Simple, Compound, and Continuous Interests
Interest is the price paid for the benefit of borrowing money for a certain period of time. Typically, the amount of interest is expressed as a certain fraction or percentage, of the principal amount of money borrowed. When the amount of time the principal is borrowed is not known in advance, the interest is typically agreed to be computed as the product of a rate of interest and the time period, where the rate of interest is expressed as a certain fraction or percentage of the principal amount, per unit time.
There are three main methods of charging the interest. When the interest is only paid at the end of the lending period, it is called simple interest. Sometimes however, the interest is charged periodically, every time a certain time interval passes. If the interest is charged and not paid, it effectively increases the amount of principal that is owed, so for the next time interval, interest will be charged on the interest from the last time interval. This method is called compound interest. Finally, continuous interest occurs when the interest is charged continuously (and constantly added to the principal).
The following examples and the demonstration illustrate different kinds of interest and how they can be useful (or harmful) to investors.
Example 1: Simple Interest
Suppose Alice wants to borrow $1000 from a bank for one year. The bank agrees to lend her the money provided that she agrees to pay the original amount plus 10% interest next year. This means she will have to pay $1000 + (10%)×($1000) = $1000 + $100 = $1100 next year.
The formula for the future value S of some investment with simple interest is:
S=P 1 + r t
where P is the principal amount, r is the interest rate, and t is the time period of the investment.
For this example it means: P = $1000, r=0.10 per year, and t=1 year. Notice that r is in decimal form rather than percent form.
S = $1000⋅1+0.10⋅1 S = $1000⋅1.10 = $1100
Example 2: Simple Interest
Suppose Alice wants to borrow $1000 for 2 years at the same simple interest rate of 10% per year. Here P = $1000, r=0.10 per year, and now t=2 years. Therefore, at the end of the 2 years she will owe:
S = $1000⋅1+0.10year⋅2 years S = $1000⋅1.20= $1200
Example 3: Compound Interest
Consider the same problem of Alice wanting to borrow $1000 from the bank for 2 years at 10% interest per year. Rather than charging simple interest on the loan, the bank can use a more widely used form of interest calculation, compound interest.
Compound interest is interest that is added to the principal of a loan such that the added interest also earns interest. The addition of interest to the principal amount is referred to as compounding.
This means that for the first year, you can borrow $1000, and adding the interest for the first year, means that you owe the bank $1000*(1+0.10) = $1100 at the end of the year.
S__1= $1000 + 0.10⋅1000 = $1100
Then for the second year, the principal owed is $1100, and subsequently you owe the bank interest on that amount at the end of the term.
S__2= $1100+0.10⋅1100= $1210
Using compounded interest, the bank receives $10 more than with simple interest.
Compound interest can also be used to your advantage. Buying guaranteed investment certificates (GICs) or government bonds, can make the bank pay you interest. GICs pay compound interest, which as you will see, is much better than simple interest for investments.
Example 4: Compound Interest
Suppose Bob buys a GIC from the bank worth $1000 at 2.5% interest compounded yearly for a period of 12 years. After the first year his investment will be worth:
S1= $1000 1 + 0.025= $1025
The next year he will make 2.5% interest on S1, so:
S2= S1 1 + 0.025= $1050.63
The difference between compound and simple interest is only $0.63 after 2 years. After the 12th year however, his initial investment has grown to:
S12= $ 1000 1+0.02512 = $1 344.89
The general formula for compound interest after n compounding periods is:
where P is the principal value of the money (initial investment) and r is the interest rate for each compounding period in decimal form. This can be further generalized for any time periods paying r interest rate per year as follows:
where m is the number of times the interest is compounded per year.
Example 5: Compound vs. Simple Interest
Suppose Bob buys two GICs from the bank worth $5000 each. Both will last 3 years and have an interest rate of 2.5% per year; however, one pays simple interest and the other is compounded monthly. After 3 years, the simple interest GIC will pay Bob:
S = $50001+0.025×3 years = $50001+0.025×3 = $5375.00
In contrast, the compound interest GIC will pay
S = $50001+0.025123⋅12 = $5389.00
which is more than the simple interest GIC.
Example 6: Continuous Interest
It is clear that the more frequent the compounding periods, the faster the investment will grow. If you take the limit as the frequency goes to infinity (or, equivalently as the duration of the compounding period goes to zero), you arrive at continuous interest. The return of continuously compounding interest is given by the formula:
S = P e r t,
where t is the duration of the investment, P is the principal value, and r is the interest rate.
Now, compare continuously compounded interest with biannually (twice a year) compounded interest. Suppose the annual interest rate is 5% and the principal value is $5000. Over 10 years, the compounded interest will give a return of:
S = $50001+0.0522⋅10 = $8193.08
whereas the continuously compounded interest will make:
S = $5000 e0.05⋅10 = $8243.61
Continuous compounding always generates more interest than discrete compounding. Some loans demand continuous interest, which makes them especially difficult to pay back if they are left to grow for too long.
Change the values in the table below to compare how much the various kinds of interest change the total amount earned (or paid) over the lifetime of the loan.
Annual Interest Rate
Frequency of Compounding
time(s) per year
Length of Investment
Final Values with:
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