Class 7
Portfolio Analysis
Risk and Uncertainty
n
n
n
Almost all business decisions are made in
the face of risk and uncertainty.
So far we have side-stepped the issue of risk
and uncertainty, except to say that
investments with greater risk should have
higher required returns.
A full consideration of risk and uncertainty
requires a statistical framework for thinking
about these issues.
Random Variables
n
A random variable is a quantity whose
outcome is not yet known.
 The high temperature on next July 1st.
 The total points scored in the next Super Bowl.
 The rate of return on the S&P500 Index over
the next year.
 The cash flows on an investment project being
considered by a firm.
Probability Distributions
n
A probability distribution summarizes the
possible outcomes and their associated
probabilities of occurrence.
 Probabilities cannot be negative and must sum to 1.0
across all possible outcomes.
 Example: Tossing a fair coin.
Outcome
Heads
Tails
Probability
50%
50%
Summary Statistics
n
Mean or Average Value
 Measures the expected outcome.
n
Variance and Standard Deviation
 Measures the dispersion of possible outcomes.
n
Covariance and Correlations
 Measures the comovement of two random
variables.
Calculating the Mean
n
n
Means or expected values are useful for
telling us what is likely to happen on
average.
The mean is a weighted average.
 List the possible outcomes.
 For each outcome, find its probability of
occurrence.
 Weight the outcomes by their probabilities and
add them up.
Calculating the Mean
n
The formula for calculating the mean is:
n
E[ X ]   X 
 X p( X )
i
i 1
i
Calculating the Mean Example
n
Suppose we flip a coin twice. The possible
outcomes are given in the table below.
Outcome
Head-Head
X
p(X)
Xp(X)
$1,000
.25
$250
Head-Tail
500
.25
125
Tail-Head
-300
.25
-75
Tail-Tail
-600
.25
-150
1.00
= $150
Total
Properties of Means
n
n
n
E(a) = a
where a is constant
E(X+Y) = E(X) + E(Y)
E(aX) = aE(X)
Calculating the Variance and
Standard Deviation
n
n
The variance and standard deviation measure the
dispersion or volatility.
The variance is a weighted average of the
squared deviations from the mean.
 Subtract the mean from each possible outcome.
 Square the difference.
 Weight each squared difference by the probability of
occurrence and add them up.
n
The standard deviation is the square root of the
variance.
Calculating the Variance and
Standard Deviation
n
The formulas for calculating the variance
and standard deviation are:
var( X )   2X 
n
2
[
X


