Future value and compounding
A dollar today is worth more than a dollar tomorrow. The time value of money is the foundation of all finance.
Why money has time value
Money available today can be invested to earn a return, and future money is subject to inflation and uncertainty. So rational people prefer money sooner rather than later. The interest rate is the price of time: what someone must be paid to wait.
Simple interest
Simple interest is earned only on the original principal: interest = \(P \times r \times t\). Investing 1,000 dollars at 5% simple interest for 3 years earns 150 dollars. Simple interest is used for some short-term loans but is rare for long-term investing.
Compound interest
Compound interest is earned on both the principal and accumulated interest. The future value of a single sum is \(FV = PV(1 + r)^n\), where r is the rate per period and n the number of periods. 1,000 dollars at 5% for 3 years grows to \(1{,}000 \times 1.05^3 = 1{,}157.63\).
FV = PV(1+r)^nThe power of compounding
Compounding grows dramatically over long periods. At 7% a year, money roughly doubles every 10 years, so 10,000 dollars could grow to about 76,000 dollars in 30 years. Starting to save early matters more than the amount saved, because of the extra time for compounding.
The rule of 72
The rule of 72 estimates doubling time: years to double is approximately 72 divided by the annual interest rate in percent. At 6%, money doubles in about 12 years; at 9%, about 8 years. It also works for inflation: at 3% inflation, prices double in about 24 years.
Compounding frequency
Interest can compound annually, semi-annually, quarterly, monthly, or daily. More frequent compounding gives slightly more growth: \(FV = PV\left(1 + \frac{r}{m}\right)^{mn}\), where m is compounding periods per year. With continuous compounding, \(FV = PVe^{rn}\).
Effective annual rate
Because compounding frequency matters, lenders and savers compare rates using the effective annual rate (EAR): \(EAR = \left(1 + \frac{r}{m}\right)^m - 1\). A 12% rate compounded monthly gives an EAR of about 12.68%. In the US, the annual percentage yield (APY) on savings accounts reports this effective rate.
Worked example
Maya invests 5,000 dollars at 6% compounded monthly for 10 years. \(FV = 5{,}000 \times (1 + 0.005)^{120} \approx 5{,}000 \times 1.8194 = 9{,}097\) dollars. With annual compounding it would be \(5{,}000 \times 1.06^{10} \approx 8{,}954\), slightly less.
Key takeaways
Money has time value because it can earn returns and faces inflation and risk. Simple interest uses only the principal; compound interest earns interest on interest: FV = PV(1 + r)^n. Long horizons make compounding powerful. The rule of 72 estimates doubling time. More frequent compounding increases growth; the effective annual rate (and APY) allows fair comparisons.
Flashcard
Money today is worth more than in the future
Check your understanding
Using the rule of 72, about how long does it take money to double at 8% per year?