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Transcript of R57 Portfolio Concepts
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CFA Level II Portfolio Management
Portfolio Concepts
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Graphs, charts, tables, examples, and figures are copyright 2012, CFA Institute. Reprod
and republished with permission from CFA Institute. All rights reserved.
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Contents
1. Introduction
2. Mean Variance Analysis
3. Practical Issues in Mean-Variance Analysis
4. Multi-Factor Models
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1. Introduction
What characteristics of a portfolio are important, and how may we quantify them?
How do we model risk?
If we could know the distribution of asset returns, how would we select an optimal p
What is the optimal way to combine risky and risk-free assets in a portfolio?
What are the limitations of using historical return data to predict a portfolios future
What risk factors should we consider in addition to market risk?
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Section Contents
1. The Minimum Variance Frontier and Related Concepts
2. Extension to the Three-Asset Case
3. Determining the Minimum-Variance Frontier for Many Assets
4. Diversification and Portfolio Size
5. Portfolio Choice with a Risk-Free Asset
6. The Capital Asset Pricing Model
7. Mean-Variance Portfolio Choice Rules: An Introduction
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2.1 Minimum Variance Frontier and Related C
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What is the expected return, variance and standard deviation for a portfolio with 70%
invested in Asset 1 and 30% invested in Asset 2?
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Table 2: Relation between Expected Return and Ris
Portfolio of Stocks and Bonds
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Figure 1 - Minimum-Variance Frontier: Large-Cap Sto
Government Bonds
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The minimum-variance fro
the minimum variance that
achieved for a given level ofreturn.
Point A represents the glob
minimum-variance portfoli
Efficient portfolio: portion
minimum-variance frontierwith the global minimum-v
portfolio and continuing ab
shows the highest expected
given level of risk.
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Figure 4 - Minimum-Variance Frontier for Varied Corre
Large-Cap Stocks and Government Bonds
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When the correlation
portfolios is less than +
offers potential benefi
correlation coefficient
other values constant,
benefits to diversificat
2 2 i h h C
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2.2 Extension to the Three-Asset Case
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T bl 6 P i h Mi i V i F i f
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Table 6 - Points on the Minimum-Variance Frontier f
Three-Asset Case
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h
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Figure 5 - Comparing Minimum-Variance Frontiers: Thr
versus Two Assets
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From this three-asset ex
draw two conclusions a
portfolio diversification.
1. We generally can im
return trade-off by e
assets in which we c
2. The composition of
variance portfolio folevel of expected ret
the expected return
correlations of those
number of assets.
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2.3 Determining the MVF for Many Ass
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The general formulas for expected return and variance of a portfolio are given below. Given n
the weights define a portfolio.
We (computers) solve for the portfolio weights (w1,
w2, w3, . . ., wn) that minimize the variance of return
for a given level of expected return z, subject to the
constraint that the weights sum to 1.
Expected
Return
i i i i i f
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Figure 6 - Minimum-Variance Frontier for Four
Classes 1970 2002
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2 4 Diversification and Portfolio Si e
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2.4 Diversification and Portfolio Size
Suppose we purchase a portfolio of n stocks and put an equalfraction of the value of the portfolio into each of the stocks. It can
be shown that the variance of return is:
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How many different stocks must we hold in order to have a well-diversified portfolio?
How does covariance or correlation interact with portfolio size in determining a portfoli
What happens to the portfolio variance as n becomes large?
Assuming:
1. Equal weight2. All stocks have the same variance
3. Stocks have same pair-wise correlation
What happens to the portfolio variance as n becomes large?
The formula
becomes:
Examples
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Examples
You have an equal-weighted portfolio consisting of
100 stocks. The pair-wise correlation between any
two stocks is 0.4. The average variance of stocks in the
portfolio is 900. What is the portfolio standarddeviation of return?
The average variance of return of all stocks in your
portfolio is 900. The correlation between the returns
of any two stocks is 0.4. What is the variance of return
of an equally weighted portfolio of 24 stocks?
What portfolio variance can be achieved given an
unlimited number of stocks, holding stock variance
and correlation constant?
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2 5 Portfolio Choice with a Risk Free As
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2.5 Portfolio Choice with a Risk-Free As
The capital allocation line (CAL) describes the combinations of expected return and stan
of return available to an investor from combining her optimal portfolio of risky assets wi
asset.
