Expert: Josh - 5/22/2010

Question
Hello:

I am having some difficulty with the following calculation for future value for compound interest for $1.00 at 1%.

Here is the calculation: (P +rP) + r(P + rP)

I saw it at the following URL: http://www.moneychimp.com/articles/finworks/fmfutval.htm

I substituted P for $1.00 and r for 1% to make the calculation easy. The correct answer is 1.030301. I believe that the solution would look like this (1 + 1%*1) + 1%(1 + 1%*1) + 1%(1 + 1%*1), but I cannot get this calculation to produce the amount of 1.030301.


The calculation will produce the correct amount for two periods or two years. (1 + 1%*1) + 1%(1 + 1%*1) = 1.0201. However, the future value for three periods or years is not exact but close: (1 + 1%*1) + 1%(1 + 1%*1) + 1%(1 + 1%*1) = 1.01 + 0.0101 + 0.0101 = 1.0302. However, the exact amount is 1.030301.


If I use a more common calculation (P + rP) + (P + rP) + (P + rP), I get the correct amount of 1.030301.

I have not seen the future value calculation with the rate "r" before the (P + rP) as in r(P + rP) as I indicated above.

Can you get the calculation from the moneychimp.com site to produce the correct amount for three years/periods or more?

I am only interested in the extended form not the more common
P(1 + r)^n.

I contacted the individual at the site with my question, but he has yet to reply.  

I thank you sincerely for your reply and assistance.

Answer
Applying the idea of compound interest once, we get
P(1+r) = P+rP

Applying twice:
P(1+r)(1+r) = (P+rP)(1+r) = P+rP +Pr+Pr*r = P+2rP+P*r^2

Applying thrice:
We get P(1+r)(1+r)(1+r) = P (1+3r+3*r^2+r^3) = P +3rP +3r^2*P+ r^3*P
Numerical evaluation gives 1+3*0.01+3*0.01*0.01+0.01*0.01*0.01 = 1.030301

The formula continues by induction using binomial expansion, not by extending the pattern in the manner  you have attempted.

Note: "(1+3r+3*r^2+r^3)" is the binomial expansion of (1+r)^3.