McGraw-Hill/Irwin Copyright © 2005 by The McGraw-Hill Companies, Inc. All rights reserved.

Chapter 8Chapter 8Chapter 8Chapter 8

Optimal Risky PortfoliosOptimal Risky Portfolios

8-28-2

Diversification and Portfolio Risk

Suppose your portfolio consists of one stock. What would be the sources of risk:

-Macroeconomic risks:Risks those come from conditions in the general economy (Business cycle, inflation, interest rates, exchange rates)

-Firm specific risks (success in R&D, personnel changes).

8-38-3

Diversification

Now consider the diversification strategy. You invest half of your funds on another stock. Because the firm-specific influences on the two stocks differ, diversification should decrease risk. If we diversify into many more securities, we continue to decrease portfolio volatility. With a large number of stocks we cannot avoid risk together since all securities are affected by the common macroeconomic factors.

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Diversification

When all risk is firm-specific, diversification can decrease the risk to low levels.

When common sources of risk affect all firms, extensive diversification cannot eliminate risk.

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The risk that cannot be diversified is called as market risk, non-diversifiable risk.

The risk that can be eliminated is called “unique risk”, firm specific risk, non-systematic risk, diversifiable risk.

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Risk Reduction with Diversification

Number of Securities

St. Deviation

Market Risk

Unique Risk

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Portfolios of Two-Risky Assets

Suppose our portfolio consists of two risky assets; debt (D) and equity (E).

The rate of return on this portfolio;

rp = wD rD + wE rE

The variance of the two-asset portfolio;

DEEDEDEEDDp wwww 222222

),cov(2),cov(),cov(2EDEDEEEEDDDDp rrwwrrwwrrww

DEEDEEDDp Covwwww 222222

EDDEED rrCov ),(

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Range of values for D,E

+ 1.0 > > -1.0

If = 1.0, the securities would be perfectly positively correlated

If = - 1.0, the securities would be perfectly negatively correlated

Correlation Coefficients: Possible Values

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In the case of perfect positive correlation;

The standard deviation of the portfolio with perfect positive correlation is the weighted average of the component standard deviations.

22 )( EEDDp ww

EEDDp ww

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In all other cases, correlation coefficient less than 1, making the standard deviation less than the weighted average of the component standard deviations.Portfolios of less than perfectly correlated assets always offer better risk return opportunities than the individual components on their own.The lower the correlation between the assets, the greater the gain in efficiency.

8-118-11

In the case of no correlation;

The weight of debt in the portfolio that minimizing the portfolio risk is;

22222EEDDp ww

22222EEDDp ww

22

2

min )(ED

EDw

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In the case of perfect negative correlation;

When ρ=-1, wD σD- wEσE=0

The weight of debt in the portfolio that makes the std

dev of the portfolio zero.

22 )( EEDDp ww )( EEDDp ww

When ρ=-1

ED

EDw

D

ED

DE ww

1

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At any correlation coefficient rate, the weight of debt that gives the minimum variance;

),(2

),()(

22

2

minEDED

EDE

rrCov

rrCovDw

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Portfolios of Two Risky Assets

Example 1: Suppose we will consider a portfolio comprised of two mutual funds, a bond portfolio (long term debt securities) denoted D, and a stock fund (equity securities), denoted E.

Debt Equity

Expected Return E(r) 8% 13%Standard Deviation (δ)12% 20%Cov(rD, rE) 72Corr. Coefficient, ρDE .30

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Portfolio Risk and Return

E(rp)= 8wD+13 wE

EDEDp wwww 3.201222012 22222

EDEDp wwww 3.201222012 22222

EDED

EDEDp

wwww

wwww

144400144

722201222

22222

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wmin(E)=1-.82=.18

),(2

),()(

22

2

minEDED

EDE

rrCov

rrCovDw

82.7222012

7220)(

22

2

min

Dw

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The return of the min variance portfolio;

Rp=8x0.82+13x0.18 = 8.9%Standard deviation of the min-variance portfolio;

=11.45%

2/12222min )]7218.82.2()2018(.)1282[(.

