The Time Value of Money

1
Chapter
8
The Time Value of Money
2
The Time Value of Money
Which would you rather have -- 1,000 today or
1,000 in 5 years?
Obviously, 1,000 today. Money received
sooner rather than later allows one to use the
funds for investment or consumption purposes.
This concept is referred to as the TIME VALUE OF
MONEY!!
3
Why TIME?
TIME allows one the opportunity to postpone
consumption and earn INTEREST. NOT having
the opportunity to earn interest on money is
called OPPORTUNITY COST.

4
How can one compare amounts in different time
periods?

• One can adjust values from different time periods
  using an interest rate.
• Remember, one CANNOT compare numbers in different
  time periods without first adjusting them using
  an interest rate.

5
Basic Definitions

• Present Value earlier money on a time line
• Future Value later money on a time line
• Interest rate exchange rate between earlier
  money and later money
• Discount rate
• Cost of capital
• Opportunity cost of capital
• Required return

6
Time Value Terminology

• Consider the time line below
• PV is the Present Value, that is, the value
  today.
• FV is the Future Value, or the value at a future
  date.
• The number of time periods between the Present
  Value and the Future Value is represented by t or
  N
• The rate of interest is called r or i.
• All time value questions involve the four values
  above
• PV, FV, r, and N. Given three of them, it is
  always possible to calculate the fourth.

7
Types of Interest

• Simple Interest
• Interest paid (earned) on only the original
  amount, or principal borrowed (lent).

• Compound Interest
• Interest paid (earned) on any previous interest
  earned, as well as on the principal borrowed
  (lent).

8
Simple Interest Formula
SI P0(i)(n)
Formula SI Simple Interest P0 Deposit
today (t0) i Interest Rate per
Period n Number of Time Periods
9
Simple Interest Example

• Assume that you deposit 1,000 in an account
  earning 7 simple interest for 2 years. What is
  the accumulated interest at the end of the 2nd
  year?

SI P0(i)(n) 1,000(.07)(2) 140
10
Simple Interest (FV)

• What is the Future Value (FV) of the deposit?

FV P0 SI 1,000 140 1,140

• Future Value is the value at some future time of
  a present amount of money, or a series of
  payments, evaluated at a given interest rate.

11
Simple Interest (PV)

• What is the Present Value (PV) of the previous
  problem?

• The Present Value is simply the 1,000 you
  originally deposited. That is the value today!
• Present Value is the current value of a future
  amount of money, or a series of payments,
  evaluated at a given interest rate.

12
Why Compound Interest?
Future Value (U.S. Dollars)
13
Future Value (Compound Interest )
Assume that you deposit 1,000 at a compound
interest rate of 7 for 2 years.
0 1 2
7
1,000
FV2
14
Future Value (Compound Interest )

• Interest 1000(.07) 70
• FV (1st year) principal interest
• 1000 70 1070
• FV1 1000(1 .07) 1070
• Since you leave the money in for another year.
  How much will you have 2 years from now?
• FV2 1000(1.07)(1.07) 1000(1.07)2 1144.90

15
Future Values General Formula
FV PV(1 r)t

• FV future value
• PV present value
• r period interest rate, expressed as a decimal
• T number of periods
• Future value interest factor FVIF (1 r)t

16
Effects of Compounding

• Consider the previous example
• FV with simple interest 1000 70 70 1140
• FV with compound interest 1144.90
• The extra 4.50 comes from the interest of .07(70)
  4.90 earned on the first interest payment

17
TVM on the Calculator

• Use the highlighted row of keys for solving any
  of the FV, PV, FVA, PVA, FVAD, and PVAD problems

N Number of periods I/Y Interest rate per
period PV Present value PMT Payment per
period FV Future value CLR TVM Clears all of
the inputs into the above TVM keys
18
Entering the FV Problem
Press
2nd
CLR TVM
2
N
7
I/Y
PV
-1000
PMT
0
FV
CPT
19
Future Values Example 2

• Suppose you invest the 1000 from the previous
  example for 5 years. How much would you have?
• FV 1000(1.05)5 1276.28
• The effect of compounding is small for a small
  number of periods, but increases as the number of
  periods increases. (Simple interest would have a
  future value of 1250, for a difference of
  26.28.)

