Question

5. Suppose that asset returns satisfy the single factor model: ri = E(ri ) + βiF + ei , and let P and Q represent two well diversified portfolios. Suppose that βp < βq and E(rp) > E(rq).

a. Show there exists an arbitrage opportunity provided that investors can borrow at a rate of interest that is below E(rq)

b. Does there exist an arbitrage opportunity if the borrowing rate is greater than E(rq )?

Answer 1

What Is APT?

APT is a multi-factor technical model based on the relationship between a financial asset's expected return and its risk. The model is designed to capture the sensitivity of the asset's returns to changes in certain macroeconomic variables. Investors and financial analysts can use these results to help price securities.

Inherent to the arbitrage pricing theory is the belief that mispriced securities can represent short-term, risk-free profit opportunities. APT differs from the more conventional CAPM, which uses only a single factor. Like CAPM, however, the APT assumes that a factor model can effectively describe the correlation between risk and return.

Three Underlying Assumptions of APT

Unlike the capital asset pricing model, arbitrage pricing theory does not assume that investors hold efficient portfolios.

The theory does, however, follow three underlying assumptions:

• Asset returns are explained by systematic factors.
• Investors can build a portfolio of assets where specific risk is eliminated through diversification.
• No arbitrage opportunity exists among well-diversified portfolios. If any arbitrage opportunities do exist, they will be exploited away by investors. (This how the theory got its name.)

Assumptions of the Capital Asset Pricing Model

We can see that these are more relaxed assumptions than those of the capital asset pricing model. That model assumes that all investors hold homogeneous expectations about mean return and variance of assets. It also assumes that the same efficient frontier is available to all investors.

For a well-diversified portfolio, a basic formula describing arbitrage pricing theory can be written as the following:

\begin{aligned} &E(R_p) = R_f + \beta_1 f_1 + \beta_2 f_2 + \dotso + \beta_n f_n \\ &\textbf{where:}\\ &E(R_p)=\text{Expected return}\\ &R_f=\text{Risk-free return}\\ &\beta_n=\text{Sensitivity to the factor of }n\\ &f_n=n^{th}\text{ factor price}\\ \end{aligned}​E(Rp​)=Rf​+β1​f1​+β2​f2​+…+βn​fn​where:E(Rp​)=Expected returnRf​=Risk-free returnβn​=Sensitivity to the factor of nfn​=nth factor price​

Rf is the return if the asset did not have exposure to any factors, that is to say all

\beta_n = 0βn​=0

Unlike in the capital asset pricing model, the arbitrage pricing theory does not specify the factors. However, according to the research of Stephen Ross and Richard Roll, the most important factors are the following:

• Change in inflation
• Change in the level of industrial production
• Shifts in risk premiums
• Change in the shape of the term structure of interest rates

According to researchers Ross and Roll, if no surprise happens in the change of the above factors, the actual return will be equal to the expected return. However, in case of unanticipated changes to the factors, the actual return will be defined as follows

\begin{aligned} &R_p = E(R_p) + \beta_1 f'_1 + \beta_2 f'_2 + \dotso + \beta_n f'_n + e \\ &\textbf{where:}\\ &\begin{aligned} f'_n=&\text{ The unanticipated change in the factor or}\\ &\ \text{ surprise factor}\end{aligned}\\ &e=\text{The residual part of actual return}\\ &7\% = 2\% + 3.45*f_1 + 0.033*f_2\\ &f_1= 1.43\%\\ &f_2= 2.47\%\\ &E(R_i) = 2\% + 1.43\%*\beta_1 + 2.47\%*\beta_2\\ \end{aligned}​Rp​=E(Rp​)+β1​f1′​+β2​f2′​+…+βn​fn′​+ewhere:fn′​=​ The unanticipated change in the factor or  surprise factor​e=The residual part of actual return7%=2%+3.45∗f1​+0.033∗f2​f1​=1.43%f2​=2.47%E(Ri​)=2%+1.43%∗β1​+2.47%∗β2​​

Note that f'n is the unanticipated change in the factor or surprise factor, e is the residual part of actual return.

(For more on the capital asset pricing model, read The Advantages and Disadvantages of the CAPM Model.)

Estimating Factor Sensitivities and Factor Premiums

How we can actually derive factor sensitivities? Recall that in the capital asset pricing model, we derived asset beta, which measures asset sensitivity to market return, by simply regressing actual asset returns against market returns. Deriving the factors' beta is pretty much the same procedure.

