1 portfolio analysis global financial management campbell r. harvey fuqua school of business duke...

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Portfolio Analysis

Global Financial Management

Campbell R. HarveyFuqua School of Business

Duke Universitycharvey@mail.duke.edu

http://www.duke.edu/~charvey

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Overview

Risk and risk aversion How to measure risk and return

» Risk measures for some classes of securities Diversification

» How to analyze the benefits from diversification» How to determine the trade-off between risk and return» Is there a limit to diversification

Minimum variance portfolios Portfolio analysis and hedging

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Toss two coins:

Outcome Gain Probability Exp. gain

H H +$600

H T +£100

T T - £400

Total

Which distribution do you prefer, safe or risky?

Risk and risk aversion

05

101520253035404550

T T H T H H

0102030405060708090

100

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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.

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?

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Relationship Between Wealth and Utility

Utility Suppose an individual has:» current wealth of W0

» 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?

Wealth

Utility Function

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Risk Aversion Example

U

W

u

dU W( )0

U W x( )0 +

W x W W x0 0 0- +

U W x( )0 -

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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.

Issues:» How do you measure risk?» How do you compare risk and return?

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US Equities: A Risky InvestmentS&P500, 1926-1995

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S&P500 Total Return %Total Return

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Equities: Distribution of Returns

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-50 -40 -30 -20 -10 0 10 20 30 40 50

Histogramm of distriubution of S&P 500 returns, 1926-1995

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Wealth IndicesUS 1926-1995

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0.1

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100

1000

10000

S&P 500

Small Company Stocks

Corporate Bonds

LT Govt Bond

IT Govt Bond

30Day TBills

Inflation

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How to put it into numbers We measure the average return RP on a portfolio in period t as:

where xj = fraction of the portfolio’s total value invested in stock j, j=1,…,N.» xj > 0 is a long position.

» xj < 0 is a short position; Sj xj = 1 Stock market indices:

» Equally weighted: x1=x2=…=xN=1/N

» Value weighted: xj= Proportion of market capitalization We measure the average return over the period as:

R x RPt j jtj

j N

1

RT

RP Ptt

t T

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Measuring Risk

The variance over time of a portfolio can be measured as:

Most of the time we shall refer to the standard deviation:

Var RT

R RP Pt Pt

t T

1

12

1

SD R Var RP Pt

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Equities: Monthly ReturnsJa

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S&P500 Total Return %Total Ret...

Mean: 1.00%Standard Deviation: 5.71%

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Bonds: Monthly ReturnsJa

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U.S. LT Gvt TR %Total Return

Mean: 0.44%Standard Deviation: 2.21%

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Risk and Return: Distributions

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-50 -40 -30 -20 -10 0 10 20 30 40 50 600

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-50 -40 -30 -20 -10 0 10 20 30 40 50 60

S&P 500SD=20.82

Small CompaniesSD=40.04

LT Government Bonds 30 Day Treasury Bills

SD=3.28SD=5.44

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Average Returns and Variabilities

Series GeometricMean

ArithmeticMean

RiskPremium

StandardDeviation

LargeStocks

10.23 12.26 8.52 20.82

SmallStocks

12.15 17.80 14.06 40.04

LT CorpBonds

5.44 5.74 2.00 8.32

LT GovtBonds

4.85 5.13 1.39 8.00

IT GovtBonds

5.11 5.24 1.50 5.44

US TBills 3.69 3.74 0.00 3.28Inflation 3.13 3.23 -0.51 4.68

Source: Ibbotson Associates/Own Calculations

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Returns and Variability

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0 5 10 15 20

Small Co’s

S&P 500

Govt ITTBills

Govt LTCorporate

Return

Variability

Variability is closely related to returns for portfolios

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InvestmentRisk Premium Variability

Stock market index 9 20

Typical individual share 9 30-40

The risk premium for individual shares is not closely related to their volatility.» Need to understand diversification

Individual Shares and the Stock Market:

A Paradox?

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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.

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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.

In the two-asset case:

E r x E r xP j jj

j njj

j n

where 1 1

1

E r x E r x E rP 1 1 1 21

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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

i

ij

212

12

22

22

1 2 12

2

2

Variance of asset i

Covariance of returns of assets i and j

var( )r x x x xp p

ij

212

12

22

22

1 2 12 1 22

Coefficient of correlation of the returns of i and j

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Fire Insurance PoliciesAn example of a two-asset portfolio

Asset 1: Your house, worth $100,000

Asset 2: Your fire insurance policy

Two states of the world :

State 1: Your house burns down and retains no value; the insurance policy pays out $100,000 (Prob. = 10%)

State 2: Your house does not burn down and retains its full value. The insurance policy does not pay out.

Question: What is the riskiness of each of these two assets individually, and together, if held as a portfolio?

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Payoffs:

State/Asset House Insurance Total

1 0 100,000 100,000

2 100,000 0 100,000

Insurance Policies: states and payoffs

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Risk Analysis:

Asset House Insurance Together

Expected 90,000 10,000 100,000

Payoff

Risk 30,000 30,000 0

Expected Values are additive, but Risk is not additive!

Perfect correlation gives perfect insurance.

Insurances: Risk Analysis

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Two Asset Case

E[r]

E[r1]

E[r2]

2 1

Asset 1

Asset 2

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Two Asset Case We want to know where the portfolios of stocks 1 and 2 plot in the

risk-return diagram.» Using (as before): xj = fraction of the portfolio’s total value

invested in stock j, j=1,2» xj > 0 is a long position.

» xj < 0 is a short position;

» x2 = 1- x1

We need to compute expectation and standard deviation of the portfolio:

We shall consider three special cases:» r12 = -1

» r12 = 1

» -1< r12 < 1

r x r x rP 1 1 2 2

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Minimum Variance Portfolio What is the upper limit for the benefits from diversification?

