Sanderson M. Smith

Home | About Sanderson Smith | Writings and Reflections | Algebra 2 | AP Statistics | Statistics/Finance | Forum

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

Home | About Sanderson Smith | Writings and Reflections | Algebra 2 | AP Statistics | Statistics/Finance | Forum

Previous Page | Print This Page

Copyright © 2003-2009 Sanderson Smith