Expert: Scott A Wilson - 11/28/2009

Question
Here is an 'Excellence' question from my son's grade 11 assignment:

A girl can save $5000 @ 5% CI, or she can invest in antiques, which appreciate @ 8% CI. However, experience shows her that there is a $2000 commission when selling antiques. After what time is the investment in antiques worth more than the money in the bank?

I can use a solver routine on a calculator or spreadsheet, but can't make the algebra work.

I have reached this stage:

5000(1.08)^t - 2000 = 5000(1.05)^t

which leads to

1.08^t - 0.4 = 1.05^t, or 1.08^t - 1.05^t = 0.4

From there, I can't get the laws of indices and logs to work!
I'm normally pretty ok with algebra, but this has me stumped ;)

(The answer is approx 8.39 years)

Many thanks for your help

Stephen

Answer
The only way to solve it that I know of is by using numerical analysis.

Bisection Method
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The simplest way that I know of is the bisection method.
Let f(t) = 1.08^t- 1.05^t - 0.4.
Choose t1 and t2 to be values that make f(t) negative and positive.
Choose t3=(t1+t2)/2 and find f(t3).  Replace t1 or t2 with t3.
The way to determine which one to replace is to see which one has the same sign for f(t).
This is a straight forward approach that always works if a solution can be found.

Newton's Method
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A faster approach is Newton's Method.  Pick any t1. As long as there are no maximums
or minimums between that choice and the root, the method will find the solution.  

In this function, I think any choice for t1 would work since f(t) is concave up.
Let t2 = t1 - f(t1)/f'(t1).  Given that f(t) = 1.08^t- 1.05^t - 0.4,
f'(t) = ln(1.08)•1.08^t + ln(1.05)•1.05^t.