L7 - Arbitrage Pricing Theory (APT) and Multifactor Models

Question 1

Why do you want to include multiple risk factors in your model?

Answer 1

• returns are multi-dimensional
  
  • there are lots of different types of risk and firms are exposed to a different amount of each risk

Question 2

Who developed the APT model?

Answer 2

• Arbitrage Pricing Theory (APT) was developed by Ross (1976).
• The APT is a multi-factor model of asset pricing.•Similar to CAPM, APT generates a linear relationship between expected return and risk.
• Unlike the CAPM, we need very few assumptions to derive the APT.
• •The APT determines asset values based on the principle of no-arbitrage and law of one price.•It is a statistical model whereas the CAPM is an equilibrium asset pricing model.
  
  • –To derive the APT, we don’t need to assume that everyone is optimizing.

Question 3

What are the assumptions of the APT model?

Answer 3

Security returns can be described by a linear (multi-) factor model. There are a few macroeconomic factors influencing returns

.•There are sufficient securities so that firm specific (idiosyncratic) risk can be diversified away.

•Well-functioning security markets do not allow for persistent arbitrage opportunities.

Question 4

What is the definition of an arbitrage opportunity?

Answer 4

Pricing restriction in the APT comes from the absence of arbitrage opportunities.

•An arbitrage opportunity exists when risk-free profits can be earned without any net initial investment

Question 5

What does the absence of arbitrage imply?

Answer 5

Absence of Arbitrage implies Law of One Price:

• –Assets with the exact same payoffs must have the same price.
• –No security can have zero price and a strictly positive payoff

.•In a well-functioning capital market, arbitrage opportunities should not exist for too long.

•Our goal is to come up with a model such that:

• –If prices/returns obey this model, there is no arbitrage.
• –If prices/returns fail to obey this model, there is arbitrage.

Question 6

Example of a single factor model?

Answer 6

excess return(Risk Premium) and shock is the same for all assets

Question 7

Why is a well-diversified portfolio important for factor models?

Answer 7

• there should be no idiosyncratic risk as it would have been diversified away - thus the value of ‘e’ in a regression should be equal to zero
• Thus if two well-diversified portfolios have the same factor sensitivity, (the same beta) they must have an equal expected return
  
  • If not you can long one portfolio and short the other to make an arbitrage profit

Question 8

How do you check if another security is under/over priced?

Answer 8

• construct a portfolio with the original asset and the risk-free rate
• compare to see if they have both different factor sensitivities and expected returns
• If the constructed portfolio has a higher expected return –> new security is overpriced
• If the constructed portfolio has a lower expected return –> new security is underpriced

Question 9

What are the fundamental differences between APT and CAPM?

Answer 9

Under the CAPM, when security is mispriced, every investor make slight changes to their portfolio –>restore equilibrium

–Under the APT, some (not all) investors who can identify the arbitrage opportunity will want an infinite position in the risk-free arbitrage portfolio–> restore equilibrium

Question 10

What are some more features of the APT Model?

Answer 10

• APT applies to well-diversified portfolios, but not necessarily to individual stocks.
  
  • –If the no-arbitrage expected return-beta relationship (simple linear model) holds for almost every well-diversified portfolio, the relationship should hold for almost all the individual securities.
• •With APT, the probability of mispricing is zero. However, it does not mean that mispricing does not exist. Some assets may be mispriced at some time point.
• •APT can have multiple risk factors.

Question 11

Example of a Two-factor model?

Answer 11

• (when people are asking for the factor model they are usually asking for the first (expected return) equation

Question 12

What is the ideal factor to include in the multifactor APT?

Answer 12

• Its ability to price the asset will depend on getting the “right” factors.
• Unfortunately, the multifactor APT gives no guidance concerning what risk factors to include.
• Ideal factors should be able to explain variation in macroeconomic performance

Question 13

Example 1: two-factor APT?

Answer 13

Question 14

What is the Fama-French Three Factor Model?

Answer 14

• Relationship between size and B/M ratio with returns
  
  • Smaller the firm and the higher the B/M ratio the larger their returns

Question 15

What are some issues with the Fama-French model?

Answer 15

It is not motivated by economic theory

• •For CAPM, the underlying risk factor is the market (macroeconomy) risk
• •What are the underlying risk factors for FF 3-factor Model? SMB and HML are risk premia for what?
• Do these firm characteristics correlate with actual (but currently unknown) systematic risk factors?