For the purpose of illustrating the technique of estimating ßn (sensitivity to the factor n) and fn (the nth factor price), let's take the S&P 500 Total Return Index and the NASDAQ Composite Total Return Index as proxies for well-diversified portfolios for which we wish to find ßn and fn. For simplicity, we'll assume that we know Rf (the risk free return) is 2 percent. We'll also assume that the annual expected return of the portfolios are 7 percent for the S&P 500 Total Return Index and 9 percent for the NASDAQ Composite Total Return Index.

Step 1: Determine Systematic Factors

We have to determine the systematic factors by which portfolio returns are explained. Let’s assume that the real gross domestic product (GDP) growth rate and the 10-year Treasury bond yield change are the factors that we need. Since we have chosen two indices with large constituents, we can be confident that our portfolios are well diversified with close to zero specific risk.

Step 2: Obtain Betas

We ran a regression on historical quarterly data of each index against quarterly real GDP growth rates and quarterly T-bond yield changes. Note that because these calculations are for illustrative purposes only, we will skip the technical sides of regression analysis.

Here are the results:

Indices (Proxies for Portfolios)	ß1 of GDP Growth Rate	ß2 of T-Bond Yield Change
S&P 500 Total Return Index	3.45	0.033
NASDAQ Composite Total Return Index	4.74	0.098

Regression results tell us that both portfolios have much higher sensitivities to GDP growth rates (which is logical because GDP growth is usually reflected in the equity market change) and very tiny sensitivities to T-bond yield change (this too is logical because stocks are less sensitive to yield changes than bonds).

Step 3: Obtain Factor Prices or Factor Premiums

Now that we have obtained beta factors, we can estimate factor prices by solving the following set of equations:

7\% = 2\% + 3.45*f_1 + 0.033*f_27%=2%+3.45∗f1​+0.033∗f2​

9\% = 2\% + 4.74*f_1 + 0.098*f_29%=2%+4.74∗f1​+0.098∗f2​
Solving these equations we get:

f_1= 1.43\%f1​=1.43% and

f_2= 2.47\%f2​=2.47%

Therefore, a general ex-ante arbitrage pricing theory equation for any i portfolio will be as follows:

E(R_i) = 2\% + 1.43\%*\beta_1 + 2.47\%*\beta_2E(Ri​)=2%+1.43%∗β1​+2.47%∗β2​

Taking Advantage of Arbitrage Opportunities

The idea behind a no-arbitrage condition is that if there is a mispriced security in the market, investors can always construct a portfolio with factor sensitivities similar to those of mispriced securities and exploit the arbitrage opportunity.

For example, suppose that apart from our index portfolios there is an ABC Portfolio with the respective data provided in the following table:

Portfolios	Expected Return	ß1	ß2
S&P 500 Total Return Index	7%	3.45	0.033
NASDAQ Composite Total Return Index	9%	4.74	0.098
ABC Portfolio (or Arbitrage Portfolio)	8%	3.837	0.0525
Combined Index Portfolio = 0.7*S&P500+0.3*NASDAQ	7.6%	3.837	0.0525

We can construct a portfolio from the first two index portfolios (with an S&P 500 Total Return Index weight of 70 percent and NASDAQ Composite Total Return Index weight of 30 percent) with similar factor sensitivities as the ABC Portfolio as shown in the last raw of the table. Let's call this the Combined Index Portfolio. The Combined Index Portfolio has the same betas to the systematic factors as the ABC Portfolio but a lower expected return.

This implies that the ABC portfolio is undervalued. We will then short the Combined Index Portfolio and with those proceeds purchase shares of the ABC Portfolio, which is also called the arbitrage portfolio (because it exploits the arbitrage opportunity). As all investors would sell an overvalued and buy an undervalued portfolio, this would drive away any arbitrage profit. This is why the theory is called arbitrage pricing theory.

The Bottom Line

Arbitrage pricing theory, as an alternative model to the capital asset pricing model, tries to explain asset or portfolio returns with systematic factors and asset/portfolio sensitivities to such factors. The theory estimates the expected returns of a well-diversified portfolios with the underlying assumption that portfolios are well-diversified and any discrepancy from the equilibrium price in the market would be instantaneously driven away by investors. Any difference between actual return and expected return is explained by factor surprises (differences between expected and actual values of factors).

The drawback of arbitrage pricing theory is that it does not specify the systematic factors, but analysts can find these by regressing historical portfolio returns against factors such as real GDP growth rates, inflation changes, term structure changes, risk premium changes and so on. Regression equations make it possible to assess which systematic factors explain portfolio returns and which do not.