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A **multifactor model** attempts to explain the observed historical return of security returns in the form of the equation below:

$$ R_i=a_i+b_{i,1}l_1+b_{i,2}l_2+\ldots+b_{i,L}l_L+C_i $$

Where:

- \(R_i\) is the return on security \(i\).
- \(a_i\) and \(C_i\) are the constant and random parts of the return component distinctive to security \(i\).
- \(l_1,…, l_{L}\) are the changes in \(L\) factors that explain the fluctuations of \(R_i\) about the expected return \(a_i\).
- \(b_{i,k}\) is security \(i\)’s sensitivity to factor \(k\).

This differs from single index models such as CAPM since it assumes that asset returns are explained by a single factor, usually the return on the market.

Multifactor models of security market returns are divided into three:

- Macroeconomic.
- Statistical.
- Fundamental factor models.

Macroeconomic factor models use observable economic time series as the factors. These factors include inflation, economic growth, interest rates, and exchange rates. Macroeconomic factor models assume that the random return of each security responds linearly to the macroeconomic shocks.

A significant limitation of the macroeconomic factor models is that they require the identification and measurement of all the pervasive shocks affecting security returns. Note that a limited number of pervasive sources of risk may exist. Additionally, these risks may not be known, or there may be insufficient data to measure them. This makes them less useful in explaining returns.

Returns are obtained by applying the following formula:

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

Where:

- \(R_i\) is the return for asset \(i\).
- \(E\left(R_i\right)\) is the expected return for asset \(i\).
- \(F_i\)’s are the surprises in the factors.
- \(b_{ij}\)’s are the surprise sensitivities of asset \(i\).
- \(\varepsilon_i\) is the firm-specific surprise, which is unrelated to the macro factors.

Assume that ABC company uses a two-factor macroeconomic model to explain returns of a stock A, \(F_1\) and \(F_2\).

- Rate of increase higher than expected for \(F_1\): 3%
- Rate of decrease lower than expected for \(F_2\): 2%
- Sensitivity for factor \(F_1\) and \(F_2\) are 1.3 and -0.3 respectively
- There was no surprise return.

If the expected return for the portfolio was 15%, the stock return for A is *closest* to:

$$ \begin{align*} R_A &=E\left(R_A\right)+b_{A1}F_1+b_{A2}F_2+\varepsilon_A \\ R_A & =0.15+1.3\left(0.03\right)-0.3\left(-0.02\right)=0.195=19.5\% \end{align*} $$

Fundamental factor models use the observed company attributes as factor betas since they explain a considerable proportion of common returns. These company attributes include firm size, book-to-market ratio, dividend yield, and industry classification. The factors are the returns estimated using cross-sectional regression.

The returns can be obtained by using the formula below:

$$ R_i=\alpha_i+\beta_{i1}F_1+\beta_{i2}F_2+\ldots+\beta_{ik}F_k+\varepsilon_i $$

Where:

- \(R_i\) is the return for stock \(i\).
- \(\beta_{ij}\)’s are the standardized sensitivities for stock \(i\) to factor \(j\).
- \(F_j\)’s are the returns for factor \(j\).
- \(\alpha_i\) is the intercept.
- \(\varepsilon_i\) is the portion of stock \(i\) return that may not be explained by the factor model.

Unlike the macroeconomic and statistical factor models, where factor betas are estimates from a time-series regression, the factor betas are calculated directly from the attributes data.

The standardized sensitivity for factor \(F_1\) of a stock A, for example, is obtained by using the formula:

$$ \beta_{A1}=\frac{F_{A1}-\bar{F_1}}{\sigma_{F_1}} $$

Where:

- \(F_{A1}\) is the \(F_1\)-the factor for stock A.
- \(\bar{F_1}\) is the average of \(F_1\) calculated across all stocks.
- \(\sigma_{F_1}\) is the standard deviation of \(F_1\) ratios across all stocks.

Statistical factor models do not specify the factors independently of the historical returns data. In contrast, a method called **principal components analysis** can be used to determine a set of indices that explain the observed variance as much as possible. However, these indices may lack a meaningful economic interpretation and may vary considerably between different data sets.

A significant limitation to the statistical and macroeconomic factor models is that, for time-series regression to accurately estimate the factor betas, a long and stable history of returns for security is required.

## Question

The following data exists for stock A of company X:

$$ \begin{array}{c|c|c|c} & \textbf{GDP} & \textbf{Interest} & \textbf{Deflation} \\ & & \textbf{Rate} & \\ \hline \text{Surprise sensitivities} & 1.1 & 0.8 & 0.1 \\ \hline \text{Surprise growth/decrease in factors} & 2\% & 1\% & (3)\% \end{array} $$

Which of the following is the

most accuratereturn for stock A calculated using a three-factor macroeconomic factor model, given that the expected return is 8% and the company experienced a 2% surprise return?

- 8.3%
- 10.3%
- 13.3%
## Solution

The correct answer is C.Using the formula:

$$ R_A=E\left(R_A\right)+\beta_{A1}F_1+\beta_{A2}F_2+\beta_{A3}F_3+\varepsilon_A $$

To obtain,

$$ R_A=8\%+1.1\times2\%+0.8\times1\%-0.1\left(-3\%\right)+2\%=13.3\% $$

Reading 40: Using Multifactor Models

*LOS 40 (d) Describe and compare macroeconomic factor models, fundamental factor models, and statistical factor models.*