Transcript ch02

Chapter 2
Valuation, Risk, Return, and
Uncertainty
1
Introduction
 Introduction
 Safe
Dollars and Risky Dollars
 Relationship Between Risk and Return
 The Concept of Return
 Some Statistical Facts of Life
2
Safe Dollars and Risky Dollars
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
3
Safe Dollars and
Risky Dollars (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
4
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?
5
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
6
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.
7
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.
8
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
9
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
10
Dispersion and Chance of Loss
(cont’d)
Investment value
Investment A
Investment B
Time
11
Dispersion and Chance of Loss
(cont’d)
 Investments
A and B have the same
arithmetic mean
 Investment
B is riskier than Investment A
12
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.
13
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”
14
Diminishing Marginal
Utility of Money (cont’d)
Utility
$
15
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?
16
St. Petersburg Paradox
(cont’d)
Number of Tails
Before First
Head
0
Probability
(1/2) = 1/2
Payoff
$1
Probability
× Payoff
$0.50
1
2
(1/2)2 = 1/4
(1/2)3 = 1/8
$2
$4
$0.50
$0.50
3
4
n
(1/2)4 = 1/16
(1/2)5 = 1/32
(1/2)n + 1
$8
$16
$2n
$0.50
$0.50
$0.50
Total
1.00

17
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
18
The Concept of Return
 Measurable
return
 Expected return
 Return on investment
19
Measurable Return
 Definition
 Holding
period return
 Arithmetic mean return
 Geometric mean return
 Comparison of arithmetic and geometric
mean returns
20
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%, compounded
continuously is worth $108.33 after one year
– The return is $8.33, or 8.33%
21
Holding Period Return

The calculation of a holding period return is
independent of the passage of time
Income  Capital Gain
Return 
Purchase price
• 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%
– The 11.58% could have been earned over one year or one week
22
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
23
Arithmetic Mean Return
(cont’d)
 Arithmetic
means are a useful proxy for
expected returns
 Arithmetic
means are not especially useful
for describing historical returns
• It is unclear what the number means once it is
determined
24
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
25
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
26
Arithmetic and Geometric
Mean Returns (cont’d)
Example (cont’d)
What is the arithmetic mean return?
Solution:
~
Ri
Arithmetic mean  
i 1 n
0.0084  0.0045  0.0021  0.0000

4
 0.0015
n
27
Arithmetic and Geometric
Mean Returns (cont’d)
Example (cont’d)
What is the geometric mean
return?
Solution:

~
Geometric mean   (1  Ri 
 i 1

n
1/ n
1
 1.0084  0.9955 1.00211.0000  1
 0.001489
1/ 4
28
Comparison of Arithmetic &
Geometric Mean Returns
 The
geometric mean reduces the likelihood
of nonsense answers
• Assume a $100 investment falls by 50% in
period 1 and rises by 50% in period 2
• The investor has $75 at the end of period 2
– Arithmetic mean = (-50% + 50%)/2 = 0%
– Geometric mean = (0.50 x 1.50)1/2 –1 = -13.40%
29
Comparison of Arithmetic &
Geometric Mean Returns
 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 and geometric mean
30
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
31
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
32
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
33
Standard Deviation and
Variance (cont’d)
 General
equation for variance:
2
n
Variance     prob( xi )  xi  x 
2
i 1
 If
all outcomes are equally likely:
n
2
1
    xi  x 
n i 1
2
34
Standard Deviation and
Variance (cont’d)
 Equation
for standard deviation:
Standard deviation     2 
2
n
 prob( x )  x  x 
i 1
i
i
35
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
36
Some Statistical Facts of Life
 Definitions
 Properties
of random variables
 Linear regression
 R squared and standard errors
37
Definitions
 Constants
 Variables
 Populations
 Samples
 Sample
statistics
38
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
39
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
40
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
41
Variables (cont’d)
 Quantitative
variables are measured by real
numbers
• E.g., numerical measurement
 Qualitative
variables are categorical
• E.g., hair color
42
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)
43
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
44
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
45
Populations (cont’d)
Positive Skewness
Negative Skewness
46
Populations (cont’d)
A
binomial distribution contains only two
random variables
• E.g., the toss of a coin
A
finite population is one in which each
possible outcome is known
• E.g., a card drawn from a deck of cards
47
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
48
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
49
Samples
A
sample is any subset of a population
• E.g., a sample of past monthly stock returns of
a particular stock
50
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
51
Properties of
Random Variables
 Example
 Central
tendency
 Dispersion
 Logarithms
 Expectations
 Correlation and covariance
52
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%
53
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 n
E ( Ri )   Ri
n i 1
54
Example (cont’d)
The expected returns for Stocks A and B are:
1 n
1
E ( RA )   Ri  (2%  1%  4%  1%)  1.50%
n i 1
4
1 n
1
E ( RB )   Ri  (3%  0%  5%  4%)  3.00%
n i 1
4
55
Dispersion
 Investors
are interest in the best and the
worst in addition to the average
 A common measure of dispersion is the
variance or standard deviation
  E  xi  x  
2
2


