The document discusses the time value of money concept. It explains that a dollar today is worth more than a dollar in the future due to factors like interest rates and the ability to earn interest on money over time. It also discusses the difference between future value, which measures the worth of cash flows after time has passed, and present value, which measures the current worth of future cash flows. Formulas are provided for calculating future value, present value, and the value of annuities over time discounted at a given interest rate. Examples are included to demonstrate calculations.

1. The Time Value of
Money
Chapter 6

2. Learning Objectives
• Explain why a dollar today is worth more than a dollar in the
future
• Define the terms future value
• Calculate the future value of an amount and an annuity

3. The Time Value of Money
• Consume Today or Tomorrow?
• TVM is based on the belief that people prefer
to consume goods today rather than wait to
consume the same goods tomorrow
•An apple we can have today is more
valuable to us than an apple we can have
in one year.
•Money has a time value because buying
an apple today is more important than
buying an apple in one year.

4. The Time Value of Money
• Consume Today or Tomorrow?
• A dollar someone has today can be spent for
consumption or loaned to earn interest
• A dollar loaned earns interest that increases
wealth and the ability to consume
• The rate of interest determines the trade-off
between consumption today and saving
(investing)

5. The Time Value of Money
• Future Value versus Present Value
• Cash-flows are evaluated based on future value or present value
• Future value measures what cash-flows are worth after a certain
amount of time has passed
• Present value measures what future cash-flows are worth
before a certain amount of time has passed

6. Comparison of Future Value and
Present Value

7. The Time Value of Money
• Future Value versus Present Value
• Compounding is the process of increasing cash-flows to a future
value
• Discounting is the process of reducing future cash-flows to a
present value

8. Future Value
• What a dollar invested today will be worth in the future depends on
 Length of the investment period
Method to calculate interest
 Interest rate
 2 types of methods to calculate interest
Simple method-calculated only on the original principal each year
Compound interest-calculated on both the original principal and on any
accumulated interest earned up to that point. Future value implies the
compound method
Interest rate
Simple interest is paid on the original principal amount only
Compound interest consists of both simple interest and interest-on-interest

11. Present Value of an Amount to be Received in the
Future
• Taking future values back to the present is called discounting
Present Value=Future Value x Present Value Factor
PV = Present value (initial investment amount)
i = the interest rate
n= no. of time periods of the investment

12. Example 1
• Suppose we place $10000 in a savings account that pays 10%
interest compounded annually. How will our savings grow?
Value at the end of year 1 = present value X (1+i)
i = interest rate
• = $10000 ퟏ + ퟎ. ퟏ ퟒ
• = $10000 (1.4641)
• = $14,641

13. Future Value: Example 2
• If we place $1,000 in a savings account paying 5% interest
compounded annually, how much will our account accrue to in 10
years?
• Future value = present value X (ퟏ + 풓)풏
• 푭푽풏 = $1,000 (ퟏ + ퟎ. ퟎퟓ)ퟏퟎ
• = $1,000 (1.62889)
• = $1,628.89
13
Refer Table B-1

14. Example 3:
• What is the future value of $10,000 with an interest rate of 16
percent and one annual period of compounding? With an
annual interest rate of 16 percent and two semiannual periods
of compounding? With an annual interest rate of 16 percent
and four quarterly periods of compounding?
Annually:
푭푽 = 푷푽 ∗ (ퟏ + 풊)풏
푭푽 = $ퟏퟎ, ퟎퟎퟎ ∗ (ퟏ+. ퟏퟔ)ퟏ
푭푽 = $ퟏퟎ, ퟎퟎퟎ ∗ (ퟏ. ퟏퟔ)ퟏ
푭푽 = $ퟏퟏ, ퟔퟎퟎ

15. Semi-annually:
푭푽 = 푷푽 푿 ퟏ +
풊
풎
풏푿풎
• 푭푽 = $ퟏퟎ, ퟎퟎퟎ 푿 ퟏ +
.ퟏퟔ
ퟐ
ퟏ푿ퟐ
• 푭푽 = $ퟏퟎ, ퟎퟎퟎ 푿 ퟏ. ퟎퟖ ퟐ
• 푭푽 = $ퟏퟎ, ퟎퟎퟎ 푿 ퟏ. ퟏퟔퟔퟒ
• 푭푽 = $ퟏퟏ, ퟔퟔퟒ
• Quarterly:
• 푭푽 = 푷푽 푿 ퟏ +
풊
풎
풏푿풎
• 푭푽 = $ퟏퟎ, ퟎퟎퟎ 푿 ퟏ +
.ퟏퟔ
ퟒ
ퟏ푿ퟒ
푭푽 = $ퟏퟎ, ퟎퟎퟎ 푿 ퟏ. ퟎퟒ ퟒ
푭푽 = $ퟏퟎ, ퟎퟎퟎ 푿 ퟏ. ퟏퟔퟗퟗ
푭푽 = $ퟏퟏ, ퟔퟗퟗ

