1/240 - 1] = .09769, or about 9.77%.

Hence, if your investment of $1,000 grows to $7,000 in 20 years, you have earned an annual interest rate of 9.77%, compounded quarterly.

When one uses the function y = P(x) as defined above, then x is the annual interest rate compounded quarterly. P is a rule that tells you what to do with the rate in order to obtain the accumulated value of an investment of $1,000 after 20 years. In this case, the interest rate (x) is the independent variable, and the accumulated value of the investment (y) is the dependent variable.

The inverse function y = P^-1(x) reverses the status of the variables. P^-1 is a rule that tells you what do do with the accumulated value in order to obtain the annual interest rate compounded quarterly. In this case, the accumulated value (x) is the independent variable, and the interest rate (y) is the dependent variable.

Inverse functions are important in finance.

"When a fellow says it hain't the money but the principle o' the thing, it's th' money."

-Frank McKinney Hubbard: Hoss Sense and Nonsense

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