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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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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