]
 i X p( X i )
i 1
var( X )   2X  E [ X 2 ]   2X
SD( X )   X   2X
Calculating Variance and
Standard Deviation Example
n
What is the variance and standard deviation
of our earlier coin tossing example?
Outcome
[X-]2
p(X)
Head-Head
722,500
.25
180,625
Head-Tail
122,500
.25
30,625
Tail-Head
202,500
.25
50,625
Tail-Tail
562,500
.25
140,625
1.00
2402,500
Total
n
[X-]2p(X)
[402,500]1/2 634.43
Properties of Variances
n
n
n
n
n
var(a) = 0 where a is constant
var(aX) = a2var(X)
var(a+X) = var(X)
var(X+Y) = var(X)+var(Y)+2cov(X,Y)
var(aX+bY) = a2var(X)+b2var(Y)
+2ab[cov(X,Y)]
Probability Distribution
Graphically
• Both distributions have the same mean.
• One distribution has a higher variance.
Covariance and Correlation
n
The covariance and correlation measure the
extent to which two random variables move
together.
 If X and Y, move up and down together, then they
are positively correlated.
 If X and Y move in opposite directions, then they
are negatively correlated.
 If movements in X and Y are unrelated, then they
are uncorrelated.
Calculating the Covariance
n
The formula for calculating the covariance is:
n
n
cov( X , Y )    [ X i   X ][Y j   Y ] p( X i , Y j )
i 1 j 1
cov( X , Y )   XY  E [ XY ]   X  Y
Calculating the Correlation
n
The correlation of two random variables is
equal to the covariance divided by the product
of the standard deviations.
 XY
corr ( X , Y)  r XY 
 X Y
n
Correlations range between -1 and 1.
 Perfect positive correlation: rXY = 1.
 Perfect negative correlation: rXY = -1.
 Uncorrelated: rXY = 0.
Calculating Covariances and
Correlations
n
Consider the following two stocks:
p
X
Y
p[X-X][Y-Y]
Boom
0.25
-20%
20%
-.0075
Normal
0.50
40%
30%
.03
Bust
0.25
-20%
-40%
.0375
=.10
=.10
.300 =.292
XY=.06
rXY=0.685
Properties of Covariances
n
n
n
cov(X+Y,Z) = cov(X,Z) + cov(Y,Z)
cov(a,X) = 0
cov(aX,bY) = ab[cov(X,Y)]
Risk Aversion
n
n
An individual is said to be risk averse if he
prefers less risk for the same expected
return.
Given a choice between $C for sure, or a
risky gamble in which the expected payoff
is $C, a risk averse individual will choose
the sure payoff.
Risk Aversion
n
Individuals are generally risk averse when it
comes to situations in which a large fraction
of their wealth is at risk.
 Insurance
 Investing
n
What does this imply about the relationship
between an individual’s wealth and utility?
Relationship Between Wealth and
Utility
Utility Function
Utility
Wealth
Risk Aversion Example
n
Suppose an individual has current wealth of
W0 and the opportunity to undertake an
investment which has a 50% chance of
earning x and a 50% chance of earning -x.
Is this an investment the individual would
voluntarily undertake?
Risk Aversion Example
U
U (W0 + x )
U (W0 )
u
d
U (W0  x)
W0  x
W0
W0 + x
W
Implications of Risk Aversion
n
n
n
Individuals who are risk averse will try to avoid
“fair bets.” Hedging can be valuable.
Risk averse individuals require higher expected
returns on riskier investments.
Whether an individual undertakes a risky
investment will depend upon three things:
 The individual’s utility function.
 The individual’s initial wealth.
 The payoffs on the risky investment relative to
those on a riskfree investment.
Diversification: The Basic Idea
n
n
n
Construct portfolios of securities that offer
the highest expected return for a given level
of risk.
The risk of a portfolio will be measured by
its standard deviation (or variance).
Diversification plays an important role in
designing efficient portfolios.
Measuring Returns
n
The rate of return on a stock is measured as:
Pt  Pt 1 + Dt
rt 
Pt
n
n
Expected return on stock j = E(rj)
Standard deviation on stock j = j
Measuring Portfolio Returns
n
The rate of return on a portfolio of stocks is:
N
rp 
x r
j j
j 1
n
xj = fraction of the portfolio’s total value invested
in stock j.
 xj > 0 is a long position.
 xj < 0 is a short position.
 Sj x j = 1
Measuring Portfolio Returns
n
The expected rate of return on a portfolio of
stocks is:
N
E ( rp ) 
 x E (r )
j
j
j 1
n
The expected rate of return on a portfolio is
a weighted average of the expected rates of
return on the individual stocks.
Measuring Portfolio Risk
n
n
The risk of a portfolio is measured by its
standard deviation or variance.
The variance for the two stock case is:
var( rp )    x  + x  + 2 x1x2 12
2
p
2
1
2
1
2
2
2
2
or, equivalently,
var(rp )   2p  x12 12 + x22 22 + 2 x1 x2 r 12 1 2
Minimum Variance Portfolio
n
n
Sometimes we are interested in the portfolio that
gives the smallest possible variance. We call
this the global minimum-variance portfolio.
For the two stock case, the global minimum
variance portfolio has the following portfolio
weights:
 22  r 12 1 2
x1  2
 1 +  22  2 r 12 1 2
x 2  1 x1
Two Asset Case
E[r]
E[r1]
Asset 1
E[r2]
Asset 2
2
1