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The relationship depends on correlation coefficient.-1.0 < < +1.0If = +1.0, no risk reduction is possible. there is no benefit from diversification.Portfolio Std Dev is simple the weighted average of the component assets standard deviations.

Correlation Effects

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If ρ=0 with no correlation between two asset, diversification is more effective and portfolio risk is lower.

Min. portfolio standard deviation when ρ=0 is 10.29% and again lower than the standard deviation of either asset.

wE=.265

22

2

min )(ED

EDw

735.2012

20)(

22

2

min

Dw

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The smaller the correlation, the greater the risk reduction potential. = -1.0, offers a perfect hedging opportunity and the maximum advantage of diversification.

We can construct a zero-variance portfolio.

wmin(E; ρ=-1)=1-.625= .375

625.2012

20)1:(min

ED

EDw

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Portfolios with Different Correlations(Portfolio Opportunity Sets)

= 1

13%

%8

E(r)

St. Dev12% 20%

= .3

= -1

= -1

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When an investor wishes to select the optimal portfolio from the opportunity set, the best portfolio will depend on risk aversion.Northeast=> provide higher rates of return but with greater risk.Southwest=> investors with greater risk aversion will prefer these portfolios with lower expected return and lower risk.The best trade-off depends on personal preference.

8-238-23

Asset Allocation with Stocks, Bonds, and Bills

We saw the asset allocation between a risk-free asset and a risky portfolio. Now two risky assets and a risk-free asset.

If we connect the portfolios which gives minimum risk at different levels of return we can have the min. variance frontier.

8-248-24

Minimum-Variance Frontier of Risky Assets

E(r)

Efficientfrontier

Globalminimum

varianceportfolio Minimum

variancefrontier

Individualassets

St. Dev.

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Alternative CALs

M

E(r)

CAL (A-Globalminimum variance)

CAL (B)CAL (P)

P

B

F

P P&F A&FM

B

A

P

M

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The first CAL is drawn from through the min-variance portfolio A, 82% in bonds and 18% in stocks. A’s expected return is 8.9% and standard deviation is 11.45%. (ρ=.30)

With a T-bill rate of 5%, the reward-to-variability ratio, slope CAL combining T-bills and the min-variance portfolio is;

34.45.11

59.8)(

A

AA

rfrES

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Now consider the CAL that was portfolio B instead of A. Portfolio B invest 70% in bonds and 30% in stocks, its expected return is 9.5% (risk premium is 4.5%) and its standard deviation is 11.70%.Reward-to-variability ratio;

Portfolio B dominates A.

38.7.11

55.9

BS

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P=> tangency portfolio is the optimal risky portfolio to mix with T-bills. The expected return & risk can be read from the graph.

E(rp)=11% σp=14.2%

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To construct optimal risky portfolios from more than two risky assets we need a spreadsheet or a computer program. We will see here a portfolio construction problem with only two risky assets and a risk free asset. We can derive a formula for the weights of each asset in the optimal portfolio.

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The objective is to find the weights wD and wE that result in the highest slope of the CAL=> the weights that result in the risky portfolio with the highest reward-to-variability ratio. Therefore the objective is to maximize the slope of the CAL for any portfolio P.

p

pp

rfrES

)(

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For the portfolio with two risky assets, the expected return and standard deviation of portfolio P are

E(rp) = wDE(rD)+ wEE(rE)

= 8 wD+ 13 wE

2/12222 )],cov(2()()[ EDEDEEDDp rrwwww

2/122 ))722(400144( EDEDp wwww

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When we maximize the objective function Sp, we have to satisfy the constraint that the portfolio weights sum to 1, wD+wE=1

subject to p

pp

wi

rfrESMax

)( 1iw

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Optimal Risky Portfolio;

wE=1-.40=.60

),cov(])()([])([])([

),cov(])([])([22

2

EDEDDEED

EDEEDD rrrfrErfrErfrErfrE

rrrfrErfrEw

40.72)51358(144)513(400)58(

72)513(400)58(

Dw

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E(rp)=(.40 x 8)+ (.6 x 13)= 11%

The CAL of this portfolio has a slope of;

(reward-variability ratio of portfolio P)

%2.14)]726.4.2()4006(.)1444[(. 2/122 p

42.2.14

511

pS

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Optimal Complete PortfolioA=4=> coefficient of risk-aversion

74.39% invest in portfolio P1-.7439=.2561 (25.61% invest in T-bills)Portfolio P

→ 40% Bonds → ywD=.4 x .7439=.2976 =>29.76%

→60% C.stock → ywE=.6 x .7439=.4463 =>44.63%

7439.)2.14(401.

