Compound Interest
Compound interest is the interest earned on both the initial principal and the accumulated interest from previous periods. Unlike simple interest, which is calculated only on the principal, compound interest causes the value of an investment or debt to grow at an accelerating rate, because each period's interest is added to the principal and earns interest in subsequent periods.
Definition
Compound interest is interest calculated on the initial principal plus all previously accumulated interest. If an amount P is invested at an annual interest rate r, compounded annually, the value after n years is:
- A = P · (1 + r)n
where A is the future value. The term (1 + r)n is the compound interest factor.
If interest is compounded more frequently—for example, m times per year—the formula becomes:
- A = P · (1 + r/m)m·n
As the frequency of compounding increases, the effective annual rate approaches a limit given by the exponential function:
- A = P · er·n
This is the case of continuous compounding.
Simple versus compound interest
The distinction between simple and compound interest is illustrated by a comparison. An investment of 1,000 units at 5 percent simple interest for 10 years grows to 1,500 units (interest of 500). The same investment at 5 percent compound interest grows to approximately 1,629 units, because interest is earned on interest.
The difference grows larger with time and with higher interest rates, which is why compound interest is often described as a powerful force in finance. The effect is sometimes illustrated with the "rule of 72," which approximates the number of years required for an investment to double by dividing 72 by the annual interest rate.
The mathematics of compounding
The essential feature of compound interest is that growth is exponential rather than linear. This has several important implications.
Time matters enormously. Because growth is exponential, the effect of a given interest rate depends critically on the length of time over which it applies. Small differences in rates or in durations can produce large differences in final values over long periods.
The frequency of compounding matters. More frequent compounding produces a higher effective rate of return, because interest is credited and begins earning returns sooner.
The effective annual rate captures the true annual cost or return when compounding occurs more than once a year. It is higher than the nominal rate whenever compounding occurs more than annually.
Applications
Compound interest is central to many areas of finance and economics.
Savings and investment: the power of compounding is why saving early and consistently can produce substantial wealth over long periods. Retirement savings, in particular, depend heavily on compounding over decades.
Loans and debt: compound interest works against borrowers, as debt can grow rapidly if not serviced. Credit card debt, which typically carries high interest rates, illustrates how compounding can cause balances to escalate.
Present value: the concept of compounding is the foundation of discounting. Because money can grow at a compound rate, a sum received in the future is worth less than the same sum today. The present value of a future payment is:
- PV = FV / (1 + r)n
This relationship is fundamental to the valuation of bonds, stocks, and other assets.
The time value of money
Compound interest formalises the principle of the time value of money, the idea that a sum of money is worth more the sooner it is received. Because money can be invested to earn a return, a unit of currency today is worth more than a unit of currency in the future.
The time value of money underlies the calculation of net present value, internal rate of return, and other techniques of capital budgeting, and it is central to the pricing of financial assets.
The rule of 72 and the doubling time
A useful approximation for the time required for an investment to double at a given compound interest rate is given by the rule of 72:
- Doubling time (years) ≈ 72 / interest rate (percent)
For example, at 6 percent annual interest, an investment doubles in approximately 12 years; at 9 percent, in approximately 8 years. The rule is an approximation but is accurate for interest rates in the range typically encountered.
More precisely, the doubling time is given by:
- Doubling time = ln(2) / ln(1 + r) ≈ 0.693 / r
which is the origin of the rule of 72 (72 is used because 69.3 is less convenient with which to calculate mentally, and 72 has more divisors).
Negative compounding and inflation
Compound interest also operates in reverse. Inflation compounds over time, eroding the purchasing power of money at an accelerating rate. An inflation rate of 3 percent per year halves the real value of money in approximately 24 years. This is why the real interest rate—the nominal rate adjusted for inflation—matters for savers.
The relationship between nominal and real rates is captured by the Fisher equation: the real interest rate equals the nominal rate minus the expected inflation rate.