FINM7008 Chap.11 Arbitrage Pricing Theory and Factor Models
Arbitrage Pricing Theory and Factor Models
Pricing from no arbitrage
Arbitrage pricing theory needs only a factor model, the absence of persistent arbitrage and enough securities to diversify.
In a large portfolio firm-specific risk vanishes, so portfolios with the same factor beta must offer the same expected return.
One factor and many
The chapter builds the one-factor model, constructs an arbitrage trade against a mispriced portfolio, connects one-factor APT with the CAPM, and extends the argument to several factors.
Why it matters next
Tests of the simple beta-return relation found that market beta explains little, while variables such as book-to-market, profitability and momentum help explain returns.
APT allows several factors, so it provides the framework for the Fama-French and Carhart models that follow, and for benchmarking funds against more than the market.
What this chapter covers
- 01
CAPM assumptions against APT propositions
- 02
The one-factor model
- 03
Diversifying away firm-specific risk
- 04
What counts as arbitrage
- 05
The no-arbitrage line
- 06
Building an arbitrage trade
- 07
Multifactor APT
Worked example · free
Spotting and exploiting a mispriced portfolio
- 1Fair return for D: 0.5 × 8% = 4%, so D offers one point too little.
- 1Replicate D's beta with half A and half bills: beta 0.5 and expected excess return 4%.
- 1Buy the mix and short D in equal amounts: no net investment, zero beta, a riskless 1% of the position.
Key terms
- Factor Model
- A model that splits each security's return into responses to common factors and a firm-specific residual.
- Well-Diversified Portfolio
- A portfolio holding enough securities that its firm-specific risk is negligible.
- Factor Beta
- A security's sensitivity to one common factor.
- Factor Risk Premium
- The expected excess return on a portfolio with a beta of one on that factor.
- No-Arbitrage Condition
- The requirement that portfolios with equal factor betas offer equal expected returns.
Arbitrage Pricing Theory and Factor Models FAQ
How does APT differ from the CAPM?
The CAPM relies on a long list of idealised assumptions about investors and markets. APT, developed by Stephen Ross in 1976, needs only a factor model, no persistent arbitrage and enough securities, and allows several factors rather than the market alone.
Why must portfolios with the same beta offer the same return?
If two well-diversified portfolios had equal betas but different expected returns, buying one and shorting the other would cancel the factor risk and earn a riskless profit with no investment. Trading would remove the gap.
What does APT leave unanswered?
It does not say how many factors there are or which ones they are. In practice factors are identified by fitting models to past trading data, which leads to the size, value and momentum factors of the next chapter.
What are the three propositions of APT?
Returns follow a factor model, the market does not let arbitrage opportunities last, and there are enough securities to diversify away firm-specific risk. From these, every well-diversified portfolio's expected excess return is its factor beta times the factor premium.
Does APT hold exactly for individual shares?
Only approximately. The exact relation applies to well-diversified portfolios, whose firm-specific risk is negligible. Individual securities may carry small alphas, but they must be small and uncommon, or the portfolios built from them would break the relation.
How does one-factor APT relate to the CAPM?
If the single factor is the market, the factor beta becomes the covariance with the market over the market variance, the CAPM beta, and the factor premium becomes the market risk premium. Different assumptions then lead to the same pricing equation.
Exam move
Learn the five-step test and trade procedure, then apply it to a one-factor and a two-factor example. Practise the diversification calculation that shows why residual risk vanishes. Finish by writing the two-factor pricing plane and testing one portfolio against it. State both conditions of a valid trade explicitly every time, zero net investment and zero factor beta, because an answer missing either is not an arbitrage.
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