» Determine 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:

The variance of the global minimum-variance portfolio is:

Note: we have not excluded short-selling here! xi<0 is possible!

x x x122

12 1 2

12

22

12 1 22 1

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Var rP

12

22

122

12

22

12

1

2

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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:

Consider the following example

x x x12

1 22 1

1

1 2

1

Stock ExpectedReturn

StandardDeviation

1 20% 40%

2 12% 20%

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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

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Perfect Positive Correlation

With perfect positive correlation, r12 = +1, it is only possible to reduce portfolio risk to zero if you can short-sell.

The portfolio weights for the global minimum variance portfolio are:

» Short sell one of the assets» Long position in the other asset.» If one asset has low risk/low return, portfolio return is below

return of the low return asset If you cannot short sell, then put all your wealth into the lower

risk asset.

x x x12

1 22 1

1 2

1 2

12

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Perfect Positive Correlation: Example

Reconsider the previous example, but assume perfect positive correlation, r12 = +1.

» Then we have portfolio weights:

» This gives an expected return of:

» Variance is reduced to zero– check this!

x x1 20 20

0 20 0 4010 1 10 2 0

.

. .. ( . ) .

E rP 10 20% 2 0 12% 4%. .

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Perfect Positive Correlation

E[r]

E[r1]

E[r2]

s2 s1 s

Asset 2

0

Minimum-variance portfolio (no short sales)

E[rp]

Portfolio of mostly Asset 2

Asset 1

Portfolio ofmostly Asset 1

Short sellingMinimum-variance with short sales

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Perfect Correlation: Examples

Derivatives have very high correlations with the underlying assets:» Futures and Forwards» Options

Use these assets (with short positions) to hedge risk

Example: You have $900 to invest into any combination of two assets:

» A stock currently trading at $100 with an expected return of 0.2% and a volatility of 2.5% for a one-week return

» A call option on the stock with an option delta of 0.4, currently trading at $4.00

» How can you minimize the risk of your portfolio for the coming week?

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Perfect Correlation: Example

First, observe that if the stock moves by one standard deviation, this is $3.00. Then the standard deviation of the call is:

Hence, the one standard deviation movement of the call is $1.00, hence sC=$1.00/$4.00=25%.

We can now use the formula for the minimum variance portfolio to give:

Hence, you write 25 call options, and invest the proceeds of $100 plus your $900 into 10 stocks.

C N d S 1 0 4 50 00. *$2. $1.

x xCall Stock

0 025

0 025 0 25

1

9

10

9

.

. .

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Imperfect Correlation

What happens in the general case where -1<r12< 1? » With less than perfect correlation, -1<r12< s2/s1 ,

diversification helps reduce risk, but risk cannot be eliminated completely.– Minimum variance portfolio has positive weights in both

assets» If correlation is large, s2/s1< r12< 1, diversification kdoes not

help to reduce risk.– Minimum variance portfolio has negative weight in one

asset

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Non-Perfect CorrelationThe Case of low correlation

E[r]

E[r1]

E[r2]

s2 s1 s

Asset 2

0

Minimum-variance portfolio

E[rp]

Portfolio of mostly Asset 2

Asset 1

Portfolio ofmostly Asset 1

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Example

Assume r12=0.25 What are the portfolio weights, expected return, and standard deviation of the global minimum variance portfolio?

Portfolio Weights

Expected Return and Standard Deviation

x

x

1

2

2 2

2

2 25 4 2

4 2 2 25 4 212 5%

1 125 87 5%

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

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

(. ) .

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

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Non-Perfect CorrelationThe Case of high correlation

E[r1]

E[r2]

s2 s1 s

Asset 2

0

Minimum-variance portfolio

E[rp]Portfolio long in asset 2, short in asset 1

Asset 1

Portfolio ofmostly Asset 1

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Example

Assume r12=0.75. What are the portfolio weights, expected return, and standard deviation of the global minimum variance portfolio?

Portfolio Weights

Expected Return and Standard Deviation

x

x

1

2

2 2

2

2 75 4 2

4 2 2 75 4 26 25%

1 0625 106 25%

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

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

( . ) .

E rp( ) ( . )( ( . )( . 0625 20%) 10625 12%) 115%

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

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

var( ) .

( ) . .

r

r

Sd r

p

p

p

0625 4 10625 2

2 0 0625 10625 75 4 2

0347

0375 18 62%

2 2 2 2

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Limits to Diversification Consider an equally-weighted portfolio. The variance of such a

portfolio is:

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.

p iji

i N

j

j N

N N

N N

211

1 1

11

1

Average

Variance

Average

Coariance

var( ) covrp

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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%

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Limits to Diversification

Market Risk

Total Risk

Firm-Specific Risk

Portfolio Risk, s

Number of Stocks

Average

Covariance

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Specific Risk and Market Risk

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.

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Efficient Portfolios with Multiple Assets

E[r]

s0

Asset 1

Asset 2Portfolios ofAsset 1 and Asset 2

Portfoliosof otherassets

EfficientFrontier

Minimum-VariancePortfolio

Investorsprefer

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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:» Invest in stocks in different industries.» Invest in both large and small company stocks.» Diversify across asset classes.

– Stocks– Bonds– Real Estate

» Diversify internationally.

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Summary It is not possible to characterize securities in terms of risk alone

» Need to understand risk Risky investments

» More risky investments have higher returns» Risk premia are not related to the risk of individual assets

Diversification benefits» Depend on correlation of assets» Possiblity of short sales» Cannot eliminate market risk

Minimum variance portfolios» Riskless if correlation perfectly negative» Applications for hedging