Chapter 8 - An Introduction to Portfolio Management

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Lecture Presentation Software to accompany

Investment Analysis and Portfolio Management

Sixth Editionby

Frank K. Reilly & Keith C. Brown

Chapter 8

mailto:[email protected]

mailto:[email protected]

SAIF ULLAH, [email protected], +923216633271

Chapter 8 - An Introduction to Portfolio Management

Questions to be answered:

• What do we mean by risk aversion and what evidence indicates that investors are generally risk averse?

• What are the basic assumptions behind the Markowitz portfolio theory?

• What is meant by risk and what are some of the alternative measures of risk used in investments?



• How do you compute the expected rate of return for an individual risky asset or a portfolio of assets?

• How do you compute the standard deviation of rates of return for an individual risky asset?

• What is meant by the covariance between rates of return and how do you compute covariance?



• What is the relationship between covariance and correlation?

• What is the formula for the standard deviation for a portfolio of risky assets and how does it differ from the standard deviation of an individual risky asset?

• Given the formula for the standard deviation of a portfolio, how and why do you diversify a portfolio?



• What happens to the standard deviation of a portfolio when you change the correlation between the assets in the portfolio?

• What is the risk-return efficient frontier?• Is it reasonable for alternative investors to select

different portfolios from the portfolios on the efficient frontier?

• What determines which portfolio on the efficient frontier is selected by an individual investor?


Background Assumptions• As an investor you want to maximize the

returns for a given level of risk.

• Your portfolio includes all of your assets and liabilities

• The relationship between the returns for assets in the portfolio is important.

• A good portfolio is not simply a collection of individually good investments.


Risk Aversion

Given a choice between two assets with equal rates of return, risk averse investors will select the asset with the lower level of risk.


Evidence ThatInvestors are Risk Averse

• Many investors purchase insurance for: Life, Automobile, Health, and Disability Income. The purchaser trades known costs for unknown risk of loss

• Yield on bonds increases with risk classifications from AAA to AA to A….


Not all investors are risk averse

Risk preference may have to do with amount of money involved - risking small amounts, but insuring large losses


Definition of Risk

1. Uncertainty of future outcomes

or

2. Probability of an adverse outcome

We will consider several measures of risk that are used in developing portfolio theory


Markowitz Portfolio Theory

• Quantifies risk

• Derives the expected rate of return for a portfolio of assets and an expected risk measure

• Shows the variance of the rate of return is a meaningful measure of portfolio risk

• Derives the formula for computing the variance of a portfolio, showing how to effectively diversify a portfolio


Assumptions of Markowitz Portfolio Theory

1. Investors consider each investment alternative as being presented by a probability distribution of expected returns over some holding period.



2. Investors minimize one-period expected utility, and their utility curves demonstrate diminishing marginal utility of wealth.



3. Investors estimate the risk of the portfolio on the basis of the variability of expected returns.



4. Investors base decisions solely on expected return and risk, so their utility curves are a function of expected return and the expected variance (or standard deviation) of returns only.



5. For a given risk level, investors prefer higher returns to lower returns. Similarly, for a given level of expected returns, investors prefer less risk to more risk.



Using these five assumptions, a single asset or portfolio of assets is considered to be efficient if no other asset or portfolio of assets offers higher expected return with the same (or lower) risk, or lower risk with the same (or higher) expected return.


Alternative Measures of Risk

• Variance or standard deviation of expected return

• Range of returns

• Returns below expectations– Semivariance - measure expected returns below

some target– Intended to minimize the damage


Expected Rates of Return

• Individual risky asset– Sum of probability times possible rate of return

• Portfolio– Weighted average of expected rates of return

for the individual investments in the portfolio


Computation of Expected Return for an Individual Risky Investment

0.25 0.08 0.02000.25 0.10 0.02500.25 0.12 0.03000.25 0.14 0.0350

E(R) = 0.1100

Expected Return(Percent)Probability

Possible Rate ofReturn (Percent)

Table 8.1


Computation of the Expected Return for a Portfolio of Risky Assets

0.20 0.10 0.02000.30 0.11 0.03300.30 0.12 0.03600.20 0.13 0.0260

E(Rpor i) = 0.1150

Expected Portfolio

Return (Wi X Ri) (Percent of Portfolio)

