I have recently learned about (implicit) factor models of the form:

$$ R = Xf + \epsilon $$

where $R \in \mathbb{R}^{n}$ are security returns, $X \in \mathbb{R}^{n \times F}$ are factor loadings for each security and each of $F$ factors and we fit a regression to get the estimated $f$.

This is also called cross-sectional regression.

Then, we compute factor covariances $\Omega := Cov(F_i,F_j)_{i,j=1,...,F}$ and can compute the volatility of the return of a portfolio $P$ with weights $w \in \mathbb{R}^{n}$ as:

$$Var(R_P) = w^T X^T \Omega X w + Var(\epsilon)$$

Someone mentioned that we could use ML-methods to fit more sophisticated models $$R = \phi(X,f) + \epsilon$$ I guess this means that $\phi$ might be a neural network or some other model class, mapping data input $x_i \in \mathbb{R}^F$ to estimated returns $R_i$ via to-be-found parameters $f$ (i.e. weights and biases of a neural network).

However, I am wondering how this might help getting a more precise estimate on $Var(R_P)$, after all $Var(\phi(X,f))$ is not easy to compute for complicated (nonlinear) $\phi$ and transparency might be an issue.

  • $\begingroup$ I think many people would be happy if this would minimize $Var(\epsilon)$ which a non-linear fit would certainly do $\endgroup$
    – LazyCat
    Sep 29, 2021 at 14:05

1 Answer 1


Not sure try to fit $\phi(X,f)$ makes sense: how would you define $X$? For a linear model $X$ is naturally defined as the beta of a linear regression.

Probably you would rather need to go "one layer deeper in the definition of factors", i.e. to use characteristics of the stock (like its market cap, P/E, dividend yield, etc) that you will name $C_1,\ldots,C_N$ and now the goal is to find a neural network (or any machine learning algo) such that

$$R = \sum_{1\leq i\leq F} X_i\cdot\psi_i(C_1,\ldots,C_N)+\epsilon.$$

This is in brief what is proposed in "Deep learning in asset pricing" by Chen, Pelger, and Zhu (2020).

That being said, what you have in mind now is specifically to focus on out of sample risk prediction. This can be done by adding a term in your loss function. What you note now is that in this better formulation, the risk of your portfolio continues to have this kind of shape

$$Var(R_P) = w^T X^T \Omega X w + Var(\epsilon)$$

where the expression of $\Omega$ is now different: $$\Omega := Cov(\psi_i(C_1,\ldots,C_N),\psi_j(C_1,\ldots,C_N))_{i,j=1,...,F}.$$

Note that the covariance is a scalar product, and hence if your learning algorithms $(\psi_i)_i$ are kernel based, you could use the "kernel trick" (I am not saying that it would be easy... I never seriously thought about doing it).

[EDIT following the comments to address the question of the best way to formulate Non-linear factors] The linear formulation of factor has the advantage of defining simultaneously the factors and the loadings (here I take a case with more than one factor to express the problem a more generic way, and I explicitly figure the time $t$): $$R_i(t)=\sum_{i,j}X_i f_j(t) +\epsilon_i(t).$$

It is a well-posed problem in the sense that under quite generic assumptions you can

  1. use a PCA on returns of a lot of stocks $(R_1,\ldots,R_N)$ to find $K$ factors (see for instance Factors That Fit the Time Series and Cross-Section of Stock Returns by Lettau and Pelger)
  2. use a linear regression in a second stage to find the loadings.

If you want to replace the factors with a non linear counterpart, my advice is

  • work on the residuals of the linear factors, ie on $\epsilon$, because at least it will guarantee you that you are doing something beyond the natural linear factors
  1. either you use any non-linear PCA like Self Organizing Maps, ie you are in a non supervised mode, and in such a case you have no inputs and the data are again $(R_1,\ldots,R_N)$. In such a case your loss function should be something like explained variance of the cross section of returns like for a PCA
  2. either you use a *supervised algorithm (the most famous being neural nets, like perceptrons) and you need exogenous inputs, like characteristics of the stock. In such a case your loss function will be the variance of the residuals of the explained returns.

It is interesting to notice

  1. that 1 and 2 respectively correspond to PCA vs characteristics of factors in the linear case (ie there is already 2 approaches that corresponds to 2 different philosophies)
  2. to really exploit the non linearity of the approach you may decide to go beyond L2 criteria. For instance for case 1 you can try to explain not only the variance of the cross section but its potential skewness or any other non L2 property.
  • $\begingroup$ In the factor regression, we fit for $f$ given data $X$, so what I meant by $\phi(X,f)$ was basically fitting a ML-model with to-be-found parameters $f$ given training data $X$. The kernel method seems interesting, but I think it will be difficult to estimate the covariance $\endgroup$ Oct 8, 2021 at 7:42
  • $\begingroup$ @ClaudioMoneo it could be good if you could elaborate, by adding details in your question. Say that $\phi$ is a neural net: do you mean that $f$ is its outputs, then, what is its input? because $X$ does not exist before training the net, no? I do not understand this. $\endgroup$
    – lehalle
    Oct 11, 2021 at 3:11
  • $\begingroup$ I hope it is clearer now. The notation $\phi(X,f)$ means that given data (input) $X$ and returns (output) $R$, we choose a parameter vector $f$ (weights and biases) to get a neural net $\phi$. Subsequently we map each $x_i$ via $\phi$ to predicted returns $R_i$ $\endgroup$ Oct 11, 2021 at 7:23
  • $\begingroup$ @ClaudioMoneo sorry there is still a lack of clarity: you want to reuse the loadings of the linear model as inputs for the neural net? or is it what I suggest in my answer (you take exogenous characteristics of the stocks)? and you do not have different "loadings" for each stock (because in your formula there is nothing in front of $\phi(X,f)$)? $\endgroup$
    – lehalle
    Oct 12, 2021 at 3:00
  • $\begingroup$ @ClaudioMoneo I added explanations in my answer $\endgroup$
    – lehalle
    Oct 12, 2021 at 3:17

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