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An Introduction to Portfolio Management

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Title: An Introduction to Portfolio Management


1
An Introduction to Portfolio Management
  • Fin 825

2
Greedy Risk Aversion
  • Greedy Given a choice between two assets with
    equal level of risk, greedy investors will select
    the asset with the higher level of risk.
  • Risk Averse Given a choice between two assets
    with equal rates of return, risk averse investors
    will select the asset with the lower level of
    risk.

3
Implications for the investment process
  • All investors are risk averse?
  • Yes.
  • All investors are risk averse?
  • Yes/No, risk preference may depends on amount of
    money involved - risking small amounts, but
    insuring large losses
  • Since most investors are risk averse, there is a
    positive relationship between expected return and
    expected risk.

4
Covariance between Returns of Two Assets
  • For two assets, i and j, the covariance of rates
    of return is a measure of the degree to which two
    variables move together relative to their
    individual mean values over time. Covariance is
    defined as
  • Covij ERi - E(Ri)Rj - E(Rj)

5
Covariance and Correlation
  • Covariance between two assets can be derived
    from their standard deviations and the
    correlation coefficient using the following
    formula

6
Markowitz portfolio optimization
  • Required inputs
  • Expected returns of all securities in the
    portfolio
  • Standard deviations of all securities in the
    portfolio
  • Covariance(s) (or correlation coefficient) among
    entire set of securities in the portfolio
  • With 100 assets, 4,950 correlation estimates

7
Portfolio Expected Return Formula
8
Portfolio Standard Deviation Formula
9
Returns Distribution for Two Perfectly Negatively
Correlated Stocks (r -1.0) and for Portfolio WM
Stock W
Stock M
Portfolio WM
.
.
.
.
25
25
25
.
.
.
.
.
.
.
15
15
15
0
0
0
.
.
.
.
-10
-10
-10
10
Returns Distributions for Two Perfectly
Positively Correlated Stocks (r 1.0) and for
Portfolio MM
Stock M
Portfolio MM
Stock M
25
15
0
-10
11
Combining Stocks with Different Returns and Risk
1 .10 .50
.0049 .07 2
.20 .50 .0100 .10
  • Case Correlation Coefficient
    Covariance
  • a 1.00
    .0070
  • b 0.50
    .0035
  • c 0.00
    .0000
  • d -0.50
    -.0035
  • e -1.00
    -.0070

12
Portfolio Risk-Return Plots for Different Weights
E(R)
2
With two perfectly correlated assets, it is only
possible to create a two-asset portfolio with
risk-return along a line
Rij 1.00
1
Standard Deviation of Return
13
Portfolio Risk-Return Plots for Different Weights
E(R)
f
2
g
With uncorrelated assets it is possible to create
a two-asset portfolio with lower risk than either
asset alone
h
i
j
Rij 1.00
k
1
Rij 0.00
Standard Deviation of Return
14
Portfolio Risk-Return Plots for Different Weights
E(R)
f
2
g
With correlated assets it is possible to create a
two- asset portfolio between the first two curves
h
i
j
Rij 1.00
Rij 0.50
k
1
Rij 0.00
Standard Deviation of Return
15
Portfolio Risk-Return Plots for Different Weights
E(R)
With negatively correlated assets it is
possible to create a two- asset portfolio with
much lower risk than either asset
Rij -0.50
f
2
g
h
i
j
Rij 1.00
Rij 0.50
k
1
Rij 0.00
Standard Deviation of Return
16
Portfolio Risk-Return Plots for Different Weights
Figure 8.7
E(R)
f
Rij -0.50
Rij -1.00
2
g
h
i
j
Rij 1.00
Rij 0.50
k
1
Rij 0.00
With perfectly negatively correlated assets it is
possible to create a two asset portfolio with
almost no risk
Standard Deviation of Return
17
Concept of Diversification
  • Combining different assets in a portfolio to
    reduce overall risks.
  • The lower the correlation between assets, the
    lower the overall portfolio risk produced.
  • Combining two assets with perfectly negative
    correlation (correlation coefficient of -1) could
    reduce the portfolio standard deviation to zero

18
Correlation Coefficient
  • Correlation coefficient is a standardized
    covariance. It varies from -1 to 1.

19
The Efficient Frontier
  • The efficient frontier represents that set of
    portfolios with the maximum rate of return for
    every given level of risk
  • Frontier will be portfolios of investments rather
    than individual securities

20
Efficient Frontier for Alternative Portfolios
Figure 8.9
Efficient Frontier
B
E(R)
A
C
Standard Deviation of Return
21
The Efficient Frontier and Investor Utility
  • An individual investors utility curve specifies
    the trade-offs he is willing to make between
    expected return and risk
  • The optimal portfolio results in the highest
    utility possible for a given investor
  • It lies at the point of tangency between the
    efficient frontier and the utility curve with the
    highest possible utility

22
Selecting an Optimal Risky Portfolio
Efficient Frontier
X
U3
U2
U1
23
Example P8-4
  • You are considering two assets with the following
    characteristics
  • E(R1).15, E(?1).10, W1.5
  • E(R2).20, E(?2).20, W1.5
  • Compute the mean and standard deviation of two
    portfolios if r1,20.4 and 0.60, respectively.

24
Solution
  • E(RP).5 x (.15) .5 x (.20) .175
  • If r1,20.4,
  • If r1,2-0.6, ?p0.08062

25
An Introduction to Asset Pricing Models
26
Risk-Free Asset
  • An asset with no risk.
  • Zero variance and zero correlation with all other
    assets
  • Provides the risk-free rate of return (RFR)
  • Will lie on the vertical axis of a portfolio
    graph
  • The combination of risk-free asset and any risky
    asset or portfolio will always have a linear
    relationship between expected return and risk.

