Transcript Valuation Risk chapter 02

Chapter 2
Valuation, Risk, Return, and
Uncertainty
Portfolio Construction, Management, & Protection, 4e, Robert A. Strong
Copyright ©2006 by South-Western, a division of Thomson Business & Economics. All rights reserved.
1
It’s what we learn after we think we know it
all that counts.
Kin Hubbard
2
Outline
 Introduction
 Valuation
 Safe
Dollars and Risky Dollars
 Relationship Between Risk and Return
 The Concept of Return
 Some Statistical Facts of Life
3
Introduction
 The
occasional reading of basic material in
your chosen field is an excellent
philosophical exercise
• Do not be tempted to conclude that you “know
it all”
– e.g., what is the present value of a growing
perpetuity that begins payments in five years?
4
Valuation
 Introduction
 Growing
Income Streams
5
Introduction
 Valuation
may be the most important part of
the study of investments
• Security analysts make a career of estimating
“what you get” for “what you pay”
• The time value of money is one of the two key
concepts in finance and is very useful in
valuation
6
Growing Income Streams
 Definition
 Growing Annuity
 Growing
Perpetuity
7
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
8
Growing Annuity
 A growing
annuity is an annuity in which
the cash flows grow at a constant rate g:
C
C (1  g ) C (1  g ) 2
C (1  g ) n
PV 


 ... 
2
3
(1  R) (1  R)
(1  R)
(1  R) n 1
N

C1
 1 g  

1  
 
R  g   1  R  
9
Growing Perpetuity
 A growing
perpetuity is an annuity where
the cash flows continue indefinitely:
C
C (1  g ) C (1  g ) 2
C (1  g )
PV 


 ... 
2
3
(1  R) (1  R)
(1  R)
(1  R)
Ct (1  g )t 1
C1


t
(1  R)
Rg
t 1

10
Safe Dollars and Risky Dollars
 Introduction
 Choosing Among
 Defining
Risky Alternatives
Risk
11
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
12
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
13
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?
14
Choosing Among
Risky Alternatives (cont’d)
A
[1–50]
[51–100]
Average
payoff
B
$110 [1–50]
$90 [51–100]
$100
Number on lottery wheel appears in brackets.
C
$200 [1–90]
$0 [91–100]
$100
D
$50 [1–99]
$550 [100]
$100
$1,000
–$89,000
$100
15
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.
16
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.
17
Defining Risk
 Risk
Versus Uncertainty
 Dispersion and Chance of Loss
 Types of Risk
18
Risk Versus Uncertainty
 Uncertainty
involves a doubtful outcome
• What birthday gift you will receive
• 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
19
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
20
Dispersion and Chance of Loss
(cont’d)
Investment value
Investment A
Investment B
Time
21
Dispersion and Chance of Loss
(cont’d)
 Investments A and
B have the same
arithmetic mean
 Investment
B is riskier than Investment A
22
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
23
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
24
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
25
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 riskless rate of interest can be earned
without bearing any risk
26
Direct Relationship (cont’d)
Expected return
Rf
0
Risk
27
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
28
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.
29
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”
30
Diminishing Marginal
Utility of Money (cont’d)
Utility
$
31
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 $2n
 What
is the expected payoff?
32
St. Petersburg Paradox
(cont’d)
Number of Tails
Before First
Head
0
1
Probability
(1/2) = 1/2
(1/2)2 = 1/4
Payoff
$1
$2
Probability
× Payoff
$0.50
$0.50
2
3
4
(1/2)3 = 1/8
(1/2)4 = 1/16
(1/2)5 = 1/32
$4
$8
$16
$0.50
$0.50
$0.50
n
Total
(1/2)n + 1
1.00
$2n
$0.50

