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CHAPTER 9

- APT AND MULTIFACTOR MODELS OF RISK AND RETURN

- Arbitrage
- Exploitation of security mispricing, risk-free

profits can be earned - No arbitrage condition, equilibrium market prices

are rational in that they rule out arbitrage

opportunities

9.1 MULTIFACTOR MODELS

Single Factor Model

- Returns on a security come from two sources
- Common macro-economic factor
- Firm specific events
- Focus directly on the ultimate sources of risk,

such as risk assessment when measuring ones

exposures to particular sources of uncertainty - Factors models are tools that allow us to

describe and quantify the different factors that

affect the rate of return on a security

Single Factor Model

- ri Return for security I
- Factor sensitivity or factor loading or

factor beta - F Surprise in macro-economic factor
- (F could be positive, negative or zero)
- ei Firm specific events
- F and ei have zero expected value, uncorrelated

Single Factor Model

- Example
- Suppose F is taken to be news about the state of

the business cycle, measured by the unexpected

percentage change in GDP, the consensus is that

GDP will increase by 4 this year. - Suppose that a stocks beta value is 1.2, if GDP

increases by only 3, then the value of F? - F-1, representing a 1 disappointment in actual

growth versus expected growth, resulting in the

stocks return 1.2 lower than previously expected

Multifactor Models

- Macro factor summarized by the market return

arises from a number of sources, a more explicit

representation of systematic risk allowing for

the possibility that different stocks exhibit

different sensitivities to its various components - Use more than one factor in addition to market

return - Examples include gross domestic product, expected

inflation, interest rates etc. - Estimate a beta or factor loading for each factor

using multiple regression. - Multifactor models, useful in risk management

applications, to measure exposure to various

macroeconomic risks, and to construct portfolios

to hedge those risks

Multifactor Models

- Two factor models
- GDP, Unanticipated growth in GDP, zero

expectation - IR, Unanticipated decline in interest rate, zero

expectation - Multifactor model Description of the factors

that affect the security returns

Factor betas

Multifactor Models

- Example
- One regulated electric-power utility (U), one

airline (A), compare their betas on GDP and IR - Beta on GDP U low, A high, positive
- Beta on IR U high, A low, negative
- When a good news suggesting the economy will

expand, GDP and IR will both increase, is the

news good or bad ? - For U, dominant sensitivity is to rates, bad
- For A, dominant sensitivity is to GDP, good
- One-factor model cannot capture differential

responses to varying sources of macroeconomic

uncertainty

Multifactor Models

- Expected rate of return13.3
- 1 increase in GDP beyond current expectations,

the stocks return will increase by 11.2

Multifactor Security Market Line

- Multifactor model, a description of the factors

that affect security returns, what determines

E(r) in multifactor model - Expected return on a security (CAPM)

Compensation for bearing the macroeconomic risk

Compensation for time value of money

Multifactor Security Market Line

- Multifactor Security Market Line for multifactor

index model, risk premium is determined by

exposure to each systematic risk factor and its

risk premium

9.2 ARBITRAGE PRICING THEORY

Arbitrage Pricing Theory

- Stephen Ross, 1976, APT, link expected returns to

risk - Three key propositions
- Security returns can be described by a factor

model - Sufficient securities to diversify away

idiosyncratic risk - Well-functioning security markets do not allow

for the persistence of arbitrage opportunities

Arbitrage Pricing Theory

- Arbitrage - arises if an investor can construct a

zero investment portfolio with a sure profit - Since no investment is required, an investor can

create large positions to secure large levels of

profit - In efficient markets, profitable arbitrage

opportunities will quickly disappear

Arbitrage

- Law of One Price
- If two assets are equivalent in all economically

relevant respects, then they should have the same

market price - Arbitrage activity
- If two portfolios are mispriced, the investor

could buy the low-priced portfolio and sell the

high-priced portfolio - Market price will move up to rule out arbitrage

opportunities - Security prices should satisfy a no-arbitrage

condition

Well-diversified portfolios

- Well-diversified portfolio, the firm-specific

risk negligible, only systematic risk remain - n-stock portfolio

Well-diversified portfolios

- The portfolio variance
- If equally-weighted portfolio , the

nonsystematic variance - N lager, the nonsystematic variance approaches

zero, the effect of diversification

Well-diversified portfolios

- This is true for other than equally weighted one
- Well-diversified portfolio is one that is

diversified over a large enough number of

securities with each weight small enough that the

nonsystematic variance is negligible, eP

approaches zero - For a well-diversified portfolio

Betas and Expected Returns

- Only systematic risk should command a risk

premium in market equilibrium - Well-diversified portfolios with equal betas must

have equal expected returns in market

equilibrium, or arbitrage opportunities exist - Expected return on all well-diversified portfolio

must lie on the straight line from the risk-free

asset

Betas and Expected Returns

- Only systematic risk should command a risk

premium in market equilibrium - Solid line plot the return of A with beta1 for

various realization of the systematic factor (Rm)

Expected rate10,completely determined by Rm

Subject to nonsystematic risk

- B E(r)8. beta1 AE(r)10. beta1
- Arbitrage opportunity exist, so A and B cant

coexist - Long in A, Short in B
- Factor risk cancels out across the long and short

positions, zero net investment get risk-free

profit - infinitely large scale until return discrepancy

disappears - well-diversified portfolios with equal betas must

have equal expected return in market equilibrium,

or arbitrage opportunities exist

- What about different betas
- A beta1,E(r)10
- C beta0.5,E(r)6
- D 50 A and 50 risk-free (4) asset,
- beta0.510.500.5, E(r)7
- C and D have same beta (0.5)
- different expected return
- arbitrage opportunity

