CHAPTER 7 - leeds-courses.colorado.eduleeds-courses.colorado.edu/FNCE4030/MISC/slides/FNCE4030-Fall-201… · INVESTMENTS | BODIE, KANE, MARCUS Fig. 7.1 Portfolio Risk as a Function

INVESTMENTS | BODIE, KANE, MARCUS

Copyright © 2011 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill/Irwin

CHAPTER 7

Optimal Risky Portfolios


The Investment Decision

Top-down process with 3 steps:

1. Capital allocation between the risky

portfolio and risk-free asset

2. Asset allocation across broad asset

classes

3. Security selection of individual assets

within each asset class

7-2

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Diversification and Portfolio Risk

• Market risk

– Risk attributable to market-wide risk

sources, and remains even after extensive

diversification

– aka Systemic or non-diversifiable

• Firm-specific risk

– Risk that can be eliminated by diversification

– aka diversifiable or non-systemic

7-3

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Fig. 7.1 Portfolio Risk as a Function of the Number of Stocks in the Portfolio

1-4

Panel A: All risk is firm specific. Panel B: Some risk is systematic or marketwide.

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Figure 7.2 Portfolio Diversification

1-5

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Covariance and Correlation

• Portfolio risk depends on the correlation between the returns of the assets in the portfolio

• Covariance and the correlation coefficient provide a measure of the way returns of two assets vary

7-6

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A Two-Security Portfolio: Return

7-7

( ) ( ) ( )p D D E EE r w E r w E r

𝑤𝐷 = Bond weight

𝑤𝐸 = Equity weight

𝑟𝐷 = Bond return

𝑟𝐸 = Equity return

𝑟𝑃 = Portfolio return

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A Two-Security Portfolio: Risk

7-8

EDEDEEDD rrCovwwww ,222222

p

= Variance of Security D

= Variance of Security E

= Covariance of returns for

Security D and Security E

2

E

2

D

ED rrCov ,

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Two-Security Portfolio: Risk

• Another way to express variance of the portfolio is to think of Covariances:

7-9

EDED

EEEE

DDDD

rrCovww

rrCovww

rrCovww

,2

,

,2

p

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Covariance

7-10

𝜌𝐷,𝐸 = Correlation coefficient of returns

𝜎𝐷 = Standard deviation of returns

for Security D

𝜎𝐸 = Standard deviation of returns

for Security E

EDDEED rrCov ,

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Table 7.2 Computation of Portfolio Variance From the Covariance Matrix

1-11

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A portfolio of 3 Assets

1-12

1 1 2 2 3 3( ) ( ) ( ) ( )pE r w E r w E r w E r

• You have three assets with weights

w1, w2, w3

• The portfolio return is simply the linear

combination of the returns with same

coefficients:

Q. is the portfolio’s variance also the

linear combination of the 3 variances?

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Bordered Matrix for 3 Assets

w1 w2 w3

w1 Cov(1,1) Cov(1,2) Cov(1,3)

w2 Cov(2,1) Cov(2,2) Cov(2,3)

w3 Cov(3,1) Cov(3,2) Cov(3,3)

7-13

Step 1: write the covariance matrix and its weights

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w1 w2 w3

w1

w2

w3

7-14

Step 2: Symmetry! baabCovbaCov ,,,

2

1 2,1 3,1

2

2

2

3

3,2

3,1 3,2

2,1

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w1 w2 w3

w1

w2

w3


7-15

Step 3: multiply by the weights around the border

2

1w

2

2w

2

3w

21ww

21ww

31ww

31ww

32ww

32ww

2

1 2,1 3,1

2

2

2

3

3,2

3,1 3,2

2,1

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1-16

3,2323,1312,121

2

3

2

3

2

2

2

2

2

1

2

1

2

222

wwwwww

wwwp

Covariance terms

Step 4: add-up all the pieces

Remember ababbabaabba ,,,,

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w1 w2 w3

w1

w2

w3

7-17

All in one step, using correlations this time

2

1 212,1 313,1

2

2

2

3

323,2

313,1 323,2

212,1

2

1w

2

2w

2

3w

31ww

33ww

21ww

33ww31ww

21ww

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1-18

323,232

313,131

212,121

2

3

2

3

2

2

2

2

2

1

2

1

2

2

2

2

ww

ww

ww

wwwp

Add-up all the pieces

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Correlation: Possible Values

7-19

Range of values for correlation

−𝟏 ≤ 𝝆 ≤ +1

If = 1.0, the securities are perfectly

positively correlated

If = - 1.0, the securities are perfectly

negatively correlated

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Two-Security Portfolio: Variance

• Remember the variance of a two-asset portfolio

7-20

EDDEED

EEDD

ww

ww

2

22222

p

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Correlation Coefficients

• When 𝜌𝐷𝐸 = 1, there is no diversification

7-21

DDEEP ww

D

ED

DE

ED

ED www

1 and

0222222

p EDEDEEDD wwww

• When 𝜌𝐷𝐸 = −1, a perfect hedge is when:

the solution (which also makes 𝑤𝐷 + 𝑤𝐸 = 1) is:

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Fig 7.3 Portfolio Expected Return as a Function of Investment Proportions

1-22

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Fig 7.4 Portfolio Standard Deviation as a Function of Investment Proportions

1-23

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The Minimum Variance Portfolio

• The minimum variance

portfolio is the portfolio

composed of the risky

assets that has the

smallest standard

deviation, the portfolio

with least risk.

