## Simple and Compound Interest

Interest is an interesting subject. We all learn about it, we touch on Simple Interest, and are then dumped into Compound Interest and Annuities .... which we more often than not use much, as a result missing some of the subtleties of interest in general.

(SIMPLE) INTEREST REVIEWED

The most fundamental formula is .....
FV = PV + Interest
FV = PV + (PV * Interest) Rate
FV = PV(1 + Interest Rate)
where .....
PV = Present Value at beginning of term
FV = Future Value at end of term
Interest = the interest amount in money terms
Interest Rate = the fraction + 1 that PV is multiplied with to give FV

Basic formula
FV = PV(1 + r*n), where....
r = interest rate per period, n = number of periods invested

General formula
FV = PV (1 + i*t), where...
i = interest rate per year, t = number of years invested

You invest (say) R10,000 for 3 years, and you are quoted an interest rate of 10% per year. At the end of the term your investment has grown to R13,000. So far, so good.

COMPOUND INTEREST

With Compound Interest, is the interest calculated and accrued, not just at the end of the term, but on a more regular basis which we shall call a compounding period. That means that the interest you earn actually also earns interest... you can say it's interest on interest.

To calculate compound interest compounded (say) yearly, would you have to do the above calculation 3 times, once for every year. Try it, you should get a final value of R13,310 !!!
That makes Compound Interest worth knowing about!

There is actually a shortcut formula, that looks like this...

Basic formula
Cl Val = Op. Val(1 + r)^n where ......
r = interest rate per compounding period, and n = the number of compounding periods invested, or

General formula
ClVal = OpVal ( 1 + i*p)^t/p, where .....
i = interest rate per year, and t = number of years invested, and p = Compounding period in years

SO WHAT ESSENTIALLY IS THE DIFFERENCE ?
As Compound Interest accrues interest on a more regular basis, and Simple Interest doesn't ... would you expect Simple Interest to actually pay out the interest at the end of each period, but it does not. So the essential difference between the two types is instead how it is specified at the onset of the contractual term, and if it is not specified can you be sure that it will be Simple Interest.

HOW INVESTMENT CO's ALSO FORGET TO USE COMPOUND INTEREST
Just as we forget about Compound Interest, so do Investment Companies... So, how do Investment companies not use Compound Interest to mislead you?
Well, let's say you have R10,000 invested which grows to R13,000 to over 3 years. They will then report that your growth was 10% ..... which seems fine using good ole Simple Interest.
But if the Compound Interest was used, which they really should since they have not just your capital but also your interest, and we flip the above formula around, we find that the effective growth if you earned interest on interest was only .....

i = 1/p[(Cl.Val/Op.Val)^p/t - 1]
i = 12 [(13,000/10,000)^1/36 - 1]
i = 0.0877 which equates to 8.77% per year!

Now, 1.23 percentage points dont seem like much, but in the process do they get a lot more from you than just their Fees. Say their fees are quoted at 1%, are they effectively taking an extra 1.23 / 3 = 0.4% thus amounting to 1.4% per year instead !