Class 7

Portfolio Analysis

Risk and Uncertainty

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

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

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 Probability

Heads 50%

Tails 50%

Summary Statistics

Mean or Average Value Measures the expected outcome.

Variance and Standard Deviation Measures the dispersion of possible outcomes.

Covariance and Correlations Measures the comovement of two random

variables.

Calculating the Mean

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

The formula for calculating the mean is:

E X X p XX i i

i

n

[ ] ( )

1

Calculating the Mean Example

Outcome X p(X) Xp(X)

Head-Head $1,000 .25 $250

Head-Tail 500 .25 125

Tail-Head -300 .25 -75

Tail-Tail -600 .25 -150

Total 1.00 = $150

Suppose we flip a coin twice. The possible outcomes are given in the table below.

Properties of Means

E(a) = a where a is constant E(X+Y) = E(X) + E(Y) E(aX) = aE(X)

Calculating the Variance and Standard Deviation

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. The standard deviation is the square root of the

variance.

Calculating the Variance and Standard Deviation

The formulas for calculating the variance and standard deviation are:

var( ) [ ] ( )

var( ) [ ]

( )

X X p X

X E X

SD X

X ii

n

X i

X X

X X

2

1

2

2 2 2

2

Calculating Variance and Standard Deviation Example

What is the variance and standard deviation of our earlier coin tossing example?

=[402,500]s 1/2=634.43

Outcome [X-]2 p(X) [X-]2p(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

Total 1.00 2402500

Properties of Variances

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

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

The formula for calculating the covariance is:

cov( , ) [ ][ ] ( , )

cov( , ) [ ]

X Y X Y p X Y

X Y E XY

ij

n

i

n

X j Y i j

XY X Y

11

Calculating the Correlation

The correlation of two random variables is equal to the covariance divided by the product of the standard deviations.

Correlations range between -1 and 1. Perfect positive correlation: rXY = 1.

Perfect negative correlation: rXY = -1.

Uncorrelated: rXY = 0.

corr X Y XYXY

X Y

( , )

Calculating Covariances and Correlations

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 XY=.06300 =.292 XY=0.685

Properties of Covariances

cov(X+Y,Z) = cov(X,Z) + cov(Y,Z) cov(a,X) = 0 cov(aX,bY) = ab[cov(X,Y)]

Risk Aversion

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

Individuals are generally risk averse when it comes to situations in which a large fraction of their wealth is at risk. Insurance Investing

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

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

W

u

dU W( )0

U W x( )0 +

W x W W x0 0 0- +

U W x( )0 -

Implications of Risk Aversion

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

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

The rate of return on a stock is measured as:

Expected return on stock j = E(rj)

Standard deviation on stock j = sj

rP P D

Ptt t t

t

1

Measuring Portfolio Returns

The rate of return on a portfolio of stocks is:

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 xj = 1

r x rp j

j

N

j

1

Measuring Portfolio Returns

The expected rate of return on a portfolio of stocks is:

The expected rate of return on a portfolio is a weighted average of the expected rates of return on the individual stocks.

E r x E rp j

j

N

j( ) ( )

1

Measuring Portfolio Risk

The risk of a portfolio is measured by its standard deviation or variance.

The variance for the two stock case is:

or, equivalently,

var( )r x x x xp p 212

12

22

22

1 2 122

var( )r x x x xp p 212

12

22

22

1 2 12 1 22

Minimum Variance Portfolio 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:

x

x x

122

12 1 2

12

22

12 1 2

2 1

2

1

Two Asset Case

E[r]

E[r1]

E[r2]

2 1

Asset 1

Asset 2

Two Asset Case

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

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:

x

x x

12

1 2

2 11

Perfect Negative Correlation

E[r]

E[r1]

E[r2]

2 1

Asset 1

Asset 2

0

Zero-variance portfolio

E[rp] Portfolio ofmostly Asset 1

Portfolio of mostly Asset 2

Example Suppose you are considering investing in a

portfolio of stocks 1 and 2.

Assume r12 = -1. What is the expected return and standard deviation of a portfolio with equal weights in each stock?

Stock ExpectedReturn

StandardDeviation

1 20% 40%

2 12% 20%

Example

Expected Return

Standard Deviation

var( ) (. ) (. ) (. ) (. ) (. )(. )(. )(. )

var( ) .

( ) . .

r

r

Sd r

p

p

p

5 4 5 2 2 5 5 4 2

01

01 10

2 2 2 2

E rp( ) (. )( (. )( 5 20%) 5 12%) 16%

Example

What are the portfolio weights, expected return, and standard deviation for the global minimum variance portfolio?

Portfolio Weights

x

x x

12

1 2

2 1

20

40 2033

1 1 33 67

.

. ..

. .

