# About the power of compound interest

A 5 minute read • Graham A

*Part of our financial management series.* Here’s a nice story. Forty years ago my friend Peter was given £100 for his birthday. Rather than spend it, Peter put the money into a savings account that provided interest at 5%. He let the interest accumulate in his account, calculating that £5 would be added each year. By now, 40 years on, he reckoned £200 would have been added. Imagine Peter’s delight when the bank said his £100 had actually grown by £600! But where had this miraculous bonus come from? It’s the power of compound interest.

^{1}

“Compound interest is the eighth wonder of the world. He who understands it, earns it … he who doesn’t … pays it.”

If you want to follow Einstein’s advice and get to understand how this phenomenon affects our borrowing and investments, read on.

## What is compound interest?

First, imagine you’re driving along the road at a steady speed. Your distance from home increases by the same amount every hour. ‘Simple’ interest works in the same way; in Peter’s example he thought his investment would increase by the same amount every year (£5, or 5% of £100). Back in your car, now imagine you keep your foot on the accelerator for the whole journey. You’re now travelling like compound interest and you’ll certainly get to your destination a lot quicker! *[Editor: along with a speeding ticket.]*

So how does compound interest come about? Let’s take a hypothetical example. Don’t worry if you get lost as we go through this; a bit later I’ll give you a much simpler ‘rule of thumb’ that you can use to gauge the power of compound interest on your own loans or investments.

### Here’s the example

Say I won £100,000 in the lottery (lucky me!) and invested it at an annual interest rate of 5%. By the end of the first year I’d still have my £100,000 but I’d also have earned £5,000 in interest. If I left that £5,000 in my account so that it earned interest too, by the end of the following year I’d have:

- a further £5,000 interest from my original £100,000
- plus £250 in interest from the £5,000 I’d re-invested

And so it will go on – provided you continue to re-invest: so in the third year I will earn:

- another £5,000 interest on the original £100,000 investment
- another £250 interest on the £5,000 earned in the first year
- £12.50 interest from the £250 earned in the second year

Here in table form:

1st year interest | + 2nd year interest | + 3rd year interest | ….. |
---|---|---|---|

£5,000 | £5,000 | £5,000 | ….. |

£250 | £250 | ….. | |

£12.50 | ….. | ||

….. |

A new row and column are added to the table each year and your total interest to date is the sum of *all* the cells in the table. This ‘interest on interest’ is the acceleration of the investment brought about by compound interest. And as each new bit of interest gets calculated, I receive that amount in each and every year of the investment. Over longer time periods this can be very powerful, as we have seen with Peter’s birthday money.

And unfortunately for investors, price inflation is the enemy of interest. If you invest £1,000 for 10 years you might double your money, but that doesn’t mean you’ll double your spending power: the price of things you want to buy will probably have risen over the same period.

## A simple compound interest tool: the Rule of 72

As you can see, compound interest can be complicated to work out precisely. Luckily there’s a quick ‘rule of thumb’ you can use to get a rough idea. It’s known as the Rule of 72^{2}.

### Loan compound interest

Here’s what you do: take the rate of interest you expect to be charged, and divide it *into* 72. The answer will be roughly how long it takes the debt to double – if you don’t make any payments over the term of the loan.

Say the interest rate is 6%. 72 divided by 6 is 12. So it will take about 12 years for the debt to double. And every subsequent 12 years it will double again.

### Investment compound interest

It works the same for investments: divide 72 by the interest rate you have been offered and the answer will be roughly how long it will take your investment to double in value – provided you don’t withdraw any interest.

But beware: this rule starts to get less accurate for interest rates over 20%.

If you prefer to skip the more complicated maths that follows, jump straight to the conclusion.

## More precise compound interest calculation

If you want a more precise way to work out how many years it will take to double your debt/investment you will need a calculator with a Natural Log (ln) function. Then you can use this formula:

T = ln(2) / ln(1 + (R/100) )

[T – years, R – annual rate of interest]

So, at an annual interest rate of 8% the time taken to double the debt/investment would be:

ln(2) / ln(1.08) = 9.006

…which is very close to what you get by dividing 8 into 72 under the Rule of 72!

## Complete compound interest calculation

In practice, although we pay (or receive) interest annually, it’s often calculated monthly. In some cases it’s even calculated daily, and settled monthly. It’s important to understand that in the periods between payments, the interest compounds, so you will pay (or receive) more than if you simply apply the interest rate to the balance left after your last payment. This is why the Annual Equivalent Rate (AER) on a loan is higher than the monthly rate.

Here is the complete formula for calculating the effect of compound interest^{3}:

FV = P x (1 + ( (r/100) / n) ) ^{nt}

[FV – Future Value; P – Principal; r – Rate; n – number of times compounded per year; t – number of years]

So, if I borrowed £1,000 (P) for 20 years (t) at an annual interest rate of 10% (r), where the interest is compounded monthly (n=12) and I didn’t make any payments, at the end of the term I would owe:

£1000 x (1 + ( (10/100) / 12) )^{12×20} = £7,328

This has been a maths-heavy article so if you’ve understood everything to this point give yourself a pat on the back. But even if you haven’t followed it all, here are the takeaways:

## In conclusion

- Here at NobodyToldUs.com we don’t give financial advice, but we do offer some tips that you can consider when deciding how best to manage your money
- If you’re considering taking on debt (or investing your cash), be aware of the power of compound interest to accelerate the value of what you owe (or what you own), particularly over long periods of time
- If you’re a borrower wanting mainly to reduce the effects of compound interest, pay your interest charges promptly; don’t let them accumulate
- If you’re an investor wanting to make compound interest work in your favour, let the interest add to your investment, and retain your investment for as long as possible

## And finally

At the time of writing, UK and US interest rates are historically low but may not stay that way. Consider this: if you borrowed £1,000 at 10% over 20 years without making any repayments you’d eventually owe about £7,000. Without compound interest it would only be £2,000. The power of compound interest that even impressed Einstein!

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##### References ^{1}Einstein Quote ^{2}Rule of 72 ^{3}Compound Interest Maths

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