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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.

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chapter 9



    • 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
single factor model
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 one’s 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 model1
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 model2
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 stock’s 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 stock’s return 1.2% lower than previously expected
multifactor models
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 models1
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

Multifactor Models

Factor betas

multifactor models2
    • 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
multifactor models3
Multifactor Models
  • Expected rate of return=13.3%
  • 1% increase in GDP beyond current expectations, the stock’s return will increase by 1%*1.2
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)

Multifactor Security Market Line

Compensation for bearing the macroeconomic risk

Compensation for time value of money

multifactor security market line1
Multifactor Security Market Line for multifactor index model, risk premium is determined by exposure to each systematic risk factor and its risk premiumMultifactor Security Market Line
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 theory1
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

  • 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
well diversified portfolios1
The portfolio variance

If equally-weighted portfolio , the nonsystematic variance

N lager, the nonsystematic variance approaches zero, the effect of diversification

Well-diversified portfolios
well diversified portfolios2
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 eachweight small enough that the nonsystematic variance is negligible, eP approaches zero

For a well-diversified portfolio

Well-diversified portfolios
betas and expected returns
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 returns1
Betas and Expected Returns

Expected rate=10%,completely determined by Rm

Subject to nonsystematic risk

Only systematic risk should command a risk premium in market equilibrium

Solid line: plot the return of A with beta=1 for various realization of the systematic factor (Rm)

B: E(r)=8%. beta=1; A:E(r)=10%. beta=1
  • Arbitrage opportunity exist, so A and B can’t 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: beta=1,E(r)=10%;
  • C: beta=0.5,E(r)=6%;
  • D: 50% A and 50% risk-free (4%) asset,
    • beta=0.5*1+0.5*0=0.5, E(r)=7%
  • C and D have same beta (0.5)
    • different expected return
    • arbitrage opportunity
an arbitrage opportunity
Expected Return %



Risk premium





Risk-free rate=4





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 beta=1
  • no-arbitrage condition to obtain an expected return-beta relationship identical to that of CAPM
  • Market index, expected return=10%;Risk-free rate=4%
  • Suppose any deviation from market index return can serve as the systematic factor
  • E, beta=2/3, expected return=4%+2/3(10%-4%)=8%
  • If E’s expected return=9%, 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
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
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 stock’s 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 apt1

Example: Suppose two factor Portfolio 1, 2,

Risk-free rate=4%

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: 9%+4%=13%

a multifactor apt2
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 apt3
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

multifactor apt1
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
multifactor apt2
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
multifactor apt3
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

Multifactor APT
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

apt and capm compared
Many of the same functions: give a benchmark for rate of return.


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 and capm compared1
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
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