21.2 Geometric Growth and Compound Interest
What you may have expected with the first example in the last section was that after you earned the $100 and it was added to your balance, then the next time interest was calculated, it would be 10% of the current balance, $1100. That did not happen using the simple interest model, but it does when we use a compound interest model. So, suppose you deposit $1000 in an account that pays interest at a rate of 10% compounded annually. How much is in the account after 1 year? After 2 years? After 3 years?
Notice that not only is the balance increasing each year, but the interest is increasing each year as well.
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Compound Interest is interest that is paid on both the original principal and accumulated interest.
In the previous example, interest was compounded once a year (or annually), but often interest is compounded multiple times a year, for example quarterly (four times per year) or monthly (12 times per year). With an interest rate of 10% per year and quarterly compounding, you get one­fourth of the rate or 2.5%, paid in interest each quarter year. The "quarter" (three months) is the compounding period, or the time elapsing before interest is paid.
The compounding period is the fundamental interval on which compounding is based, within which no compounding is done.
Compound Interest Formula for an Annual Rate
An initial principal P in an account that pays interest at a nominal annual rate r, compounded m times per year, grows after t years to
Notation for Savings
A
P
r
t
m
amount accumulated, sometimes denoted FV for "future value"
initial principal, sometimes denoted PV for "present value"
nominal annual rate of interest
number of years
number of compounding periods per year
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Use the compound interest formula below to find the amount accumulated for each of the following.
1. $5000 is invested for 20 years at an annual interest rate of 7%, compounded semi­annually.
2. $400 is invested for 3 years at an annual interest rate of 5.5%, compounded monthly.
3. $10,000 is invested for 8 years at an annual interest rate of 2.5%, compounded weekly.
4. $7500 is invested for 50 years at an annual interest rate of 4.25%, compounded annually.
To find the amount of interest earned, subtract the principal P, from the amount accumulated, A. I = A ­ P
Determine the amount of interest earned in problems 1­ 4 above.
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Suppose you invest $1000 at a rate of 10% annual interest, compounded quarterly. How much have you accumulated after 1 year?
How much total interest was earned during the first year?
What percent of $1000 is this?
This shows that when interest is compounded, there is a difference between the nominal interest rate and the effective rate. These two terms are defined below.
A nominal rate is any stated rate of interest for a specified length of time. By itself, a nominal rate does not indicate or take into account whether or how often interest is compounded.
The effective rate is the rate of simple interest that would produce exactly the same amount of interest over the same length of time. For a year, the effective rate is called the annual percentage yield (APY).
So in our example, the nominal rate was 10%, but the effective rate (APY) was 10.381%.
The APY is always as large or larger
than the nominal rate.
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For an interest rate r, a principal of $1 grows to (1 + r/m)m in 1 year, so the interest earned on that dollar in one year is the APY (annual percentage yield), which is the total amount minus the original $1. Hence the formula for APY is given as follows:
where APY = annual percentage yield (effective annual rate)
r = nominal interest rate
m = number of compounding periods per year
Examples:
5. Determine the APY if the nominal interest rate is 4.5%, compounded weekly.
6. Determine the APY if the nominal interest rate is 3.2%, compounded quarterly.
7. Determine the APY if the nominal interest rate is 1.82%, compounded daily.
8. Determine the APY if the nominal interest rate is 5%, compounded monthly.
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The amount added each compounding period is proportional to the amount present at the time of compounding; this type of growth is called geometric growth.
Geometric Growth (also called exponential growth) is growth proportional to the amount present.
Contrast this to arithmetic or linear growth, which simply adds the same amount each period.
The table and graph below show the difference in growth for simple (linear) interest and compound (exponential) interest.
the balance is multiplied
by 1.1 each year
$100 is added each year
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The distinction between linear growth and exponential growth is fundamental to the major theory of demographer and economist Thomas Robert Malthus (1766­1834). He claimed that human populations grow geometrically (exponentially) but food supplies grow arithmetically (linearly), so that populations tend to outstrip their ability to feed themselves. Homework: Read Spotlight 21.1 (p. 777), do HW #20 (handout).
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