Chapter 10: Compound Interest, Future Value, and Present Value Find the future value and compound

interest by compounding manually.

Find the future value and compound interest by using a $1.00 future value table.

Find the future value and compound interest using a formula (optional).

Find the effective interest rate.

Find the interest compounded daily using a table.

Key TermsInterest period: the amount of time which

interest is calculated and added to the principal.

Compound interest: the total interest that accumulated after more than one interest period.

Future value, maturity value, compound amount: the accumulated principal and interest after one or more interest periods.

Period interest rate: the rate for calculating interest for one interest period-the annual interest rate is divided by the number of periods per year.

Find the Future Value and Compound Interest by Compounding Manually

Dividing the annual interest rate by the annual number of interest periods gives us the period interest rate.

12% annual interest rate divided by 2 interest periods yields a period interest rate of 6%, for example.

Using I = P x R x T, we can calculate the interest per period, simplifying the formula to I = P x R, since T is one period.

Find the period interest rate

Period interest rate =

Annual interest rate Number of interest periods per

year

Find the period interest rate for:

A 12% annual interest rate with 4 interest periods per year.

Annual interest rate

Number of interest periods per year

12/4 = 3%

An 18% annual rate with 12 interest periods per year.

18/12 = 1 ½ %

An 8% annual rate with 4 interest periods per year.

8/4 = 2%

Find the future value Using the simple interest formula method:

1. Find the end of period principal: multiply the original principal by the sum of 1 and the period interest rate.

2. For each remaining period in turn, find the next end of period principal: multiply by the previous end of period principal by the sum of 1 and the period interest rate.

3. Identify the last end-of-period principal as the future value.

Look at this example

Find the future value of a loan of $800 at 13% for three years.

The period interest rate is 13% since it is calculated annually.

First end-of-year = $800 x 1.13 = $904Second end-of-year = $904 x 1.13 = $1021.52Third end-of-year = $1021.52 x 1.13 =

$1,154.32

The FV of this loan is $1,154.32

Find the compound interest

Compound interest =

future value – original principal

In the previous example, the compound interest is equal to the future value – original principal.

CI = $1,154.32 - $800 = $354.32

The compound interest = $354.32

Compare the compound interest amount to the simple interest

CI = $354.32Simple interest for the same loan would

be:I = PRTI = $800 x 0.13 x 3 = $312.00Simple interest would be $312.00The difference between compound

interest and simple interest for this loan = $354.32 - $312.00 = $43.32

Find the FV of an investment

Principal = $10,0008% annual interest rate, compounded

semi-annuallyFind the FV at the end of three years.Find the period interest rate: 8% ÷ 2 =

4%Determine number of periods: 3 x 2 = 6Calculate each end-of-period principal.Period 1 = 10,000 x 1.04 = $10,400

Find the FV of an investmentSecond end-of-period principal =

$10,400 x 1.04 = $10,816

Calculate each end-of-principal through the sixth end-of-period principal.

What is the final end-of-principal amount?

$12,653.19

Using a $1.00 FV Table

Since it would be tedious and time-consuming to calculate a large number of periods with the previous method, we can use Table 10-1, which is the future value or compound amount of $1.00.

Find the number of periods and the rate per period to identify the value by which the principal is multiplied.

Calculating Compound Amount by Table Lookup

Step 1. Find the periods: Years multiplied by number of times interest is compounded in 1 year.

Step 2. Find the rate: Annual rate divided by number of times interest is compounded in 1 year.

Step 3. Go down the period column of the table to the number desired; look across the row to find the rate. At the intersection is the table factor.

Step 4. Multiply the table factor by the amount of the loan.

Look at this example

Using Table 10-1, find the compound interest on $500 for six years compounded annually at 8%.