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Tangency portfolio
is the optimal risky
portfolio
Expected
Return
Risk (P)
Best risk-return
tradeoff
Example 5B CAL Calculations (1/2)
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Example 5B CAL Calculations (1/2)
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Example 5B CAL Calculations (2/2)
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Example 5B CAL Calculations (2/2)
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Capital Market Line (CML)
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Capital Market Line (CML)When investors share identical expectations about the mean returns, variance of return
correlations of risky assets, the CAL for all investors is the same and is known as the CM
With identical expectations, the tangency portfolio must be the same portfolio for all inv
It is called the market portfolio. It contains all risky assets in proportions reflecting their
weights.
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2 6 The Capital Asset Pricing Model
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2.6 The Capital Asset Pricing Model
CAPM Assumptions
Investors need only know the expected returns, the variances, and the covariances o
determine which portfolios are optimal for them. (This assumption appears througho
variance theory.)
Investors have identical views about risky assets mean returns, variances of returns,
correlations.
Investors can buy and sell assets in any quantity without affecting price, and all asset
marketable (can be traded).
Investors can borrow and lend at the risk-free rate without limit, and they can sell sh
any quantity.
Investors pay no taxes on returns and pay no transaction costs on trades.
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CAPM
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CAPM
The CML represents the efficient frontier when the assumptions of the CAPM hold. In a C
therefore, all investors can satisfy their investment needs by combining the risk-free asse
identical tangency portfolio, which is the market portfolio of all risky assets (no risky asse
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CAPM: expected returns of assets are based on systematic risk
Ri = Rf+ i [E(RM) RF] where i = Cov(Ri, RM)/Var(RM)
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2 7 MeanVariance Portfolio Choice Rule
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2.7 MeanVariance Portfolio Choice Rule
Introduction
Comparisons of Portfolios as Stand-Alone Investments
Given a choice between Portfolio A and B, prefer A if:RA RB but A has a smaller
RA > RB but A and B have the same
If we can lend/borrow at the risk free rate then we should use the Sharpe ratio
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Decision to Add an Investment to an Existing Po
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Decision to Add an Investment to an Existing Po
Example 6:
Your portfolio has a Sharpe ratio of 0.25.
Which of the following asset classes
should you add to your portfolio?
Eurobonds: predicted Sharpe ratio = 0.10;predicted correlation with existing
portfolio = 0.42.
Non-US equities: predicted Sharpe ratio =
0.30; predicted correlation with existing
portfolio = 0.67.
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If the following relationship holds
then add the new investment.
3 Practical Issues in Mean-Variance Ana
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3. Practical Issues in Mean Variance Ana
1. Estimating Inputs for Mean-Variance Optimization
1. Historical Estimates2. Market Model Estimates: Historical Beta
3. Market Model Estimates: Adjusted Beta
2. Instability in the Minimum Variance Frontier
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Market Model Estimates: Historical Beta
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Market Model Estimates: Historical Beta
Asset returns may be related to each other through their correlation with a limited set o
factors. The market model explains the return on a risky asset as a linear regression wit
the market as the independent variable.
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Assumptions:
The expected value of the error term is 0
The market return is uncorrelated with the error term
The error terms are uncorrelated among different assets
Given these assumptions, the market model makes the following three predictions:
Systematic
risk of asset i.Non-systematic
risk of asset i.
We can use t
covariance fro
greatly simpli
the covarianc
the minimum
Example 7 Computing Stock Correlations Using the Ma
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p p g g
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Example 7 - Solution
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Example 7 Solution
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Market Model Estimates: Adjusted Beta
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j
Adjusted beta is a historical beta adjusted to reflect the tendency of beta to be mean r
One common adjustment is:
Adjusted beta = 0.33 + 0.67 Historical beta
An adjusted beta tends to predict future beta better than historical beta does.
If the historical beta = 1.0, then adjusted beta = 0.333 + 0.667(1.0) = 1.0
if the historical beta = 1.5, then adjusted beta = 0.333 + 0.667(1.5) = 1.333
if the historical beta = 0.5, then adjusted beta = 0.333 + 0.667(0.5) = 0.667
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3.2 Instability in the Minimum-Variance Fr
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3.2 Instability in the Minimum Variance Fr
A problem with standard meanvariance optimization is that small changes in inputs fre
large changes in the weights of portfolios that appear on the minimum-variance frontie
problem of instability.
