This post outlines the methodology behind building a fundamental factor model.

In a multi-factor model, the return of a stock can be broken out into multiple factors.

$$r_i = \alpha_i + \sum_{j=1}^{K} \beta_{ij} f_j + \epsilon_i$$

The term $\epsilon_i$ is the error term which is assumed to have a distribution with zero mean. Now we can write the average return as

$$E(r_i) = E(\alpha_i) + \sum_{j=1}^{K} \beta_{ij} E(f_j)$$

This can be expressed concisely using the matrix representation:

$$r_i = \mathbf{\beta}^T_i \mathbf{f} + \epsilon_i$$

where:

\begin{eqarray} \mathbf{\beta}i &=& (\alpha_i, \beta, \beta_{i2}, ..., \beta_{iK})^T \ \mathbf(f) &=& (1, f_1, f_2, ..., f_K)^T \end{eqarray}

Using this notation, the average return can then be written as:

$$E(r_i) = \mathbf{\beta}_i^T E(\mathbf{f})$$

We see that the average return on the stock is the product of factor exposures ($\mathbf{\beta}_i$) and factor premium ($\mathbf{f}$). In a fundamental factor model, the factor exposures are characteristics of an investment (such as a stock) that is known, such as the Price to Earnings ratio, or momentum of the stock, or market capitalization. The factor premium is the unknown that we wish to determine empirically.

One of the tenets of quantitative portfolio management is: the average return of an investment is the payoff for taking risk. The risk of a stock has two components:

Total Risk = Non-Diversifiable Risk + Diversifiable Risk

Risk can be measured using the variance of the returns:

$$V(r_i) = V(\alpha + \beta_{i1}f_1 + ... + \beta_{iK}f_K) + V(\epsilon_i) + V(\epsilon_i)$$.

This can be expressed in the matrix notation as:

$$V(r_i) = \mathbf{\beta}_i^T V(\mathbf{f}) \mathbf{\beta}_i + V(\epsilon_i)$$

where $V(\mathbf{f})$ is a $(K+1) \times (K+1)$-dimensional variance-covariance matrix. The diversifiable component which an investor can mitigate by diversification is the term $V(\epsilon_i)$. The non-diversifiable componenent which the market rewards is obtained from factor exposures $\mathbf{\beta}_i$ and factor premium risk $V(\mathbf{f})$.

## Preparatory Work

There are a series of steps that one has to do before we can run a multi-variable regression to determine the factor premiums.

### Choosing the factors

We need to choose the factors that we wish to include in the model. This step is usually driven by the intuition an investor has on the market, and characteristics of the stock that are important. For instance, some fundamental factors that would make sense for equity investments include:

1. Valuation Factors: Market capitalization (size), price-to-book ratio (P/B), price-to-earnings ratio (P/E), price-to-sales ratio (P/S).
2. Solvency Factors: Debt-to-equity ratio (D/E), current ratio (CUR, a metric of short term solvency), interest coverage ratio.
3. Growth Potential Factors: Capital-expenditure-to-sales ratio, R&D-expenditure-to-sales ratio, advertising-expenditure-to-sales ratio.
4. Technical factors: Momentum, trading volume, short interest shares shorted
5. Analyst factors: Analyst rating changes, earnings revision etc.

Similarly some fundamental factors that would make sense for fixed income investments include:

1. Credit Factors: Credit quality, different vendor ratings
2. Interest Rate Sensitivity Factors: Effective duration, key rate duration at various tenors, convexity
3. Asset Type Factors: Some factors can be asset specific, such as:
1. Municipals:
1. State or issuer
2. Purpose such as revenue or general obligation
2. Mortgages:
1. Coupon
2. Amortization type
3. Issuer

### Treatment of the Risk-Free Rate

Since we are building a model to measure the payoff for undertaking risk, it is conventional to subtract the risk-free rate from the returns of the investment. The rational is that the risk-free rate is the return that can be obtained on any investment without any risk. In order to do this, we need to define what is considered risk-free. In the U.S. markets, the U.S. treasury bills are used as a proxy for the risk free rate. In other markets, one has to exercise care in identifying a risk free rate. All sovereign bond/bill rates do not immediately translate into risk-free rates because a lot of countries (developing and developed) have risk of default as well.

Once the risk-free rate $r_{ft}$ has been identified, the excess return can be defined as

$$r_{it}' = r_{it} - r_{ft}$$

### Building the Data Sample

One of the crucial components in the factor model building process is to create a data sample that can then be used to deduce the model parameters. For example, to model the U.S. equity market one can use the top 2000 to 3000 stocks by market cap. Depending on where we cutoff, this will include stocks in the large-cap and mid-cap range. This would mean that the model may not precisely model the effects in the small-cap stocks. On the other hand, limiting to stocks in the mid-cap or large-cap range can provide more stability to the model, and that could be a desirable thing.