20
Future Values Example 3

• Suppose you had a relative deposit 10 at 5.5
  interest 200 years ago. How much would the
  investment be worth today?
• FV 10(1.055)200 447,189.84
• What is the effect of compounding?
• Simple interest 10 200(10)(.055) 210.55
• Compounding added 446,979.29 to the value of the
  investment

21
Future Value as a General Growth Formula

• Suppose your company expects to increase unit
  sales of widgets by 15 per year for the next 5
  years. If you currently sell 3 million widgets in
  one year, how many widgets do you expect to sell
  in 5 years?
• FV 3,000,000(1.15)5 6,034,072

22
Quick Quiz Part 1

• What is the difference between simple interest
  and compound interest?
• Suppose you have 500 to invest and you believe
  that you can earn 8 per year over the next 15
  years.
• How much would you have at the end of 15 years
  using compound interest?
• How much would you have using simple interest?

23
Present Values

• How much do I have to invest today to have some
  amount in the future?
• FV PV(1 r)t
• Rearrange to solve for PV FV / (1 r)t
• When we talk about discounting, we mean finding
  the present value of some future amount.
• When we talk about the value of something, we
  are talking about the present value unless we
  specifically indicate that we want the future
  value.

24
Present Value One Period Example

• Suppose you need 10,000 in one year for the down
  payment on a new car. If you can earn 7
  annually, how much do you need to invest today?
• PV 10,000 / (1.07)1 9345.79
• Calculator
• 1 N
• 7 I/Y
• 10,000 FV
• CPT PV -9345.79

25
Present Values Example 2

• You want to begin saving for you daughters
  college education and you estimate that she will
  need 150,000 in 17 years. If you feel confident
  that you can earn 8 per year, how much do you
  need to invest today?
• PV 150,000 / (1.08)17 40,540.34

26
Present Values Example 3

• Your parents set up a trust fund for you 10 years
  ago that is now worth 19,671.51. If the fund
  earned 7 per year, how much did your parents
  invest?
• PV 19,671.51 / (1.07)10 10,000

27
Present Value Important Relationship I

• For a given interest rate the longer the time
  period, the lower the present value
• What is the present value of 500 to be received
  in 5 years? 10 years? The discount rate is 10
• 5 years PV 500 / (1.1)5 310.46
• 10 years PV 500 / (1.1)10 192.77

28
Present Value Important Relationship II

• For a given time period the higher the interest
  rate, the smaller the present value
• What is the present value of 500 received in 5
  years if the interest rate is 10? 15?
• Rate 10 PV 500 / (1.1)5 310.46
• Rate 15 PV 500 / (1.15)5 248.58

29
Present Value of 1 for Different Periods and
Rates
30
Quick Quiz Part 2

• What is the relationship between present value
  and future value?
• Suppose you need 15,000 in 3 years. If you can
  earn 6 annually, how much do you need to invest
  today?
• If you could invest the money at 8, would you
  have to invest more or less than at 6? How much?

31
The Basic PV Equation - Refresher

• PV FV / (1 r)t
• There are four parts to this equation
• PV, FV, r and t
• If we know any three, we can solve for the fourth
• If you are using a financial calculator, be sure
  and remember the sign convention or you will
  receive an error when solving for r or t

32
Discount Rate

• Often we will want to know what the implied
  interest rate is in an investment
• Rearrange the basic PV equation and solve for r
• FV PV(1 r)t
• r (FV / PV)1/t 1
• If you are using formulas, you will want to make
  use of both the yx and the 1/x keys

33
Discount Rate Example 1

• You are looking at an investment that will pay
  1200 in 5 years if you invest 1000 today. What
  is the implied rate of interest?
• r (1200 / 1000)1/5 1 .03714 3.714
• Calculator the sign convention matters!!!
• N 5
• PV -1000 (you pay 1000 today)
• FV 1200 (you receive 1200 in 5 years)
• CPT I/Y 3.714