511

01.

)(22

p

p

A

rfrEy

=> % invest in portfolio P

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The steps to find the complete portfolio;

1. Specify the return characteristics of all securities ( expected return, variances, covariances)

2. Establish the risky portfolioa. Calculate optimal risky portfolio, p( equation 8.7)b. Calculate the properties of portfolio P using the weights determined

pg.239In step a and equation (8.1) and (8.2)

3. Allocate funds between the risky portfolio and the risk-free asset.a. Calculate the fraction of the complete portfolio allocated to portfolio P and T-bills.b. Calculate the share of the complete portfolio invested in each asset and T-bills.

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The Markowitz Portfolio Selection Model

Harry Markowitz (1952) Model of portfolio selection 1990=> Nobel Prize for economics

We can generalize the portfolio construction problem to the case of many risky assets and a risk free asset. As in the two risky assets example the portfolio construction problem has 3 parts.1. Identify the risk-return combinations available from the set of risky assets.2. Identify the optimal portfolio of risky assets by finding the portfolio weights that result in the steepest CAL.3. Choose an appropriate complete portfolio by mixing the risk-free asset with the optimal risky portfolio.

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The optimal combinations result in lowest level of risk for a given return. (Min variance frontier of risky assets)The optimal trade-off is described as the efficient frontier.These portfolios are dominant.All the portfolios lie on the min-variance frontier from the global min-variance portfolio and upward provide the best risk-return combinations and candidates for the optimal portfolio. This portfolio is the efficient frontier of risky assets. The bottom part of the min-variance frontier is inefficient.

1. Detemining the risk-return opportunities available to the investor

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Minimum-Variance Frontier of Risky Assets

E(r)

Efficientfrontier

Globalminimum

varianceportfolio Minimum

variancefrontier

Individualassets

St. Dev.

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Now our portfolio optimization problem involves the risk free asset. We search for CAL with the highest reward-to-variability ratio. The CAL that is supported by the optimal portfolio, P, is tangent to the efficient frontier. The CAL dominates all alternative feasible lines.Portfolio P is the optimal risky portfolio. The optimal combination becomes linear.A single combination of risky and riskless assets will dominate.

2. Identify the optimal portfolio of risky assets by finding the portfolio weights that result in the steepest CAL.

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The opportunity set of debt and equity funds with the optimal CAL and the optimal risky portfolio

E(r)

Frf

A

P

Q

B

CAL

St. Dev

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3. Choose an appropriate complete portfolio by mixing the risk-free asset with the optimal risky portfolio.

Individual investor chooses the appropriate mix between the optimal risky portfolio P and T-bills

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Determination of the optimal overall portfolio

E(r)

Frf

A

P

Q

B

CAL

St. Dev

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Capital Allocation and the Separation Property

Portfolio P is the point at which CAL tangent to the efficient frontier. Portfolio P maximize the reward-to-variability ratio, the slope of the line from F to portfolio on the efficient frontier. Portfolio P is the optimal risky portfolio that a portfolio manager will offer to all clients regardeless of their degree of risk aversion.

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Seperation Property

The degree of risk aversion (A) comes into play only in the selection of the desired point along the CAL. More risk averse investor will invest more in rf and less in optimal risky portfolio than will a less risk averse investor. However both will use portfolio P as their optimal risky portfolio.

This is called a separation property (J. Tobin)

8-468-46

Portfolio Selection without Rf & Risk Aversion

E(r)

Efficientfrontier ofrisky assets

Morerisk-averseinvestor

U’’’ U’’ U’

Q

PS

St. Dev

Lessrisk-averseinvestor

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Efficient Frontier with Lending & Borrowing

E(r)

Frf

A

P

Q

B

CAL

St. Dev

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HOMEWORK

Problems 10, 14, and 31 (a,b,c,d,e).