1. Chapter 2
Time Value of Money

2. Opportunity cost
Opportunity cost is the value of the ______ best choice forgone.
next
This also refers to the _________________ option forgone.
highest-valued
It helps in making the best decision when we choose among ______________.
alternatives
For example, Mabel is considering three activities on this Saturday afternoon.
Singing karaoke
Exercising
Going shopping

3. Opportunity cost She ranks her priorities as: Exercising
Going shopping
Singing karaoke
Since Mabel ranked going shopping as her first priority, she decides to _____________ on Saturday.
go shopping
Mabel’s opportunity cost of going shopping is the time she has to give up ____________.
exercising
This is the ___________ choice among the forgone alternatives.
next best

4. Opportunity cost
Only the highest-valued alternative forgone is relevant
The opportunity cost of going shopping is not both of the other two alternatives forgone.
If Mable does not choose to go shopping, she can only choose to __________ or ______________.
exercise
sing karaoke
 Most likely, Mabel will choose to __________ if she does not go shopping because this is her ___________ choice.
exercise
next-best
 Only the highest-valued alternative forgone (do exercises) is the opportunity cost. or

5. Opportunity cost Opportunity cost may vary from person to person
Given the same set of alternatives, the opportunity cost may ______ from one person to another.
vary
For example, Patrick is also considering three activities:
Singing karaoke
Exercising
Going shopping

6. Opportunity cost Opportunity cost may vary from person to person
Assume he ranks his priorities as:
Exercising
Singing karaoke
Going shopping
In this example, Patrick will choose to _________. Thus, his opportunity cost is the time he gives up __________________.
exercise
singing karaoke
Opportunity cost =

7. Opportunity cost Reflect the value of time
The opportunity cost of doing something is usually measured as the ________ that we cannot earn from the _______________ alternative.
income
highest-paid
The ________ the income forgone, the ________ the opportunity cost of a person’s time.
higher
higher
Opportunity cost and the value of ______ tend to be _______.
time
equal
My consultation fee is $1,500 per 30 minutes.
That means 30 minutes of your time is worth $1,500.

8. Time value of money
Time value of money means that a dollar received today is worth ______ than a dollar received in the future.
more
A dollar received today can be ___________ and earn __________.
invested
interest
Invest
Today
Future
Money _______ over time.
grows
The additional amount we receive reflects the ____________ of money.
time value

9. Time value of money Compounding
Compounding means that the profit made on an investment is ____________ to make even more profit.
reinvested
This is the process of finding ______________.
future value
For example, most investors will receive ____________ interest if they deposit money in banks.
compound
Future value is the value at the end of a
period from a sum of money today.
 __________ earned in the current period becomes part of the _________ and the total sum will earn __________ in subsequent periods.
Interest
deposit
interest

10. Time value of money + = Compounding Example
Martin has just deposited $1,000 in a savings account at an interest of 5% per annum.
Martin has just deposited $1,000 in a savings account at an interest of 5% per annum.
Martin has just deposited $1,000 in a savings account at an interest of 5% per annum.
One year later
Principal
Interest + =
$1,050
$1,000  5%
$1,000
= $50

11. Time value of money + = Compounding Example
Martin has just deposited $1,000 in a savings account at an interest of 5% per annum.
Two years later
Principal +
1st year’s interest
Interest + =
$1,102.5
$1,050  5%
$1,050
= $52.5

12. Present value is a sum of money in today’s value.
Time value of money
Discounting
Discounting is the process of finding _______________.
present value
This is useful when we need to know how much money is needed ____________ to grow (compound) into a certain amount in future.
Present value is a sum of money in today’s value.
originally
How much do my parents
need to save now, so that
it can grow to the amount
needed for my university tuition fees?

13. Calculating the time value of money
Future value (FV)
As mentioned before, future value is the value at the _____ of a period from a sum of money _______.
end
today
Present value
Future value

14. Calculating the time value of money
Future value (FV)
Example 1:
Today, Mr Lee decides to deposit $100 in a bank for 3 years.
Today, Mr Lee decides to deposit $100 in a bank for 3 years.
The annual interest rate offered by the bank is 10%.
The interest is compounded annually.
How much will Mr Lee get after three years?
One year from now
Two years from now
Three years from now
Today
PV = $100
FV = ?

