When money is placed on deposit to earn interest, the interest can be paid out periodically as it is earned, or it can be left on deposit. If interest left on deposit does not itself earn interest, the deposit is said to earn simple interest.
Measure time in years. With simple interest, at any time t, the value of the deposit is given by the product
a(1 + rs t )
[1]
where a is the amount of the initial deposit, and rs is the simple rate of interest. For example, if USD 100 is left on deposit to earn an rs = .06 rate of simple interest, at the end three years, the deposit will be worth
100 (1 + (.06) 3) = 118
[2]
Formula [1] means that, when a deposit earns simple interest, its value grows linearly with time.
Exhibit 1: The value of a deposit earning simple interest grows linearly over time.
In practice, simple interest is rarely used for deposits held more than a year. An alternative is to credit interest, not based upon the initial value of the deposit, but based on its accumulated value. This approach is called compound interest. With it, interest is earned on both the initial deposit and on any interest that has already been earned but left on deposit—interest is earned on interest. At any time t, the value of the deposit is given by
[3]
where n is the compounding frequency—the number of times per year that interest is credited. The constant rn is the interest rate. Typical values for n include
- 1 for annual compound interest,
- 2 for semiannual compound interest,
- 4 for quarterly compound interest, and
- 12 for monthly compound interest.
For example, if USD 100 is left on deposit to earn an r2 = .06 rate of semiannually compounded interest, at the end of three years, the deposit will be worth
[4]
Formula [3] means that, when a deposit earns compound interest, its value grows exponentially with time.
Exhibit 2: The value of a deposit earning compound interest grows exponentially over time.
With compounding, larger values of n correspond to interest being credited more and more frequently. The limiting case of this is called continuous compounding where interest is credited on a continuous basis. The distinction is like the difference between getting water from a hand pump and getting water from a faucet. With the hand pump, the water flow is broken. With the faucet, it is continuous. The faucet does not necessarily deliver water any faster than the pump. It just delivers it continuously.
With continuous compounding, at any time t, the value of a deposit is given by
a exp(rc t )
[5]
where rc is the continuously compounded interest rate exp is the exponential function with base e =2.71828…
For example, if USD 100 is left on deposit to earn an rc = .06 rate of continuously compounded interest, at the end three years, the deposit will be worth
100exp(.06(3)) =100(2.718280.18) = 119.72
[6]
Interest is rarely compounded continuously in practice. Continuous compounding is more of a theoretical notion. It is used frequently in theoretical finance because it simplifies many formulas.