Multifactor Models
What you'll learn
- LOS 37.aDescribe arbitrage pricing theory (APT), including its underlying assumptions and its relation to multifactor models.
- LOS 37.bDefine arbitrage opportunity and determine whether an arbitrage opportunity exists.
- LOS 37.cCalculate the expected return on an asset given an asset’s factor sensitivities and the factor risk premiums.
Arbitrage pricing theory (APT) was developed as an alternative to the capital asset pricing model. It is a linear model with multiple systematic risk factors priced by the market. However, unlike CAPM, APT does not identify the specific risk factors (or even the number of factors).
Assumptions of Arbitrage Pricing Theory (APT)
- Unsystematic risk can be diversified away in a portfolio. Investors have the choice of a large number of assets such that unsystematic risk can be diversified by forming portfolios of assets.
This is a reasonable assumption and is supported by empirical evidence.
- Returns are generated using a factor model. Unfortunately, the APT provides little practical guidance for the identification of the risk factors.
The lack of clarity for the risk factors is a major weakness of the APT.
- No arbitrage opportunities exist. An arbitrage opportunity is defined as an investment opportunity that bears no risk and no cost, but provides a profit.
This assumption implies that investors will undertake infinitely large positions (long and short) to exploit any perceived mispricing, causing asset prices to adjust immediately to their equilibrium values.
The asset pricing model developed by the arbitrage pricing theory is called the arbitrage pricing model.
The APT Equation
The APT describes the equilibrium relationship between expected returns for well- diversified portfolios and their multiple sources of systematic risk.
E(RP) = RF + βP,1(λ1) + βP,2(λ2) + ... + βP,k(λk)
Each λ stands for the expected risk premium associated with each risk factor. λj equals the risk premium for a portfolio (called a pure factor portfolio) with factor sensitivity equal to 1 to factor j and factor sensitivities of zero for the remaining factors. Remember that a risk premium is the difference between the expected return and the risk free rate (RF). It is the extra expected return from taking on more risk.
Each β represents the factor sensitivity of Portfolio P to that risk factor. Each factor in the arbitrage pricing model is “priced,” meaning that each risk premium is statistically and economically significant. Unlike the CAPM, the APT does not require that one of the risk factors is the market portfolio.
This is a major advantage of the arbitrage pricing model.
The method for exploiting arbitrage opportunities in the APT framework is detailed in the following example.
- Suppose your investment firm uses a single factor model to evaluate assets. Consider the following data for Portfolios A, B, and C:
- Calculate the arbitrage opportunity from the data provided.
- Answer:
- By allocating 50% of our funds to Portfolio A and 50% to Portfolio B, we can obtain a Portfolio (D) with beta equal to the Portfolio C beta (1.5):
Beta for Portfolio D = 0.5(1) + 0.5(2) = 1.5
- While the betas for Portfolios D and C are identical, the expected returns are different:
Expected return for Portfolio D = 0.5(0.10) + 0.5(0.20) = 0.15 = 15%
- Therefore, we have created Portfolio D that has the same risk as Portfolio C (beta = 1.5) but has a higher expected return than Portfolio C (15% versus 13%). By purchasing Portfolio D and short selling Portfolio C, we expect to earn a 2% return (15% minus 13%).
Recall that a portfolio beta equals the weighted average of the individual asset betas, and, likewise, the portfolio expected return equals the weighted average of the individual asset expected returns.
- The portfolio that is long Portfolio D and short Portfolio C is called the arbitrage portfolio. We have invested nothing upfront because we can use the proceeds of the short sale on Portfolio C to purchase Portfolio D, and we have undertaken no net systematic risk. The overall beta of our investment equals the difference in betas between our long and short positions: 1.5 − 1.5 = 0.
- As investors exploit the arbitrage opportunity, prices of assets in Portfolio C will drop and the (future) expected return for Portfolio C will rise to its equilibrium value.
The APT assumes there are no market imperfections preventing investors from exploiting arbitrage opportunities. As a result, extreme long and short positions are permitted and mispricing will disappear immediately. Therefore, all arbitrage opportunities such as the one described in the previous example would be exploited and eliminated immediately.
Given a portfolio’s factor exposures (betas) and factor risk premiums, we can easily compute the portfolio’s expected return as shown in the following example.
- An investment firm employs a two factor APT model. The risk free rate equals 5%. Determine the expected return for the Invest Fund using the following data:
- Answer:
- Using the two factor APT model, the expected return for the Invest Fund (IF) equals:
E(RIF) = 0.05 + 1.5(0.03) + 2(0.0125) = 0.12 = 12%
We can also use factor models to compute the parameter values given expected returns and factor exposures.
- Given a one factor model and the following information, calculate the risk free rate and the factor risk premium.
- Verify that Portfolio C with an expected return of 6.2% and factor sensitivity of 0.8 is priced correctly.
- Answer:
- Expected return = risk free rate + factor sensitivity × risk premium
Therefore, given the information for portfolios A and B:
0.07 = Rf + 1.0 × λ; Rf = 0.07 − λ
Substituting this information for Portfolio B:
0.078 = (0.07 − λ) + 1.2λ; λ = 0.04 or 4% Rf = 0.07 − λ = 0.07 − 0.04 = 0.03 or 3%
- Expected return for Portfolio C = 0.03 + (0.8 × 0.04) = 6.2%. Hence, Portfolio C is correctly priced.