What are the factor models, factor exposures, and factor returns in quantitative investing? how to use?

It’s normal to be confused, I can guarantee that many people are still confused but think they understand it.

I personally feel that factor models can distinguish the following concepts:

Factor, Factor Value, Factor Exposure, Factor Return, Factor Portfolio

In the field of asset pricing, the assumption is that the performance of an asset is driven by a number of drivers, also known as factors. In other words, there is a causal relationship between the performance of factors and assets.

Here, “performance” and “factor” are very abstract concepts that have nothing to do with mathematics. If we want to quantify this view, we need to do two things:

1. Propose an appropriate mathematical model (model)

2. Find a reasonable proxy variable (proxy)

As a proxy variable for the “performance” of an asset, the most straightforward idea is to use the return on the asset.

How to quantify the “factor” of the drive varies greatly, and the quantized factor will get a “factor value”.

For example, I think the return on an asset like a stock is closely related to the financial quality of the company behind it. Here, the company’s financial quality is an impact factor, but in actual analysis, we need to find a reasonable variable to quantify the company’s financial quality, such as ROE and ROA. ROE and ROA are factor values that can be specifically calculated from financial data.

Usually, in the articles we see, factors and factor values are not distinguished, and factor value is just a term I put forward under this answer, in order to distinguish.

A “proper model” is an ordinary linear model. For the sake of writing simplicity, I assume that there is a single factor, for the ith stock

$E(R_i)=\alpha_i + \beta_i * \lambda$

(The returns we are talking about are excess returns, so the risk-free rate is removed)

$\alpha_i$ Indicates pricing error

$\beta_i$ Indicates factor exposure

$\lambda$ Indicates the expected factor return

Notice that both sides of the equation are expectations, this is not a regression equation, this is an equation.

If we think in terms of random variables, we can examine a sample at time t:

$R_{i,t}=\alpha_i + \beta_i * \lambda_t+\epsilon_{i,t}$

The role of mathematical statistics is mainly to do two things, using sample data for estimation and hypothesis testing . We have T period data and N stocks constitute a sample, in which we can directly observe the return on assets. But factor exposures and factor returns require us to use some statistical methods to estimate them and test whether they are significant.

When learning CAPM, we are not confused about exposure and factor returns, because the CAPM model is a single-factor model and uses market factors. The return of market factors can be directly observed from the market, and then we can estimate the sensitivity of each stock to market factors using regression analysis.

What is factor exposure? It is elasticity and sensitivity. To put it bluntly, when the return of the factor changes a little, how much will it affect the return of the stock. We have asset return data and factor return data, and we can estimate elasticity by doing a regression. But for most cases, factor exposure and factor return are not observable at all wow! As long as one of these two goods can be observed, the other can be estimated, so this is a chicken-and-egg problem.

To understand this, let’s start with factor returns. Since the concept of factor returns is still too abstract, we make them concrete and propose the concept of factor portfolio, that is, at each moment t, an asset portfolio is constructed according to the “guidance” of the factor.

The combination at each moment t may be different, but the principle of constructing the combination is the same. For example, the market portfolio is easy to understand, that is, each period is weighted according to the market value of the stock to construct a portfolio. If I think the more auspicious the stock code, the better, I will buy more stocks with 8, 6, or consecutive numbers in the stock code. This kind of investment portfolio can reflect the role of the factor of “code auspicious”.

The easiest way to do this is to use the sorting method to construct a long-empty combination. The specific method is to sort the cross-section of the assets according to the size of the factor value at time t, and then divide them into several groups (for example, into 5 groups), and then take the first group and the fifth group, and use equal weight or market value in each group. The combination is constructed in a weighted way, and then the fifth group is long and the first group is short to obtain the factor return R_group5-R_group1 in the t period. With the factor series, taking the average directly is an estimate of the expected factor return.

I think there is an implicit assumption here. The larger the factor value, the greater the exposure. Although it is not necessarily a linear relationship, the basic monotonicity is still there. Stocks in Groups 1 and 5 should represent two groups of stocks with maximum and minimum exposure to factors, and the long-short portfolio constructed by the two groups of stocks should, in a sense, have strong factor exposures.