Two Asset Case
n
n
We want to know where the portfolios of
stocks 1 and 2 plot in the risk-return diagram.
We shall consider three special cases:
 r12 = -1
 r12 = 1
 1<r12 < 1
Perfect Negative Correlation
n
n
With perfect negative correlation, r12 = -1, it is
possible to reduce portfolio risk to zero.
The global minimum variance portfolio has a
variance of zero. The portfolio weights for the
global minimum variance portfolio are:
2
x1 
1 + 2
x2  1  x1
Perfect Negative Correlation
E[r]
Zero-variance
portfolio
E[r1]
Asset 1
E[rp]
Portfolio of
mostly Asset 1
E[r2]
Asset 2
Portfolio of
mostly Asset 2
0
2
1

Example
n
Suppose you are considering investing in a
portfolio of stocks 1 and 2.
Stock
1
2
n
Expected Standard
Return
Deviation
20%
40%
12%
20%
Assume r12 = -1. What is the expected return and
standard deviation of a portfolio with equal
weights in each stock?
Example
n
Expected Return
E ( rp )  (.5)(20%) + (.5)(12%)  16%
n
Standard Deviation
var( rp )  (.5) 2 (.4) 2 + (.5) 2 (.2) 2  2(.5)(.5)(.4)(.2)
var( rp ) .01
Sd ( rp )  .01 .10
Example
n
n
What are the portfolio weights, expected return,
and standard deviation for the global minimum
variance portfolio?
Portfolio Weights
2
.20
x1 

.33
 1 +  2 .40+.20
x 2  1  x1  1.33 .67
Example
n
Expected Return
E ( rp )  .3320%  + .6712%   14.67%
n
Standard Deviation
var( rp )  (.33) (.4) + (.67) (.2)  2(.33)(.67)(.4)(.2)
2
var( rp )  0
Sd ( rp )  0
2
2
2
Perfect Positive Correlation
E[r]
Minimum-variance
portfolio
E[r1]
Asset 1
E[rp]
E[r2]
Asset 2
Portfolio of
mostly Asset 2
0
2
Portfolio of
mostly Asset 1
1

Perfect Positive Correlation
n
n
With perfect positive correlation, r12 = 1, there
are no benefits to diversification. This means
that it is not possible to reduce risk without also
sacrificing expected return.
Portfolios of stocks 1 and 2 lie along a straight
line running through stocks 1 and 2.
Perfect Positive Correlation
n
n
With perfect positive correlation, r12 = 1, it is
still possible to reduce portfolio risk to zero, but
this requires a short position in one of the
assets.
The portfolio weights for the global minimum
variance portfolio are:
2
x1 
 2 1
x 2  1 x1
Example
n
Consider again stocks 1 and 2.
Stock
1
2
n
Expected Standard
Return
Deviation
20%
40%
12%
20%
Assume now that r12 = 1. What is the expected
return and standard deviation of an equallyweighted portfolio of stocks 1 and 2?
Example
n
Expected Return
E (rp )  (.5)(20%) + (.5)(12%)  16%
n
Standard Deviation
var( rp )  (.5) 2 (.4) 2 + (.5) 2 (.2) 2 + 2(.5)(.5)(.4)(.2)
var( rp ) .09
Sd ( rp )  .09 .30
Example
n
n
What are the portfolio weights, expected return,
and standard deviation of the global minimum
variance portfolio?
Portfolio Weights
.20
x1 
 10
.
.20.40
x 2  1  ( 10
. )  2.0
Example
n
Expected Return
E ( rp )  ( 1.0)( 20%) + (2.0)(12%)  4.0%
n
Standard Deviation
var( rp )  ( 1) (.4) + (2) (.2) + 2( 1)(2)(.4)(.2)
2
var( rp )  0
Sd ( rp )  0
2
2
2
Non-Perfect Correlation
E[r]
Minimum-variance
portfolio
E[r1]
Asset 1
E[rp]
Portfolio of
mostly Asset 1
E[r2]
Portfolio of
mostly Asset 2
0
2
Asset 2
1