• If correlation < +1

the portfolio standard

deviation may be

smaller than that of

either of the individual

component assets.

• If correlation = -1

the standard deviation

of the minimum

variance portfolio is

zero. 7-24

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Fig 7.5 Portfolio Expected Return as a Function of Standard Deviation

1-25

Portfolio

opportunity

set for given

correlation

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Correlation Effects

• The amount of possible risk reduction through diversification depends on the correlation.

• The risk reduction potential increases as the correlation approaches -1. – If 𝜌 = +1.0, no risk reduction is possible.

– If 𝜌 = 0, 𝜎𝑃 may be less than the standard

deviation of either component asset.

– If 𝜌 = -1.0, a riskless hedge is possible.

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Fig 7.6 The Opportunity Set of the Debt and Equity Funds and Two Feasible CALs

1-27

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The Sharpe Ratio

• Maximize the slope of the CAL for any possible portfolio, P.

• The objective function is the slope:

• The slope is also the Sharpe ratio.

7-28

P

fP

P

rrES

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INVESTMENTS | BODIE, KANE, MARCUS 7-29

Fig 7.7 The Opportunity Set of Debt and Equity Funds with the Optimal CAL and the Optimal Risky Portfolio

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Fig 7.8 Determination of the Optimal Overall Portfolio

1-30

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Markowitz Portfolio Selection Model

• Security Selection

– The first step is to determine the risk-return

opportunities available

– All portfolios that lie on the minimum-variance

frontier from the global minimum-variance

portfolio and upward provide the best risk-return

combinations

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Fig 7.10 The Minimum-Variance Frontier of Risky Assets

1-32

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• We now search for the CAL with the highest reward-to-variability ratio

Q. How do we measure that?

• That means to find that optimal line that stems from the risk-free point and is tangent to the efficient frontier

7-33

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Fig 7.11 The Efficient Frontier of Risky Assets with the Optimal CAL

1-34

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• Everyone invests in P, regardless of their degree of risk aversion. However:

– More risk averse investors put more in the

risk-free asset.

– Less risk averse investors put more in P.

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Capital Allocation, Separation Property

• The separation property tells us that the portfolio choice problem may be separated into two independent tasks:

1. Determination of the optimal risky portfolio

(purely technical / mathematical)

2. Allocation of the complete portfolio to risk

free asset versus the risky portfolio

(depends on personal preference)

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Fig 7.13 Capital Allocation Lines with Various Portfolios from the Efficient Set

1-37

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The Power of Diversification

1-38

• Remember:

• Consider an equally weighted portfolio:

n

i

n

j

jijiP rrCovww1 1

2 ,

nwi

1

• Look at covariance the matrix structure:

2

P

n

i

in1

2

2

1

n

i

n

ijj

ji rrCovnn1 1

,11

• Rewrite covariance as:

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1-39

• Rearrange:

• Define avg variance and avg covariance as:

n

i

in 1

22 1

terms1

1 1

,1

1

nn

n

i

n

ijj

ji rrCovnn

Cov

n

i

n

ijj

ji

n

i

iP rrCovnnnn 1 11

22 ,1111

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7-40

• Then we can rewrite portfolio variance:

terms1

1 11

22 ,1111

2

nn

n

i

n

ijj

ji

n

i

iP rrCovnnnn

Covn

n

nP

11 22

as:

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7-41

Study case where all assets have same

standard deviation and one correlation for all

Q. What happens for very large n?

222 11

n

n

nP

Q. What happens when correlation = 0?

22

P

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Table 7.4 Risk Reduction of Equally Weighted Portfolios in Correlated and Uncorrelated Universes

1-42

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Optimal Portfolios and Non-normal Returns

• The optimal portfolio approach we just studied assumes normal returns.

• Fat-tailed distributions can result in extreme values of Value-at-Risk (VaR) and Expected Shortfall (ES) and encourage smaller allocations to the risky portfolio.

• If other portfolios provide sufficiently better VaR and ES values than the mean-variance efficient portfolio, we may prefer these when faced with fat-tailed distributions.

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Risk Pooling - Insurance Principle

• Risk pooling: merging (adding) uncorrelated, risky projects as a means to reduce risk. – increases the scale of the risky investment by

adding additional uncorrelated assets.

• The insurance principle: risk increases less than proportionally to the number of policies insured when the policies are uncorrelated

– Sharpe ratio increases

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Risk Sharing

• As risky assets are added to the portfolio, a portion of the pool is sold to maintain a risky portfolio of fixed size.

• Risk sharing combined with risk pooling is the key to the insurance industry.

• True diversification means spreading a portfolio of fixed size across many assets, not merely adding more risky bets to an ever-growing risky portfolio.

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Investment for the Long Run

Long Term Strategy

• “Invest in the risky

portfolio for 2 years”

• Long-term strategy is

riskier.

• Risk can be reduced by

selling some of the

risky assets in year 2.

• “Time diversification” is

not true diversification.

Short Term Strategy

• “Invest in the risky

portfolio for 1 year and

in the risk-free asset for

the second year”

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Documents

CHAPTER 7 - leeds-courses.colorado.eduleeds-courses.colorado.edu/FNCE4030/MISC/slides/FNCE4030-Fall-201… · INVESTMENTS | BODIE, KANE, MARCUS Fig. 7.1 Portfolio Risk as a Function