Chapter 2
The Two Key Concepts in Finance
1

It’s what we learn after we think we know it all that counts.
- Kin Hubbard
2

Outline
 Introduction
 Time value of money
 Safe dollars and risky dollars
 Relationship between risk and return
3

Introduction
 The occasional reading of basic material in your chosen field is an excellent philosophical exercise
• Do not be tempted to include that you “know it all”
– E.g., what is the present value of a growing perpetuity that begins payments in five years
4

Time Value of Money
 Introduction
 Present and future values
 Present and future value factors
 Compounding
 Growing income streams
5

Introduction
 Time has a value
•
If we owe, we would prefer to pay money later
•
If we are owed, we would prefer to receive money sooner
• The longer the term of a single-payment loan, the higher the amount the borrower must repay
6

Present and Future Values
 Basic time value of money relationships:
PV

FV

DF
FV

PV

CF where PV = present value;
FV = future value;
DF = discount factor = 1/(1

R ) t
CF = compounding factor = (1

R ) t
R = interest rate per perio d; and
t = time in periods
7

Present and Future Values
(cont’d)
 A present value is the discounted value of one or more future cash flows
 A future value is the compounded value of a present value
 The discount factor is the present value of a dollar invested in the future
 The compounding factor is the future value of a dollar invested today
8

Present and Future Values
(cont’d)
 Why is a dollar today worth more than a dollar tomorrow?
•
The discount factor:
– Decreases as time increases
•
The farther away a cash flow is, the more we discount it
– Decreases as interest rates increase
• When interest rates are high, a dollar today is worth much more than that same dollar will be in the future
9

Present and Future Values
(cont’d)
 Situations:
•
Know the future value and the discount factor
– Like solving for the theoretical price of a bond
• Know the future value and present value
–
Like finding the yield to maturity on a bond
•
Know the present value and the discount rate
– Like solving for an account balance in the future
10

Present and Future Value
Factors
 Single sum factors
 How we get present and future value tables
 Ordinary annuities and annuities due
11

Single Sum Factors
 Present value interest factor and future value interest factor :
PV

FV

PVIF
FV

PV

FVIF where
PVIF

1
(1

R ) t
FVIF
 
R ) t
12

Single Sum Factors (cont’d)
Example
You just invested $2,000 in a three-year bank certificate of deposit (CD) with a 9 percent interest rate.
How much will you receive at maturity?
13

Single Sum Factors (cont’d)
Example (cont’d)
Solution: Solve for the future value:
FV



$2, 590
3
14

How We Get Present and
Future Value Tables
 Standard time value of money tables present factors for:
•
Present value of a single sum
•
Present value of an annuity
•
Future value of a single sum
•
Future value of an annuity
15

How We Get Present and
Future Value Tables (cont’d)
 Relationships:
•
You can use the present value of a single sum to obtain:
–
The present value of an annuity factor (a running total of the single sum factors)
–
The future value of a single sum factor (the inverse of the present value of a single sum factor)
16

Ordinary Annuities and Annuities Due
 An annuity is a series of payments at equal time intervals
 An ordinary annuity assumes the first payment occurs at the end of the first year
 An annuity due assumes the first payment occurs at the beginning of the first year
17

Ordinary Annuities and Annuities Due (cont’d)
Example
You have just won the lottery! You will receive $1 million in ten installments of $100,000 each. You think you can invest the $1 million at an 8 percent interest rate.
What is the present value of the $1 million if the first
$100,000 payment occurs one year from today? What is the present value if the first payment occurs today?
18

Ordinary Annuities and Annuities Due (cont’d)
Example (cont’d)
Solution: These questions treat the cash flows as an ordinary annuity and an annuity due, respectively:
PV of ordinary annuity
  
$671, 000
PV of annuity due
   
$724, 680
19

Compounding
 Definition
 Discrete versus continuous intervals
 Nominal versus effective yields
20

Definition
 Compounding refers to the frequency with which interest is computed and added to the principal balance
• The more frequent the compounding, the higher the interest earned
21

Discrete Versus
Continuous Intervals
 Discrete compounding means we can count the number of compounding periods per year
• E.g., once a year, twice a year, quarterly, monthly, or daily
 Continuous compounding results when there is an infinite number of compounding periods
22

