Let F = future value of the investment after t years;
P = present value of the investment (initial principal invested);
r = annual compound interest rate;
n = number of compounding periods per year;
t = time (expressed in years)
The formula that governs the future value of the investment is:
F = P(1 + r/n)nt
Note: The formula for simple annual compounding is F = P(1 + r)t.
For example, according to US CIA* figures, the real growth rate of the US GDP (Gross Domestic Product) was an estimated 2.2% in 2012, while China's growth rate was 7.8% in the same year. If the US economy is now 127% of the Chinese economy, how long at current growth rates will it take for the Chinese GDP to equal that of the US?
1.27 = (1 + 0.078 − 0.022)t ⇒ ln 1.27 = t·ln 1.056
⇒ t = (ln 1.27)/(ln 1.056) = 0.239/0.0545 ≈ 4.4 years
We use the fact that lima→∞(1 + 1/a)a = e = limb→0(1 + b)1/b
F = limn→∞P(1 + r/n)nt
= P[limn→∞(1 + r/n)n]t
= P[limn→∞[(1 + r/n)n/r]tr
= P[limn/r→∞[(1 + r/n)n/r]tr
To double one's money, we need tr = ln 2 ≈ 0.693. That is, the product of years invested and interest rate must be about 0.693.
*See The World Factbook