Chapter 5

Section 5.2

Compound Interest

Compound Interest

Many interest investment will pay a portion of the interest that has been earned of certain regular fixed time periods. This type of interest is called compound interest and the time period it is paid over is called a compounding period. There is a definite advantage to this type of investment in that it allows the interest to begin earning interest.

There are many different ways to name compounding periods. The chart below shows several of the most common. Keep in mind the prefix "bi" means 2 and the prefix "semi" means ½. What you want to be able to find is the number of compounding periods in 1 year. (Some of the most common ones are in red.)

Compouding

Period

Number of Periods in 1 year

annually 1

semiannually 2

quarterly 4

bimonthly 6

monthly 12

Compouding

Period

Number of Periods in 1 year

biweekly 26

weekly 52

semiweekly 104

daily 365

semidaily 730

Periodic Rate

The importance of knowing how many compounding periods are in a year is to be able to compute the periodic rate (i) for the investment which is the yearly rate (r) divided by the number of periods in 1 year.

yearinperiodsofnumber

rateyearlyrateperiodic

Investment Periods

The term of a compound interest investment is not measured in years, but is measured by the number of compounding periods over the duration of the investment. The variable n stands for the number of periods in the entire investment. For example, if it compounded weekly then n is the number of weeks it is invested, if monthly then n is the number of months, if semiannually then n is the number of 6 month intervals etc.

Compound Interest Future Value

If a principal P is invested in a compound interest investment with a periodic rate i for a duration of n compounding periods the future value FV is given by:

niPFV 1

It is a matter of algebra to see how you get this formula. Recall the future value formula for a simple interest investment in 1 period. Notice that i = rt, since t will be the number of periods in 1 year.

In the 1st period we have : iPFV 1

In the 2nd period we have : 2111 iPiiPFV In the 3rd period we have : 32 111 iPiiPFV

In the nth period we have : niPFV 1Example

A grandfather invests $10,000 in a trust fund for his grandson's college education when he is born. The fund pays 4% compounded monthly. How much will he have for college on his 18th birthday.

The questions wants to know the future value (FV) of this investment.

P = 10,000 12

04.i

2161812 n

75.519,20

1000,10

1216

1204.

niPFV

Present Value

The present value (P or PV) of an investment is the amount of money need now to have a certain future value. A formula for this can be found by using some algebra to solve for P in the compound interest formula.

niFV

P

1

Example

A person wants to invest a certain amount of money now in an account that pays 6.4% compounded quarterly so that they have $30,000 in 5 years. How much money should they invest?

The question wants to know the present value for the investment.

FV = 30,0004

064.i

2054 n

72.839,21

1

000,30

1

20

4064.

ni

FVP

Annual Yield

The annual yield (or yield) of a compound interest investment is the interest that is earned on $1 for a term of 1 year. If the compounding period is annual it is exactly the same as the rate (r). The purpose of the annual yield is to provide an even (fair) comparison of different terms of compound interest investments to determine which is best.The formula for the annual yield is obtained by computing the future value of the investment for 1 year and subtracting a $1. The value of n is the number of periods in a year.

11YieldAnnual

11YieldAnnual

n

nr

ni

Find the annual yield for an investment of 2.56% compounded semiannually.

n = 2 and r = .0256

0257638.

11YieldAnnual 2

20256.

Find the annual yield for an investment of 2.55% compounded daily.

n = 365 and r = .0255

025827.

11YieldAnnual 365

3650255.

Which of the two investments above is better?

The investment that is 2.55% compounded daily is better even though it has a smaller rate! In 1 year it will earn more for each dollar invested.