Compound Interest Formula
The Geometric mean is most frequently used in the determination of average percent of change. For e.g., if a city has a population of 2, 00, 000 in a given year and 2.40.000 after ten years, we may be interested in finding out the annual percent of change. The increase is 2, 40,000 - 2,00,0000 i.e. 40,000 over a period of 10 years and so one may say that the annual percent increase is 2. However, if we compute 2 per cent increase every year over the preceding year the population figure turns out to be 2, 43,800. This means that the correct figure is little less than 2 percent as we are actually compounding. The average annual percent increase is computed by applying the formula:
pn = po (1 +r)n
Where,
po = the value at the beginning of the period
pn = the value at the end of the period n
R = rate of change
N = length of time period
It follows from the above formula that r = n √(pn - 1)/po
For the above data 2, 40,000 = 2, 00,000 (1 + r)10
Taking logarithms 5.38802 = 5.301 + 10 log (1 + r)
Log (1 + 1) = 0.00792
(1 + r) = antilogarithms 0.0079
1 +r + 0.0184 or r = 0.0184 = 1.84 percent.
The expression Pn = Po (1+ r) n is termed as compound interest formula because it is extremely useful in problems involving compound interest. In the case above we have used it to determine the average annual percent of growth. However, if we know any 3 values of the 4 used in the formula we can find out the fourth one. Therefore, we may determine:
The annual average percent of change, r.
The Population in the given number of years later pn assuming the constant relative change.
The Number of years, n, later which the given population will be, attained again assuming a constant relative change.
Population the given number of years earlier po
It may be pointed out that the assumption of a constant relative change for population is not valid over extended time period for any country except possible' new'' countries.