Expert: Ahmed Salami - 9/1/2010

Question
QUESTION: Hello:

How can (P + rP)^5 equal the future value amount on $100.00 at a rate of 5% for five years without using the calculation:
P(1 + 0.05)^5 = $100(1.05)(1.05)(1.05)(1.05)(1.05)?

(P + rP)(P + rP)(P + rP)(P + rP)(P + rP) will not provide the correct amount or will the following:

(P + rP) + (P + rP) + (P + rP) + (P + rP) + (P + rP), as in, (100 + 0.05*$100)(100 + 0.05*$100)(100 + 0.05*$100)(100 + 0.05*$100)(100 + 0.05*$100)(100 + 0.05*$100) or (100 + 0.05*$100) + (100 + 0.05*$100) + (100 +
0.05*$100) + (100 + 0.05*$100) + (100 + 0.05*$100).

I thank you for your reply.


ANSWER: Hi Kenneth,
The future value A of an amount P at an annual rate r for n years is given by
A = P(1 + r)^n
and it would be wrong, as you've wondered, to be written as
(P + rP)^n
since this is equivalent to
P^n . (1 + r)^n

In essence i'm saying, in relation to your situation, that (P + rP)^5 is not the same as P(1 + r)^5. The latter is the correct form. You can easily see this if you realise that the amount at the end of any year is multiplied by the factor (1 + r) which has nothing to do with P.

Regards

---------- FOLLOW-UP ----------

QUESTION: Hello:

I want to thank you for your reply.

So, the correct use of (P + rP) is to replace the principal by a new principal.

For example, ($100 + 5%*$100) = $105
This in the new principal ($105 + 5%*105) = $110.25
Again, there is a new principal ($110.25 + 5%*$110.25) = $115.7625, etc.
Is my understanding correct?

After three years the $100.00 will be $115.76.

I thank you for your follow-up reply.

Answer
Hi Kenneth,
Yes, you're correct. Its always good to understand the basics.
But remember though that the formula
A = P(1 + r)^n
proves to be time-saving say, for instance, you wanted to compute the future value after 20 years which would be tedious to do repeatedly in the above manner.
Happy to help out.

Regards