## 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 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

= P·etr

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.

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