27
Portfolio Possibilities Combining the Risk-Free
Asset and Risky Portfolios on the Efficient
Frontier
Figure 9.1
D
M
C
B
A
RFR
28
Portfolio Possibilities Combining the Risk-Free
Asset and Risky Portfolios on the Efficient
Frontier
CML
Borrowing
Lending
M
RFR
29
The Market Portfolio
  • Portfolio M lies at the point of tangency, it has
    the highest slope of trade-off between expected
    return and risk.
  • All investors will want to invest in Portfolio M
    and borrow or lend to be somewhere on the CML
  • Therefore this portfolio must include ALL RISKY
    ASSETS in proportion to their market values.
  • M is a completely diversified portfolio, which
    means that all the unique risk of individual
    assets is diversified away

30
Systematic Risk
  • Only systematic risk remains in the market
    portfolio, M
  • Systematic risk is the variability in all risky
    assets caused by macroeconomic variables
  • Systematic risk is measured by the standard
    deviation of returns of the market portfolio

31
Examples of Macroeconomic Factors Affecting
Systematic Risk
  • Variability in growth of money supply
  • Interest rate volatility
  • Inflation
  • Fiscal and Monetary policy changes
  • War and political events

32
Portfolio Standard Deviations
33
Portfolio Diversification
Diversification Spreading an investment across a
number of assets will eliminate some, but not
all, of the risk.
34
Portfolio Diversification
35
The CML and the Separation Theorem
  • The CML leads all investors to invest in the M
    portfolio (the investment decision)
  • The decision to borrow or lend to obtain a point
    on the CML is based on individual risk
    preferences (the financing decision)
  • Tobin refers to this separation of the investment
    decision from the financing decision as the
    Separation Theorem

36
CML and the Separation Theorem
CML
Borrowing
Lending
Figure 9.2
M
RFR
37
The Capital Asset Pricing Model Expected Return
and Risk
  • The existence of a risk-free asset resulted in
    capital market line (CML) that became the
    relevant frontier
  • An assets covariance with the market portfolio
    (systematic risk) is the relevant risk measure
  • Systematic risk can be used to determine an
    appropriate expected rate of return on a risky
    asset

38
Graph of Security Market Line
Figure 9.5
SML
M
RM
RFR
39
The Security Market Line (SML)
  • The equation for the risk-return line is

We then define as beta
40
Plot of Estimated Returnson SML Graph
.22 .20 .18 .16 .14 .12 Rm .10 .08 .06 .04 .02
C
SML
A
E
B
D
.20 .40 .60 .80
1.20 1.40 1.60 1.80
-.40 -.20
41
Calculating Systematic Risk The Characteristic
Line
where Ri,t the rate of return for asset i
during period t RM,t the rate of return for the
market portfolio M during t
42
Scatter Plot of Rates of Return
Figure 9.8
The characteristic line is the regression line of
the best fit through a scatter plot of rates of
return
Ri-rf
RM-rf
43
Arbitrage Pricing Theory (APT)
  • Assumptions
  • - Capital markets are perfectly competitive.
  • - Investors always prefer more wealth to less
    wealth with certainty.
  • - The stochastic process generating asset
    returns can be represented as a K factor model.

44
Arbitrage Pricing Theory (APT)
  • Assumptions do not Required
  • - Quadratic utility function.
  • - Normally distributed security returns.
  • - A market portfolio that contains all risky
    assets and is mean-variance efficient.

45
Return Generating Process
  • Ri E(Ri) bi1d1 bi2d2 ... bikdk Îi for
    i 1 to n
  • where
  • Ri return on asset i during a specified time
    period
  • E (Ri) expected return for asset i
  • bik reaction in asset is returns to movements
    in the common factor k
  • dk a common factor k with a zero mean that
    influences the returns on all assets
  • Îi a unique effect on asset is return that is
    completely diversifiable in large portfolios and
    has a mean of zero
  • n number of assets

46
Expected Return for Any Asset
  • E(Ri) l0 l1bi1, l2bi2 ... l kbik
  • where
  • l0 the expected return on an asset with zero
    systematic risk where l0 E(R0)
  • l1 the risk premium related to each of the
    common factors
  • bi the pricing relationship between the risk
    premium and asset i

47
2 Assets, 2-Factor Model
  • Factors
  • ?1 Changes in the rate of inflation
  • ? 2 percent growth in industrial production
  • l1 0.01, the risk premium associated with ?1
  • l2 0.015, the risk premium associated with ? 2
  • l0 0.04, rate of return on a zero-systematic-ris
    k asset

48
2 Assets, 2-Factor Model (Cont.)
  • Response Coefficients (B) for Assets F G
  • bF1 response of asset F to changes in the rate
    of inflation (0.5)
  • bF2 response of F to changes in level of
    industrial production (1.25)
  • bG1 response of asset G to changes in rate of
    inflation (1.75)
  • bG2 response of G to changes in level of
    industrial production (2.00)

49
E(Ri) l0 l1bi1 l2bi2
  • E(RF) 0.04 (0.01)(0.5) (0.015)(1.25)
  • 0.06375, or 6.38
  • E(RG) 0.04 (0.01)(1.75) (0.015)(2.00)
  • 0.0875, or 8.75

50
APT and CAPM Compared
  • APT applies to well diversified portfolios and
    not necessarily to individual stocks
  • With APT it is possible for some individual
    stocks to be mispriced - not lie on the SML
  • APT is more general in that it gets to an
    expected return and beta relationship without the
    assumption of the market portfolio
  • APT can be extended to multifactor models
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