Compounding and Annuities

Most of us would have heard the cliched "magic of compounding". The magic is  in the fact that the growth of an investment is exponential and not quite the linear grwoth that is generally experienced where  simple interest is worked out . The reason lies in the fact that the Interest is added to the base principal at regular intervals and that forms the basis for further Interest calcuation

Say at interest of 10% per annum, an investment of Re 1 at the beginning of year 1 , would becomes 1.1 at the end of year 1, in generic way it can be expressed as (1+i)

And at the end of say 3 years the same would be (1+i)^3 and at the end of n years, the same would be 

A =P(1+i)^n

A bit more complicated would be to ascertain the value of an  Annuity or the maturity value of a regular payment at a regular intervals at a regular rate of interest . The derivation is through a summation of a geometric expression and the simplified form is as follows

Sn     = R *{(1+i)^n  - 1}/i   where

Sn is the Annuity or the maturity value

R is the regular payment at regular intervals and 

i is the rate of interest at the intervals specified

Of course one assumption in this is that the amounts are either invested at the end of the regular interval and say the first investment earns interest for (n-1) year and the last regular investment - no interest.

A variation of this is where the regular payment is at the beginning of the interval and earns interest for the full quantum of n years/months as the case may be and the last one also earns interest for on year would be as follows.

Sn   = R * { {(1+i)^(n+1)}-(1+i)}/i