__Lesson: Compound interest__

A compound interest means a return on the initial investment and all previous interest that have been made previously on the investment. Basically, if you invest $ 1000 for a 10 % interest after one period, you have $ 1 100 at the end of that period. Now, if you were offered a compound interest of 10 % for several periods it means that in the next period the interest is not calculated on $ 1 000 but on the $ 1 100 because this sum includes the previously earned interest.

__1. Finite number of compounds__

For any person on this forum it is important to recognize one of the simplest relationships there is on economics. You know that if you invest X for a compunded interest rate of R, then your investment will be worth A after N periods. This is given by the mathematical relationship:

**X(1+R)^N = A.**

R is the percentage factor of the interest rate. E.g. 5 % is expressed as 5/100 = 0.05.

This simple equations lets us easily solve for missing values, given that we have values for the other variables:

**X = A / [(1+R)^N]**

**R = Nth_Root(A/X) - 1**

**N = log(A/X)/[log(1+R)]**

**A = X(1+R)^N**

**Example 1:**Say you can get a 1 % monthly compounding interest on a managed forex account for a year and you would like to have $ 10 000 available to be credited at the end of those twelve months. You are wondering how much you would have to deposit today. Well, then we need to solve the equation above for X:

X = A / [(1+R)^N] = 10 000 / (1.01^12) = 8 874.49

This means you would have to deposit $ 8 874.49 for them to be $ 10 000 in one year's time. You can validate this by inserting the values into the first formula.

**Example 2:**You want to invest $ 1 000 in a forex account and you want it to have increased 50 % in 15 months' time. What require?

R = Nth_Root(A/X) - 1 = 15th_Root(1500/1000) - 1 ~=2.74 % monthly interest

**Example 3: Debunking HYIP promises**"I made $ 50 000 the first week", "After two weeks I quit my day job", "I made $ 1 402 312 the first year", "This system generates 5 % weekly on autopilot". These are typical phrases you will find on sales letters that try to convince you into buying their forex robot (EA) or other HYIP (High Yield Investment Products). Now, you can easily use some reasoning and the mathematical formula provided above to make up your mind.

Assume some sales letter is offering you a 5 % weekly compounded interest and that it is trying to give the impression that this can be sustained. You have $ 1 dollar you are interested in investing in that project. According to Forbe's world billionaire overview the richest persons in the world are Carlos Slim Helu and his family with a net worth of USD 53.5 BN dollars. $ 53.5 billion - sick, eh? Well, let us find out how long you would have to be invested with this product starting with $1 in order to achieve the same amount of wealth (in nominal terms).

N = log(A/X)/[log(1+R)] = log(53 500 000 000 / 1)/[log(1.05)] = 506 weeks = almost 10 years.

So, according to this sales letter, with a $1 investment you can be equally rich as Carlos Slim Helu and his family are now (in nominal terms) in 10 years!

Of course, there are other concerns. For example, as your account to amazingly large amounts, your broker cannot offer you similarly geared positions anymore as before. However, it is still a suitable tool to debunk ridiculous claims. Many sales letters are written on the assumption that the reader does not know enough about compounded interest rates to have an intuition about the validity of the claims.

__2. Continuously compound interest__

For a continuously compounded investment X with interest rate R over N periods, you can calculate the relationship using Euler's number e:

**X * e^(R*N) = A**

The variables are calculated this way:

**X = A*e^(-R*N)**

R = ln(A/X) / N

N = ln(A/X) / R

A = X * e^(R*N)

R = ln(A/X) / N

N = ln(A/X) / R

A = X * e^(R*N)