15. Calculating the time value of money
Future value (FV)
FV = PV  (1 + i ) n
Three years later, Mr Lee can get back:
FV = $100  (1 + 10% ) 3
= $100  1.331
= $133.1
One year from now
Two years from now
Three years from now
Today
PV = $100
FV = $133.1
FV = ?

16. Calculating the time value of money
Future value (FV)
Example 2:
Today, Ms Wong invests $3,000 for 5 years.
Today, Ms Wong invests $3,000 for 5 years.
The annual interest rate is 8%.
The annual interest rate is 8%.
The interest is compounded annually.
How much will Ms Wong get after five years?
FV = PV  (1 + i ) n
FV = $3,000  (1 + 8% ) 5
= $3,000  1.469
= $4,407

17. Calculating the time value of money
Future value (FV)
Future Value Interest Factor (FVIF)
FVIF = (1 + i)n
Since
FV = PV  (1 + i ) n 
FV = PV  FVIF
It simplifies the calculation of future value
An extract of FVIF table:

18. Calculating the time value of money
Future value (FV)
Future Value Interest Factor (FVIF)

19. Calculating the time value of money
Future value (FV)
Future Value Interest Factor (FVIF)
Refer to Mr Lee’s example above:
Today, Mr Lee decides to deposit $100 in a bank for 3 years.
Today, Mr Lee decides to deposit $100 in a bank for 3 years.
The annual interest rate offered by the bank is 10%.
The annual interest rate offered by the bank is 10%.
The interest is compounded annually.
How much will Mr Lee get after three years?
FV = PV  FVIF i,n
FV = $100  FVIF 10%, 3
= $100  1.331
= $133.1

20. Calculating the time value of money
Present value (PV)
As mentioned before, present value is the _________ value of a ________ sum of money.
current
future
It is the amount needed _______ to grow into a specific sum in the ________.
today
future
Present value
Future value

21. Calculating the time value of money
Present value (PV)
Example 1:
Today, Mrs Lo invests in a financial product which promises her $133.1 three years later.
The opportunity cost of this investment is 10%.
The investment return is compounded once a year.
How much should Mrs Lo invest today so that she can get the promised return of $133.1 three years later?
One year from now
Two years from now
Three years from now
Today
PV = ?
FV = $133.1

22. Calculating the time value of money
Present value (PV) FV
(1 + i ) n
PV =
$133.1
(1 + 10% ) 3
PV =
= $133.1  1.331
= $100
 Mrs Lo should invest $100 today.
One year from now
Two years from now
Three years from now
Today
PV = $100
PV = ?
FV = $133.1

23. Calculating the time value of money
Present value (PV)
Present Value Interest Factor (PVIF)
PVIF = 1
(1 + i ) n FV
(1 + i ) n
PV =
Since 
PV = FV  PVIF
It simplifies the calculation of present value
An extract of PVIF table:

24. Calculating the time value of money
Present value (PV)
Present Value Interest Factor (PVIF)

25. Calculating the time value of money
Present value (PV)
Refer to Mrs Lo’s example above:
Today, Mrs Lo invests in a financial product which promises her $133.1 three years later.
The opportunity cost of this investment is 10%.
The investment return is compounded once a year.
How much should Mrs Lo invest today so that she can get the promised return of $133.1 three years later?
PV = FV  PVIF i, n
PV = $133.1  PVIF 10%, 3
= $133.1  0.751
= $100

26. Calculating the time value of money
Future value of an annuity (FVA)
Future value of an annuity is the ________ value of a _______ of cash flows.
future
series
Future value of an annuity
Present value

27. Calculating the time value of money
Future value of an annuity (FVA)
Example:
You decide to deposit $100 in a bank at the end of each year for the next three years.
The annual interest rate is 10%.
Interest is compounded annually.
How much will you get back after three years?
This amount can be calculated by finding the future value of an annuity.