    E  xi  x  
2
2


56
Example (cont’d)
The variance ad standard deviation for Stock A are:
2
 2  E  xi  x  


1
(2%  1.5%) 2  (1%  1.5%) 2  (4%  1.5%) 2  (1%  1.5%) 2 
4
1
 (0.0013)  0.000325
4

   2  0.000325  0.018  1.8%
57
Example (cont’d)
The variance ad standard deviation for Stock B are:
2
 2  E  xi  x  


1
(3%  3.0%)2  (0%  3.0%)2  (5%  3.0%)2  (4%  3.0%)2 
4
1
 (0.0014)  0.00035
4

   2  0.00035  0.0187  1.87%
58
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
 Logarithms
make other statistical tools
more appropriate
• E.g., linear regression
59
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
60
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)
61
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 )
62
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
63
Correlations and Covariance
(cont’d)
COV ( A, B)   AB  E ( A  A)( B  B ) 
 AB 
COV ( A, B)
 A B
64
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)
65
Correlations and Covariance
 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
66
A
B
C
D
E
F
G
H
I
J
K
CORRELATION +1
Adams Farm and Morgan Sausage Stocks
2
Year
3 1990
4 1991
5 1992
6 1993
7 1994
8 1995
9 1996
10 1997
11 1998
12 1999
13
14 Correlation
15
rMorgan Sausage,t = 3% + 0.6*rAdams Farm,t
Adams
Farm stock
return
30.73%
55.21%
15.82%
33.54%
14.93%
35.84%
48.39%
37.71%
67.85%
44.85%
Morgan
Sausage
stock
return
21.44% <-- =3%+0.6*B3
36.13%
12.49%
23.12%
11.96%
24.50%
32.03%
25.63%
43.71%
29.91%
1.00 <-- =CORREL(B3:B12,C3:C12)
Annual Stock Returns, Adams Farm and Morgan
Sausage
50%
45%
40%
Morgan Sausage
1
35%
30%
25%
20%
15%
10%
5%
0%
10%
20%
30%
40%
50%
Adams Farm
60%
70%
67
A
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
B
C
D
E
F
G
H
I
CALCULATING THE RETURNS
Month
0
1
2
3
4
5
6
7
8
9
10
11
12
Stock A
Price
Return
25.00
24.12
-3.58%
23.37
-3.16%
24.75
5.74%
26.62
7.28%
26.50
-0.45%
28.00
5.51%
28.88
3.09%
29.75
2.97%
31.38
5.33%
36.25
14.43%
37.13
2.40%
36.88
-0.68%
Monthly mean
Monthly variance
Monthly stand. dev.
3.24%
0.23%
4.78%
Annual mean
Annual variance
Annual stand. dev.
38.88%
2.75%
16.57%
Stock B
Price
Return
45.00
44.85
-0.33%
46.88
4.43% <-- =LN(E23/E22)
45.25
-3.54%
50.87
11.71%
53.25
4.57%
53.25
0.00%
62.75
16.42%
65.50
4.29%
66.87
2.07%
78.50
16.03%
78.00
-0.64%
68.23 -13.38%
3.47% <-- =AVERAGE(F22:F33)
0.65% <-- =VARP(F22:F33)
8.03% <-- =STDEVP(F22:F33)
41.62% <-- =12*F35
7.75% <-- =12*F36
27.83% <-- =SQRT(F40)
68
A
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
B
C
D
E
COVARIANCE AND VARIANCE CALCULATION
Stock A
Stock B
Return Return-mean
Return Return-mean
-0.0358
-0.0316
0.0574
0.0728
-0.0045
0.0551
0.0309
0.0297
0.0533
0.1443
0.0240
-0.0068
-0.0682
-0.0640
0.0250
0.0404
-0.0369
0.0227
-0.0015
-0.0027
0.0209
0.1119
-0.0084
-0.0392
-0.0033
0.0443
-0.0354
0.1171
0.0457
0.0000
0.1642
0.0429
0.0207
0.1603
-0.0064
-0.1338
-0.0380
0.0096
-0.0701
0.0824
0.0110
-0.0347
0.1295
0.0082
-0.0140
0.1257
-0.0411
-0.1685
Covariance
Correlation
F
G
H
I
J
=D48-$F$35
Product
0.00259 <-- =E48*B48
-0.00061
-0.00175
0.00333
-0.00041
-0.00079
-0.00019
-0.00002
-0.00029
0.01406
0.00035
0.00660
0.00191
0.00191
0.49589
0.49589
<-- =AVERAGE(G48:G59)
<-- =COVAR(A48:A59,D48:D59)
<-- =G62/(F37*C37)
<-- =CORREL(A48:A59,D48:D59)
69
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
 Regression
finds the equation of a line
through the points that gives the best
possible fit
70
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
71
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)
72
R Squared and
Standard Errors
 Application
R
squared
 Standard Errors
73
Application
 R-squared
and the standard error are used
to assess the accuracy of calculated
statistics
74
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
75
A
C
D
E
F
G
H
I
J
K
L
SIMPLE REGRESSION EXAMPLE IN EXCEL
Date
Jan-97
Feb-97
Mar-97
Apr-97
May-97
Jun-97
Jul-97
Aug-97
Sep-97
Oct-97
Nov-97
Dec-97
Jan-98
Feb-98
Mar-98
Apr-98
May-98
Jun-98
Jul-98
Aug-98
Sep-98
Oct-98
Nov-98
Dec-98
S&P 500
Mirage
Index
Resorts
SPX
MIR
6.13%
16.18%
0.59%
0.00%
-4.26% -15.42%
5.84%
-5.29%
5.86%
18.63%
4.35%
5.76%
7.81%
5.94%
-5.75%
0.23%
5.32%
12.35%
-3.45% -17.01%
4.46%
-5.00%
1.57%
-4.21%
1.02%
1.37%
7.04%
-0.54%
4.99%
5.99%
0.91%
-9.25%
-1.88%
-5.67%
3.94%
2.40%
-1.16%
0.88%
-14.58% -30.81%
6.24%
12.61%
8.03%
1.12%
5.91% -12.18%
5.64%
0.42%
MIR Returns vs S&P500 Returns
30%
Monthly Returns, 1997-1998
MIR
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
B
20%
10%
0%
-20%
-15%
-10%
-5%
-10%
0%
5%
10%
S&P500
-20%
-30%
-40%
Slope
Intercept
y = 1.4693x - 0.0424
R2 = 0.5001
1.469256 <-- =SLOPE(C3:C26,B3:B26)
1.469256 <-- =COVAR(C3:C26,B3:B26)/VARP(B3:B26)
-0.042365 <-- =INTERCEPT(C3:C26,B3:B26)
-0.042365 <-- =AVERAGE(C3:C26)-B28*AVERAGE(B3:B26)
R-squared 0.500072 <-- =RSQ(C3:C26,B3:B26)
0.500072 <-- =CORREL(C3:C26,B3:B26)^2
76
Standard Errors
 The
standard error is the standard deviation
divided by the square root of the number of
observations:
Standard error 

