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Multifactor models are mainly used for return attribution, risk attribution, and portfolio construction.

**Return attribution** is a set of techniques used in the identification of the excess return of a portfolio, relative to its benchmark. This helps to understand the consequences of active investment decisions. Return attribution is mainly used for quantifying portfolio managers’ active decisions, monitoring consistency, and informing the senior management and clients. It is a feedback mechanism used to evaluate investment decisions and identify the decisions that have added or reduced portfolio value, relative to the benchmark.

Remember that:

$$ \text{Active return} =R_p – R_b. $$

Active return can be decomposed into two components: factor return and security selection. **Factor return** typically arises from a manager’s decision to take on factor exposures that are different from those of the benchmark.

**Security selection**, on the other hand, arises from a manager’s choice of different weights for specific securities compared to those of the securities in the benchmark.

$$ \text{Active return} = \text{Factor return} + \text{Security selection return} $$

Where:

$$ \text{Factor return} =\sum_{i=1}^{k}{\left(\beta_{pi}-\beta_{bi}\right)\lambda_i} $$

Where:

- \(\beta_{pi}\) is the factor sensitivity for factor \(i\) in the active portfolio
- \(\beta_{bi}\) is the factor sensitivity for the factor \(i\) in the benchmark portfolio
- \(\lambda_i\) is the factor risk premium for factor \(i\).

The security selection return is, therefore, obtained by:

$$ \text{Security selection return} = \text{Active return} – \text{Factor return} $$

ABC Fund generated a return of 10.68% in the year 2019. The benchmark portfolio returned 10.54%. Attribute the cause of the difference in returns using a macroeconomic factor model with two factors, as shown in the table below. Also, describe the manager’s apparent skill in factor bets as well as in security selection.

$$ \begin{array}{c|c|c} & \textbf{GDP} & \textbf{Inflation} \\ \hline \text{Portfolio sensitivities } ({\beta}_{p}) & 0.80 & 1.50 \\ \hline \text{Benchmark sensitivities } ({\beta}_{b}) & 1.20 & 1.20 \\ \hline \text{Factor risk premium } ({\lambda}) & 4.00\% & -2.00\% \end{array} $$

We now calculate the return from factor tilts as follows:

$$ \text{Factor return} =\sum_{i=1}^{k}{\left(\beta_{pi}-\beta_{bi}\right)\lambda_i} $$

Therefore,

$$ \text{Factor return} =\left(0.8-1.2\right)\left(4\%\right)+\left(1.5-1.2\right)\left(-2\%\right)=-2.2\% $$

Recall,

$$ \text{Security selection} = \text{Active return} – \text{Factor return} $$

and

$$ \text{Active return} = R_P-R_b $$

Thus,

$$ \text{Security selection} = \left(10.68\%-10.54\%\right)-\left(-2.2\%\right)=2.34\% $$

The active manager’s regrettable factor bets resulted in a return of –2.2% relative to the benchmark. However, his superior security selection return of +2.34% resulted in a total active return of +0.14% relative to the benchmark.

**Risk attribution** is a methodology used to decompose the total risk of a portfolio into smaller units, which correspond to individual securities, or the subsets of securities in a portfolio. Active risk can be divided into two components, i.e., active factor risk, and active specific risk, as discussed in risk decomposition.

The **active risk**, also called the **tracking error** of a portfolio, can be separated into two components:

$$ {\text{(Active risk)}}^{2} =\text {Active factor risk} + \text{Active specific risk} $$

**Active factor risks** are risks from active factor tilts. The risks are consequences of deviations of a portfolio’s sensitivities from a benchmark’s sensitivities to corresponding factors.

**Active specific risk** or asset selection risk emanates from active asset selection. The risk is occasioned by deviations of a portfolio’s individual asset weightings from the benchmark’s individual asset weightings.

Active specific risk is obtained by:

$$ \text{Active specific risk} =\sum_{i=1}^{n}{\left(w_{pi}-w_{bi}\right)^2\sigma_{\varepsilon i}^2} $$

Where:

- \(w_{pi}\) is the \(i\)
^{th}security in the active portfolio. - \(w_{bi}\) is the \(i\)
^{th}security in the active benchmark. - \(\sigma_{\varepsilon i}^2\) is the residual risk in the \(i\)
^{th}asset.