Once we have a filter for deciding how we want to build the basket of instruments for model building, we also have to decide the frequency and period for which we need to collect the data. What do we mean by that? The regression step involved in the model building will fit the stock returns $r_{it}$ for a period (such as one day returns or monthly returns) as a function of various factors $f$. The return data is collected over multiple periods, such as 3 years or 5 years. For instance, our data sample could be top 3000 stocks by market cap with monthly returns collected for 3 years. A rule of thumb is that the frequency of the investor's re-balancing strategy should determine the frequency of the returns used in the model building. If the re-balances monthly, then one should use monthly returns in the model fitting. There should be sufficient time periods included in order to avoid spurious fitting effects and to improve robustness of fit. It is conventional to use 3 to 5 years worth of data while using monthly returns, and 2 to 3 years while using weekly returns.

Another factor that one should keep in mind is to include companies that are not only alive at the time of model building but also to include companies that may have gotten delisted or bankrupt. Failure to do so will introduce what is called as survivorship bias in your model.

## Fitting the Model

Once we have the data sample assembled, one can use regression utilities in statistical packages such as R, Python's Statsmodels package, Matlab, SAS to perform a panel regression. The goal of the panel regression is to estimate factor premium $\mathbf{f}$ given by the equation

$$r_i = \mathbf{\beta}^T_i \mathbf{f} + \epsilon_i$$

where the returns $r_i$ and the factor exposures $\mathbf{\beta}_i$ are known. A simple way to estimate factor premiums is by using Ordinary Least Squares (OLS). This will also give us an estimate of the sample variance-covariance matrix $V(\mathbf{\hat{f}})$.

\begin{eqnarray} \mathbf{f} &=& \left( \sum_{t=1}^T \sum_{i=1}^N (\mathbf{\beta}_{it} - \bar{\mathbf{\beta}} ) ( \mathbf{\beta}_{it} - \bar{\mathbf{\beta}} )^T \right)^{-1} \sum_{t=1}^{T} \sum_{i=1}^{N} (\mathbf{\beta}_{it} - \bar{\mathbf{\beta}} ) (r_{it} - \bar{r}) \end{eqnarray} where

$$\bar{\mathbf{\beta}} = \frac{1}{NT} \sum_{t=1}^{T} \sum_{i=1}^{N} \mathbf{\beta}_{it}$$ and $$\bar{r} = \frac{1}{NT} \sum_{t=1}^{T} \sum_{i=1}^{N} r_{it}$$

The variance-covariance matrix of the factor premium estimate is given as:

\begin{eqnarray} \hat{V}(\hat{f}) &=& \hat{\sigma}^2 \left( \sum_{t=1}^T \sum_{i=1}^N
(\mathbf{\beta}_{it} - \bar{\mathbf{\beta}} ) ( \mathbf{\beta}_{it} - \bar{\mathbf{\beta}} )^T \right) \end{eqnarray}

where $\hat{\sigma}^2$ is the variance of the error term $\epsilon_{it}$ $$\hat{\sigma}^2 = \frac{1}{NT} \sum_{t=1}^T \sum_{i=1}^N (r_{it} - \mathbf{\beta}_{it}\hat{f})$$

## Robustness Considerations

It is important that the estimates of the factor premium and the variance of factor premiums are robust. One way to accomplish that is by using generalized least squares (GLS). Corrections for heteroskedasticity and autocorrelation can be incorporated by using the Newey-West Estimator to correct for heteroskedacity and auto correlation. The statistical package R has an implementation of Newey-West in the sandwich package. Covariance Shrinkage was suggested by Ledoit and Wolf as a way to reduce estimation error in the Covariance Matrix. This technique helps reduce the extremes in the covariance matrix coefficients towards a central value, and is necessary in the portfolio optimization process.

## Conclusions

Here we gave an introduction to the process of building a factor model. This text is informative for some one to get started. There are lots of nuances that one needs to consider to incorporate the intuition of investment managers and follow the wind as the market currents change.

1. Ludwig B. Chincarini & Daehwan Kim, Quantitative Equity Portfolio Management, McGraw Hill
2. Olivier Ledoit & Michael Wolf, Honey, I Shrunk the Sample Covariance Matrix, http://www.ledoit.net/honey.pdf

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I am Goutham Balaraman, and I explore topics in quantitative finance, programming, and data science. You can follow me @gsbalaraman.

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