Calculating CAGR for Ordinary annuity

Question

Basic compounding interest question:

I paid 5000 every month for 12 months and got 67500 in return, what was the annual compounding intereset rate?

I considered formular as follows:

FV = PMT(((1+r)^n-1)/r)

equation image

where

FV = Future value

PMT = Regular Payment amount

r = Annual Interest rate

n = number of paymments

Solving for this I got r = 0.021. Solving screenshot

Is it 21% or 2.1% per annum?

Am I complete wrong here?l

Answer 1

First illustrating the method with a small example: 3 annual payments of £100 at 10% per annum.

c = 100
r = 0.1

The first £100 compounds for 3 years, the second £100 for 2 years and the third for 1 year:-

a = c(1 + r)^3 + c(1 + r)^2 + c(1 + r)^1 = 364.10

This can be expressed as a summation (Σ) and converted to a formula:-

  c = 100
  r = 0.1
  n = 3

∴ a = c (((1 + r)^n - 1)/r) (1 + r) = 364.10

Now the OP's figures:-

c = 5000
n = 12
a = 67500

a = c (((1 + r)^n - 1)/r) (1 + r)

67500 = 5000 (((1 + r)^12 - 1)/r) (1 + r)

Solve for r

r = 0.0179891 per month

Annualising for an effective rate

(1 + r)^12 - 1 = 0.238561 = 23.8561 % per annum

or for a nominal rate

12 r = 21.5869 % per annum compounded monthly