Example

Expected Return

Standard Deviation

E rp( ) . . . 33 20% 67 12% 14 67%

var( ) (. ) (. ) (. ) (. ) (. )(. )(. )(. )

var( )

( )

r

r

Sd r

p

p

p

33 4 67 2 2 33 67 4 2

0

0

2 2 2 2

Perfect Positive Correlation

E[r]

E[r1]

E[r2]

2 1

Asset 2

0

Minimum-variance portfolio

E[rp]

Portfolio of mostly Asset 2

Asset 1

Portfolio ofmostly Asset 1

Perfect Positive Correlation

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

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:

x

x x

12

2 1

2 11

Example

Consider again stocks 1 and 2.

Assume now that r12 = 1. What is the expected return and standard deviation of an equally-weighted portfolio of stocks 1 and 2?

Stock ExpectedReturn

StandardDeviation

1 20% 40%

2 12% 20%

Example

Expected Return

Standard Deviation

E rp( ) (. )( (. )( 5 20%) 5 12%) 16%

var( ) (. ) (. ) (. ) (. ) (. )(. )(. )(. )

var( ) .

( ) . .

r

r

Sd r

p

p

p

5 4 5 2 2 5 5 4 2

09

09 30

2 2 2 2

Example

What are the portfolio weights, expected return, and standard deviation of the global minimum variance portfolio?

Portfolio Weights

x

x

1

2

20

20 4010

1 10 2 0

.

. ..

( . ) .

Example

Expected Return

Standard Deviation

E rp( ) ( . )( ( . )( . 10 20%) 2 0 12%) 4 0%

var( ) ( ) (. ) ( ) (. ) ( )( )(. )(. )

var( )

( )

r

r

Sd r

p

p

p

1 4 2 2 2 1 2 4 2

0

0

2 2 2 2

Non-Perfect Correlation

E[r]

E[r1]

E[r2]

2 1

Asset 2

0

Minimum-variance portfolio

E[rp]

Portfolio of mostly Asset 2

Asset 1

Portfolio ofmostly Asset 1

Non-Perfect Correlation

With non-perfect correlation, -1<r12<1, diversification helps reduce risk, but risk cannot be eliminated completely.

Most stocks have positive, but non-perfect correlation with each other.

The global minimum variance portfolio will have a lower variance than either asset 1 or asset 2 if:

r < s2/s1,

where s2<s1.

Example

Consider again stocks 1 and 2.

Assume now that r12 = .25. What is the expected return and standard deviation of an equally-weighted portfolio of stocks 1 and 2?

Stock ExpectedReturn

StandardDeviation

1 20% 40%

2 12% 20%

Example

Expected Return

Standard Deviation

E rp( ) (. )( (. )( 5 20%) 5 12%) 16%

var( ) (. ) (. ) (. ) (. ) (. )(. )(. )(. )(. )

var( ) .

( ) . .

r

r

Sd r

p

p

p

5 4 5 2 2 5 5 25 4 2

06

06 24 49%

2 2 2 2

Example

What are the portfolio weights, expected return, and standard deviation of the global minimum variance portfolio?

Portfolio Weights

x

x

1

2

2 2

2

2 25 4 2

4 2 2 25 4 212 5%

1 125 87 5%

(. ) (. )(. )(. )

(. ) (. ) (. )(. )(. ).

(. ) .

Example

Expected Return

Standard Deviation

E rp( ) (. )( (. )( . 125 20%) 875 12%) 13 0%

var( ) (. ) (. ) (. ) (. )

(. )(. )(. )(. )(. )

var( ) .

( ) . .

r

r

Sd r

p

p

p

125 4 875 2

2 125 875 25 4 2

0375

0375 19 36%

2 2 2 2

Multiple Assets

The variance of a portfolio consisting of N risky assets is calculated as follows:

var( )

var( )

r x x

r x x x

p j k ijk

N

j

N

p jj

N

j j kk

k j

N

j

N

jk

11

2

1

2

11

Limits to Diversification Consider an equally-weighted portfolio. The

variance of such a portfolio is:

p ii

N

i

N

j

N

i j

ijN N2

22

1 1 1

21 1

FHGIKJ F

HGIKJ

p N N2 1

11

LNM

OQP FHG

IKJLNM

OQP

Average

Variance

Average

Coariance

Limits to Diversification

As the number of stocks gets large, the variance of the portfolio approaches:

The variance of a well-diversified portfolio is equal to the average covariance between the stocks in the portfolio.

var( ) covrp

Limits to Diversification What is the expected return and standard deviation

of an equally-weighted portfolio, where all stocks have E(rj) = 15%, sj = 30%, and rij = .40?

N xj=1/N E(rp) p

1 1.00 15% 30.00%10 0.10 15% 20.35%25 0.04 15% 19.53%50 0.02 15% 19.26%

100 0.01 15% 19.12%1000 0.001 15% 18.99%

Limits to Diversification

Market Risk

Total Risk

Firm-Specific Risk

Portfolio Risk, s

Number of Stocks

Average

Covariance

Examples of Firm-Specific Risk

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

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]

s0

Asset 1

Asset 2Portfolios ofAsset 1 and Asset 2

Portfoliosof otherassets

EfficientFrontier

Minimum-VariancePortfolio

Efficient Portfolios with Multiple Assets

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

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

Diversify internationally.