Expected Security

Return (Ri)

Weight (Wi)

Table 8.2

iasset for return of rate expected the )E(Riasset in portfolio theofpercent theW

:where

RW)E(R

i

i

1ipor

n

iii


Variance (Standard Deviation) of Returns for an Individual Investment

Standard deviation is the square root of the variance

Variance is a measure of the variation of possible rates of return Ri, from the expected rate of return [E(Ri)]



n

i 1i

2ii

2 P)]E(R-R[)( Variance

where Pi is the probability of the possible rate of return, Ri



n

i 1i

2ii P)]E(R-R[)(

Standard Deviation



Possible Rate Expected

of Return (Ri) Return E(Ri) Ri - E(Ri) [Ri - E(Ri)]2 Pi [Ri - E(Ri)]

2Pi

0.08 0.11 0.03 0.0009 0.25 0.0002250.10 0.11 0.01 0.0001 0.25 0.0000250.12 0.11 0.01 0.0001 0.25 0.0000250.14 0.11 0.03 0.0009 0.25 0.000225

0.000500

Table 8.3

Variance ( 2) = .0050

Standard Deviation ( ) = .02236


Variance (Standard Deviation) of Returns for a Portfolio

Computation of Monthly Rates of Return

Table 8.4

Coca - Cola ExxonClosing Closing

Date Price Dividend Return (%) Price Dividend Return (%)

Dec-97 66.688 0.14 61.188 0.41Jan-98 64.750 -2.91% 59.313 -3.06%Feb-98 68.625 5.98% 63.750 0.41 8.17%Mar-98 77.438 0.15 13.06% 67.625 6.08%Apr-98 75.875 -2.02% 73.063 8.04%May-98 78.375 3.29% 70.500 0.41 -2.95%Jun-98 85.500 0.15 9.28% 71.375 1.24%Jul-98 80.500 -5.85% 70.250 -1.58%

Aug-98 65.125 -19.10% 65.438 0.41 -6.27%Sep-98 57.625 0.15 -11.29% 70.625 7.93%Oct-98 67.563 17.25% 71.625 1.42%Nov-98 70.063 0.15 3.92% 75.000 0.41 5.28%Dec-98 67.000 -4.37% 73.125 -2.50%

E(RCoca-Cola)= 0.61% E(RExxon)= 1.82%


Time Series Returns forCoca-Cola: 1998

-25.00%

-20.00%

-15.00%

-10.00%

-5.00%

0.00%

5.00%

10.00%

15.00%

20.00%

J F M A M J J A S O N D

Figure 8.1


Times Series Returns forExxon: 1998

-8.00%-6.00%-4.00%-2.00%0.00%2.00%4.00%6.00%8.00%

10.00%


Figure 8.2


Times Series Returns forCoca-Cola and Exxon: 1998

-20.00%

-15.00%

-10.00%

-5.00%

0.00%

5.00%

10.00%

15.00%



Variance (Standard Deviation) of Returns for a Portfolio

For two assets, i and j, the covariance of rates of return is defined as:

Covij = E{[Ri - E(Ri)][Rj - E(Rj)]}


Computation of Covariance of Returns for Coca-Cola and Exxon: 1998

Rate of Rate of Coca-Cola Exxon Coca-Cola ExxonDate Return (%) Return (%) Ri - E(Ri) Rj - E(Rj) [Ri - E(Ri)] X [Rj - E(Rj)]

Jan-98 -2.91% -0.0306 -0.0352 -0.049 17.18Feb-98 5.98% 8.17% 0.0537 0.064 34.10Mar-98 13.06% 6.08% 0.1245 0.043 53.04Apr-98 -2.02% 8.04% -0.0263 0.062 -16.36

May-98 3.29% -2.95% 0.0268 -0.048 -12.78Jun-98 9.28% 1.24% 0.0867 -0.006 -5.03Jul-98 -5.85% -1.58% -0.0646 -0.034 21.96

