Chapter 6 An Introduction to Portfolio Management
description
Transcript of Chapter 6 An Introduction to Portfolio Management
Chapter 6An Introduction to
Portfolio Management
2/46
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
3/46
The relationship between the returns for assets in the portfolio is important.A good portfolio is not simply a collection of individually good investments.
4/46
Risk Aversion
Given a choice between two assets with equal rates of return, most investors will select the asset with the lower level of risk.
5/46
Evidence ThatEvidence ThatInvestors are Risk AverseInvestors 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….
6/46
Definition of RiskDefinition of Risk
1. Uncertainty of future outcomesor2. Probability of an adverse
outcome
7/46
Markowitz Portfolio TheoryMarkowitz Portfolio Theory
Quantifies riskDerives the expected rate of return and expected risk for a portfolio of assets.
8/46
Shows that the variance of the rate of return is a meaningful measure of portfolio riskDerives the formula for computing the variance of a portfolio, showing how to effectively diversify a portfolio
9/46
Assumptions of Assumptions of Markowitz Portfolio TheoryMarkowitz Portfolio Theory
1. Investors consider each investment alternative as being presented by a probability distribution of expected returns over some holding period.
10/46
Assumptions of Assumptions of Markowitz Portfolio TheoryMarkowitz Portfolio Theory
2. Investors maximize one-period expected utility, and their utility curves demonstrate diminishing marginal utility of wealth.
11/46
Assumptions of Assumptions of Markowitz Portfolio TheoryMarkowitz Portfolio Theory
3. Investors estimate the risk of the portfolio on the basis of the variability of expected returns.
12/46
Assumptions of Assumptions of Markowitz Portfolio TheoryMarkowitz 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.
13/46
Assumptions of Assumptions of Markowitz Portfolio TheoryMarkowitz 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.
14/46
Markowitz Portfolio TheoryMarkowitz Portfolio TheoryUsing 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.
15/46
Alternative Measures of RiskAlternative Measures of RiskVariance or standard deviation of expected returnRange of returnsReturns below expectations
Semivariance – a measure that only considers deviations below the mean
These measures of risk implicitly assume that investors want to minimize the damage from returns less than some target rate
16/46
Expected ReturnExpected Return for an for an IndividualIndividual Risky InvestmentRisky 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)
Exhibit 6.1
17/46
Expected Return for a Expected Return for a PortfolioPortfolio of Risky Assets 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)
Exhibit 6.2
iasset for return of rate expected the )E(Riasset in portfolio theofpercent theW
:where
RW)E(R
i
i
1ipor
n
iii
18/46
Variance (Standard Deviation) of Variance (Standard Deviation) of Returns for an Returns for an IndividualIndividual Investment Investment
n
i 1i
2ii
2 P)]E(R-R[)( Variance
where Pi is the probability of the possible rate of return, Ri
19/46
Variance (Standard Deviation) of Variance (Standard Deviation) of Returns for an Returns for an IndividualIndividual Investment 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
Exhibit 6.3
Variance ( 2) = .0050
Standard Deviation ( ) = .02236
20/46
Variance (Standard Deviation) of Returns Variance (Standard Deviation) of Returns for a for a PortfolioPortfolio
Computation of Monthly Rates of ReturnExhibit 6.4
Closing ClosingDate Price Dividend Return (%) Price Dividend Return (%)
Dec.00 60.938 45.688 Jan.01 58.000 -4.82% 48.200 5.50%Feb.01 53.030 -8.57% 42.500 -11.83%Mar.01 45.160 0.18 -14.50% 43.100 0.04 1.51%Apr.01 46.190 2.28% 47.100 9.28%May.01 47.400 2.62% 49.290 4.65%Jun.01 45.000 0.18 -4.68% 47.240 0.04 -4.08%Jul.01 44.600 -0.89% 50.370 6.63%
Aug.01 48.670 9.13% 45.950 0.04 -8.70%Sep.01 46.850 0.18 -3.37% 38.370 -16.50%Oct.01 47.880 2.20% 38.230 -0.36%Nov.01 46.960 0.18 -1.55% 46.650 0.05 22.16%Dec.01 47.150 0.40% 51.010 9.35%
E(RCoca-Cola)= -1.81% E(Rhome Depot)=E(RExxon)= 1.47%
21/46
Covariance of ReturnsCovariance of Returns
A measure of the degree to which two variables “move together” relative to their individual mean values over time
22/46
Covariance of ReturnsCovariance of Returns