34
Discount Rate Example 2

• Suppose you are offered an investment that will
  allow you to double your money in 6 years. You
  have 10,000 to invest. What is the implied rate
  of interest?
• r (20,000 / 10,000)1/6 1 .122462 12.25

35
Discount Rate Example 3

• Suppose you have a 1-year old son and you want to
  provide 75,000 in 17 years towards his college
  education. You currently have 5000 to invest.
  What interest rate must you earn to have the
  75,000 when you need it?
• r (75,000 / 5,000)1/17 1 .172688 17.27

36
Quick Quiz Part 3

• What are some situations where you might want to
  compute the implied interest rate?
• Suppose you are offered the following investment
  choices
• You can invest 500 today and receive 600 in 5
  years. The investment is considered low risk.
• You can invest the 500 in a bank account paying
  4.
• What is the implied interest rate for the first
  choice and which investment should you choose?

37
Double Your Money!!!
Quick! How long does it take to double 5,000 at
a compound rate of 12 per year (approx.)?
We will use the Rule-of-72.
38
The Rule-of-72
Approx. Years to Double 72 / i
72 / 12 6 Years Actual Time is 6.12 Years
39
Finding the Number of Periods

• Start with basic equation and solve for t
  (remember your logs)
• FV PV(1 r)t
• You can use the financial keys on the calculator
  as well, just remember the sign convention.

t ln(FV / PV) / ln(1 r)
40
Double Your Money!!!
Quick! How long does it take to double 5,000 at
a compound rate of 12 per year (approx.)?
72 / 12 6 Years
t ln(10,000/ 5,000) / ln(1 0.12)
t 0.693 / 0.113 6.12 years
41
Number of Periods Example 1

• You want to purchase a new car and you are
  willing to pay 20,000. If you can invest at 10
  per year and you currently have 15,000, how long
  will it be before you have enough money to pay
  cash for the car?
• t ln(20,000 / 15,000) / ln(1.1) 3.02 years

42
Number of Periods Example 2

• Suppose you want to buy a new house. You
  currently have 15,000 and you figure you need to
  have a 10 down payment plus an additional 5 in
  closing costs. If the type of house you want
  costs about 150,000 and you can earn 7.5 per
  year, how long will it be before you have enough
  money for the down payment and closing costs?

43
Number of Periods Example 2 Continued

• How much do you need to have in the future?
• Down payment .1(150,000) 15,000
• Closing costs .05(150,000 15,000) 6,750
• Total needed 15,000 6,750 21,750
• Compute the number of periods
• PV -15,000
• FV 21,750
• I/Y 7.5
• CPT N 5.14 years
• Using the formula
• t ln(21,750 / 15,000) / ln(1.075) 5.14 years

44
Quick Quiz Part 4

• When might you want to compute the number of
  periods?
• Suppose you want to buy some new furniture for
  your family room. You currently have 500 and the
  furniture you want costs 600. If you can earn
  6, how long will you have to wait if you dont
  add any additional money?

45
Types of Annuities

• An Annuity represents a series of equal payments
  (or receipts) occurring over a specified number
  of equidistant periods.

• Ordinary Annuity Payments or receipts occur at
  the end of each period.
• Annuity Due Payments or receipts occur at the
  beginning of each period.

46
Examples of Annuities

• Student Loan Payments
• Car Loan Payments
• Insurance Premiums
• Mortgage Payments
• Retirement Savings

