Arbitrage Pricing Theory

Chap 9 of Reilly and Brown and Chap. Of Prasanna Chandra

Limitations of CAPM

• The CAPM has been one of the most useful-and frequently used- financial theories ever developed.

• Still, the model has some deficiencies• Some tests of CAPM indicated that the beta

for individual securities were not stable, but portfolio betas were stable (with some conditions)

Limitations of CAPM

• Some studies find that firms give more return than suggested with beta, for example low P/E firms give more return than high P/E firms after adjusting for risk as measured by beta– High book-to-market price ratio firms generated

more return than low book-to-market price ratio firms

• In efficient markets such return differential should not exist

Limitations cont..

• The explanations of such difference could be:– Either the markets are not efficient– Or the model (CAPM) that predicts such returns is

not accurate• Markets are efficient in the long run, as

suggested by number of studies, so the only logical conclusion is : CAPM is not accurate.

Limitations cont…

• The CAPM captures only one form of risk, i.e. systematic risk in terms of Beta

• CAPM is based on many assumptions which are not true in practical lifeTo overcome these limitations, Stephen Ross

developed Arbitrage Pricing Theory (APT) which requires less assumptions and it allows multiple risk factors

Assumptions of APT

• Capital markets are perfectly competitive• Investors always prefer more wealth to less

wealth with certainty• The stochastic process generating asset

returns can be expressed as a linear function of a set of K risk factors (or indices)

APT- Return generating Process• The APT assumes that the return on asset is linearly

related to a set of risk factors as shown below:

Ri is the actual return on asset i during a specific time period (i=1,2,…n)

E(Ri) is expected return on asset i if all the risk factors have zero changes

bij is the sensitivities of asset i’s return to the common risk factor j

ƍj is a set of common factors, with a zero mean, that influences the return on all assets

ei is the random error term (a unique effect on i’s return)

APT- cont

• ƍj are multiple risk factors expected to have an impact on the returns of all assets.

• For example: GDP, inflation, political upheavals, change in interest rate etc

• APT considers these all factors that may have impact on return compared to CAPM which considers only covariance of the asset with market portfolio (beta)

APT- cont

• Given those common factors, i.e. ƍ, the bij determine how each asset reacts to the jth particular common factor

• For example: Interest rate change will affect all securities in the economy, but some securities (like banking stocks) will have more impact than other securities (like FMCG stocks)

APT- Equilibrium risk-return relationship• APT requires that in equilibrium the return on a zero-

investment, zero-systematic risk portfolio is zero when the unique effects (εi) are diversified away [this is the basic concept of arbitrage) – The key idea that guides the development of equilibrium risk-return

relationship is the law of one price which says that two identical things cannot sell at different prices

– Similarly, two portfolios having same risk, cannot offer different returns

– If they offer so, arbitrageurs will step in and one price will be established

• This assumption implies that the expected return on any asset i can be expressed as

APT-cont

Where:λ0= the expected return on an asset with zero

systematic riskλj= the risk premium related to the jth common

risk factor, ƍjbij= the pricing relationship between the risk

premium and the asset; or the responsiveness of the asset i to the jth common factor

Comparing APT and CAPMCAPM APT

Form of equation Linear Linear

Number of risk factors 1 K≥1

Factor risk premium E(Rm) – Rf Λj

Factor risk sensitivity βj Bij

Zero beta return (when no systematic risk is present)

Rf λ0

Example 1

• Consider the following data for two risk factors (1 and 2) and two securities (J and L):

– Compute the expected return for both the securities,– Suppose that J is priced at $22.50 while L is at $15.00. It is

expected that both securities with pay $0.75 as dividend. What is the expected price of both the securities?

– Suppose, somehow you know that after one year price of J will be $24 and L will be $17. How can you benefit from the situation?

λ0= 0.05 bJ1=0.8

λ1=0.02 bJ2=1.40

λ2=0.04 bL1=1.6

bL2=2.25

Example 2Consider the following data for two stocks D and E and two risk factors 1 and 2

Stock bi1 bi2 E(Ri)

D 1.2 3.4 13.1%

E 2.6 2.6 15.4%

What should the price of each stock be today to be consistent with the expected rate of return levels given?

c. Suppose now that the risk premium for Factor 1 that you calculated in part a suddenly increases by 0.25%. What are the new expected returns for stocks D and E?

d. If the increase in the Factor 1 risk premium in Part c does not cause you to change your opinion about what the stock price be in one year, what adjustment be necessary in the current price?

a. Assuming that the risk free rate is 5%, calculate the levels of the factor risk premia that are consistent with the reported values for the factor betas and the expected returns for the two stocks.

b. You expect that in one year the prices for stock D and E will be $55 and $36 respectively. Also, neither stock is expected to pay a dividend over the next year.

Example 3• Suppose that three stocks (A, B and C) and two common

risk factors (1 and 2) have the following relationship:

– a. If λ1=4% and λ2=2%, what are the prices expected next year for each of the stocks? Assume that all three stocks currently sell for $30 and will not pay a dividend in the next year

– b. Suppose you know that next year the prices for stock A, B and C will actually be $31.50, $35.00 and $30.50. Create and demonstrate a riskless, arbitrage investment to take advantage of these mispriced securities. What is the profit from your investment?