The problem of instability is practically important because the inputs to mean-variance o
are often based on sample statistics, which are subject to random variation.
The minimum-variance frontier is not stable over time. Two major reasons:
1. Estimation error in means, variances, and covariance
2. Shifts in the distribution of asset returns between sample time periods
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Example 8 Time Instability of the MVF
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p y
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4. Multifactor Models
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4. Multifactor Models
In this section we will cover:
1. Factors and Types of Multifactor Models
2. The Structure of Macroeconomic Factor Models
3. Arbitrage Pricing Theory and Factor Model
4. The Structure of Fundamental Factor Models
5. Multifactor Models in Current Practice
6. Applications
7. Concluding Remarks
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Multifactor models have gained importance for the practical business of portfolio management fo
reasons. First, multifactor models explain asset returns better than the market model does. Secon
models provide a more detailed analysis of risk than does a single factor model.
4.1 Factors and Types of Multifactor Mo
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ypA factor is a common or underlying element with which several variables are correlated
factors affect the average returns of a large number of different assets. These factors re
risk, risk for which investors require an additional return for bearing.
Types of multifactor models:
Macroeconomic factor models: the factors are surprises in macroeconomic variables th
explain equity returns.
Fundamental factor models: the factors are attributes of stocks or companies that are im
explaining cross-sectional differences in stock prices.
Statistical factor models: statistical methods are applied to a set of historical returns to
portfolios that explain historical returns in one of two senses. In factor analysis models,
the portfolios that best explain (reproduce) historical return covariances. In principal-co
models, the factors are portfolios that best explain (reproduce) the historical return vari
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4.2 The Structure of Macroeconomic Factor
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A factor sensitivity is a measure of the response of return to each unit of increase in a f
holding all other factors constant.
Example 10 Factor Sensitivities for a Two-Stock
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Example 10 - Solution
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4.3 Arbitrage Pricing Theory and the Factor M
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APT describes the expected return on an asset (or portfolio) as a linear function of the ri
(or portfolio) with respect to a set of factors.
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The factor risk premium (or factor price) j represents the expected return in excess ofrate for a portfolio with a sensitivity of 1 to factor j and a sensitivity of 0 to all other fac
portfolio is called a pure factor portfolio for factor j.
APT and CAPM
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If the market is the factor in a single-factor model, APT is consistent with the CAPM.
Like the CAPM, the APT describes a financial market equilibrium. However, the APT mak
assumptions than the CAPM. The APT relies on three assumptions:
1. A factor model describes asset returns.
2. There are many assets, so investors can form well-diversified portfolios that eliminaspecific risk.
3. No arbitrage opportunities exist among well-diversified portfolios.
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Example 11 Parameters in a One-Factor ATP
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p
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Example 12 Checking Whether Portfolio Retu
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Consistent with No Arbitrage
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If Portfolio D actually had an expected
investors would bid up its price until th
and the arbitrage opportunity vanishe
restores equilibrium relationships amoIf the return on D is 8%. Is there an arbitrage opportunity?
Example 13 - Parameters in a Two-Factor Mo
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4.4 Structure of Fundamental Factor Mo
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Economic factor models and fundamental factor models have the same form but there a
differences. In fundamental factor models:
The factors are stated as returns rather than return surprises in relation to predicted
they do not generally have expected values of zero. This approach changes the interp
of the intercept, which we no longer interpret as the expected return.
The factor sensitivities are attributes of the security.
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The Fama-French model is a fundamental factor model:ri = RF + i
mkt RMRF + isize SMB + i
value HML
4.5 Multifactor Models in Current Pract
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A specific example of macroeconomic factor models is the five-factor BIRR
ri = T-bill rate
+ (Sensitivity to confidence risk 2.59%)
(Sensitivity to time horizon risk 0.66%)
(Sensitivity to inflation risk 4.32%)
+ (Sensitivity to business-cycle risk 1.49%)
+ (Sensitivity to market-timing risk 3.61%)
Example 14 Expected Return in Macroeconomic Facto
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The BIRR model includes five factors:
1. Confidence risk: the unanticipated change in the return difference between risky cor
and government bonds, both with maturities of 20 years. Risky corporate bonds bea
default risk than does government debt. Investors attitudes toward this risk should a
average returns on equities. To explain the factors name, when their confidence is hare willing to accept a smaller reward for bearing this risk.
