Chapter 10:Compound Interest, Future Value,and Present Value
Find the future value and compound interestby compounding manually.
Find the future value and compound interestby using a $1.00 future value table.
Find the future value and compound interestusing a formula (optional).
Find the effective interest rate.
Find the interest compounded daily using atable.
Key Terms
Interest period: the amount of time which interest is calculated andadded to the principal.
Compound interest: the total interest that accumulated after morethan one interest period.
Future value, maturity value, compound amount: theaccumulated principal and interest after one or more interest periods.
Period interest rate: the rate for calculating interest for one interestperiod-the annual interest rate is divided by the number of periods peryear.
Find the Future Value and CompoundInterest by Compounding Manually
Dividing the annual interest rate by the annual number ofinterest periods gives us the period interest rate.
12% annual interest rate divided by 2 interest periods yieldsa 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 oneperiod.
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 principalby the sum of 1 and the period interest rate.
2.For each remaining period in turn, find the next end of periodprincipal: multiply by the previous end of period principal bythe 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 threeyears.
The period interest rate is 13% since it is calculatedannually.
First end-of-year = $800 x 1.13 = $904
Second end-of-year = $904 x 1.13 = $1021.52
Third 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 tothe future value – original principal.
CI = $1,154.32 - $800 = $354.32
The compound interest = $354.32
Compare the compound interest amountto the simple interest
CI = $354.32
Simple interest for the same loan would be:
I = PRT
I = $800 x 0.13 x 3 = $312.00
Simple interest would be $312.00
The difference between compound interest and simpleinterest for this loan = $354.32 - $312.00 = $43.32
Find the FV of an investment
Principal = $10,000
8% annual interest rate, compounded semi-annually
Find the FV at the end of three years.
Find the period interest rate: 8% ÷ 2 = 4%
Determine number of periods: 3 x 2 = 6
Calculate each end-of-period principal.
Period 1 = 10,000 x 1.04 = $10,400
Find the FV of an investment
Second 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 tocalculate a large number of periods with the previousmethod, we can use Table 10-1, which is the futurevalue or compound amount of $1.00.
Find the number of periods and the rate perperiod to identify the value by which the principal ismultiplied.
Calculating Compound Amount byTable Lookup
Step 1. Find the periods: Years multiplied by numberof times interest is compounded in 1 year.
Step 2. Find the rate: Annual rate divided by numberof times interest is compounded in 1 year.
Step 3. Go down the period column of the table tothe number desired; look across the row to find therate. At the intersection is the table factor.
Step 4. Multiply the table factor by the amount of theloan.
Look at this example
Using Table 10-1, find the compound interest on $500 for six yearscompounded annually at 8%.
Determine the number of periods: 6
Determine the interest rate per period: 8%
Locate the value in the intersecting cell: 1.58687
Multiply the principal, $500, x 1.58687 = $793.44
The FV of the loan is $793.44.
Compound interest = $793.44- $500 = $293.44
Table 10-1
Future Value or Compound Amount of $1.00
Try this example
Using Table 10-1, find the future value and compoundinterest on $2,000 invested for four years compoundedsemiannually 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 actuallyforegoing 10¢ per year for the rest of your life (assuming a10% interest rate). That annual dime of savings builds tomuch more because of interest that is earned on theinterest!
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An amount of $1 at 10% interest
Year1 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 InterestUsing a Formula (optional)
The future value formula is:
FV =
where FV is the future value, P is the principal, R is theperiod interest rate, and N is the number of periods.
The formula for finding future value will require acalculator that has a power function.
Try this example
Find 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
Find the Effective Interest Rate
Effective interest rate is also called the annual percentageyield or APY when identifying rate of earning on aninvestment.
It is called APR, annual percentage rate, when identifyingthe rate of interest on a loan.
Effective rate: the equivalent simple interest rate that isequivalent to a compound rate
Look at this example
Marcia borrowed $600 at 10% compoundedsemiannually. What is the effective interest rate?
