1. Compound Interest, Future Value, and Present Value
CHAPTER 13
Compound Interest, Future Value, and Present Value

2. Find the future value and compound interest by compounding manually.
13-1
Learning Outcomes
Find the future value and compound interest by compounding manually.
Find the future value and compound interest using a $1.00 future value table.
Find the future value and compound interest using a formula or a calculator application (optional).
Find the effective interest rate.
Find the interest compounded daily using a table.

3. Interest
Section 13-1
Compound Interest and Future Value
In some loans, interest is computed once during the life of the loan, using the simple interest formula.
In other loans, interest is computed more than once during the life of the loan or investment.
The interest is added to the principal, the new amount becoming the principal for the next interest calculation.
This process is called compounding interest.

4. Interest period Compound interest Key Terms…
Section 13-1
Compound Interest and Future Value
Interest period
The amount of time after which interest is calculated and added to the principal.
Compound interest
The total interest that has accumulated after more than one interest period.

5. Future value, maturity value, compound amount
Key Terms…
Section 13-1
Compound Interest and Future Value
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 divided by the number of interest periods per year.

6. Using I = P x R x T, we can calculate the interest per period.
Find the Future Value and Compound Interest by Compounding Manually
13-1-1
Section 13-1
Compound Interest and Future Value
Dividing the annual interest rate by the annual number of interest periods gives us the period interest rate.
For example, 12% annual interest rate divided by 2 interest periods yields a period interest rate of 6%.
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.

7. Find the period interest rate
HOW TO:
Find the period interest rate
Section 13-1
Compound Interest and Future Value
Period interest rate =

8. A 12% annual interest rate with 4 interest periods per year.
Examples…
Section 13-1
Compound Interest and Future Value
A 12% annual interest rate with 4 interest periods per year. 3%
An 18% annual rate with 12 interest periods per year.
1 ½ %
An 8% annual rate with 4 interest periods per year. 2%

9. Using the simple interest formula method:
HOW TO:
Find the future value
Section 13-1
Compound Interest and Future Value
Using the simple interest formula method:
STEP 1
Find the end of period principal—multiply the original principal by the sum of 1 and the period interest rate.
STEP 2
For each remaining period in turn, find the next end of period principal: multiply the previous end of period principal by the sum of 1 and the period interest rate.
STEP 3
Identify the last end-of-period principal as the future value.

10. Find the future value of a loan of $800 at 13% for three years.
An Example…
Section 13-1
Compound Interest and Future Value
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 = $904
Second end-of-year =$904 x 1.13 = $
Third end-of-year = $ x 1.13 = $1,154.32
The FV of this loan is $1,

11. Find the compound interest
HOW TO:
Find the compound interest
Section 13-1
Compound Interest and Future Value
Compound interest = future value – original principal
First end-of-year = $800 x 1.13 = $904
Second end-of-year =$904 x 1.13 = $
Third end-of-year = $ x 1.13 = $1,154.32
The FV of this loan is $1,
In this example, the compound interest is equal to:
CI = $1, – $800 = $354.32
The compound interest is $

12. Compare the compound interest amount to the simple interest HOW TO:
Section 13-1
Compound Interest and Future Value
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 simple interest for this loan = $ – $312
The difference is $43.32.

13. 8% annual interest rate, compounded semi-annually
HOW TO:
Find the FV of an investment
Section 13-1
Compound Interest and Future Value
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

14. Find the FV of an investment
HOW TO:
Find the FV of an investment
Section 13-1
Compound Interest and Future Value
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
Period 2 = $10,400 x 1.04 = $10,816
Calculate through sixth end-of-period principal.
Final end-of-principal amount = $12,653.19

15. Find the future value and compound interest using a $1
Find the future value and compound interest using a $1.00 future value table.
13-1-2
Section 13-1
Compound Interest and Future Value
It would be tedious and time-consuming to calculate a large number of periods with the previous method.
Use Table 13-1, 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.
See page 461

16. Determine the number of periods: 6
Find the future value and compound interest using a $1.00 future value table.
HOW TO:
Section 13-1
Compound Interest and Future Value
Using Table 13-1, find the compound interest on $500 for six years compounded annually at 8%.
Determine the number of periods: 6
Determine the interest rate per period: 8%
Locate the value in the intersecting cell:
Multiply the principal: $500 x = $793.44
The FV of the loan is $
Compound interest = $ – $500 = $293.44

17. What would the simple interest be for the same loan?
An Example…
Section 13-1
Compound Interest and Future Value
Using Table 13-1, 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

18. The future value formula is:
Find the Future Value and Compound Interest Using a Formula (optional)
13-1-3
Section 13-1
Compound Interest and Future Value
The formula for finding future value will require a calculator that has a power function.
The future value formula is:
FV = P(1 + R)N
…where FV is the future value, P is the principal, R is the period interest rate, and N is the number of periods.