28. Calculating the time value of money
Future value of an annuity (FVA)
Method 1: Add up the FVs
Year 1
Year 2
Year 3
$100
FV3 = $100
$100
FV2 = $110
FV2 = $100  (1 + 10%)1
$100
FV1 = $121
FV1 = $100  (1 + 10%)2
Therefore, FVA = FV1 + FV2 + FV3
= $121 + $110 + $100
= $331
You will get back $331 after three years.

29. Calculating the time value of money
Future value of an annuity (FVA)
Method 2: Use the FVA equation
(1 + i) n - 1 i
FVA = Pmt 
Refer to the above example that you deposit $100 per year for three years.
(1 + 10%) 3 - 1
10%
FVA = $100 
= $100  3.31
= $331
The result is the same as that calculated by Method 1.

30. Calculating the time value of money
Future value of an annuity (FVA)
Method 3: Use the FVIFA
FVIFA refers to Future Value Interest Factor of an Annuity.
FVIFA =
(1 + i) n - 1 i
Since
(1 + i) n - 1 i
FVA = Pmt  
FVA = Pmt  FVIFA
Refer to the above example:
FVA = $100  FVIFA 10%, 3
= $100  3.310
= $331
The result is the same as that calculated by Methods 1 and 2.

31. Calculating the time value of money
Present value of an annuity (PVA)
Present value of an annuity is the _________ value of a _______ of future cash flows.
current
series
Present value of an annuity
Future value

32. Calculating the time value of money
Present value of an annuity (PVA)
Example:
You decide to borrow money from a bank and will repay $100 at the end of each year for the next three years.
The annual interest rate is 10%.
Interest is compounded annually.
What is the amount you can get from the bank now?
This amount can be calculated by finding the present value of an annuity.

33. Calculating the time value of money
Present value of an annuity (PVA)
Method 1: Add up the PVs
Year 1
Year 1
Year 2
Year 3
$100
(1 + 10%) 1
PV1 =
$100
PV1 = $90.9
$100
(1 + 10%) 2
PV2 =
$100
PV2 = $82.6
$100
(1 + 10%) 3
PV3 =
$100
PV3 = $75.1
Therefore, PVA = PV1 + PV2 + PV3
= $ $ $75.1
= $248.6
You can get a loan of $248.6 now.

34. Calculating the time value of money
Present value of an annuity (PVA)
Method 2: Use the PVA equation
PVA = Pmt  1
(1 + i) n
1 - i
Refer to the above example that you repay $100 per year for three years.
PVA = $100  1
(1 + 10%) 3
1 -
10%
= $100  2.487
= $248.7
The result is the same as that calculated by Method 1.

35. Calculating the time value of money
Present value of an annuity (PVA)
Method 3: Use the PVIFA
PVIFA refers to Present Value Interest Factor of an Annuity.
PVIFA = 1
(1 + i) n
1 - i
Since
PVA = Pmt  1
(1 + i) n
1 - i 
PVA = Pmt  PVIFA
Refer to the above example:
PVA = $100  PVIFA 10%, 3
= $100  2.487
= $248.7
The result is the same as that calculated by Methods 1 and 2.

36. Using net present value to make investment decisions
Net present value is the ____ of future cash _________, less ________ outlay. PV
inflows
initial
This is useful in making _____________ decisions.
investment
Example:
The owner of a manufacturing firm is considering buying a new machine which costs $10,000.
He expects that the machine has a useful life of 10 years.
He also expects that the machine will be sold for $1,000 at the end of Year 10.
The cost of capital is 10% per annum.
Should the owner buy the machine?

37. Using net present value to make investment decisions
To answer the question, we should calculate the NPV.
Step  Determine the cost of the machine.
The cost of the machine in the above example is $10,000.
Step  Estimate annual cash inflows by using this machine for the next 10 years.
The owner estimates that the machine can help generate $1,500 in cash inflows per year for the next 10 years.
Step  Estimate the market value of the machine in Year 10.
The owner estimates that the market value of the machine in Year 10 will bw $1,000.

