chapter 8 - an introduction to portfolio management
DESCRIPTION
Investment Analysis and Portfolio Mangement By Reilly and BrownChapter No. 8 An Introduction to Portfolio ManagementTRANSCRIPT
Saif [email protected]
[email protected]+923216633271
Lecture Presentation Software to accompany
Investment Analysis and Portfolio Management
Sixth Editionby
Frank K. Reilly & Keith C. Brown
Chapter 8
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?
SAIF ULLAH, [email protected], +923216633271
Chapter 8 - An Introduction to Portfolio Management
• 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?
SAIF ULLAH, [email protected], +923216633271
Chapter 8 - An Introduction to Portfolio Management
• 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?
SAIF ULLAH, [email protected], +923216633271
Chapter 8 - An Introduction to Portfolio Management
• 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?
SAIF ULLAH, [email protected], +923216633271
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.
SAIF ULLAH, [email protected], +923216633271
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.
SAIF ULLAH, [email protected], +923216633271
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….
SAIF ULLAH, [email protected], +923216633271
Not all investors are risk averse
Risk preference may have to do with amount of money involved - risking small amounts, but insuring large losses
SAIF ULLAH, [email protected], +923216633271
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
SAIF ULLAH, [email protected], +923216633271
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
SAIF ULLAH, [email protected], +923216633271
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.
SAIF ULLAH, [email protected], +923216633271
Assumptions of Markowitz Portfolio Theory
2. Investors minimize one-period expected utility, and their utility curves demonstrate diminishing marginal utility of wealth.
SAIF ULLAH, [email protected], +923216633271
Assumptions of Markowitz Portfolio Theory
3. Investors estimate the risk of the portfolio on the basis of the variability of expected returns.
SAIF ULLAH, [email protected], +923216633271
Assumptions of Markowitz Portfolio Theory
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.
SAIF ULLAH, [email protected], +923216633271
Assumptions of Markowitz Portfolio Theory
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.
SAIF ULLAH, [email protected], +923216633271
Assumptions of Markowitz Portfolio Theory
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.
SAIF ULLAH, [email protected], +923216633271
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
SAIF ULLAH, [email protected], +923216633271
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
SAIF ULLAH, [email protected], +923216633271
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
SAIF ULLAH, [email protected], +923216633271
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
SAIF ULLAH, [email protected], +923216633271
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)]
SAIF ULLAH, [email protected], +923216633271
Variance (Standard Deviation) of Returns for an Individual Investment
n
i 1i
2ii
2 P)]E(R-R[)( Variance
where Pi is the probability of the possible rate of return, Ri
SAIF ULLAH, [email protected], +923216633271
Variance (Standard Deviation) of Returns for an Individual Investment
n
i 1i
2ii P)]E(R-R[)(
Standard Deviation
SAIF ULLAH, [email protected], +923216633271
Variance (Standard Deviation) of Returns for an Individual Investment
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
SAIF ULLAH, [email protected], +923216633271
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%
SAIF ULLAH, [email protected], +923216633271
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
SAIF ULLAH, [email protected], +923216633271
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%
J F M A M J J A S O N D
Figure 8.2
SAIF ULLAH, [email protected], +923216633271
Times Series Returns forCoca-Cola and Exxon: 1998
-20.00%
-15.00%
-10.00%
-5.00%
0.00%
5.00%
10.00%
15.00%
J F M A M J J A S O N D
SAIF ULLAH, [email protected], +923216633271
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)]}
SAIF ULLAH, [email protected], +923216633271
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
SAIF ULLAH, [email protected], +923216633271
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
SAIF ULLAH, [email protected], +923216633271
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
SAIF ULLAH, [email protected], +923216633271
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
SAIF ULLAH, [email protected], +923216633271
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
SAIF ULLAH, [email protected], +923216633271
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
SAIF ULLAH, [email protected], +923216633271
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
SAIF ULLAH, [email protected], +923216633271
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
SAIF ULLAH, [email protected], +923216633271
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
SAIF ULLAH, [email protected], +923216633271
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
SAIF ULLAH, [email protected], +923216633271
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
SAIF ULLAH, [email protected], +923216633271
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 = 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
SAIF ULLAH, [email protected], +923216633271
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 = 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
SAIF ULLAH, [email protected], +923216633271
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 = 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
SAIF ULLAH, [email protected], +923216633271
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 = 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
SAIF ULLAH, [email protected], +923216633271
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
SAIF ULLAH, [email protected], +923216633271
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
SAIF ULLAH, [email protected], +923216633271
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
SAIF ULLAH, [email protected], +923216633271
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
SAIF ULLAH, [email protected], +923216633271
Efficient Frontier for Alternative Portfolios
Efficient Frontier
A
B
C
Figure 8.9
E(R)
Standard Deviation of Return
SAIF ULLAH, [email protected], +923216633271
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
SAIF ULLAH, [email protected], +923216633271
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
SAIF ULLAH, [email protected], +923216633271
Selecting an Optimal Risky Portfolio
)E( port
)E(R port
X
Y
U3
U2
U1
U3’
U2’ U1’
Figure 8.10
SAIF ULLAH, [email protected], +923216633271
The InternetInvestments Online
www.pionlie.com
www.investmentnews.com
www.micropal.com
www.riskview.com
www.altivest.com
SAIF ULLAH, [email protected], +923216633271
End of Chapter 8–An Introduction to Portfolio Management
SAIF ULLAH, [email protected], +923216633271
Future topicsChapter 9
• Capital Market Theory
• Capital Asset Pricing Model
• Beta
• Expected Return and Risk
• Arbitrage Pricing Theory