19. FV = P(1 + R)N FV = 5,000(1 + 0.005)36 An Example… FV = $5,983.40
Section 13-1
Compound Interest and Future Value
Find the future value and compound interest for a 3-year $5,000 investment that earns 6% compounded monthly.
FV = P(1 + R)N
FV = 5,000( )36
FV = $5,983.40
CI = $5, – $5,000 = $983.40

20. Same Example, using a calculator…
Section 13-1
Compound Interest and Future Value
Scientific:
BA II Plus:

21. Same Example, using a calculator…
Section 13-1
Compound Interest and Future Value
TI-84
Highlight END

22. 13-1-4
Find the effective interest rate
Section 13-1
Compound Interest and Future Value
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.

23. Effective rate Key Terms…
Section 13-1
Compound Interest and Future Value
Effective rate
The equivalent simple interest rate that is equivalent to a compound rate.

24. Marcia borrowed $600 at 10% compounded semiannually.
An Example…
Section 13-1
Compound Interest and Future Value
Marcia borrowed $600 at 10% compounded semiannually.
What is the effective interest rate?
Using the manual compound interest method:
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
Period rate interest = = 5% = 0.05

25. Effective interest rate
HOW TO:
Effective interest rate
Section 13-1
Compound Interest and Future Value
Annual effective interest rate =
Multiplied by 100%: x 100% = 10.25%
Using the table method (Table 13-1)
The table value is
Subtract 1.00 and multiply by 100%.
The effective rate is 10.25%.

26. Find the Interest Compounded Daily Using a Table
13-1-5
Section 13-1
Compound Interest and Future Value
Table 13-2 gives compound interest for $100 compounded daily (using 365 days as a year).
Exactly like using Table 11-2, which gives the simple interest on $100.
Table 13-2 uses $100 as the principal amount; other tables may use $1, $10 or other amounts.
See page 469

27. The compounded interest is $4.62.
An Example…
Section 13-1
Compound Interest and Future Value
Find the interest on $800 at 7.5% annually, compounded daily for 28 days.
Find the corresponding value by intersecting the number of days (28) and annual interest rate (7.5%): value =
Multiply this value by 8 to get $4.62.
The compounded interest is $4.62.
Divide the principal by $100 as you are using Table ($800 ÷ $100 = 8)
See page 469

28. An Example…
Section 13-1
Compound Interest and Future Value
Find the interest on $1,000 for 30 days compounded at a 6% annual rate.
Divide $1,000 ÷ 100 = 10 Locate the cell where 30 days and 6% intersect to determine the value:
Multiply by 10.
The interest is $4.94.
See page 466

29. Find the present value based on annual compounding for one year.
13-2
Learning Outcomes
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 or a calculator application (optional).

30. Find the Present Value Based on Annual Compounding for One Year
13-2-1
Section 13-2
Present Value
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.

31. The simplest case is annual compounding interest for one year.
HOW TO:
Present value
Section 13-2
Present Value
The simplest case is 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) =
* denotes decimal equivalent

32. An Example…
Section 13-2
Present Value
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.
An investment of $9, at 4% would have a value of $10,000 in one year.
PV =

33. Examples…
Section 13-2
Present Value
Calculate 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 an account that pays 3% annual interest?
$145.63

34. 13-2-2
Find Present Value Using a $1.00 PV Table
Section 13-2
Present Value
Using a present value table is the most efficient way to calculate the money needed now for a future expense or investment.
Table 13-3 shows the present value of $1.00 at different interest rates for different periods.
See page 475

35. STEP 1 STEP 2 HOW TO: Use table 13-3
Section 13-2
Present Value
STEP 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
STEP 2
Find the interest rate—divide the annual interest rate by the number of interest periods per year.
Period interest rate =

36. STEP 3 STEP 4 STEP 5 STEP 6 HOW TO: Use table 13-3
Section 13-2
Present Value
STEP 3
Select the periods row corresponding to the number of interest periods.
STEP 4
Select the rate per period column corresponding to the period interest rate.
STEP 5
Locate the value in the cell where the periods row intersects the rate-per-period column.
STEP 6
Multiply the future value by value from step 5.

37. How much should be invested at 4% interest compounded annually?
An Example…
Section 13-2
Present Value
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
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.

38. Examples…
Section 13-2
Present Value
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?
$14,881.80
How much money would you have to invest for 3 years at 10% paid semi-annually to purchase an automobile that costs $20,000?
$14,924.40

39. The present value formula:
Find the Present Value Using a Formula (optional)
13-2-3
Section 13-2
Present Value
The present value formula:
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.