Discrete Versus
Continuous Intervals (cont’d)
 Mathematical adjustment for discrete compounding:
FV

PV (1

/ ) mt
R

annual interest rate m

number of compounding periods per year t

time in years
23

Discrete Versus
Continuous Intervals (cont’d)
 Mathematical equation for continuous compounding:
FV

PVe
Rt e

2.71828
24

Discrete Versus
Continuous Intervals (cont’d)
Example
Your bank pays you 3 percent per year on your savings account. You just deposited $100.00 in your savings account.
What is the future value of the $100.00 in one year if interest is compounded quarterly? If interest is compounded continuously?
25

Discrete Versus
Continuous Intervals (cont’d)
Example (cont’d)
Solution: For quarterly compounding:
FV

PV (1

/ ) mt


$103.03
4
26

Discrete Versus
Continuous Intervals (cont’d)
Example (cont’d)
Solution (cont’d): For continuous compounding:
FV

PVe
Rt

$100.00
 e
0.03

$103.05
27

Nominal Versus
Effective Yields
 The stated rate of interest is the simple rate or nominal rate
•
3.00% in the example
 The interest rate that relates present and future values is the effective rate
•
$3.03/$100 = 3.03% for quarterly compounding
• $3.05/$100 = 3.05% for continuous compounding
28

Growing Income Streams
 Definition
 Growing annuity
 Growing perpetuity
29

Definition
 A growing stream is one in which each successive cash flow is larger than the previous one
• A common problem is one in which the cash flows grow by some fixed percentage
30

Growing Annuity
 A growing annuity is an annuity in which the cash flows grow at a constant rate g :
PV

(1
C

R )

C
(1
(1


R ) g
2
)

C (1
(1


R g
)
3
)
2

C
1
R
 g



1


1

 1
 g
R 
N


C (1
 g ) n
(1

R ) n

1
31

Growing Perpetuity
 A growing perpetuity is an annuity where the cash flows continue indefinitely:
PV

(1
C

R )

C
(1
(1


R ) g
2
)

C
(1
(1


R g
)
3
)
2
 t



1
C (1 g ) t
 t
(1

R ) t

1

C
1
R
 g
C (1
 g )

(1

R )

32

Safe Dollars and Risky Dollars
 Introduction
 Choosing among risky alternatives
 Defining risk
33

Introduction
 A safe dollar is worth more than a risky dollar
•
Investing in the stock market is exchanging bird-in-the-hand safe dollars for a chance at a higher number of dollars in the future
34

Introduction (cont’d)
 Most investors are risk averse
•
People will take a risk only if they expect to be adequately rewarded for taking it
 People have different degrees of risk aversion
•
Some people are more willing to take a chance than others
35

Choosing Among
Risky Alternatives
Example
You have won the right to spin a lottery wheel one time.
The wheel contains numbers 1 through 100, and a pointer selects one number when the wheel stops. The payoff alternatives are on the next slide.
Which alternative would you choose?
36

[1-50]
Choosing Among
Risky Alternatives (cont’d)
A B C D
$110 [1-50] $200 [1-90] $50 [1-99] $1,000
[51-100] $90 [51-100] $0 [91-100] $500 [100] -$89,000
Avg. payoff $100 $100 $100 $100
37

Choosing Among
Risky Alternatives (cont’d)
Example (cont’d)
Solution:
 Most people would think Choice A is “safe.”

Choice B has an opportunity cost of $90 relative to Choice A.

People who get utility from playing a game pick
Choice C.

People who cannot tolerate the chance of any loss would avoid Choice D.
38

Choosing Among
Risky Alternatives (cont’d)
Example (cont’d)
Solution (cont’d):

Choice A is like buying shares of a utility stock.
 Choice B is like purchasing a stock option.

Choice C is like a convertible bond.

