## compound interest

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)

### simple compounding

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

### continuous compounding

We use the fact that lim_{a→∞}(1 + 1/a)^{a} = e = lim_{b→0}(1 + b)^{1/b}

F = lim_{n→∞}P(1 + r/n)^{nt}

= P[lim_{n→∞}(1 + r/n)^{n}]^{t}

= P[lim_{n→∞}[(1 + r/n)^{n/r}]^{tr}

= P[lim_{n/r→∞}[(1 + r/n)^{n/r}]^{tr}

= P·e^{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