# Compound Interest

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.

### 9 Responses to Compound Interest

1. George Hafiz at #

Would the ‘compounding frequency’ be what the banks (in the UK at least) refer to as the frequency the interest is calculated? E.g. 3% AER (calculated monthly) would mean a compounding frequency of 12?

Sometimes banks say they calculate interest daily. Dare I ask what happens on leap years in this case? Do we make earn more interest?

• You would want to ask the banks what they mean by “frequency the interest is calculated”. There is a difference between compounding frequency, which relates to the formula used to calculate interest, and the frequency with which interest is credited, which relates to how frequently a formula is used to credit interest to your account. For example, you could credit interrest to an account daily using an annual compounding formula for each day’s interest. Compounding frequency relates to how much interest you earn. The frequency with which interest is credited relates to when it is credited to your account. If your bank credits interest daily, it probably credits interest on the 366th day of a leap year. However, it may ignore that day in calculating how much interest you earn overall. For each financial product there is a day-count convention. Some count actual days, so the 366th day is included. Some assume a 365 day year, so the 366th day is excluded. Some assume 30 days in every month, which is simple but excludes several days each year.

2. Jolaine Greaux at #

Some savings and loan companies advertise that they pay interest continuously. Explain what this means and contrast it compound interest. I would appreciate your help please…

Thank you,

Jolaine Greaux

• Are you a student seeking help with homework, or have you actually seen continuously compound interest advertised somewhere? I have never seen it offered or advertised, so I would be interested to hear of real-world instances.

3. Jolaine Greaux at #

I am a student looking help with home work

• The above article gives you plenty to say. Your professor will likely appreciate the hand pump vs faucet analogy.

The reality is that different compounding methods — annual, quarterly, continuous, etc — are all just conventions for indicating the same process.

With some mathematics, you can easily show that all the compounding processes (excluding simple interest, which is not compounded) can be expressed as a(c^t) for some constant c. For example, with quarterly compounding c = (1 + r/4)^4. With continuous compounding c = e^r. The different compounding methods are just different conventions for specifying that constant c.

If you quote me a certain quarterly interest rate, I can calculate an equivalent continuous rate. That would be a rate that, using the continuous compounding formula, produces the same constant c that you would obtain with your quarterly rate using the formula for quarterly compounding.

• Jolaine Greaux at #

Thank you every much

4. Fillipus Shimhanda at #

What formula do I use if I want to calculate interest compounded bi-annually?

5. Bi-annual would be interest compounded twice a year, also known as semiannual. Use formula [3] with n = 2.

A less common situation would be interest compounded once every two years. In that case, you would use the same formula [3] but with n = 1/2.