484. Einstein on Interest

According to Albert Einstein, Compound Interest is the 8th Wonder of the World. Here is the formula :
A = P(1 + r/q)nq

The nq outside the brackets above does not mean "multiplied by nq" but "to the power of nq".

P is the principal (the money you start with, your first deposit)

r is the annual rate of interest as a decimal (5% means r = 0.05)

n is the number of years you leave it on deposit.

q is the number of times per year that the interest is to be calculated.



Sweet's modern business arithmetic: A treatise on modern and practical methods of arithmetical calculations for the use of business and commercial colleges, ... in other educational institutions

185. The Formula for Compound Interest

When considering investments, consider the power of compounding. Compound interest is where the interest you have earned is added to the investment and it itself generates more interest : The formula is as follows :

A = P(1 + r/q)nq

The nq outside the brackets above does not mean "multiplied by nq" but "to the power of nq".

P is the principal (the money you start with, your first deposit)

r is the annual rate of interest as a decimal (5% means r = 0.05)

n is the number of years you leave it on deposit

A is how much money you've accumulated after n years, including interest.

If the interest is compounded once a year:

A = P(1 + r)n

If the interest is compounded q times a year:

A = P(1 + r/q)nq


As a good exercise you should work out dozens of examples comparing the returns when interest is compounded and when it is just simple interest (SI = PRT/100).

Formula for the present value (discounted value) of a future amount

P = the present value of amount A, due n years from now

r = the rate of interest

For example, someone contracts to pay you $100,000 in ten years. What's that worth right now, if they changed their mind and decided to paid you upfront? Say the interest rate is 5%.

At simple interest:

P = A/(1 + nr)

If A = 100,000 and n = 10 and r = 0.05 (which is to say, 5%), then

P = 100,000/(1 + 10x0.05) = 100,000/1.5 = 66,667

At interest compounded annually:

P = A/(1 + r)n

Using the same example as for simple interest, this gives

P = 100,000/(1 + .05)10 = 100,000/1.62889 = 61,391

At interest compounded q times a year:

P = A/(1 + r/q)nq

Or in the same example but compounding monthly (q = 12)

P = 100,000/(1 + 0.05/12)120 = 100,000/1.64701 = 60716