Choice D is like writing out-of-the-money call options.
39

Defining Risk
 Risk versus uncertainty
 Dispersion and chance of loss
 Types of risk
40

Risk Versus Uncertainty
 Uncertainty involves a doubtful outcome
•
What you will get for your birthday
•
If a particular horse will win at the track
 Risk involves the chance of loss
•
If a particular horse will win at the track if you made a bet
41

Dispersion and Chance of Loss
 There are two material factors we use in judging risk:
•
The average outcome
•
The scattering of the other possibilities around the average
42

Dispersion and Chance of Loss
(cont’d)
Investment value
Investment A
Investment B
Time
43

Dispersion and Chance of Loss
(cont’d)
 Investments A and B have the same arithmetic mean
 Investment B is riskier than Investment A
44

Types of Risk
 Total risk refers to the overall variability of the returns of financial assets
 Undiversifiable risk is risk that must be borne by virtue of being in the market
•
Arises from systematic factors that affect all securities of a particular type
45

Types of Risk (cont’d)
 Diversifiable risk can be removed by proper portfolio diversification
•
The ups and down of individual securities due to company-specific events will cancel each other out
•
The only return variability that remains will be due to economic events affecting all stocks
46

Relationship Between Risk and
Return
 Direct relationship
 Concept of utility
 Diminishing marginal utility of money
 St. Petersburg paradox
 Fair bets
 The consumption decision
 Other considerations
47

Direct Relationship
 The more risk someone bears, the higher the expected return
 The appropriate discount rate depends on the risk level of the investment
 The risk-less rate of interest can be earned without bearing any risk
48

Direct Relationship (cont’d)
Expected return
R f
0 Risk
49

Direct Relationship (cont’d)
 The expected return is the weighted average of all possible returns
•
The weights reflect the relative likelihood of each possible return
 The risk is undiversifiable risk
•
A person is not rewarded for bearing risk that could have been diversified away
50

Concept of Utility
 Utility measures the satisfaction people get out of something
•
Different individuals get different amounts of utility from the same source
– Casino gambling
– Pizza parties
– CDs
–
Etc.
51

Diminishing Marginal
Utility of Money
 Rational people prefer more money to less
•
Money provides utility
•
Diminishing marginal utility of money
– The relationship between more money and added utility is not linear
– “I hate to lose more than I like to win”
52

Utility
Diminishing Marginal
Utility of Money (cont’d)
$
53

St. Petersburg Paradox
 Assume the following game:
•
A coin is flipped until a head appears
•
The payoff is based on the number of tails observed ( n ) before the first head
• The payoff is calculated as $2 n
 What is the expected payoff?
54

St. Petersburg Paradox
(cont’d)
Number of Tails
Before First
Head Probability Payoff
Probability x Payoff
Total
2
3
0
1
4 n
(1/2)1 = 1/2
(1/2)
2
= 1/4
(1/2)
3
= 1/8
(1/2)
4
= 1/16
(1/2)
5
= 1/32
(1/2) n + 1
1.00
$1
$2
$4
$8
$16
$2 n
$0.50
$0.50
$0.50
$0.50
$0.50
$0.50

55

St. Petersburg Paradox
(cont’d)
 In the limit, the expected payoff is infinite
 How much would you be willing to play the game?
•
Most people would only pay a couple of dollars
• The marginal utility for each additional $0.50 declines
56

Fair Bets
 A fair bet is a lottery in which the expected payoff is equal to the cost of playing
•
E.g., matching quarters
•
E.g., matching serial numbers on $100 bills
 Most people will not take a fair bet unless the dollar amount involved is small
•
Utility lost is greater than utility gained
57

The Consumption Decision
 The consumption decision is the choice to save or to borrow
•
If interest rates are high, we are inclined to save
– E.g., open a new savings account
•
If interest rates are low, borrowing looks attractive
–
E.g., a higher home mortgage
58

The Consumption
Decision (cont’d)
 The equilibrium interest rate causes savers to deposit a sufficient amount of money to satisfy the borrowing needs of the economy
59

Other Considerations
 Psychic return
 Price risk versus convenience risk
60

Psychic Return
 Psychic return comes from an individual disposition about something
•
People get utility from more expensive things, even if the quality is not higher than cheaper alternatives
–
E.g., Rolex watches, designer jeans
61

Price Risk Versus
Convenience Risk
 Price risk refers to the possibility of adverse changes in the value of an investment due to:
•
A change in market conditions
• A change in the financial situation
•
A change in public attitude
 E.g., rising interest rates and stock prices, a change in the price of gold and the value of the dollar
62

Price Risk Versus
Convenience Risk (cont’d)
 Convenience risk refers to a loss of managerial time rather than a loss of dollars
• E.g., a bond’s call provision
– Allows the issuer to call in the debt early, meaning the investor has to look for other investments
63