An arbitrage opportunity

A/C/D, well-diversified portfolio, D 50 A and

50 risk-free asset, C and D have same beta

(0.5), different expected return, arbitrage

opportunity

- M, market index portfolio, on the line and beta1

- no-arbitrage condition to obtain an expected

return-beta relationship identical to that of

CAPM

- EXAMPLE
- Market index, expected return10Risk-free

rate4 - Suppose any deviation from market index return

can serve as the systematic factor - E, beta2/3, expected return42/3(10-4)8
- If Es expected return9, arbitrage opportunity
- Construct a portfolio F with same beta as E,
- 2/3 in M, 1/3 in T-bill
- Long E, short F

One-Factor SML

- M, market index portfolio, as a well-diversified

portfolio, no-arbitrage condition to obtain an

expected return-beta relationship identical to

that of CAPM - three assumptions a factor model, sufficient

number of securities to form a well-diversified

portfolios, absence of arbitrage opportunities - APT does not require that the benchmark portfolio

in SML be the true market portfolio

9.3 A MULTIFACTOR APT

Multifactor APT

- Use of more than a single factor
- Several factors driven by the business cycle that

might affect stock returns - Exposure to any of these factors will affect a

stocks risk and its expected return - Two-factor model
- Each factor has zero expected value, surprise
- Factor 1, departure of GDP growth from

expectations - Factor 2, unanticipated change in IR
- e, zero expected ,firm-specific component of

unexpected return

A MULTIFACTOR APT

- Requires formation of factor portfolios
- Factor portfolio
- Well-diversified
- Beta of 1 for one factor
- Beta of 0 for any other
- Or Tracking portfolio the return on such

portfolio track the evolution of particular

sources of macroeconomic risk, but are

uncorrelated with other sources of risk - Factor portfolios will serve as the benchmark

portfolios for a multifactor SML

A MULTIFACTOR APT

- Example Suppose two factor Portfolio 1, 2,
- Risk-free rate4
- Consider a well-diversified portfolio A ,with

beta on the two factors - Multifactor APT states that the overall risk

premium on portfolio A must equal the sum of the

risk premiums required as compensation for each

source of systematic risk - Total risk premium on the portfolio A
- Total return on the portfolio A 9413

A MULTIFACTOR APT

- Factor Portfolio 1 and 2, factor exposures of any

portfolio P are given by its and - Consider a portfolio Q formed by investing in

factor portfolios with weights - in portfolio 1
- in portfolio 2
- in T-bills
- Return of portfolio Q

A MULTIFACTOR APT

- Suppose return on A is 12 (not 13), then

arbitrage opportunity - Form a portfolio Q from the factor portfolios

with same betas as A, with weights - 0.5 in factor 1 portfolio
- 0.75 in factor 2 portfolio
- -0.25 in T-bill
- Invest 1 in Q, and sell in A, net investment

is 0, but with positive riskless profit - Q has same exposure as A to the two sources of

risk, their expected return also ought to be

equal

9.4 WHERE TO LOOK FOR FACTORS

Multifactor APT

- Two principles when specify a reasonable list of

factors - Limit ourselves to systematic factors with

considerable ability to explain security returns - Choose factors that seem likely to be important

risk factors, demand meaningful risk premiums to

bear exposure to those sources of risk

Multifactor APT

- Chen, Roll, Ross 1986
- Chose a set of factors based on the ability of

the factors to paint a broad picture of the

macro-economy - IP change in industrial production
- EI change in expected inflation
- UI change in unexpected inflation
- CG excess return of long-term corporate bonds

over long-term government bonds - GB excess return of long-term government bonds

over T-bill - Multidimensional SCL, multiple regression,

residual variance of the regression estimates the

firm-specific risk

Multifactor APT

- Fama, French, three-factor model
- Use firm characteristics that seem on empirical

grounds to proxy for exposure to systematic risk - SMB return of a portfolio of small stocks in

excess of the return on a portfolio of large

stocks - HML return of a portfolio of stocks with high

book-to-market ratio in excess of the return on a

portfolio of stocks with low ratio - Market index is expected to capture systematic

risk

- Fama, French, three-factor model
- Long-standing observations that firm size and

book-to-market ratio predict deviations of

average stock returns from levels with the CAPM - High ratios of book-to-market value are more

likely to be in financial distress, small stocks

may be more sensitive to changes in business

conditions - The variables may capture sensitivity to

risk-factors in macroeconomy

9.5 THE MULTIFACOTOR CAPM AND THE APT

APT and CAPM Compared

- Many of the same functions give a benchmark for

rate of return. - APT
- highlight the crucial distinction between factor

risk and diversifiable risk - APT assumption rational equilibrium in capital

markets precludes arbitrage opportunities (not

necessarily to individual stocks) - APT yields expected return-beta relationship

using a well-diversified portfolio (not a market

portfolio)

APT and CAPM Compared

- APT applies to well diversified portfolios and

not necessarily to individual stocks - 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

The Multifactor CAPM and the APM

- A multi-index CAPM
- Derived from a multi-period consideration of a

stream of consumption - will inherit its risk factors from sources of

risk that a broad group of investors deem

important enough to hedge, from a particular

hedging motive - The APT is largely silent on where to look for

priced sources of risk

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