33
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
34
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
35
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 bigger home mortgage
36
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
37
Other Considerations
 Psychic
Return
 Price Risk versus Convenience Risk
38
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
39
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 influence stock prices,
and a change in the price of gold can affect the
value of the dollar
40
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
41
The Concept of Return
 Introduction
 Measurable
Return
 Return on Investment
42
Introduction
 “Return”
can mean various things, and it is
important to be clear when discussing an
investment
43
Measurable Return
 Definition
 Holding
Period Return
 Arithmetic Mean Return
 Geometric Mean Return
 Comparison of Arithmetic and Geometric
Mean Returns
 Expected Return
44
Definition
definition of return is “the
benefit associated with an investment”
 A general
• In most cases, return is measurable
• e.g., a $100 investment at 8 percent,
compounded continuously is worth $108.33
after one year
– The return is $8.33, or 8.33 percent
45
Holding Period Return
 The
calculation of a holding period return
is independent of the passage of time
• e.g., you buy a bond for $950, receive $80 in
interest, and later sell the bond for $980
– The return is ($80 + $30)/$950 = 11.58 percent
– The 11.58 percent could have been earned over one
year or one week
46
Arithmetic Mean Return
 The
arithmetic mean return is the
arithmetic average of several holding period
returns measured over the same holding
period:
n ~
Ri
Arithmetic mean  
i 1 n
~
Ri  the rate of return in period i
47
Arithmetic Mean Return
(cont’d)
 Arithmetic
means are a useful proxy for
expected returns
 Arithmetic
means are not especially useful
for describing historical return data
• It is unclear what the number means once it is
determined
48
Geometric Mean Return
 The
geometric mean return is the nth root
of the product of n values:

~ 
Geometric mean   (1  Ri )
 i 1

n
1/ n
1
49
Arithmetic and
Geometric Mean Returns
Example
Assume the following sample of weekly stock returns:
Week
Return
Return Relative
1
2
0.0084
–0.0045
1.0084
0.9955
3
0.0021
1.0021
4
0.0000
1.000
50
Arithmetic and Geometric
Mean Returns (cont’d)
Example (cont’d)
What is the arithmetic mean return?
Solution:
n ~
Ri
Arithmetic mean  
i 1 n
0.0084  0.0045  0.0021  0.0000

4
 0.0015
51
Arithmetic and Geometric
Mean Returns (cont’d)
Example (cont’d)
What is the geometric mean return?
Solution:
1/ n
n

