Expert: Josh - 1/18/2007

Question
Hello:

Can you explain the compound interest formula?

Let's say that an investor has $1000.00 to invest for 1 year at rate of 5% compound interest.

How was this formula M = P( 1 + i )^n created?
This formula P( 1 + i )^n is a factored version from something like (1000 + 1000 X 5%) or maybe (1000 + 5%). I do not know which one or maybe even a different calculation.


I thank you for your reply.


Answer
Hello,

For this derivation, let P be the principal, r be the interest.
(If interest is R%, r=R/100)

Write A(n) for the amount accumulated after n instalments.

Assuming interest is compounded annually, after n=1 year,
A(1)=P+P*r, where P*r is the interest earned.

We can factorize this as A(1)=P(1+r).
By the end of the second year,
A(2)=A(1)+A(1)*r
   =P(1+r)+P(1+r)*r
   =P(1+2r+r^2)
   =P(1+r)^2

By the end of the third year,
A(3)=A(2)+A(2)*r
   =P(1+r)^2+P(1+r)^2*r
   =P(1+2r+r^2)+P(1+2r+r^2)*r
   =P(1+3r+3r^2+r^3)
   =P(1+r)^3
and so forth.

FORMAL PROOF: (This is not easy to understand)
Assuming that A(k)=P(1+r)^k, ......[#1]
A(k+1)=A(k)+A(k)*r
    =P(1+r)^k + P(1+r)^k*r
    =P[1+C(k,1)*r+C(k,2)*r^2+...+r^k]
     + P[1+C(k,1)*r+C(k,2)*r^2+...+r^k]*r
    =P[1+C(k,1)*r+C(k,2)*r^2+...+r^k]
     + P[r+C(k,1)*r^2+C(k,2)*r^3+...+r^(k+1)]
     P[1+{C(k,0)+C(k,1)}*r+{C(k,1)+C(k,2)}*r^2+{C(k,2)+C(k,3)}*r^3+...+r^(k+1)] ...Note 1
    =P(1+r)^(k+1) for all integer k>=0

Hence, by the process of mathematical induction, the formula is proven for all non-negative integer k.

Note 1: Here, we applied the formula for binomial expansion, where C(n,k)=n!/[(n-k)!*k!] represents the kth coefficient in row n of the Pascal triangle. Don't worry too much if you don't follow this.