16. Present Value and Discounting
• Present Value concepts
• A present value calculation takes end-of-the-period cash flows
and reverses the effect of compounding to determine the
equivalent beginning-of-the-period cash flows
•Present Value Equation
•P풓풆풔풆풏풕 푽풂풍풖풆 = 푭푽풏 푿 푷푽푭풊,풏
•P풓풆풔풆풏풕 푽풂풍풖풆 = 푭푽풏 [
ퟏ
(ퟏ+풓)풏]
•This is discounting and the interest rate i is called the discount rate.
•Present value (PV) is often referred to as the discounted value of
future cash-flows.

17. Present Value and Discounting
• Time and the discount rate affect present value
•The greater the amount of time before a cash flow is to occur,
the smaller the present value of the cash-flow.
•The higher the discount rate, the smaller the present value of a
future cash-flow.

18. Example 4
What is the value of $500 to be received 10 years from today if our
discount rate is 6%.
P풓풆풔풆풏풕 푽풂풍풖풆 = 푭푽풏 [
ퟏ
(ퟏ+풓)풏]
FV = $500, n = 10, r = 6% or 0.06
= $500 [
ퟏ
(ퟏ+ퟎ.ퟎퟔ)ퟏퟎ]
= $500 (0.558)
= $279.20
18
Refer Table B -3

19. Example 5
What is the present value of an investment that yields $1,000 to be
received in 7 years and $1,000 to be received in 10 years if the discount
rate is 6 percent?
ퟏ
풑풓풆풔풆풏풕 풗풂풍풖풆 = 푭푽풏 [
(ퟏ+풓)풏] + 푭푽풏 [
ퟏ
(ퟏ+풓)풏]
PV= $ퟏ, ퟎퟎퟎ [
ퟏ
(ퟏ+ퟎ.ퟎퟔ)ퟕ] + $1,000 [
ퟏ
(ퟏ+ퟎ.ퟎퟔ)ퟏퟎ]
• = $1,000 (0.665) + $1,000 (0.558)
• = $665 + $558
• = $1223

20. Future and Present Values of Annuities
• Annuity- A series of equal payment made or received at
regular time intervals
• Ordinary Annuity- A series of equal annuity payments made
or received at the end of each period
• Future value of an annuity-What an equal series of payments
will be worth at some future date
• Future Value Factor of an Annuity (FVFA)-A factor that when
multiplied by a stream of equal payments equals the future
value of that stream
• Future Value of an Annuity Table- Table of factors that shows
the future value of equal flows at the end of each period,
given a particular interest rate

21. Example 6
Carlos Menendez is planning to invest $3,500 every year for the next six years in
an investment paying 12 percent annually. What will be the amount he will have at
the end of the six years? (Round to the nearest dollar.)
A) $21,000
B) $28,403
C) $24,670
D) $26,124

23. Example 7
You plan to save $1,250 at the end of each of the next three years to pay for a
vacation. If you can invest it at 7 percent, how much will you have at the end of
three years? (Round to the nearest dollar.)
A) $3,750
B) $3,918
C) $4,019
D) $4,589

25. Example 8:
Maricela Sanchez needs to have $25,000 in five years. If she can earn 8 percent on
any investment, what is the amount that she will have to invest every year at the
end of each year for the next five years? (Round to the nearest dollar.)
A) $5,000
B) $4,261
C) $4,640
D) $4,445

27. Example 9
Jane Ogden wants to save for a trip to Australia. She will need $12,000 at the end of
four years. She can invest a certain amount at the beginning of each of the next four
years in a bank account that will pay her 6.8 percent annually. How much will she
have to invest annually to reach her target? (Round to the nearest dollar.)
A) $3,000
B) $2,980
C) $2,538
D) $2,711

