Sanderson M. Smith
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CONCEPT OF INVERSE FUNCTION AND FINANCE
The concept of inverse function is very important, in finance and in other areas of our capitalistic society.
Here is an attempt to demonstrate the power (and the language) of mathematics.
OK, I invest $1,000 now, and I want to let it accumulate, with interest, for 25 years.
If I now clearly define x to be the annual interest rate, then
y = F(x) = 1000(1+x)25
would represent the accumulation, y, after 25 years. Note that I have introduced the notation of a function, F. OK, F is a rule that says "take the interest rate, add 1, raise this to the 25th power, and multiply by 1000." (The order of operations is important here. Also, note that x is a dummy variable. If I write F(w) = 1000(1+w)25, the accumulation rule does not change.)
Now, if x = 9%, then F(.09) = 1000(1.09)25 = 8623.08. That is, if the interest rate is 9%, and I apply the rule F (clearly defined) to it, I learn that I will have $8,623.80 after 25 years.
I would now like to come up with a rule that will take the amount that I will have after 25 years (that is, an input for y), and produce the interest rate. OK, we are going to have to reverse the role of the variables. I am going to take
y = 1000(1+x)25
and solve for x. You now have the MATH POWER to do this.
y = 1000(1+x)25 ==> (1+x)25 = y/1000 ==> 1 + x = (y/1000)1/25 ==>x = (y/1000)1/25 - 1.
OK, I now have F-1, the inverse rule.
If I now clearly define x (or whatever variable you choose to use) to be the accumulated amount after 25 years, then rule F-1 says
F-1(x) = (x/1000)1/25 - 1
In words, F-1 says "take the accumulated amount, divide it by 1000, take the 25th root, then subtract 1." You will then have the annual interest rate. For example, if your $1,000 grows to $15,000 in 25 years, then we let x = 15,000 and calculate
F-1(15000) = (15000/1000)1/25 - 1 = 0.1144, or about 11.44%.
Now, the point (15000, 0.1144) is on the graph of F-1. Hence, the point (0.1144, 15000) should be on the graph of F. Let's check this out...
F(0.1144) = 1000(1.1144)25 = 14997.80 (Close enough... the interest rate was rounded to two places.)
This is MATH POWER at it's best. Since returning from break, we have been introduced to fractional exponents, and to the concept of the inverse of a function. I hope that you are becoming aware of the doors of financial knowledge that are being opened for you if you will learn (not memorize) the amazing language of mathematics that we are presently studying. And, this is only the beginning...
"I advise my students to listen carefully the moment they decide to take no more mathematics courses. They might be able to hear the sound of closing doors.
(CAIP Quarterly 2, Fall, 1989)
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