Aug-98 -19.10% -6.27% -0.1971 -0.081 159.45Sep-98 -11.29% 7.93% -0.1190 0.061 -72.71Oct-98 17.25% 1.42% 0.1664 -0.004 -6.66Nov-98 3.92% 5.28% 0.0331 0.035 11.45Dec-98 -4.37% -2.50% -0.0498 -0.043 21.51

E(RCoca-Cola)= 0.61% E(RExxon)= 1.82% Sum = 205.16Covij = 205.16 / 12 = 17.10

Table 6.5


Scatter Plot of Monthly Returns for Coca-Cola and Exxon: 1998

-8.00%

-6.00%

-4.00%

-2.00%

0.00%

2.00%

4.00%

6.00%

8.00%

10.00%

-8.00% -6.00% -4.00% -2.00% 0.00% 2.00% 4.00% 6.00% 8.00% 10.00% 12.00% 14.00% 16.00%

Monthly Returns for Coca-Cola

Mo

nth

ly R

etu

rn f

or

Ex

xo

n

Figure 8.3


Covariance and Correlation

Correlation coefficient varies from -1 to +1

jt

iti

ij

R ofdeviation standard the

R ofdeviation standard the

returns oft coefficienn correlatio ther

:where

Covr

j

ji

ijij


Computation of Standard Deviation of Returns for Coca-Cola and Exxon: 1998

Date Ri - E(Ri) [Ri - E(Ri)]2 Rj - E(Rj) [Rj - E(Rj)]

2

Jan-98 -3.63% 13.18 -5.27% 27.77 Feb-98 5.47% 29.92 6.61% 43.69 Mar-98 12.54% 157.25 3.84% 14.75 Apr-98 -2.72% 7.40 6.48% 41.99 May-98 2.78% 7.73 -4.50% 20.25

Jun-98 8.77% 76.91 -0.90% 0.81 Jul-98 -6.53% 42.64 -3.13% 9.80

Aug-98 -19.62% 384.94 -7.82% 61.15 Sep-98 -11.80% 139.24 5.70% 32.49 Oct-98 16.42% 269.62 -0.14% 0.02 Nov-98 3.41% 11.63 3.73% 13.91 Dec-98 -5.09% 25.91 -4.59% 21.07

1,166.37 287.70

Variancei= 1166.37 / 12 = 97.20 Variancej= 287.70 / 12 = 23.98

Standard Deviationi = 97.20 1/2 = 9.86 Standard Deviationj = 23.98

1/2 = 4.90

These figures have not been rounded to two decimals at each step as was in the book Table 8.6


Portfolio Standard Deviation Formula

ji

ijij

ij

2i

i

port

n

1i

n

1iijj

n

1ii

2i

2iport

rCov where

j, and i assetsfor return of rates ebetween th covariance theCov

iasset for return of rates of variancethe

portfolio in the valueof proportion by the determined are weights

whereportfolio, in the assets individual theof weightstheW

portfolio theofdeviation standard the

:where

Covwww


Portfolio Standard Deviation Calculation

• Any asset of a portfolio may be described by two characteristics:– The expected rate of return– The expected standard deviations of returns

• The correlation, measured by covariance, affects the portfolio standard deviation

• Low correlation reduces portfolio risk while not affecting the expected return


Combining Stocks with Different Returns and Risk

Case Correlation Coefficient Covariance

a +1.00 .0070

b +0.50 .0035

c 0.00 .0000

d -0.50 -.0035

e -1.00 -.0070

W)E(R Asset ii2

ii 1 .10 .50 .0049 .07

2 .20 .50 .0100 .10


Combining Stocks with Different Returns and Risk

• Assets may differ in expected rates of return and individual standard deviations

• Negative correlation reduces portfolio risk

• Combining two assets with -1.0 correlation reduces the portfolio standard deviation to zero only when individual standard deviations are equal


Constant Correlationwith Changing Weights

Case W1 W2E(Ri)

f 0.00 1.00 0.20 g 0.20 0.80 0.18 h 0.40 0.60 0.16 i 0.50 0.50 0.15 j 0.60 0.40 0.14 k 0.80 0.20 0.12 l 1.00 0.00 0.10