To sum it up, the simplest process looks like this:

Calculate the factor value (for example, use ROE to represent the quality factor) -> use the cross-sectional ranking of factor values to construct a long-short portfolio -> calculate the factor return sequence -> perform time series regression on historical data to estimate the sensitivity of each stock to the quality factor, namely factor exposure.

Consider the following 3 factors to understand the difference and connection between factors, factor values, factor returns and factor exposures:

1. The traditional style factor is generally represented by company characteristics (quality growth value), which is characterized by a factor value that can be calculated for each stock n at each moment t. At the same time t, the factor values of each stock may be different.

2. Market factors, non-traditional style factors. We directly skip the step of factor value calculation, and treat the market capitalization-weighted stock portfolio as a factor portfolio, which can directly observe the market’s return.

3. Macro factors, for each time t, each stock shares the same factor value, such as GDP, CPI, etc. However, we know that each stock may have different sensitivities to the same macro indicator, that is, different exposures. For example, we believe that there is a relationship between national GDP growth and stock returns. If there are three kinds of companies, the national disaster stock, the national wealth stock and the soy sauce stock. National distress stocks are stocks with a negative exposure to the GDP factor, Guofu stocks are stocks with positive GDP exposure, and soy sauce stocks are exposed to 0. When GDP declines, national distress stocks will drive some positive returns, while Guofu stocks will drive some positive returns. May fall, but soy sauce stocks and GDP are basically irrelevant.

Consider a stricter definition of factor combinations:

1. This combination only has exposure to the factor we want, and the exposure to other factors is 0

2. Therefore, in the combination that satisfies condition 1, the specificity variance of the factor combination is the smallest.

Unfortunately, the long-short combination obtained by our sorting method should not strictly meet the above two requirements. Moreover, the method of combining long and short is only applicable to traditional style factors. This pure factor combination can be solved for, for example, using cross-sectional regression or Fama-Macbeth regression. The specific process is not helpful for understanding the concept, so I will not repeat it here.

Many modern multi-factor models directly use the factor values of company characteristics (ROA ROE and the like) (with some normalization) as the proxy for the exposure of assets to factors, and found that the effect is better (although no one seems to be able to explain it clearly. why). For example, in the common BARRA model, the exposure of the factor and the signal value of the factor calculation are the same thing. It is also intuitively understandable that if I think that the financial quality of a company is one of the driving forces for the stock return, then the higher the financial quality index of a company, the more financial quality returns it will get, that is, the greater the exposure.

Of course, it can be seen that a factor can be instantiated by a combination of factors. This combination naturally has risks (fluctuations) in addition to returns. If an asset is exposed to a factor, it will not only bring benefits, but also bring risks.

in conclusion:

We believe that the performance of an asset is driven by several factors, assuming for the time being that each factor driver is independent of each other.

The rate of return of a factor indicates how much rate of return a factor can drive independently.

The factor exposure of an asset indicates how many shares of the factor’s driving force it has undertaken, that is, how much driving force it has undertaken.

Therefore, the rate of return of an asset is equal to the sum of the number of shares in the factor * the rate of return driven by the unit factor. This asset can be a single stock or a combination of stocks.

The so-called pure factor portfolio is an asset portfolio that is only driven by a single factor and not driven by other factors. So the return of a factor is equal to the sum of the pure factor portfolio stock weight * stock return.

This seems to be stuck in a vicious circle. Stock returns are decomposed into factor returns, and factor returns can be decomposed into stock returns.

The above statement has nothing to do with factor values and how factors are calculated! Due to the current multi-factor models (such as BARRA), the factor values of traditional style factors are often used as factor exposures, which leads to a sense of confusion about the concept when people read different books and papers.

Asset returns are apparent, factor returns are latent. We can calculate the factor value to quantify the factor, and then use the factor value to guide the construction of the factor simulation portfolio (ranking method or use the factor value instead of factor exposure) to estimate the factor return, so that the factor return can be represented, thus breaking this vicious circle.

Source: Zhihu www.zhihu.com

Author: dear dragon

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