Determine the number of periods: 6Determine the interest rate per period: 8%Locate the value in the intersecting cell: 1.58687Multiply the principal, $500, x 1.58687 = $793.44The FV of the loan is $793.44.Compound interest = $793.44- $500 =

$293.44

Table 10-1Future Value or Compound Amount of $1.00

Try this example

Using Table 10-1, find the future value and compound interest on $2,000 invested for four years compounded semiannually at 8%.

FV = $2,737.14 CI = $737.14

What would the simple interest be for the same loan?

$640

When you spend a dollar on a soft drink, you are actually foregoing 10¢ per year for the rest of your life (assuming a 10% interest rate). That annual dime of savings builds to much more because of interest that is earned on the interest!

An amount of $1 at 10% interestYear 1 2 3 n

$1.10 $1.21 $1.33 $1(1 + r)n

Formula: FV = P(1 + r) n

Find the Future Value and Compound Interest Using a Formula (optional)

The future value formula is:

FV =

where FV is the future value, P is the principal, R is the period interest rate, and N is the number of periods.

The formula for finding future value will require a calculator that has a power function.

(1 )NP R

Try this exampleFind the future value and compound interest

of a 3-year $5,000 investment that earns 6% compounded monthly.

R = 6%/12 or .06/12 = 0.005

N = 3(12) = 36

FV =

FV =

FV = $5,983.40

CI = $5,983.40 – $5,000 = $983.40

(1 )NP R365,000(1 .005)

Find the Effective Interest Rate

Effective interest rate is also called the annual percentage yield or APY when identifying rate of earning on an investment.

It is called APR, annual percentage rate, when identifying the rate of interest on a loan.

Effective rate: the equivalent simple interest rate that is equivalent to a compound rate

Look at this example

Marcia borrowed $600 at 10% compounded semiannually. What is the effective interest rate?

Using the manual compound interest method:

Period rate interest = 10% / 2 = 5% = 0.05First end-of-period principal = $600 x 1.05 =

$630Second end-of-principal = $630 x 1.05

=$661.50Compound interest after first year =

$61.50

Effective interest rateAnnual effective interest rate =

$61.50 $600

Multiplied by 100% = 0.1025 x 100%

= 10.25%Using the table method (Table 10-1):

The table value is 1.10250. Subtract 1.00 and multiply by 100%. The effective

rate is 10.25%

Find the Interest Compounded Daily Using a Table

Table 10-2 gives compound interest for $100 compounded daily (using 365 days as a year.)

Pay attention to the table value given. Table 10-2 uses $100 as the principal amount; other tables may use $1, $10 or other amounts.

Using Table 10-2 is exactly like using Table 8-2 which gives the simple interest on $100.

Look at this exampleFind the interest on $800 at 7.5% annually,

compounded daily for 28 days.

Divide the principal by $100 as you are using Table 10-2. [$800 ÷ 100 = 8]

Find the corresponding value by intersecting the number of days (28) and annual interest rate (7.5%) = 0.5769413

Multiply this value by 8 = $4.62

The compounded interest is $4.62

Table 10-2 Compound Interest on $100, Compounded Daily (365 Days) (Exact Time, Exact Interest Basis)—Continued

Try this example

Find the interest on $1,000 for 30 days compounded at a 6% annual rate.

Answer:Divide $1,000 ÷ 100 = 10Locate the cell where 30 days and 6% intersect to determine the value: 0.4943279Multiply by 10.The interest is $4.94

Table 10-2 Compound Interest on $100, Compounded Daily (365 Days) (Exact Time, Exact Interest Basis)—Continued

Present Value

Find the present value based on annual compounding for one year.

Find the present value using a $1.00 present value table.

Find the present value using a formula (optional).

Find the Present Value Based on Annual Compounding for

One Year

Suppose you want to go on a long vacation in a couple of years…or pay for your child’s college education. How much money would you have to invest right now to be able to pay for it?

Using the concepts of compound interest, you can determine amounts needed now to cover expenses in the future.

The amount of money you set aside now is called present value.