40. Examples…
Section 13-2
Present Value
Find the present value required at 5.2% compounded monthly to total $8,000 in three years.
PV =
Period int. rate = =
PV = = $6,846.78

41. The same example using calculator keystrokes
Section 13-2
Present Value
Scientific:
BA II Plus:

42. Highlight END The same example using calculator keystrokes TI-84:
Section 13-2
Present Value
TI-84:
Highlight END

43. Exercises Set A

44. Rate of compound interest
EXERCISE SET A 2.
Use Table 13-1 or the appropriate formula for the Exercise.
Number of periods = 4(4) = 16
Period rate = 4%/4 = 1%
Table value =
Compound amount = 5,000( ) = 5,862.90
Compound interest = 5, ,000 =
Principal
Term (years)
Rate of compound interest
Compounded
Compound amount
Compound interest
$5,000 4 4%
quarterly

45. Rate of compound interest
EXERCISE SET A 4.
Use Table 13-1 or the appropriate formula for the Exercise.
Number of periods = 4(2) = 8
Period rate = 1%/2 = 0.5%
Table value =
Compound amount = 8,000( ) = 8,325.68
Compound interest = 8, ,000 =
Principal
Term (years)
Rate of compound interest
Compounded
Compound amount
Compound interest
$8,000 4 1%
semiannually

46. Investment time (years)
EXERCISE SET A 6.
Find the amount that should be set aside today to yield the desired future amount. Use Table 13-3 or the present value formula.
Number of periods = 6(4) = 24 periods
Period rate = 6%/4 = 1.5%
Table value =
Present value = 8,000( ) = 5,596.32
Future amount needed
Interest rate
Compounded
Investment time (years)
$8,000 6%
quarterly 6

47. Investment time (years)
EXERCISE SET A 8.
Find the amount that should be set aside today to yield the desired future amount. Use Table 13-3 or the present value formula.
Number of periods = 20(1) = 20
Period rate = 3%/1 = 3%
Table value =
Present value = 14,700( ) = 8,139.10
Future amount needed
Interest rate
Compounded
Investment time (years)
$14,700 3%
annually 20

48. EXERCISE SET A
10.
Manually calculate the compound interest on a 13% loan of $1,600 for three years if the interest is compounded annually.
1,600(1.13) = 1,808
1,808(1.13) = 2,043.04
2,043.04(1.13) = 2, compound amount
2, ,600 = interest

49. 5 years(2 periods per year) = 10 periods
EXERCISE SET A
12.
Use Table 13-1 or the appropriate formula to find the interest on a certificate of deposit (CD) of $10,000 for five years at 4% compounded semiannually.
5 years(2 periods per year) = 10 periods
4%  2 periods per year = 2% per period
Table value =
10,000( ) = 12, compound amount
12, ,000 = 2,189.90

50. EXERCISE SET A
14.
Find the compound interest on a loan of $5,000 for two years if the interest is compounded quarterly at 12%.
2(4) = 8 periods
12%  4 = 3% per period
Table value =
5,000( ) = 6, compound amount
Compound interest = 6, ,000 = 1,333.85

51. 2(2) = 4 periods, 2% , 2 = 1% per period
EXERCISE SET A
Lauren McAnally invests $2,000 at 2% compounded semiannually for two years, and Inez Everett invests an equal amount at 2% compounded quarterly for 18 months. Use Table 13-1 to determine which investment yields the greater interest.
2(2) = 4 periods, 2% , 2 = 1% per period
2,000( ) = 2, compound amount
2, ,000 = compound interest
compounded semiannually for two years
(continued next slide)
16.

52. EXERCISE SET A
Lauren McAnally invests $2,000 at 2% compounded semiannually for two years, and Inez Everett invests an equal amount at 2% compounded quarterly for 18 months. Use Table 13-1 to determine which investment yields the greater interest.
18 months  3 months per quarter = 6 quarters or periods
2%  4 = 0.5% per period
2,000( ) = 2, future value
2, ,000 = compound interest compounded
quarterly for 18 months (6 quarters)
The two-year investment yields the greater amount of interest.
16.

53. EXERCISE SET A
18.
Use Table 13-2 to find the amount of interest on $100 invested for 10 days at 8.5% compounded daily.
Table value =
compound interest = $0.23.