n
77
Standard Errors (cont’d)
 The
standard error enables us to determine
the likelihood that the coefficient is
statistically different from zero
• About 68% of the elements of the distribution
lie within one standard error of the mean
• About 95% lie within 1.96 standard errors
• About 99% lie within 3.00 standard errors
78
Runs Test
A
runs test allows the statistical testing of
whether a series of price movements
occurred by chance.
 A run is defined as an uninterrupted
sequence of the same observation. Ex: if the
stock price increases 10 times in a row, then
decreases 3 times, and then increases 4
times, we then say that we have three runs.
79
Notation
R = number of runs (3 in this example)
 n1 = number of observations in the first category.
For instance, here we have a total of 14 “ups”, so
n1=14.
 n2 = number of observations in the second
category. For instance, here we have a total of 3
“downs”, so n2=3.
 Note that n1 and n2 could be the number of
“Heads” and “Tails” in the case of a coin toss.

80
Statistical Test
The z statistic computed is:
Rx
z
(thus z is a standard normal variable)

where
2n1n2
x
1
n1  n2
2n1n2 (2n1n2  n1  n2 )
 
(n1  n2 ) 2 (n1  n2  1)
2
81
Example
 Let
the number of runs R=23
 Let the number of ups n1=20
 Let the number of downs n2=30
Then the mean number of runs x  25
The standard deviation   3.36
Yielding a z statistic of: z  0.595
82
About 2.5% of the area under the
normal curve is below a z score of 1.96.
83
Interpretation
 Since
our z-score is not in the lower tail
(nor is it in the upper tail), the runs we have
witnessed are purely the product of chance.
 If, on the other hand, we had obtained a zscore in the upper (2.5%) or lower (2.5%)
tail, we would then be 95% certain that this
specific occurrence of runs didn’t happen
by chance. (Or that we just witnessed an
extremely rare event)
84