29. Annuities Continued
• Present Value of an Annuity- What the series of payments in
the future is worth today
• Present Value Factor of an Annuity (PVFA)- A factor that
when multiplied by a stream of equal payments equals the
present value of that stream
• Present Value of an Annuity Table- Table of factors that
shows the value today of equal flows at the end of each
future period, given a particular interest rate

30. How to calculate present value of an
annuity
(6.1) 30
o 푷푽풐풇 풂풏 풂풏풏풖풊풕풚 = 푷푴푻
PVA CF PVFA
PVFA
1
 
(1 )
1
 

1
0














 


 
i
i
CF
i
CF
n
ퟏ−풑풓풆풔풆풏풕 풗풂풍풖풆 풇풂풄풕풐풓
풓
o 푷푽풐풇 풂풏 풂풏풏풖풊풕풚 = 푷푴푻
ퟏ−(ퟏ+풓)−풏
풓

31. Level Cash Flows: Annuities and
• A contract will pay $2,000 at the end of each year for three
years and the appropriate discount rate is 8%. What is a fair
price for the contract?
$5,154.19
Perpetuities
• Present Value of an Annuity Example
(1 1/(1 0.08)
  
0.08
PVA $2000
3

3 


 
Refer Table B-4 31

32. Example 1
What is the value of 10-year $1,000 annuity discounted back to
the present at 5 percent?

33. 푷푽풐풇 풂풏 풂풏풏풖풊풕풚 = 푷푴푻
ퟏ − (ퟏ + 풓)−풏
풓
푷푽 = $ퟏ, ퟎퟎퟎ
ퟏ − (ퟏ + ퟎ. ퟎퟓ)−ퟏퟎ
ퟎ. ퟎퟓ
• = $1,000 (7.722)
• = $7,722

34. Example 2
• Transit Insurance Company has made an investment in another company that will
guarantee it a cash flow of $37,250 each year for the next five years. If the company uses a
discount rate of 15 percent on its investments, what is the present value of this
investment? (Round to the nearest dollar.)
A) $101,766
B) $124,868
C) $251,154
D) $186,250
Annual payment = PMT = $37,250
No. of payments = n = 5
Required rate of return = 15%
Present value of investment = PVA5

35.  
   
  
5
1
1
(1 )
n
1
1
(1.15)
$37,250 $37,250 3.3522
0.15
n
i
PVA PMT
i
 
 
 
  
     
 
 
 $124,867.78

36. Example 3
Herm Mueller has invested in a fund that will provide him a cash flow of $11,700
for the next 20 years. If his opportunity cost is 8.5 percent, what is the present value
of this cash flow stream? (Round to the nearest dollar.)
A) $234,000
B) $132,455
C) $110,721
D) $167,884

37.  
   
  
20
1
1
(1 )
1
1
n
(1.085)
$11,700 $11,700 9.4633
0.085
n
i
PVA PMT
i
 
 
 
  
     
 
 
 $110,721.04

38. Example 4
Myers, Inc., will be making lease payments of $3,895.50 for a 10-year period,
starting at the end of this year. If the firm uses a 9 percent discount rate, what is
the present value of this annuity? (Round to the nearest dollar.)
A) $23,250
B) $29,000
C) $25,000
D) $20,000

39.  
   
  
10
1
1
(1 )
1
1
n
(1.09)
$3,895.50 $3,895.50 6.4177
0.09
n
i
PVA PMT
i
 
 
 
  
     
 
 
 $24,999.99

40. Ordinary Annuity versus Annuity
Due
• Present Value of Annuity Due
•Cash flows are discounted for one period less than in an ordinary
annuity.
• Future Value of Annuity Due
•Cash flows are earn compound interest for one period more than
in an ordinary annuity.
40
PVA PVA (1 ) (6.4)
1
Due
1
Due
i
  
FVA FVA (1 i
)
  

41. Ordinary Annuity versus Annuity Due
41

42. Perpetuities
• A stream of equal cash flows that goes on forever
• Preferred stock and some bonds are perpetuities
• Equation for the present value of a perpetuity can be derived
from the present value of an annuity equation
42
PVP CF esent value factor for an annuity
1


1 1 0
( i)
( . )
 
Pr 0
CF
i
( )
i
CF
i
CF
6 3

1


 











 


