## Compound Interest

In design work, there is a business side of things that depends on loans and investments.  There is also a business side of things where people invest different amounts for different periods of time in helping you — and you might have to figure out their share of things in the future.

There is also the idea of inflation and cost of living — as you have to design to a cost, a budget, you have to sometimes predict the future costs involved.

This is a small introduction to the basics of compound interest, included here in the hope that it may help.

Example One.

The scene: you are thinking of borrowing £100 at an interest rate of 9% compound (constant) per year over a three year loan period, but you want to know how much you will have to pay every month to see if you can afford the loan, and you want to know the totals — how much it actually cost.

Well, 9% is shorthand for (9/100), which is 0.09.

You have to repay the amount borrowed as well as the interest over this period, so you are paying 100% back plus 9% interest, which is like saying 109% or (9/100 +1).

The general formula for the amount to repay in total:

£ . Tn = £ . A . rn

The letters stand for things: T stands for Total to repay, n stands for the number of years of the loan period (3), r stands for the rate of interest (9%), and A stands for the amount you borrowed.

Tip: Do each step and always consider the units to be like numbers, multiply the pounds, months and so forth.  This approach will keep you on track as you aim toward an answer and units cancel out.

£ . T3 = (£ . 100) . (9/100+1)3
=> £ 129.50 total repay (obviously the £ 100 loan plus £29.50 interest).

The next bit figures out the monthly fixed repayment

£ 129.5
_______________________
(n . years . 12. months/year)

which is

£ 129.5
_______________________
3. 12 . months

which is

£ 129.5
_______________________
36 . months

=> £3.60/month

Example Two.

Do what you just did, but backwards.  How much of a loan do you need to ask for when you can only afford a fixed monthly repayment of a tenner-a-month? Say the interest rate is fixed throughout the term at 9%, and the term is 3 years.

(£10/month) . 3 . year . 12 . months/year = £360 = £T3 = £A . r3

£360 =  £A . (9/100 +1)3

£360 =  £ A. (1.295 029)

£360
______ = £A  = £277.99

1.295

=£268 more or less

Example Three.

The same thing can be done in a different way. In this example, we don’t want to borrow, we want to save — to invest, using the same figures.  We could therefore ask…

How much do I have to invest now to give me an account with £129.50 in it in three years’ time (assuming a fixed interest rate of 9%).

The formula we used above was:

£Tn = £A . rn

We just rearrange the equation, but bear in mind that the letters now stand for different things: T now stands for Total in account at the end, n stands for the number of years of the savings period (3), r stands for the rate of interest (9%), and A stands for the amount you invest.

£Tn = £129.50 is what we want,

so

£129.50 = £A . r3

Rearranging for £A

£A = £129.50/(1.295 029)

£A = £100