Using the manual compound interest method:
Period rate interest = 10% / 2 = 5% = 0.05
First end-of-period principal = $600 x 1.05 = $630
Second end-of-principal = $630 x 1.05 =$661.50
Compound interest after first year = $61.50
Effective interest rate
Annual 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 by100%. The effective rate is 10.25%
Find the Interest Compounded DailyUsing a Table
Table 10-2 gives compound interest for $100compounded daily (using 365 days as a year.)
Pay attention to the table value given. Table10-2 uses $100 as the principal amount; othertables may use $1, $10 or other amounts.
Using Table 10-2 is exactly like using Table 8-2 whichgives the simple interest on $100.
Look at this example
Find the interest on $800 at 7.5% annually, compoundeddaily 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 ofdays (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 a6% annual rate.
Answer:Divide $1,000 ÷ 100 = 10Locate the cell where 30 days and 6% intersect to determinethe 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 annualcompounding for one year.
Find the present value using a $1.00present value table.
Find the present value using a formula(optional).
Find the Present Value Based onAnnual Compounding for One Year
Suppose you want to go on a long vacation in a couple ofyears…or pay for your child’s college education. Howmuch money would you have to invest right now to be ableto pay for it?
Using the concepts of compound interest, you candetermine amounts needed now to cover expenses in thefuture.
The amount of money you set aside now is called presentvalue.
Present value
The simplest case would be annual compounding interest forone year:the number of interest periods is 1 and the period interestrate is the annual interest rate.
Principal (present value) = future value
1 + annual interest rate*
* denotes decimal equivalent
Look at this example
•Find the amount of money that The 7th Inning needs to setaside today to ensure that $10,000 will be available to buy anew large screen plasma television in one year if the annualinterest rate is 4% compounded annually.
•PV = 10,000 1.04= $9,615.38
•An investment of $9,615.38 at 4% would have avalue of $10,000 in one year.
Try these examples
Calculate the amount of money needed now to purchase alaptop computer and accessories valued at $2,000 in a yearif you invest the money at 6%.
$1,886.79
John wants to replace a tool valued at $150 in a year. Howmuch money will he have to put into a savings account thatpays 3% annual interest?
$145.63
Use a $1.00Present Value Table
Using a present value table is the mostefficient way to calculate the money needednow for a future expense or investment.
Table 10-3 shows the present value of $1.00at different interest rates for differentperiods.
How to use the table
1.Find the number of interest periods: multiply the timeperiod in years by number of interest periods per year.Interest periods = number of years xnumber of interest periods per year
2.Find the interest rate: divide the annual interest rate bythe number of interest periods per year.Period interest rate = annual interest rateNumber of interest periods per year
Using the table (continued)
3. Select the periods row corresponding to the number ofinterest periods.
4. Select the rate per period column corresponding to theperiod interest rate.
5. Locate the value in the cell where the periods row intersectsthe 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 framingequipment. How much should be invested at 4% interestcompounded 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% compoundedannually for four years to have $35,000.
Table 10-3
Present 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 ona 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 downpayment on a house.
Try these examples
How much money would you have to invest for 3 years at10% paid semi-annually to purchase an automobile thatcosts $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% compoundedsemi-annually for three years to have a $20,000 topurchase a car in three years.
$14,924.40
Find the Present ValueUsing a Formula (optional)
The present value formula is:
PV =
where PV is the present value, FV is the future value, Ris the period interest rate, and N is the number ofperiods.
Try this example
Find the present value required at 5.2% compounded monthlyto total $8,000 in three years.
PV =
Period int. rate = 5.2%/12 = .0043333333
PV = = $6,846.78
Things to Note
An increase in the interest rate causes present valueto fall.
Higher rates of interest mean smaller amounts can grow toequal some fixed amount during a specified period of time.
A decrease in the interest rate causes present value torise.
Lower rates of interest mean larger amounts are needed toreach 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 purchaseland
• How much do we need to invest today in haveenough money to purchase a lot in 1 year?
• Answer the future value and present valueinheritance 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 10years
• Figure finance charges.
• Determine how long it will take to double yourmoney
• Determine the interest rate to double your moneyin 10 years
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