47
Parts of an Annuity
End of Period 2
(Ordinary Annuity) End of Period 1
End of Period 3
0 1 2
3
100 100
100
Equal Cash Flows Each 1 Period Apart
Today
48
Parts of an Annuity
Beginning of Period 2
(Annuity Due) Beginning of Period 1
Beginning of Period 3
0 1 2
3
100 100 100
Equal Cash Flows Each 1 Period Apart
Today
49
Whats the difference between an ordinary annuity
and an annuity due?
Ordinary Annuity
0
1
2
3
i
PMT
PMT
PMT
Annuity Due
0
1
2
3
i
PMT
PMT
PMT
PV
FV
50
Overview of an Ordinary Annuity -- FVA
Cash flows occur at the end of the period
0 1 2
n n1
. . .
i
R R R
R Periodic Cash Flow
FVAn
FVAn R(1i)n-1 R(1i)n-2 ... R(1i)1
R(1i)0
51
Example of anOrdinary Annuity -- FVA
Cash flows occur at the end of the period
0 1 2
3 4
7
1,000 1,000 1,000
1,070
1,145
3,215 FVA3
FVA3 1,000(1.07)2 1,000(1.07)1
1,000(1.07)0 1,145 1,070
1,000 3,215
52
Ordinary Annuity FVA General Equation
FVA PMT (FVIFAi.n)
53
Future Value (HP 17 B II Calculator)
Exit until you get Fin Menu. 2nd, Clear
Data. Choose Fin, then TVM
1,000 /-
PMT
3
N
7
IYr
CPT, FV
3,214.90
54
Hint on Annuity Valuation
The future value of an ordinary annuity can be
viewed as occurring at the end of the last cash
flow period, whereas the future value of an
annuity due can be viewed as occurring at the
beginning of the last cash flow period.
55
Overview View of anAnnuity Due -- FVAD
Cash flows occur at the beginning of the period
0 1 2
3 n-1 n
. . .
i
R R R
R R
FVADn
FVADn R(1i)n R(1i)n-1 ... R(1i)2
R(1i)1 FVAn (1i)
56
Example of anAnnuity Due -- FVAD
Cash flows occur at the beginning of the period
0 1 2
3 4
7
1,000 1,000 1,000
1,070
1,145
1,225
3,440 FVAD3
FVAD3 1,000(1.07)3 1,000(1.07)2
1,000(1.07)1 1,225 1,145 1,070
3,440
57
Annuity Due FVAD General Equation
FVAD PMT (FVIFAi.n)(1i)
58
Solving the FVAD Problem
Inputs
3 7 0 -1,000
N
I/Y
PV
PMT
FV

3,439.94
Compute
Complete the problem the same as an ordinary
annuity problem, except you must change the
calculator setting to BGN first. Dont forget
to change back! Step 1 Press keys Step
2 Press keys Step 3 Press keys
BGN
2nd
2nd
SET
2nd
QUIT
59
Present Value of anOrdinary Annuity -- PVA
Cash flows occur at the end of the period
0 1 2
n n1
. . .
i
R R R
R Periodic Cash Flow
PVAn
PVAn R/(1i)1 R/(1i)2 ... R/(1i)n
60
Example of PV of anOrdinary Annuity -- PVA
Cash flows occur at the end of the period
0 1 2
3 4
7
1,000 1,000 1,000
934.58 873.44 816.30
2,624.32 PVA3
PVA3 1,000/(1.07)1 1,000/(1.07)2
1,000/(1.07)3 934.58 873.44 816.30
2,624.32
61
Ordinary Annuity PVA General Equation
PVA PMT (PVIFAi.n)
62
Present Value (HP 17 B II Calculator)
Exit until you get Fin Menu. 2nd, Clear
Data. Choose Fin, then TVM
PMT
1,000
3
N
7
I Yr
CPT, PV
-2,624.32
63
Hint on Annuity Valuation
The present value of an ordinary annuity can be
viewed as occurring at the beginning of the first
cash flow period, whereas the present value of an
annuity due can be viewed as occurring at the end
of the first cash flow period.
64
Overview of anAnnuity Due -- PVAD
Cash flows occur at the beginning of the period
0 1 2
n-1 n
. . .
i
R R R
R
R Periodic Cash Flow
PVADn
PVADn R/(1i)0 R/(1i)1 ... R/(1i)n-1
PVAn (1 i)
65
Example of anAnnuity Due -- PVAD
Cash flows occur at the beginning of the period
0 1 2
3 4
7
1,000.00 1,000 1,000
934.58
873.44
2,808.02 PVADn
PVADn 1,000/(1.07)0 1,000/(1.07)1
1,000/(1.07)2 2,808.02
66
Annuity Due PVAD General Equation
PVAD PMT (PVIFAi.n)(1i)
67
Solving the PVAD Problem
Inputs
3 7 -1,000
0
N
I/Y
PV
PMT
FV
2,808.02
Compute
Complete the problem the same as an ordinary
annuity problem, except you must change the
calculator setting to BGN first. Dont forget
to change back! Step 1 Press keys Step
2 Press keys Step 3 Press keys
BGN
2nd
2nd
SET
2nd
QUIT
68
Multiple Cash Flows Example
Suppose an investment promises a cash flow of
500 in one year, 600 at the end of two years
and 10,700 at the end of the third year. If the
discount rate is 5, what is the value of this
investment today?
0 1 2 3
5
500 600 10,700
PV0
69
Multiple Cash Flow Solution
0 1 2 3
5
500 600 10,700
476.19 544.22 9,243.06
10,263.47 PV0 of the Multiple Cash Flows
70
Multiple Cash Flow Solution (HP 17 B II
Calculator)
Exit until you get Fin Menu. 2nd, Clear Data.
FIN
CFLO
Flow(0)?
0
Input
Flow(1)?
500
Input
Times (1) 1
Input
Flow(2)?
600
Times (2) 1
Input
Input
Flow(3)?
10,700
Input
Exit
Calc
I
5
NVP
71
Mixed Flows Example
Julie Miller will receive the set of cash flows
below. What is the Present Value at a discount
rate of 10?
0 1 2 3 4 5
10
600 600 400 400 100
PV0
72
How to Solve?