~
Geometric mean   (1  Ri   1
 i 1

 1.0084  0.9955 1.00211.0000  1
1/ 4
 0.001489
52
Comparison of Arithmetic and
Geometric Mean Returns
 The
geometric mean reduces the likelihood
of nonsense answers
• Assume a $100 investment falls by 50 percent
in period 1 and rises by 50 percent in period 2
• The investor has $75 at the end of period 2
– Arithmetic mean = [(0.50) + (–0.50)]/2 = 0%
– Geometric mean = (0.50 × 1.50)1/2 – 1 = –13.40%
53
Comparison of Arithmetic and
Geometric Mean Returns (Cont’d)
 The
geometric mean must be used to
determine the rate of return that equates a
present value with a series of future values
 The
greater the dispersion in a series of
numbers, the wider the gap between the
arithmetic mean and geometric mean
54
Expected Return
 Expected
return refers to the future
• In finance, what happened in the past is not as
important as what happens in the future
• We can use past information to make estimates
about the future
55
Return on Investment (ROI)
 Definition
 Measuring
Total Risk
56
Definition
 Return
on investment (ROI) is a term that
must be clearly defined
• Return on assets (ROA)
• Return on equity (ROE)
– ROE is a leveraged version of ROA
57
Measuring Total Risk
 Standard
Deviation and Variance
 Semi-Variance
58
Standard Deviation and
Variance
 Standard
deviation and variance are the
most common measures of total risk
 They
measure the dispersion of a set of
observations around the mean observation
59
Standard Deviation and
Variance (cont’d)
 General
equation for variance:
2
n
Variance   2   prob( xi )  xi  x 
i 1
 If
all outcomes are equally likely:
n
2
1
    xi  x 
n i 1
2
60
Standard Deviation and
Variance (cont’d)
 Equation
for standard deviation:
Standard deviation     2 
2
n
 prob( x )  x  x 
i 1
i
i
61
Semi-Variance
 Semi-variance
considers the dispersion only
on the adverse side
• Ignores all observations greater than the mean
• Calculates variance using only “bad” returns
that are less than average
• Since risk means “chance of loss,” positive
dispersion can distort the variance or standard
deviation statistic as a measure of risk
62
Some Statistical Facts of Life
 Definitions
 Properties
of Random Variables
 Linear Regression
 R Squared and Standard Errors
63
Definitions
 Constants
 Variables
 Populations
 Samples
 Sample
Statistics
64
Constants
 A constant
is a value that does not change
• e.g., the number of sides of a cube
• e.g., the sum of the interior angles of a triangle
 A constant
can be represented by a numeral
or by a symbol
65
Variables
 A variable
has no fixed value
• It is useful only when it is considered in the
context of other possible values it might assume
 In
finance, variables are called random
variables
• Designated by a tilde
– e.g.,
x
66
Variables (cont’d)
 Discrete
random variables are countable
• e.g., the number of trout you catch
 Continuous
random variables are
measurable
• e.g., the length of a trout
67
Variables (cont’d)
 Quantitative
variables are measured by
real numbers
• e.g., numerical measurement
 Qualitative
variables are categorical
• e.g., hair color
68
Variables (cont’d)
 Independent
variables are measured
directly
• e.g., the height of a box
 Dependent
variables can only be measured
once other independent variables are
measured
• e.g., the volume of a box (requires length,
width, and height)
69
Populations
 A population
is the entire collection of a
particular set of random variables
 The nature of a population is described by
its distribution
• The median of a distribution is the point where
half the observations lie on either side
• The mode is the value in a distribution that
occurs most frequently
70
Populations (cont’d)
 A distribution
can have skewness
• There is more dispersion on one side of the
distribution
• Positive skewness means the mean is greater
than the median
– Stock returns are positively skewed
• Negative skewness means the mean is less than
the median
71
Populations (cont’d)
Positive Skewness
Negative Skewness
72
Populations (cont’d)
 A binomial
distribution contains only two
random variables
• e.g., the toss of a coin (heads or tails)
 A finite
population is one in which each
possible outcome is known
• e.g., a card drawn from a deck of cards
73
Populations (cont’d)
 An
infinite population is one where not all
observations can be counted
• e.g., the microorganisms in a cubic mile of
ocean water
 A univariate
population has one variable
of interest
74
Populations (cont’d)
 A bivariate
population has two variables
of interest
• e.g., weight and size
 A multivariate
population has more than
two variables of interest
• e.g., weight, size, and color
75
Samples
 A sample
is any subset of a population
• e.g., a sample of past monthly stock returns of a
particular stock
76
Sample Statistics
 Sample
statistics are characteristics of
samples
• A true population statistic is usually
unobservable and must be estimated with a
sample statistic
– Expensive
– Statistically unnecessary
77
Properties of
Random Variables
 Example
 Central
Tendency
 Dispersion
 Logarithms
 Expectations
 Correlation and Covariance
78
Example
Assume the following monthly stock returns for Stocks A
and B:
Month
Stock A
Stock B
1
2
3
2%
–1%
4%
3%
0%
5%
4
1%
4%
79
Central Tendency
Central tendency is what a random variable
looks like, on average
 The usual measure of central tendency is the
population’s expected value (the mean)

• The average value of all elements of the population
1
~
E ( Ri ) 
n
n

~
Ri
i 1
80
Example (cont’d)
The expected returns for Stocks A and B are:
1
~
E ( RA ) 
n
1
~
E ( RB ) 
n
n

1
~
R A  (2%  1%  4%  1%)  1.50 %
4
i 1
n

1
~
RB  (3%  0%  5%  4%)  3.00 %
4
i 1
81
Dispersion
 Investors
are interested in the best and the
worst in addition to the average
 A common measure of dispersion is the
variance or standard deviation
82
Example (cont’d)
The variance and standard deviation for Stock A are:


