# Calculating CAGR for Ordinary annuity

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

• Note that CAGR doesn't apply here since CAGR measures the growth of a single initial investment - here your initial investment is zero. What are you actually trying to calculate? The equivalent interest rate if compounded annually instead of monthly? Apr 14, 2023 at 16:25
• Yess .. I guess I am not well versed with precise financial terminology. I know terms are very important in finance. Pardone me.
– RajS
Apr 14, 2023 at 16:59
• No pardon needed - just trying to make sure I understand what you're asking. Apr 14, 2023 at 17:03
• The formula above is for the future value of an ordinary annuity. The appropriate formula would be the one for an 'annuity due'. In an ordinary annuity the payment is made at the end of each period while in an annuity due it is made at the start of each period. Note the formula in my answer is for an annuity due, and has an extra compounding period `(1 + r)` Apr 15, 2023 at 11:56
• In these formulas `r` applies to the compounding period, which here is monthly. So `r` is the interest per month. `r` can be annualised for an effective or nominal annual rate as shown in my answer. Apr 15, 2023 at 12:09

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:-

$a=\sum_{k=1}^{n}c(1+r)^{k}=c\left ( \frac{\left ( 1+r \right )^{k}-1}{r} \right )(1+r)$

``````  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
``````
• OP has "ordinary annuity" in the question title. While I am not sure if that is what's actually needed (if OP only has the task description, they may have just guessed/googled that it is an ordinary annuity, but maybe the title or topic for the excercise is "ordinary annuity"), you should probably mention/emphasize in your answer that you take a different formula (e.g. not just in a comment to the question that someone might overlook) Apr 15, 2023 at 10:40
• @Solarflare You are correct. I commented that an annuity due would be appropriate and I will comment further here: An ordinary annuity would not make sense because with payment at the end of each period, for the first month nothing would happen, then the first payment would be made at the end of the month. Likewise the last payment would not gain any interest, so again pointless. Apr 15, 2023 at 11:52