)E(R Asset i

1 .10 r ij = 0.00

2 .20


Constant Correlationwith Changing Weights

Case W1 W2 E(Ri) E( port)

f 0.00 1.00 0.20 0.1000g 0.20 0.80 0.18 0.0812h 0.40 0.60 0.16 0.0662i 0.50 0.50 0.15 0.0610j 0.60 0.40 0.14 0.0580k 0.80 0.20 0.12 0.0595l 1.00 0.00 0.10 0.0700


Portfolio Risk-Return Plots for Different Weights

-

0.05

0.10

0.15

0.20

0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.11 0.12

Standard Deviation of Return

E(R)

Rij = +1.00

1

2With two perfectly correlated assets, it is only possible to create a two asset portfolio with risk-return along a line between either single asset



-

0.05

0.10

0.15

0.20

0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.11 0.12


E(R)

Rij = 0.00

Rij = +1.00

f

gh

ij

k1

2With uncorrelated assets it is possible to create a two asset portfolio with lower risk than either single asset



-

0.05

0.10

0.15

0.20

0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.11 0.12


E(R)

Rij = 0.00

Rij = +1.00

Rij = +0.50

f

gh

ij

k1

2With correlated assets it is possible to create a two asset portfolio between the first two curves



-

0.05

0.10

0.15

0.20

0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.11 0.12


E(R)

Rij = 0.00

Rij = +1.00

Rij = -0.50

Rij = +0.50

f

gh

ij

k1

2

With negatively correlated assets it is possible to create a two asset portfolio with much lower risk than either single asset



-

0.05

0.10

0.15

0.20

0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.11 0.12


E(R)

Rij = 0.00

Rij = +1.00

Rij = -1.00

Rij = +0.50

f

gh

ij

k1

2

With perfectly negatively correlated assets it is possible to create a two asset portfolio with almost no risk

Rij = -0.50

Figure 8.7


Estimation Issues

• Results of portfolio allocation depend on accurate statistical inputs

• Estimates of– Expected returns – Standard deviation– Correlation coefficient

• Among entire set of assets• With 100 assets, 4,950 correlation estimates

• Estimation risk refers to potential errors


Estimation Issues

• With assumption that stock returns can be described by a single market model, the number of correlations required reduces to the number of assets

• Single index market model:imiii RbaR

bi = the slope coefficient that relates the returns for security i to the returns for the aggregate stock market

Rm = the returns for the aggregate stock market


Estimation IssuesIf all the securities are similarly related to

the market and a bi derived for each one, it can be shown that the correlation coefficient between two securities i and j is given as:

marketstock aggregate

for the returns of variancethe where

bbr

2m

i

2m

jiij

j


The Efficient Frontier• The efficient frontier represents that set of

portfolios with the maximum rate of return for every given level of risk, or the minimum risk for every level of return

• Frontier will be portfolios of investments rather than individual securities– Exceptions being the asset with the highest

return and the asset with the lowest risk


Efficient Frontier for Alternative Portfolios

Efficient Frontier

A

B

C

Figure 8.9

E(R)



The Efficient Frontier and Investor Utility

• An individual investor’s utility curve specifies the trade-offs he is willing to make between expected return and risk

• The slope of the efficient frontier curve decreases steadily as you move upward

• These two interactions will determine the particular portfolio selected by an individual investor


The Efficient Frontier and Investor Utility

• The optimal portfolio has the highest utility for a given investor

• It lies at the point of tangency between the efficient frontier and the utility curve with the highest possible utility


Selecting an Optimal Risky Portfolio

)E( port

)E(R port

X

Y

U3

U2

U1

U3’

U2’ U1’

Figure 8.10


The InternetInvestments Online

www.pionlie.com

www.investmentnews.com

www.micropal.com

www.riskview.com

www.altivest.com

http://www.pionlie.com/

http://www.investmentnews.com/

http://www.micropal.com/

http://www.riskview.com/

http://www.altivest.com/


End of Chapter 8–An Introduction to Portfolio Management


Future topicsChapter 9

• Capital Market Theory

• Capital Asset Pricing Model

• Beta

• Expected Return and Risk

• Arbitrage Pricing Theory

Chapter 8 - An Introduction to Portfolio Management

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