chapter 7 slides fin 435

7/31/2019 Chapter 7 Slides FIN 435

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Investment Analysis &Investment Analysis &

Portfolio ManagementPortfolio Management

FIN 435 (Instructor- Saif Rahman)

Chapter 7Chapter 7

Optimal Risky PortfoliosOptimal Risky Portfolios


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Market risk vs. Unique riskMarket risk vs. Unique risk

Stand-alone risk = Market risk + Firm-specific risk

The risk that remains even after extensive diversification is called

market risk, risk that is attributable to market wide risk sources.


Such risk is also called systematic risk or nondiversifiable risk.

Measured by beta. (e.g. War, Inflation, High Interest Rates)

In contrast, the risk that can be eliminated by diversification is

called unique risk, firm-specific risk, nonsystematic risk or

diversifiable risk.


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Risk Reduction with DiversificationRisk Reduction with Diversification

Company-Specific Risk

Stand-Alone Risk

p (%)

35

FIN 435 (Instructor- Saif Rahman)2

# Stocks in Portfolio10 20 30 40 2,000+

Market Risk

20

0


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TwoTwo--Security Portfolio: ReturnSecurity Portfolio: Return

Portfolio Return

Bond Weight

Bond Return

p D ED E

P

D

r

r

w

r

w wr r


Equity Weight

Equity Return

E

E

w

r

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

1

n

1iiw


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= Variance of Security D

TwoTwo--Security Portfolio: RiskSecurity Portfolio: Risk

2 2 2 2 2 2 ( , )P D D E E D E D E w w w Cov r r

2D


= Variance of Security E

= Covariance of returns forSecurity D and Security E

2

E

( , )D ECov r r


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Range of values for 1,2

+ 1.0 > r > -1.0

Correlation Coefficients: Possible ValuesCorrelation Coefficients: Possible Values


r= . , e secur es wou e per ec ypositively correlated

Ifr= - 1.0, the securities would be perfectlynegatively correlated


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Importance of CorrelationImportance of Correlation

Correlation is important because it affects the

degree to which diversification can be achieved

using various assets.

FIN 435 (Instructor- Saif Rahman)7

Theoretically, if two assets returns are perfectly

negatively correlated, it is possible to build a

riskless portfolio with a return that is greater thanthe risk-free rate.


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2

p = W12

1

2

+ W22

1

2

rp = W1r1 + W2r2 + W3r3

+ W32

3

2

ThreeThree--Security PortfolioSecurity Portfolio


+ 2W1W2 Cov(r1r2) Cov(r1r3)+ 2W1W3Cov(r2r3)+ 2W2W3

p2 = w1

212 + w2

22

2 + 2W1W2 1 2 12


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rp = Weighted average of the nsecurities

p2 = (Consider all pair-wise

In General, For an nIn General, For an n--Security PortfolioSecurity Portfolio


covariance measures)


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TwoTwo--Security Portfolios with Different CorrelationsSecurity Portfolios with Different Correlations

13

E(r)

-

p2 = w1

212 + w2

22

2 + 2W1W2 1 2 12


= 1%

8

St. Dev12% 20%

= .3

= -1


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The relationship depends on correlation coefficient.

-1.0 < < +1.0

The smaller the correlation, the greater the risk reduction

Portfolio Risk/Return Two Securities:Portfolio Risk/Return Two Securities:

Correlation EffectsCorrelation Effects


Ifr = +1.0, no risk reduction is possible.


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What is the minimum level to which portfolioWhat is the minimum level to which portfolio

standard deviation can be held?standard deviation can be held?

),cov(2

),cov()(

22

2

min

EDED

EDE

rr

rrDw


The optimal combinations result in lowest level

of risk for a given return.

The optimal trade-off is described as the efficient

frontier.

These portfolios are dominant.


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Markowitz Portfolio Selection ModelMarkowitz Portfolio Selection Model

Security Selection

First step is to determine the risk-return


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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The MinimumThe Minimum--Variance FrontierVariance Frontier

of Risky Assetsof Risky AssetsE(r)

Efficientfrontier


Globalminimum

variance

portfolio Minimumvariancefrontier

Individualassets

St. Dev.


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Alternative CALsAlternative CALs (CAL with the highest reward(CAL with the highest reward--toto--

variability ratio)variability ratio)

M

E(r)

CAL (A)CAL (P)

P

M


o aminimum variance)

A

F

P P&F A&FM

A

G


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Capital Allocation and the Separation PropertyCapital Allocation and the Separation Property

The most striking conclusion is that a portfolio manager will offer the same

risky portfolio, P, to all clients regardless of their degree of risk aversion

The result is called a separation property, it tells us that the portfolio choice

problem may be separated into two independent tasks:


1) First determine the optimal risky portfolio

2) Then choose the allocation of the complete portfolio to risk-free

assets

An example of a situation when there is more than one optimal risky

portfolio: Risk-free lending only


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Portfolio Selection & Risk AversionPortfolio Selection & Risk Aversion

E(r)

Efficient

U U U


risky assets

Morerisk-averseinvestor

Q

P

St. Dev

Less

risk-averseinvestor

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