1
 (2%  1.5%)
4
 2  E ( ~xi  x ) 2

2
 (1%  1.5%) 2  (4%  1.5%) 2  (1%  1.5%) 2
1
(0.0013 )  0.000325
4
   2  0.000325  0.018  1.8%
83

Example (cont’d)
The variance and standard deviation for Stock B are:


1
 (3%  3.0%)
4
 2  E ( ~xi  x ) 2

2
 (0%  3.0%) 2  (5%  3.0%) 2  (4%  3.0%) 2

1
(0.0014 )  0.00035
4
   2  0.00035  0.0187  1.87 %
84
Logarithms
 Logarithms
reduce the impact of extreme
values
• e.g., takeover rumors may cause huge price
swings
• A logreturn is the logarithm of a return relative
 Logarithms
make other statistical tools
more appropriate
• e.g., linear regression
85
Logarithms (cont’d)
 Using
logreturns on stock return
distributions:
• Take the raw returns
• Convert the raw returns to return relatives
• Take the natural logarithm of the return
relatives
86
Expectations
 The
expected value of a constant is a
constant:
E (a )  a
 The
expected value of a constant times a
random variable is the constant times the
expected value of the random variable:
E (ax)  aE ( x)
87
Expectations (cont’d)
 The
expected value of a combination of
random variables is equal to the sum of the
expected value of each element of the
combination:
E ( x  y )  E ( x)  E ( y )
88
Correlations and Covariance
 Correlation
is the degree of association
between two variables
 Covariance
is the product moment of two
random variables about their means
 Correlation
and covariance are related and
generally measure the same phenomenon
89
Correlations and Covariance
(cont’d)
COV ( A, B)   AB  E ( A  A)( B  B ) 
 AB 
COV ( A, B)
 A B
90
Example (cont’d)
The covariance and correlation for Stocks A and B are:
 AB
1
  (0.5%  0.0%)  (2.5%  3.0%)  (2.5%  2.0%)  (0.5% 1.0%) 
4
1
 (0.001225)
4
 0.000306
 AB 
COV ( A, B)
 A B
0.000306

 0.909
(0.018)(0.0187)
91
Correlations and Covariance
(cont’d)
 Correlation
ranges from –1.0 to +1.0.
• Two random variables that are perfectly
positively correlated have a correlation
coefficient of +1.0
• Two random variables that are perfectly
negatively correlated have a correlation
coefficient of –1.0
92
Linear Regression
 Linear
regression is a mathematical
technique used to predict the value of one
variable from a series of values of other
variables
• e.g., predict the return of an individual stock
using a stock market index
 Linear
regression finds the equation of a
line through the points that gives the best
possible fit
93
Linear Regression (cont’d)
Example
Assume the following sample of weekly stock and stock
index returns:
Week
Stock Return
Index Return
1
2
0.0084
–0.0045
0.0088
–0.0048
3
4
0.0021
0.0000
0.0019
0.0005
94
Linear Regression (cont’d)
Return (Stock)
Example (cont’d)
0.01
Intercept = 0
0.008
Slope = 0.96
R squared = 0.99 0.006
0.004
0.002
0
-0.01
-0.005 -0.002 0
0.005
0.01
-0.004
-0.006
Return (Market)
95
R Squared and
Standard Errors
 Application
R
Squared
 Standard Errors
96
Application
 R-squared
and the standard error are used to
assess the accuracy of calculated statistics
97
R Squared

R squared is a measure of how good a fit we get
with the regression line
• If every data point lies exactly on the line, R squared is
100%

R squared is the square of the correlation
coefficient between the security returns and the
market returns
• It measures the portion of a security’s variability that is
due to the market variability
98
Standard Errors
 The
standard error is equal to the standard
deviation divided by the square root of the
number of observations:
Standard error 

n
99
Standard Errors (cont’d)
 The
standard error enables us to determine
the likelihood that the coefficient is
statistically different from zero
• About 68 percent of the elements of the
distribution lie within one standard error of the
mean
• About 95 percent lie within 1.96 standard errors
• About 99 percent lie within 3.00 standard errors
100