Present value

The simplest case would be annual compounding interest for one year: the number of interest periods is 1 and the period interest rate is the annual interest rate.

Principal (present value) = future value1 + annual interest rate*

* denotes decimal equivalent

Look at this example• Find the amount of money that The 7th Inning

needs to set aside today to ensure that $10,000 will be available to buy a new large screen plasma television in one year if the annual interest rate is 4% compounded annually.

• PV = 10,000 1.04 = $9,615.38

• An investment of $9,615.38 at 4% would have a value of $10,000 in one year.

Try these examplesCalculate the amount of money needed

now to purchase a laptop computer and accessories valued at $2,000 in a year if you invest the money at 6%.

$1,886.79

John wants to replace a tool valued at $150 in a year. How much money will he have to put into a savings account that pays 3% annual interest?

$145.63

Use a $1.00 Present Value Table

Using a present value table is the most efficient way to calculate the money needed now for a future expense or investment.

Table 10-3 shows the present value of $1.00 at different interest rates for different periods.

How to use the table

1. Find the number of interest periods: multiply the time period in years by number of interest periods per year.Interest periods = number of years x

number of interest periods per year

2. Find the interest rate: divide the annual interest rate by the number of interest periods per year.Period interest rate = annual interest rate

Number of interest periods per year

Using the table (continued)

3. Select the periods row corresponding to the number of interest periods.

4. Select the rate per period column corresponding to the period interest rate.

5. Locate the value in the cell where the periods row intersects the rate-per-period column.

6. Multiply the future value by value from step 5.

Look at this example

The 7th Inning needs $35,000 in 4 years to buy new framing equipment. How much should be invested at 4% interest compounded annually?

4 periods at 4% shows a value of 0.85480

Multiply this value by $35,000

The result is $29,918

They must invest $29,918 at 4% compounded annually for four years to have $35,000.

Table 10-3Present Value of $1.00

Try these examples

How much money would you have to invest for 5 years at 6% paid semi-annually to make a down payment of $20,000 on a house?

R = 6%/2 (semi-annually) = 3%

N = 5 x 2 (5years semi-annually) = 10 periods

PV = 20,000 x .74409 = $14,881.80

They must invest $14,881.80 at 6% compounded semi-annually for five years to have a $20,000 down payment on a house.

Try these examplesHow much money would you have to invest for 3

years at 10% paid semi-annually to purchase an automobile that costs $20,000?

R = 10%/2 (semi-annually) = 5%

N = 3 x 2 (3 years semi-annually) = 6 periods

PV = 20,000 x .74622 = $14,924.40

They must invest $14,924.40 at 10% compounded semi-annually for three years to have a $20,000 to purchase a car in three years.

$14,924.40

Find the Present Value Using a Formula (optional)

The present value formula is:

PV =

where PV is the present value, FV is the future value, R is the period interest rate, and N is the number of periods.

(1 )NFV

R

Try this example

Find the present value required at 5.2% compounded monthly to total $8,000 in three years.

PV =

Period int. rate = 5.2%/12 = .0043333333

PV = = $6,846.78

36

8,000

(1.0043333333)

(1 )NFV

R

Things to Note

An increase in the interest rate causes present value to fall.Higher rates of interest mean smaller

amounts can grow to equal some fixed amount during a specified period of time.

A decrease in the interest rate causes present value to rise.Lower rates of interest mean larger amounts

are needed to reach some fixed amount during a specified period of time.

Assignment Checklist BU250 - Unit 7: 

Challenge Problem •    Should the land be purchased today or

in 2 years? •    Problems and advantages of waiting to

purchase land •    How much do we need to invest today

in have enough money to purchase a lot in 1 year?

•    Answer the future value and present value inheritance questions 

Case Study 10-1: How Fast does your money grow?:  

•    Find the future value of an investment•    Figure the value of the investment if

delayed 10 years •    Figure finance charges. •    Determine how long it will take to

double your money•    Determine the interest rate to double

your money in 10 years