For two assets, i and j, the covariance of rates of return is defined as:
Covij = E{[Ri - E(Ri)][Rj - E(Rj)]}
23/46
Computation of Covariance of Returns for Coca Computation of Covariance of Returns for Coca cola and Home Depot: 2001cola and Home Depot: 2001
Exhibit 6.7
24/46
Covariance and Covariance and CorrelationCorrelation
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
25/46
Portfolio Standard Deviation Portfolio Standard Deviation FormulaFormula
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
26/46
Portfolio Standard Deviation Portfolio Standard Deviation CalculationCalculation
Any asset of a portfolio may be described by two characteristics:
The expected rate of returnThe expected standard deviations of returns
The correlation, measured by covariance, affects the portfolio standard deviationLow correlation reduces portfolio risk while not affecting the expected return
27/46
Time Patterns of Returns for Two Assets with Time Patterns of Returns for Two Assets with Perfect NegativePerfect Negative Correlation Correlation
Exhibit 6.10
28/46
Risk-Return Plot for Portfolios with Risk-Return Plot for Portfolios with Equal Returns and Standard Equal Returns and Standard DeviationsDeviations but Different Correlations but Different Correlations (page 182-183)(page 182-183)
Exhibit 6.11Correlation affects portfolio risk
A: correlation=1B: correlation=0.5C: correlation=0D: correlation=-0.5E: correlation=-1
29/46
Combining Stocks with Combining Stocks with Different Different Returns and RiskReturns 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
30/46
Combining Stocks with Different Combining Stocks with Different Returns and RiskReturns and Risk
Negative correlation reduces portfolio riskCombining two assets with -1.0 correlation reduces the portfolio standard deviation to zero only when individual standard deviations are equal.
31/46
Risk-Return Plot for Portfolios with Different Risk-Return Plot for Portfolios with Different Returns, Standard Deviations, and CorrelationsReturns, Standard Deviations, and Correlations
Exhibit 6.12
32/46
ConstantConstant Correlation Correlationwith Changing Weightswith Changing Weights
Case W1 W2 E(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 rij = 0.00
2 .20
33/46
Constant CorrelationConstant Correlationwith Changing Weightswith 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
34/46
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
35/46
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
36/46
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
37/46
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
38/46
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
Exhibit 6.13
39/46
Estimation IssuesEstimation Issues
Results of portfolio allocation depend on accurate statistical inputsEstimates of
Expected returns Standard deviationCorrelation coefficient
n(n-1)/2 correlation estimates: with 100 assets, 4,950 correlation estimates
40/46
Estimation IssuesEstimation Issues
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
41/46
Estimation IssuesEstimation Issues
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
42/46
The Efficient FrontierThe 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 returnFrontier will be portfolios of investments rather than individual securities
Exceptions being the asset with the highest return and the asset with the lowest risk
43/46
Efficient Frontier Efficient Frontier for Alternative Portfoliosfor Alternative Portfolios
Efficient Frontier
A
B
C
Exhibit 6.15
E(R)
Standard Deviation of Return
44/46
The Efficient Frontier The Efficient Frontier and Investor Utilityand Investor Utility
An individual investor’s utility curve specifies the trade-offs he is willing to make between expected return and riskThe slope of the efficient frontier curve decreases steadily as you move upwardThese two interactions will determine the particular portfolio selected by an individual investor
45/46
The Efficient Frontier The Efficient Frontier and Investor Utilityand Investor Utility
The optimal portfolio has the highest utility for a given investorIt lies at the point of tangency between the efficient frontier and the utility curve with the highest possible utility
46/46
Selecting an Optimal Risky PortfolioSelecting an Optimal Risky Portfolio
)E( port
)E(R port
X
Y
U3
U2
U1
U3’
U2’ U1’
Exhibit 6.16