Interpreting Factor Betas Through Stochastic Discount Factors

# Philosophical Question about Factor Models


# Philosophical Question about Factor Models












This might be a dumb question, but on a purely understanding level it is hard for me to wrap my head around the basic interpretation of some factor models. With the CAPM the interpretation of the $\beta$ coefficient is that this is nondiversifiable risk. Since we cannot diversify this risk away due to covariance with the market, we are getting a reward in terms of return for more risk being taken on. This is my understanding of CAPM and it seems natural (if this interpretation is wrong please feel free to say).

However, even though I understand the mathematics (factor models involving state variables other than just market return) line by line and the regressions, I am having trouble understanding the $\beta$ coefficient interpretation on different factors than just market return. Let's say we have a size factor, so $smb$ in the Fama French 3 factor model. How should I interpret the beta coefficient on that? If I extrapolate from the interpretation above I would say that $\beta_{smb}$ is a measure of non-diversifiable risk related to the size of companies in the sample, but this string of words together in a sentence in that order just seems silly to me. I think there must be something I am not understanding about the basic interpretation of factor models.

## Answer by Matthew Gunn (score 16, accepted)

https://quant.stackexchange.com/a/34066

This is actually a rather involved question, and different interpretations exist.

- A narrow, linear algebra based interpretation is that the stochastic discount factor lies in the linear span of the factors. (Recall that a linear asset pricing function implies the existence of a stochastic discount factor.)

- You can take an expansive, economics based interpretation of the stochastic discount factor as reflecting the marginal rate of substitution between various states of the world. Hence, in the words of Eugene Fama, factors are "variables of special hedging concern to investors." This is the macroeconomic risk based interpretation of asset pricing.

- Another interpretation is that the expected returns to various factors reflect irrational investors, various psychological anomalies, overreaction or underreaction to information. In theory, variation in expected returns due to macroeconomic risk could be distinguished from variation in expected returns due to investor psychology based upon whether expected returns are determined by when cash flows occur (i.e. in high or low marginal utility states of the world) or by firm characteristics (e.g. is it a glamour stock). This theoretical distinction is hard to measure and quite controversial in practice, partially because firms with similar characteristics tend to have returns that covary with each other.

Perhaps it makes sense to quickly review the basics of academic asset pricing theory.

#### A linear pricing function implies the existence of a stochastic discount factor

- Let $X$ and $Y$ random variables denoting payoffs in the future.

- Let $p(x)$ be a function returning the current price of a future payoff.

A reasonable assumption is that the pricing function $p(x)$ is a linear functional:

$$ p(\alpha X + \beta Y) = \alpha p(X) + \beta p(y)$$

(Note that with various psychology, behavioral effects, the pricing function need not be linear.)

By the Riesz Representation Theorem you get that $p(X)$ can be written as the inner product with some stochastic discount factor $S$.

$$ p(X) = \operatorname{E}[SX]$$

#### The SDF equation can be written as a 1 factor model

A return is a payoff with the price of 1. If there is a risk free rate $R^f$ then $1 = \operatorname{E}[S R^f] = \operatorname{E}[S] R^f$ since the payoff of the risk free rate is known before hand. So for an arbitrary return: $$ 1 = \operatorname{E}[S R]$$ Rewrite as a covariance: $$ 1 = \operatorname{Cov}(S, R) + \operatorname{E}[S]\operatorname{E}[R]$$ Divide by $\operatorname{E}[S]$, apply $\operatorname{E}[S] = 1/R^f$, and after some more algebra: $$ \operatorname{E}[R] - R^f = -\frac{\operatorname{Cov}(S, R)}{\operatorname{Var}(S)} \frac{\operatorname{Var}(S)}{\operatorname{E}[S]}$$

Finally we have

$$ \operatorname{E}[R] - R^f = \beta_{R, S} \lambda_S$$

where $\beta_{R,S}$ is a regression beta of the returns on the discount factor and $\lambda_S$ is the factor premium.

You can further show that for a multi-factor model to work, the stochastic discount factor $S$ must be a linear function of the factors. (There is a close relation here too to more classic notions in finance of the mean-variance frontier.)

The whole game of academic asset pricing is to figure out what the stochastic discount factor $S$ is! It's debatable how much progress has been made. (Or if behavioral effects dominate, whether it will ever be successful.) It's trivial to construct an $S$ that overfits the data and perfectly predicts the past. The challenge is explaining variation in returns going forward.

#### Economics and SDF theory

In some sense, this is where the bold statements start getting made.

In a simple economic model with additive separable utility, you get that the stochastic discount factor reflects the ratio of marginal utilities between the various (possible) states of the world.

Anyway, this is getting too long...

## Concluding Thoughts

A perhaps obvious point is that none of these factor models work perfectly. It's not hard to find test assets / portfolios that lead to overwhelmingly statistical rejection of most any asset pricing model.

To the extent factor models do work, you can interpret the factors in a way that leans more or less on economics. In pure linear algebra terms, the SDF may lie in the linear span of the factors. There's a growing, recent literature around identifying factors by using principal components analysis, the underlying idea being that the basis that explains the most variance in returns may be a basis that explains cross-sectional variation in expected returns (this is not obvious / automatically true).

Applying more economics and taking the risk based interpretation, the factors may proxy for variables of hedging concern for investors. (Another name for the stochastic discount factor is a marginal rate of substitution process.)

The Fama-French factors (and several others) are based upon long-short portfolios formed on firm characteristics. An interpretation that flows from the factors' method of construction is that firms sorted by certain characteristics (eg. high book to market ratio) tend to do well or poorly at the same time. Thus, you can even take a behavioral finance view/twist that characteristics matter rather than risk, but that risk based model still does ok because firms with similar characteristics move together.

References

Cochrane, John, Asset Pricing: Revised Edition, 2005, Princeton University Press

Duffie, Darrel, Dynamic Asset Pricing Theory, 1992, Princeton University Press

## Answer by vonjd (score 2)

https://quant.stackexchange.com/a/34072

For a mathematical interpretation see my answer here: https://quant.stackexchange.com/a/31456/12

The classic "philosophical" interpretation is that they are additional risk factors (like $\beta$) for which the investor will be compensated, e.g. small companies are riskier than large companies.

Another interpretation is that they represent anomalies which can be used by savvy investors.

For some more details on both interpretations see my answer here: https://quant.stackexchange.com/a/15959/12

You decide which interpretation suits you best.