Transcript Document

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
Outline
Introduction
 Valuation
 Safe Dollars and Risky Dollars
 Relationship Between Risk and Return
 The Concept of Return
 Some Statistical Facts of Life

2
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?
3
Valuation
Introduction
 Growing Income Streams

4
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
5
Growing Income Streams
Definition
 Growing Annuity
 Growing Perpetuity

6
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
7
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  
8
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

R
)
t 1

t 1
C1

Rg
9
Safe Dollars and Risky
Dollars
Introduction
 Choosing Among Risky Alternatives
 Defining Risk

10
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
11
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
12
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?
13
Choosing Among
Risky Alternatives (cont’d)
A
[1–50]
[51–100]
Average
payoff
B
$110 [1–50]
$90 [51–100]
$100
C
$200 [1–90]
$0 [91–
100]
$100
D
$50 [1–99]
$550 [100]
$100
$1,000
–
$89,000
$100
Number on lottery wheel appears in brackets.
14
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.
15
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.
16
Defining Risk
Risk Versus Uncertainty
 Dispersion and Chance of Loss
 Types of Risk

17
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
18
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
19
Dispersion and Chance of
Loss
(cont’d)
Investment value
Investment A
Investment B
Time
20
Dispersion and Chance of
Loss (cont’d)

Investments A and B have the same
arithmetic mean

Investment B is riskier than Investment A
21
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
22
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
23
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

24
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

25
Direct Relationship (cont’d)
Expected return
Rf
0
Risk
26
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
27
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.

28
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”
29
Diminishing Marginal
Utility of Money (cont’d)
Utility
$
30
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?
31
St. Petersburg Paradox
(cont’d)
Number of
Tails Before
First
Head
0
Probability
Payoff
Probability
× Payoff
(1/2) = 1/2
$1
$0.50
1
2
3
4
(1/2)2 = 1/4
(1/2)3 = 1/8
(1/2)4 = 1/16
(1/2)5 = 1/32
$2
$4
$8
$16
$0.50
$0.50
$0.50
$0.50
n
Total
(1/2)n + 1
1.00
$2n
$0.50

32
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
33
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
34
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
35
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
36
Other Considerations
Psychic Return
 Price Risk versus Convenience Risk

37
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
38
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
39
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
40
The Concept of Return
Introduction
 Measurable Return
 Return on Investment

41
Introduction

“Return” can mean various things, and it is
important to be clear when discussing an
investment
42
Measurable Return
Definition
 Holding Period Return
 Arithmetic Mean Return
 Geometric Mean Return
 Comparison of Arithmetic and Geometric
Mean Returns
 Expected Return

43
Definition

A general definition of return is “the
benefit associated with an investment”
 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
44
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

45
Arithmetic Mean Return

The arithmetic mean return is the
arithmetic average of several holding
period returns measured over the same
holding period:
n ~
Ri
Arithmeticmean  
i 1 n
~
Ri  the rat eof ret urnin periodi
46
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
47
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
48
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
49
Arithmetic and Geometric
Mean Returns (cont’d)
Example (cont’d)
What is the arithmetic mean return?
n ~
Solution:
Ri
Arithmeticmean  
i 1 n
0.0084 0.0045 0.0021 0.0000

4
 0.0015
50
Arithmetic and Geometric
Mean Returns (cont’d)
Example (cont’d)
What is the geometric mean return?
1/ n
Solution:
n

~
Geometricmean   (1  Ri   1
 i 1

 1.0084 0.99551.00211.0000  1
1/ 4
 0.001489
51
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%
1/2
 Geometric mean = (0.50 × 1.50)
– 1 = –13.40%

52
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
53
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
54
Return on Investment (ROI)
Definition
 Measuring Total Risk

55
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
56
Measuring Total Risk
Standard Deviation and Variance
 Semi-Variance

57
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
58
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
59
Standard Deviation and
Variance (cont’d)

Equation for standard deviation:
Standard deviation     2 
2
n
 prob( x )  x  x 
i 1
i
i
60
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
61
Some Statistical Facts of Life
Definitions
 Properties of Random Variables
 Linear Regression
 R Squared and Standard Errors

62
Definitions
Constants
 Variables
 Populations
 Samples
 Sample Statistics

63
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
64
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
x
 Designated

by a tilde
e.g.,
65
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
66
Variables (cont’d)

Quantitative variables are measured by
real numbers
 e.g.,

numerical measurement
Qualitative variables are categorical
 e.g.,
hair color
67
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)
68
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
69
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
70
Populations (cont’d)
Positive Skewness
Negative Skewness
71
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
72
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
73
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
74
Samples

A sample is any subset of a population
 e.g.,
a sample of past monthly stock returns of
a particular stock
75
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

76
Properties of
Random Variables
Example
 Central Tendency
 Dispersion
 Logarithms
 Expectations
 Correlation and Covariance

77
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%
78
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
79
Example (cont’d)
The expected returns for Stocks A and B are:
1
~
E( RA ) 
n
1
~
E ( RB ) 
n
n

i 1
n

i 1
1
~
R A  (2%  1%  4%  1%)  1.50%
4
1
~
RB  (3%  0%  5%  4%)  3.00%
4
80
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

81
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%
82

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%
83
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
84
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
85
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)
86
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 )
87
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
88
Correlations and Covariance
(cont’d)
COV ( A, B)   AB  E ( A  A)( B  B ) 
 AB 
COV ( A, B)
 A B
89
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)
90
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
91
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
92
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
93
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)
94
R Squared and
Standard Errors
Application
 R Squared
 Standard Errors

95
Application

R-squared and the standard error are
used to assess the accuracy of calculated
statistics
96
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
97
Standard Errors

The standard error is equal to the
standard deviation divided by the square
root of the number of observations:
Standard error 

n
98
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
99
errors