Therefore, we can obtain the active factor risk by:

$$ \text{Active factor risk} =\text{Active risk}^{2}- \text{Active specific risk} $$

XYZ company uses a two-factor model to analyze the performance of its actively managed portfolios. The following table shows the results of active risk decomposition:

$$ \begin{array}{c|c|c|c|c|c} \textbf{Portfolio} & \textbf{Factor 1} & \textbf{Factor 2} & \textbf{Total} & \textbf{Active specific} & \textbf{Active} \\ & & & \textbf{factor} & \textbf{risk} & \textbf{risk} \\ \hline A & 3.2 & 12.6 & 15.8 & 4.2 & 20.0 \\ \hline B & 8.3 & 0.2 & 8.5 & 16.5 & 25.0 \\ \hline C & 10.1 & 15.1 & 25.2 & 4.8 & 30.0 \end{array} $$

- Which portfolio assumes the highest level of factor 2 risk relative to active risk?
- Which portfolio assumes the lowest level of active specific risk relative to active risk?

- We calculate the percentage of factor 2 risk as a proportion of active risk as follows:
$$ \begin{align*} \text{For A: } & \frac{12.6}{20.0}\times100=63.0\% \\ \text{For B: } & \frac{0.2}{25}\times100=0.8\% \\ \text{For C: } & \frac{15.1}{30}\times100=50.3\% \end{align*} $$

Therefore, portfolio A has the highest level of factor 2 risk relative to active risk.

- We calculate the percentage of active specific risk relative to active risk as follows:
$$ \begin{align*} \text{For A: } & \frac{4.2}{20.0}\times100=21.0\% \\ \text{For B: } & \frac{16.5}{25}\times100=66.0\% \\ \text{For C: } & \frac{4.8}{30}\times100=16.0\% \end{align*} $$

Therefore, portfolio C has the lowest level of active specific risk relative to active risk.

Managers seek to track a benchmark to construct a tracking portfolio. A **tracking portfolio** is deliberately created to have the same set of factor exposures tracking a predetermined benchmark. Multifactor models are used in the construction of portfolios that obtain a consistent result on the characteristics of the benchmark. This helps identify active decisions relative to the benchmark and measure the sizing of those decisions.

Multifactor models help in establishing intended exposures to various risk factors. Active managers, therefore, can use multifactor models to make particular bets on desired factors while hedging or even remaining neutral on other factors.

We illustrate the interpretation of the output of the multifactor model’s analysis using the illustration below.

Consider the following table of the factor exposures of three portfolios of Company ABC. The portfolios are evaluated based on a Fama-French three-factor model.

$$ \begin{array}{c|c|c|c} \textbf{Portfolio} & \textbf{RMRF} & \textbf{SMB} & \textbf{HML} \\ \hline X & 1.5 & 0.1 & 0.0 \\ \hline Y & 1.2 & 0.0 & 0.0 \\ \hline Z & 0.0 & 0.0 & -1.0 \end{array} $$

If a manager expects a higher RMRF, the most appropriate strategy would be going long on portfolio Y, which is generated to have exposure only to the RMRF factor. Portfolio X would not be ideal since it provides unwanted exposures to the SMB and HML factors as well.

If a manager expects growth stocks to outperform value stocks, the most appropriate strategy would be to go long on Z, which is a sure bet for value stocks.

## Question

Consider the following table for the factor exposures of three portfolios of XYZ company. The portfolios are evaluated based on a Carhart four-factor model.

$$ \begin{array}{c|c|c|c} \textbf{Portfolio} & \textbf{RMRF} & \textbf{SMB} & \textbf{HML} & \textbf{WML}\\ \hline A & 1.0 & 0.0 & 0.0 & 0.0 \\ \hline B & 0.0 & 1.0 & 0.0 & 0.0 \\ \hline C & 1.3 & 0.0 & 0.3 & 0.0 \end{array} $$

Which of the following would be the

most appropriatestrategy if a manager expects large-cap stocks to outperform small-cap stocks?

- Go long on portfolio B.
- Go long on portfolio B and short on portfolios A and C.
- Go short on portfolio B.
## Solution

The correct answer is C.The most appropriate strategy would be to go short on factor portfolio B, which is a sure bet for small-cap stocks.

Reading 41: Using Multifactor Models

*LOS 41 (f) Describe the uses of multifactor models and interpret the output of analyses based on multifactor models.*