43. Valuing Perpetuity
Example 1
• Suppose you decide to endow a chair in finance. The goal of the
endowment is to provide $100,000 of financial support per year
forever. If the endowment earns a rate of 8%, how much money
will you have to donate to provide the desired level of support?
43
$1,250,000
$100,000
CF
0   
0.08
i
PVP

44. Example 2
• What is the value of a $500 perpetuity discounted back to
the present at 8 percent?

45. 푷푽 =
푪푭
풊
푷푽 =
$ퟓퟎퟎ
ퟎ.ퟎퟖ
= $6250

46. Example 3
Your father is 60 years old and wants to set up a cash flow stream that would be
forever. He would like to receive $20,000 every year, beginning at the end of this
year. If he could invest in account earning 9 percent, how much would he have to
invest today to receive his perpetual cash flow? (Round to the nearest dollar.)
A) $222,222
B) $200,000
C) $189,000
D) $235,200

47. Annual payment needed = PMT = $20,000
Investment rate of return = i = 9%
Term of payment = Perpetuity
Present value of investment needed = PV
PMT $20,000
PV of Perpetuity
 
i 0.09
 $222,222.22

48. Example 4
A lottery winner was given a perpetual payment of $11, 444. She could invest the
cash flows at 7 percent. What is the present value of this perpetuity? (Round to
the nearest dollar.)
A) $112,344
B) $163,486
C) $191,708
D) $201,356

49. Annual payment needed = PMT =
$11,444
Investment rate of return = i = 7%
Term of payment = Perpetuity
Present value of investment needed = PV
PMT $11,444
PV of Perpetuity
 
i 0.07
 $163,485.71

50. Cash Flows That Grow at a Constant
Rate
• Growing Annuity
• equally-spaced cash flows that increase in size at a constant rate
for a finite number of periods
• Multiyear product or service contract with periodic cash flows
that increase at a constant rate for a finite number of years
• Growing Perpetuity
• equally-spaced cash flows that increase in size at a constant rate
forever
• Common stock whose dividend is expected to increase at a
constant rate forever

51. Cash Flows That Grow at a Constant
Rate
• Growing Annuity
• Calculate the present value of growing annuity (only) when the
growth rate is less than the discount rate.
51
CF
 
(6.5)
1 g

1 i

1
i - g
PVA
n
1



n 








  

52. Growing Annuity: Example
• A coffee shop will operate for fifty more years. Cash flow was
$300,000 last year and increases by 2.5% each year. The discount
rate for similar firms is 15%. Estimate the value of the firm.
1 $300,000 (1 0.025) $307,500
   
$307,500
CF
$2,460,000 0.9968
 
$2,452,128
1.025
1.15

1
0.15 0.025
50
0








 


PVA

53. Cash Flows That Grow at a Constant Rate
• Growing Perpetuity
• Use Equation 6.6 to calculate the present value of growing
perpetuity (only) when the growth rate is less than discount
rate.
• It is derived from equation 6.5 when the number of periods
approaches infinity
53
CF
PVP 1
 
(6.6)
i - g
0 

54. Cash Flows That Grow at a Constant Rate
• Growing perpetuity example
• A firm’s cash flow was $450,000 last year. You expect the cash
flow to increase by 5% per year forever. If you use a discount
rate of 18%, what is the value of the firm?
54
1 $450,000 (1 0.05) $307,500
   
$472,500
$3,634,615
$472,500
0.13
0.18 0.05
0




CF
PVP

55. The Effective Annual Interest Rate
• Describing interest rates
• The most common way to quote interest rates is in terms of
annual percentage rate (APR). It does not incorporate the
effects of compounding.
• The most appropriate way to quote interest rates is in terms of
effective annual rate (EAR). It incorporates the effects of
compounding.
55

56. The Effective Annual Interest Rate
• Calculate Annual Percentage Rate (APR)
APR = (periodic rate) x m
m is the # of periods in a year
• APR does not account for the number of compounding periods
or adjust the annualized interest rate for the time value of
money
• APR is not a precise measure of the rates involved in borrowing
and investing
56

57. Future and Present Value Calculations and
Excel Functions for Special Situations
• The more frequent the compounding for any given interest
level and time period, the higher the future value.
• In the Excel RATE and NPER functions, the Payment box or
loan payment box must be a negative value to represent cash
outflows

58. Summary
• Future values determine the value of dollar payments in the
future
• Present value indicates the current value of future dollars
• Formulas are used to calculate both future and present
values
• All calculations can be made using tables or spreadsheets