54. EXERCISE SET A
20.
Find the amount of money that Keaton and Jana Smith must set aside today so that they will have $5,000 available to buy a home security system in one year if the annual interest rate is 5% compounded annually.
= 1.05
$5,000  1.05 = $4,761.90

55. $1,000 in seven years at 8% compounded quarterly
EXERCISE SET A
22.
Find the amount of money that should be invested (present value) at the stated interest rate to yield the given amount (future value) after the indicated amount of time. Use Table 13-3 or the appropriate formula.
$1,000 in seven years at 8% compounded quarterly
7(4) = 28 periods, 8%  4 = 2% per period,
Table value =
$1.000( ) = $574.37

56. $500 in 15 years at 4% annual interest compounded semiannually
EXERCISE SET A
24.
Find the amount of money that should be invested (present value) at the stated interest rate to yield the given amount (future value) after the indicated amount of time. Use Table 13-3 or the appropriate formula.
$500 in 15 years at 4% annual interest compounded semiannually
15(2) = 30 periods, 4%  2 = 2% per period,
Table value =
$500( ) = $276.04

57. 4(4) = 16 periods, 6%  4 = 1.5% per period, Table value = 0.78803
EXERCISE SET A
26.
Find the amount of money that should be invested (present value) at the stated interest rate to yield the given amount (future value) after the indicated amount of time. Use Table 13-3 or the appropriate formula.
Myrna Lewis wishes to have $4,000 in four years to tour Europe. How much must she invest today at 6% annual interest compounded quarterly to have $4,000 in four years?
4(4) = 16 periods, 6%  4 = 1.5% per period,
Table value =
$4,000( ) = $3,152.12

58. Practice Test

59. 3,598.39(1.0625) = 3,823.29 (4th year) compound amount
PRACTICE TEST 2.
Manually calculate the compound interest on a 6.25% annual interest loan of $3,000 for four years if interest is compounded annually.
3,000(1.0625) = 3, (1st year)
3,187.50(1.0625) = 3, (2nd year)
3,386.72(1.0625) = 3, (3rd year)
3,598.39(1.0625) = 3, (4th year) compound amount
3,  3,000 = compound interest

60. 7(4) = 28 periods, 6%  4 = 1.5% per period, Table value = 1.51722
PRACTICE TEST 4.
Use Table 13-1 or the appropriate formula to find the future value on an investment of $12,000 for seven years at 6% annual interest compounded quarterly.
7(4) = 28 periods, 6%  4 = 1.5% per period,
Table value =
12,000( ) = 18,206.64

61. PRACTICE TEST 6.
Use Table 13-1 to find the compound interest on a loan of $3,000 for one year at 12% annual interest if the interest is compounded quarterly.
4 periods, 3% per period,
Table value =
3,000( ) = 3, compound amount
3, ,000 = compound interest

62. PRACTICE TEST 8.
Use Table 13-2 to find the interest compounded on an investment of $2,000 invested at 5.75% for 28 days compounded daily.

63. $3,400 in four years at 4% annual interest compounded annually
PRACTICE TEST
10.
Find the amount that should be invested today (present value) at the stated interest rate to yield the given amount (future value) after the indicated amount of time.
$3,400 in four years at 4% annual interest compounded annually
4 periods, 4% per period,
Table value =
3,400( ) = 2,906.32

64. $8,000 in 12 years at 5% annual interest compounded annually
PRACTICE TEST
12.
Find the amount that should be invested today (present value) at the stated interest rate to yield the given amount (future value) after the indicated amount of time.
$8,000 in 12 years at 5% annual interest compounded annually
12 periods, 5% per period,
Table value =
8,000( ) = 4,454.72

65. 10(2) = 20 periods, 2%  2 = 1% per period, Table value = 0.81954
PRACTICE TEST
14.
Jamie Juarez needs $12,000 in 10 years for her daughter’s college education. How much must be invested today at 2% annual interest compounded semiannually to have the needed funds?
10(2) = 20 periods, 2%  2 = 1% per period,
Table value =
12,000( ) = 9,834.48

66. the required amount in four years?
PRACTICE TEST
16.
Derek Anderson plans to buy a house in four years. He will make an $8,000 down payment on the property. How much should he invest today at 6% annual interest compounded quarterly to have
the required amount in four years?
4(4) = 16 periods, 6%  4 = 1.5 % per period,
Table value =
8,000( ) = 6,304.24

67. 3(4) = 12 periods, 6%  4 = 1.5% per period, Table value = 1.19562
PRACTICE TEST
18.
If you invest $2,000 today at 6% annual interest compounded quarterly, how much will you have after three years? ( Table 13-1 or appropriate formula or calculator application)
3(4) = 12 periods, 6%  4 = 1.5% per period,
Table value =
2,000( ) = 2,391.24

68. PRACTICE TEST
20.
How much money should Bryan Trailer Sales set aside today to have $15,000 in one year to purchase a forklift if the interest rate is 2.95% compounded annually?