## Compounding is relevant for almost every financial evaluation for periods of longer than a year. A sound understanding of compounding and its effects is an essential building block of your financial awareness.

This article shows you how to:

1. Perform compounding calculations period by period.

2. Sense-check and explain your answers.

3. Apply short cut formulae with full understanding.

## 1. Compounding period by period

Compound interest is based on interest on interest, as well as interest on the original amount borrowed or invested.

Let’s say we invest $100m for two years, at a compound interest rate of 20% per year.

**Year 1 interest calculation**

Our interest for the first year is:

$100m x 20% **= $20m**

The total value of our investment has grown to:

$100m + $20m **= $120m**

For the first year, compounding hasn’t had any effect yet. There was no accumulated interest brought forward, on which to calculate any interest on interest.

**Year 2 interest calculations**

In the second year, compounding starts to make a difference to our results.

Interest for the second year is:

(1) A further 20% on our original amount:

$100m x 20% **= $20m**

(2) Plus compound interest of 20%, on our $20m interest brought forward from Year 1:

$20m x 20% **= $4m**

Our total interest for Year 2:

$20m + $4m = **$24m**

By the end of Year 2, the total value of our investment has grown to:

$120m + $24m **= $144m**

Compounding has resulted in an additional $4m of interest so far (= $100m x 20% x 20%).

**Interest on our original amount accumulates steadily at $20m per year**

**Year by year and cumulative totals – Years 1 to 4**

**Year 3 calculations**

Now let’s check the calculations in our table for Year 3.

Interest for the third year is:

(1) A further 20% on our original amount:

$100m x 20% **= $20m**

(2) Compound interest of 20% on our cumulative interest of $44m at the end of Year 2:

$44m x 20% **= $8.8m**

Total interest for Year 3:

$20m + $8.8m = $28.8m

By the end of Year 3, the total value of our investment has grown to:

$144m + $28.8m **= $172.8m**

**Interest on interest accumulates by ever-increasing annual amounts**

**Interest on interest is why the future total grows by an ever-increasing value per year**

**Interest on interest grows as a proportion of the total – for example Years 5 to 20**

Exactly the same relationships apply whether we are working with interest, other forms of return, or any other kind of growth.

**It’s your turn now**

Please confirm your understanding by recalculating the interest amounts and our future total for Year 4. (The answer is at the end.)

**The lower the rates, the smaller the compounding effects – for example 2%**

At a rate of 2% per year, illustrated above, the interest on interest amounts - and proportions of the total - are much smaller than they were at the rate of 20% per year.

**At just 0.2% per year, compounding effects are negligible, even after 20 years**

## 2. Summarising what we’ve learned so far – sense-checks

- When rates of interest, growth or return are positive, compounding results in greater total values.
- The greater the rates of interest, growth or return, the larger the compounding effects.
- The more periods that compounding is applied for, the greater the compounding effects.
- When lower rates are applied for smaller numbers of periods, compounding effects are small.
- Compounding effects are greater when rates are higher, or the number of periods is more, or both.

## 3. Is there a short-cut formula for compounding?

There is.

To work out the total future amount by formula, the Future Value (FV):**FV = PV x (1 + r)^n**

Where:

PV = Present value, the value today

r = interest rate per period

n = number of periods

**Example – 3 years at 20% per year**

PV = $100m

r = 20% = 0.20

n = 3

FV = PV x (1 + r)^n

= 100 x (1 + 0.20)^3

= 100 x 1.728

= **$172.8m**, as before

**Are there short cut formulae for the interest amounts?**

Yes indeed, there are three of them.**(1) Total cumulative interest = FV - PV**

Where:

FV = Future value, the future amount

PV = Present value, the value today

**Using the same example – 3 years at 20% per year**

FV = $172.8m, calculated above

PV = $100m

Total cumulative interest = FV - PV

= 172.8 – 100

= **$72.8m**, as in our table above

**(2) Cumulative interest on original amount = PV x r x n**

Where:

PV = Present value, the value today

r = interest rate per period

n = number of periods

**Continuing the same example – 3 years at 20% per year**

PV = $100m

r = 20% = 0.20

n = 3

Cumulative interest on original amount = PV x r x n

= 100 x 0.20 x 3

= **$60m**, as in our table above

**(3) Cumulative interest on interest = Total interest – interest on original amount**

**Completing the same example – 3 years at 20% per year**

Cumulative interest on interest = Total interest – interest on original amount

= 72.8 – 60

= **$12.8m**, as in the table

## Answer - It’s your turn now - Year 4 calculations

Confirming the calculations in our table for Year 4.

Interest for the fourth year is:

(1) A further 20% on the original amount:

$100m x 20% = **$20m**

(2) Compound interest of 20% on the $72.8m cumulative interest at the end of Year 3:

$72.8m x 20% = **$14.56m**

Total interest for Year 3:

$20m + $14.56m = $34.56m

The total value of the investment has grown to:

$172.8m + $34.56m

= **$207.36m**, as in the table.

Many congratulations!

You have mastered a very important and challenging financial concept.

**___________________**

**Author****: Doug Williamson, FCT**