Non-Perfect Correlation
With non-perfect correlation, -1<r12<1,
diversification helps reduce risk, but risk cannot
be eliminated completely.
n Most stocks have positive, but non-perfect
correlation with each other.
n The global minimum variance portfolio will have
a lower variance than either asset 1 or asset 2 if:
r < 2/1,
where 2 <1.
n
Example
n
Consider again stocks 1 and 2.
Stock
1
2
n
Expected Standard
Return
Deviation
20%
40%
12%
20%
Assume now that r12 = .25. What is the
expected return and standard deviation of an
equally-weighted portfolio of stocks 1 and 2?
Example
n
Expected Return
E (rp )  (.5)(20%) + (.5)(12%)  16%
n
Standard Deviation
var( rp )  (.5) (.4) + (.5) (.2) + 2(.5)(.5)(.25)(.4)(.2)
2
2
var( rp ) .06
Sd ( rp )  .06  24.49%
2
2
Example
n
n
What are the portfolio weights, expected return,
and standard deviation of the global minimum
variance portfolio?
Portfolio Weights
(.2) 2  (.25)(.4)(.2)
x1 
 12.5%
2
2
(.4) + (.2)  2(.25)(.4)(.2)
x 2  1  (.125)  87.5%
Example
n
Expected Return
E (rp )  (.125)(20%) + (.875)(12%)  13.0%
n
Standard Deviation
var( rp )  (.125) 2 (.4) 2 + (.875) 2 (.2) 2
+2(.125)(.875)(.25)(.4)(.2)
var( rp ) .0375
Sd ( rp )  .0375  19.36%
Multiple Assets
n
The variance of a portfolio consisting of N
risky assets is calculated as follows:
N
N
var(rp )    x j x k  ij
j 1 k 1
N
N
N
var(rp )   x 2j  2j +   x j x k 
j 1
j 1 k 1
k j
jk
Limits to Diversification
n
Consider an equally-weighted portfolio. The
variance of such a portfolio is:

1I
F
  GJ
HN K
2
N
2
p
i 1
1I
F
+   GJ
H
NK



N
2
i
N
i 1
2
ij
j 1
i j
L
M
N
O
F
G
P
QH
1 Average
1
 
+ 1
N Variance
N
2
p
Average O
IJL
P
KM
Coariance
N
Q
Limits to Diversification
n
As the number of stocks gets large, the variance of
the portfolio approaches:
var( rp )  cov
n
The variance of a well-diversified portfolio is
equal to the average covariance between the
stocks in the portfolio.
Limits to Diversification
n
What is the expected return and standard deviation
of an equally-weighted portfolio, where all stocks
have E(rj) = 15%, j = 30%, and rij = .40?
N
1
10
25
50
100
1000
xj=1/N
1.00
0.10
0.04
0.02
0.01
0.001
E(rp)
15%
15%
15%
15%
15%
15%
p
30.00%
20.35%
19.53%
19.26%
19.12%
18.99%
Limits to Diversification
Portfolio Risk, 
Total Risk
Average
Covariance
Firm-Specific Risk
Market Risk
Number of Stocks
Examples of Firm-Specific Risk
n
n
n
n
A firm’s CEO is killed in an auto accident.
A wildcat strike is declared at one of the
firm’s plants.
A firm finds oil on its property.
A firm unexpectedly wins a large
government contract.
Examples of Market Risk
n
n
n
n
Long-term interest rates increase unexpectedly.
The Fed follows a more restrictive monetary
policy.
The U.S. Congress votes a massive tax cut.
The value of the U.S. dollar unexpectedly
declines relative to other currencies.
Efficient Portfolios with
Multiple Assets
E[r]
Efficient
Frontier
Asset
Portfolios
Asset 1
of other
Portfolios of
assets
2 Asset 1 and Asset 2
Minimum-Variance
Portfolio
0

Efficient Portfolios with
Multiple Assets
n
n
n
With multiple assets, the set of feasible portfolios
is a hyperbola.
Efficient portfolios are those on the thick part of
the curve in the figure. They offer the highest
expected return for a given level of risk.
Assuming investors want to maximize expected
return for a given level of risk, they should hold
only efficient portfolios.
Common Sense Procedures
n
n
n
n
Hold a well-diversified portfolio.
Invest in stocks in different industries.
Invest in both large and small company stocks.
Diversify across asset classes.
 Stocks
 Bonds
 Real Estate
n
Diversify internationally.