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)

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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