2. Time horizon risk: the unanticipated change in the return difference between 20-yea
bonds and 30-day Treasury bills. This factor reflects investors willingness to invest fo
3. Inflation risk: the unexpected change in the inflation rate. Nearly all stocks have nega
to this factor, as their returns decline with positive surprises in inflation.
4. Business cycle risk: the unexpected change in the level of real business activity. A pos
or unanticipated change indicates that the expected growth rate of the economy, me
constant dollars, has increased.
5. Market timing risk: the portion of the S&P 500s total return that remains unexplaine
four risk factors. Almost all stocks have positive sensitivity to this factor.
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4.6 Applications
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Multifactor models can help us understand in detail the sources of a m
returns relative to a benchmark.
Active return = Rp RB
Portfolio managers active return has two components
1. The return from factor tilts: product of the portfolio managers factor tilts (acti
sensitivities) and the factor returns
2. The return from asset selection: part of active return reflecting the managers
individual asset selection
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Example 18 - Active Return Decomposition of anPortfolio Manager
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Portfolio Manager
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Active Risk
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Active risk (tracking error, tracking risk) is the standard deviation of active returns.
TE = s(Rp RB)
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Variance of active risk is called active risk squared: s2(Rp RB)
Active risk squared = Active factor risk + Active specific risk
Risk due to portfoliosdifferent-than-benchmark
exposures relative to factors
specified in the risk model.
Risks resulting from the portfoliosactive weights on individual
assets. Also called asset selection
risk.
Example: Portfolio has a sample
benchmark has a sample mean r
Portfolios tracking error is 6%. W
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Questions:
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1. Contrast the active risk de
Portfolios A and B.
2. Contrast the active risk de
Portfolios B and C.
3. Characterize the investme
Portfolio D.
Factor and Tracking Portfolios
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A factor portfolio is a portfolio with unit sensitivity to a factor and zero se
other factors.
A tracking portfolio is a portfolio with factor sensitivities that match those
benchmark portfolio or other portfolio.
Factor and tracking portfolios can be constructed using as many assets as constraints on the portfolio.
Example 22 Factor Portfolios
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1. A portfolio manager wants to place a bet that real
business activity will increase.
A. Determine and justify the portfolio among the
six given that would be most useful to the
manager.
B. What type of position would the manager take
in the portfolio chosen in Part A?
2. A portfolio manager wants to hedge an existing
positive exposure to time horizon risk.
A. Determine and justify the portfolio among the
six given that would be most useful to the
manager.
B. What type of position would the manager takein the portfolio chosen in Part A?
Example 23 Tracking PortfolioThe portfolio manager determines that the benchmark has a sensitivity of 1.3 to the surprise in
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The portfolio manager determines that the benchmark has a sensitivity of 1.3 to the surprise in
sensitivity of 1.975 to the surprise in GDP.
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There are three constraints:
1. portfolio weights sum to 1
2. weighted sum of sensitivities to the inflation factor = 1.3
3. the weighted sum of sensitivities to the GDP factor = 1.975
Thus we need three investments
4.6 Concluding RemarksFrom a CAPM perspective investors should allocate their money between the risk free a
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From a CAPM perspective, investors should allocate their money between the risk-free a
broad-based index fund.
With multiple sources of systematic risk, when an investors factor risk exposures to oth
income and risk aversion differ from the average investors, a tilt away from an indexed
be optimal.
The average investor is exposed to and negatively affected by cyclical risk, which is a pric
Investors who hold jobs want lower cyclical risk and create a cyclical risk premium. Inves
labor income will accept more cyclical risk to capture a premium for a risk that they do n
As a result, an investor who faces lower-than-average recession risk optimally tilts towa
average exposure to the business cycle factor, all else equal.
Investors should know which priced risks they face and analyze the extent of their expos
Compared with single-factor models, multifactor models offer a rich context for investor
ways to improve portfolio selection.
Summary
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Mean-variance analysis
Efficient frontier
Instability of the efficient frontier
Diversification benefit using the two-asset portfolio
Variance of an equally weighted
portfolio of n stocks
CAL and CML
CAPM and its assumptions SML
Adjusted beta
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Multifactor Models
Macroeconomic factor mo
Fundamental factor mode
Difference between the tw
Statistical factor models
APT and its assumptions
Active return
Active risk
Information ratio
Factor and tracking portfo
ConclusionL i bj i
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Learning objectives
Summary
Examples
Practice problems
Problems from other sources
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