• Solve a piece-at-a-time by discounting each
  piece back to t0.
• Solve a group-at-a-time by first breaking
  problem into groups of annuity streams and any
  single cash flow group. Then discount each
  group back to t0.

73
Piece-At-A-Time
0 1 2 3 4 5
10
600 600 400 400 100
545.45 495.87 300.53 273.21 62.09
1677.15 PV0 of the Mixed Flow
74
Group-At-A-Time (1)
0 1 2 3 4 5
10
600 600 400 400 100
1,041.60 573.57 62.10
1,677.27 PV0 of Mixed Flow Using Tables
600(PVIFA10,2) 600(1.736)
1,041.60 400(PVIFA10,2)(PVIF10,2)
400(1.736)(0.826) 573.57 100 (PVIF10,5)
100 (0.621) 62.10
75
Group-At-A-Time (2)
0 1 2 3 4
400 400 400 400
1,268.00
0 1 2
PV0 equals 1677.30.
Plus
200 200
347.20
0 1 2 3 4
5
Plus

100
62.10
76
Solving the Mixed Flows Problem using CF Registry

• Use the highlighted key for starting the process
  of solving a mixed cash flow problem

• Press the CF key and down arrow key through a few
  of the keys as you look at the definitions on the
  next slide

77
Solving the Mixed Flows Problem using CF Registry

• Defining the calculator variables
• For CF0 This is ALWAYS the cash flow occurring
  at time t0 (usually 0 for these problems)
• For Cnn This is the cash flow SIZE of the nth
  group of cash flows. Note that a group may
  only contain a single cash flow (e.g., 351.76).
• For Fnn This is the cash flow FREQUENCY of the
  nth group of cash flows. Note that this is
  always a positive whole number (e.g., 1, 2, 20,
  etc.).

nn represents the nth cash flow or frequency.
Thus, the first cash flow is C01, while the tenth
cash flow is C10.
78
Solving the Mixed Flows Problem using CF Registry

• Steps in the Process
• Step 1 Press CF key
• Step 2 Press 2nd keys
• Step 3 For CF0 Press 0 keys
• Step 4 For C01 Press 600 keys
• Step 5 For F01 Press 2 keys
• Step 6 For C02 Press 400 keys
• Step 7 For F02 Press 2 keys

CF
CLR Work
2nd
?
Enter
0
?
Enter
600
?
Enter
2
?
Enter
400
?
Enter
2
79
Solving the Mixed Flows Problem using CF Registry

• Steps in the Process
• Step 8 For C03 Press keys
• Step 9 For F03 Press keys
• Step 10 Press keys
• Step 11 Press key
• Step 12 For I, Enter keys
• Step 13 Press key
• Result Present Value 1,677.15

?
Enter
100
?
Enter
1
?
?
NPV
?
Enter
10
CPT