# Calculating Expected Returns from the Arbitrage Pricing Model

## Single-factor Model

The single-factor model assumes that there is just one macroeconomic factor, for example, the return on the market. Therefore:

$$R_i=E(R_i)+\beta_iF+\varepsilon_i$$

Where:

• $$E(R_i)$$ is the expected return on stock $$i$$.
• $$R_i$$ is the return for stock $$i$$.
• $$F$$ is the macroeconomic factor.
• $$B_i$$ is the factor-beta.
• $$\varepsilon_i$$ is the surprise return.

In case the macroeconomic factor has a value of zero in any particular period, then the return on the security will equal its initial expected return $$E(R_i)$$ plus the effects of firm-specific events.

#### Example: Calculating Expected Returns Using a Single-Factor Model

Assume that the common stock of Blue Ray Limited (BRL) is examined with a single-factor model, using unexpected percent changes in GDP as the single factor. Further, assume that the following data is provided:

$$\begin{array}{c|c} & \textbf{GDP} \\ \hline \text{Factor betas} & 1.5 \\ \hline \text{Growth for the factors} & 4\% \end{array}$$

Compute the required rate of return of BRL stock given that the expected return is 10% and assuming that there’s no new information regarding firm-specific events.

#### Solution

Using the single index factor model,

\begin{align*} & R_i =E(R_i)+\beta_iF+\varepsilon_i \\ & 10\%+1.5\times4\%=16\% \end{align*}

## Multifactor Pricing Model (MPM)

The rate of return in a general multifactor model is given by:

$$R_i=E\left(R_i\right)+\beta_{i1}F_1+\beta_{i2}F_2+\ldots+\beta_{ik}F_k+\varepsilon_i$$

Where:

• $$R_i$$ is the rate of return on stock $$i$$.
• $$E(R_i)$$ is the expected return on stock $$i$$.
• $$\beta_{ik}$$ is the sensitivity of the factor $$k$$.
• $$F_k$$ is the macroeconomic factor $$k$$.
• $$\varepsilon_i$$ is the firm-specific return or portion of the stock’s return unexplained by macro factors.

#### Example: Calculating Expected Return Using a Multifactor Model

Imagine that the common stock of BRL is examined using a multifactor model, based on two factors: unexpected percent change in GDP and interest rates. Further, assume that the following data is provided:

$$\begin{array}{c|c|c} & \textbf{GDP} & \textbf{Interest Rate} \\ \hline \text{Factor betas} & 1.5 & 2 \\ \hline \text{Surprise growth for the factors} & 2\% & 1\% \end{array}$$

Compute the required rate of return on BRL stock, assuming that the expected return for BRL is 10%, and there is no new information regarding firm-specific events.

### Solution

\begin{align*} R_i &=E\left(R_i\right)+\beta_{i1}F_1+\beta_{i2}F_2 \\ & =10\%+1.5\times2\%+2.0\times1\% \\ & =15\% \end{align*}

## The APT Equation(s)

For a well-diversified portfolio with several sources of systematic risk, the expected return is given by:

$$E(R_i)=E(R_Z)+\beta_{i1}\lambda_1+\cdots+\beta_{iK}\lambda_K$$

Where,

• $$E(R_i)$$ is the expected return on a well-diversified portfolio $$i$$.
• $$\beta_{ij}$$ is the sensitivity for a portfolio $$i$$ relative to factor $$j$$.
• $$E(R_Z)$$ is the expected rate of return on a portfolio with zero betas (such as risk-free rate of return).
• $$\lambda_{j}$$ is the risk premium relative to factor $$j$$.

#### Example: Calculating Expected Returns from the Arbitrage Pricing Theory

Assume the following data exists for portfolio A which has a risk-free rate of 6%:

$$\begin{array}{c|c|c} & \textbf{Factor 1} & \textbf{Factor 2} \\ \hline \text{Factor sensitivities} & 1.5 & 1.2 \\ \hline \text{Factor risk premium} & 0.02 & 0.03 \end{array}$$

Calculate the expected return for portfolio A using a two-factor APT model.

#### Solution

Using the formula:

\begin{align*} E(R_A) &=E(R_Z)+\beta_{A1}\lambda_1+\cdots+\beta_{AK}\lambda_K \\ E(R_A) & =0.06+1.5(0.02)+1.2(0.03)=0.126=12.6\% \end{align*}

## Question

The following data exists for a portfolio A:

$$\begin{array}{c|c|c|c} & \textbf{GDP} & \textbf{Interest Rate} & \textbf{Inflation} \\ \hline \text{Factor betas} & 0.5 & 0.4 & 0.6 \\ \hline \text{Expected growth in factors} & 2\% & 1\% & 3\% \end{array}$$

Which of the following is the most accurate return for portfolio A calculated using a three-factor general multifactor model, given that the expected return is 12%?

1. 13.2%
2. 14.2%
3. 15.2%

#### Solution

Using the formula:

\begin{align*} R_A& =E\left(R_A\right)+\beta_{A1}F_1+\beta_{A2}F_2+\beta_{A3}F_3 \\ R_A& =12\%+0.5\times2\%+0.4\times1\%+0.6\times3\%=15.2\% \end{align*}

LOS 41 (c) Calculate the expected return on an asset given an asset’s factor sensitivities and the factor risk premiums.

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