38. Using net present value to make investment decisions
To answer the question, we should calculate the NPV.
Step  Calculate the PVs of all the annual net cash inflows and the machine in Year 10.
Total PV of annual cash inflows:
($1,500  1.1) + ($1,500  1.12) + ($1,500  1.13) + ($1,500  1.14) + ($1,500  1.15) + ($1,500  1.16) + ($1,500  1.17) + ($1,500  1.18) + ($1,500  1.19) + ($1,500  1.110)
= $9,216
PV of the machine in Year 10:
$1,000  1.110
= $386

39. Using net present value to make investment decisions
To answer the question, we should calculate the NPV.
Step  Add up all the PVs (total PV), i.e., PVs of the annual net cash inflows and the machine in Year 10.
Total PVs:
$9,216 + $386
= $9,602
Step  Find the NPV.
Total PV – Initial cost of the machine
$9,602 - $10,000
= -$398

40. Using net present value to make investment decisions
To answer the question, we should calculate the NPV.
Step  (i) If NPV is positive, the firm should buy the machine as it will bring financial benefit to the firm
(ii) If NPV is negative, the firm should not buy the machine as the firm will suffer a loss.
(iii) If NPV is equal to zero, the firm is indifferent about buying the machine or not.
Since the NPV is -$398, it is negative. The firm should not buy the machine as it will suffer a loss of $398.

41. Nominal versus effective rate of return
Nominal rate of return
The nominal rate of return is the rate of interest ___________ financial instruments.
stated on
It is also called the _________________________ or _______________________.
nominal rate of return
quoted interest rate
For example, in Mr Lee’s example:
Today, Mr Lee decides to deposit $100 in a bank for 3 years.
The annual interest rate offered by the bank is 10%.
The nominal rate of return is 10%.

42. Nominal versus effective rate of return
Nominal rate of return
The nominal rate of return is the rate of interest ___________ financial instruments.
stated on
It is also called the _________________________ or _______________________.
nominal rate of return
quoted interest rate
For example, in Ms Wong’s example:
Today, Ms Wong invests $3,000 for 5 years.
The annual interest rate is 8%.
The nominal rate of return is 8%.

43. Nominal versus effective rate of return
Effective rate of return (ERR)
The effective rate of return can reflect the effect of the ___________ of compounding on the ________ investment return.
frequency
actual
It is also called the ______________________________ or ________________________.
effective annual rate (EAR)
effective interest rate
The more ____________ interest is compounded, the ________ the sum we receive at the end of the period.
frequently
larger
the interest grows ________.
faster
It is commonly used to __________ different investment plans.
compare

44. Nominal versus effective rate of return
Effective rate of return (ERR)
The annual nominal rate of 10% may yield ______ than 10% after one year, depending on how ____________ the interest is compounded.
more
frequently
For example,
Interest rate of 10% per annum

45. Nominal versus effective rate of return
Effective rate of return (ERR)
The annual nominal rate of 10% may yield ______ than 10% after one year, depending on how ____________ the interest is compounded.
more
frequently
For example,
If the interest is compounded once a year
After one year, a $100 deposit will become:
At the beginning of the year
$100  1.1
At the end of the year
$100
$110

46. Nominal versus effective rate of return
Effective rate of return (ERR)
The annual nominal rate of 10% may yield ______ than 10% after one year, depending on how ____________ the interest is compounded.
more
frequently
For example,
If the interest is compounded twice a year
After one year, a $100 deposit will become:
At the beginning of the year
At the middle of the year
At the end of the year
$100  1.05
$105  1.05
$100
$105
$110.25

47. Nominal versus effective rate of return
Effective rate of return (ERR)
The annual nominal rate of 10% may yield ______ than 10% after one year, depending on how ____________ the interest is compounded.
more
frequently
For example,
If the interest is compounded four times a year
After one year, a $100 deposit will become:
$110.38
At the end of the year
$105.06
After 3 months
At the beginning of the year
$  1.025
$100
$102.5  1.025
$  1.025
$100  1.025
$107.69
$102.5
After 3 months
After 3 months

48. Nominal versus effective rate of return
Effective rate of return (ERR)
The annual nominal rate of 10% may yield ______ than 10% after one year, depending on how ____________ the interest is compounded.
more
frequently
Compounded once a year
Compounded twice a year
Compounded four times a year
Amount at the end of one year (initial amount of $100 at 10% p.a.)
Frequency of compounding per year
$110
$110.25
$110.38
Therefore, the more frequently the interest is compounded, the ________ the final amount will be.
larger