Estimating Asset Pricing Models in the Presence of Cross-Sectionally Correlated Pricing Errors

Abstract

In this study, we propose an adversarial learning approach to the asset pricing model estimation problem which aims to find estimates of factors and loadings that capture time-series covariations while minimizing the worst-case cross-sectional pricing errors. The proposed estimator is defined by a novel min-max optimization problem in which finding a solution is known to be difficult. This contrasts with other related estimators that admit a well-defined analytic solution but do not effectively account for correlations among the pricing errors. To this end, we propose an approximate algorithm based on the alternating optimization procedure and empirically demonstrate that our proposed adversarial estimation framework outperforms other existing factor models, especially when the explanatory power of the pricing model is limited.

1. Introduction

In this paper, we study one of the central problems in the field of finance, namely the estimation of the multifactor model for asset pricing that explains how prices of various assets, e.g., stocks and bonds, are set to the current values by using a small number of factors compared to the number of assets. In this section, we explain the importance of the estimation problem and discuss how the problem can be cast into a problem of unsupervised learning. To begin with, we describe factor pricing models in its simplest form.

1.1. Factor Pricing Models

Let us consider a bivariate linear mapping

$$E\left(R^{ei}\right)={\beta }_{i}E\left(f\right)+{\alpha }_i\mathrm{for}i\in \{1,2,\cdots ,N\}$$

(1)

from a pair of real numbers $({\alpha }_i,{\beta }_i)$ to a real number $E\left(R^{ei}\right)$ where a natural number N denotes the number of all assets and $E(·)$ denotes the expectation operator defined on the set of all possible random variables $R^{ei}$ and f that represent excess return of asset i and pricing factor, respectively. The second term in Equation (1), ${\alpha }_i$, represents the pricing error. Moreover, let us consider N constraints where ${\alpha }_i$ in Equation (1) is equal to zero for all i, i.e.,

$${\alpha }_i=0\mathrm{for}i\in \{1,2,\cdots ,N\}.$$

(2)

Here, we explicitly write the constraint on ${\alpha }_i$’s instead of directly applying it to Equation (1) in order to clarify that finding a model with smaller pricing errors, ${\alpha }_i$’s, is one of the two objectives that an estimator proposed in this paper pursues. Given a set of realizations $R_1^{ei},R_2^{ei},\cdots ,R_T^{ei}\in \mathbb{R}$ of the random variable $R^{ei}$ generated at time $t\in \{1,2,\cdots ,T\}$ for every $i\in \{1,2,\cdots ,N\}$, the estimation of our interest consists of finding ${\beta }_i$ and $E\left(f\right)$ that satisfy Equations (1) and (2) as closely as possible.

The factor pricing model, described by Equations (1) and (2), postulates that an asset i’s expected excess return linearly depends on its factor loading ${\beta }_i$. The model exhibits how the expected excess return of an arbitrary asset is valued based on the asset’s factor loading. If we acknowledge that ${\alpha }_i$’s differ significantly from 0, we could decide either of two things. First, the factor pricing model is wrong. For example, the underlying factor f is chosen in a wrong way. This leads us to conclude that a better pricing factor should be chosen or more pricing factors be added to the model. That is, the model needs to be improved. Second, the world is wrong. This means that the expected excess returns are not explained by the model since the corresponding assets are “mis-priced”. This might present practical trading opportunities for shrewd investors [1]. This is one of many reasons that correctly estimating the underlying pricing factor is a central problem in the field of finance that led to more than 300 published pricing factor candidates [2]. Furthermore, the exploration of new factors driving asset returns continues to be an active area of interest in both academic research and practical applications [3,4,5,6,7].

1.2. Definition of Multifactor Models

In this paper, we consider multifactor models for the following reason. Note first that we can generalize the factor pricing models described above by adding more pricing factors. It is therefore more sensible to use $K(>1)$ pricing factors instead of a single factor to explain expected excess returns of a large number of arbitrary assets [8,9]. The multifactor models amount to a $(K+1)$-variate linear mapping

$$E\left(R^{ei}\right)=\sum _{k=1}^K{\beta }_{i,k}E\left(f^k\right)+{\alpha }_i,\mathrm{for}i\in \{1,2,\cdots ,N\}$$

(3)

from a tuple of real numbers $({\alpha }_i,{\beta }_{i,1},{\beta }_{i,2},\cdots ,{\beta }_{i,K})$ to a real number $E\left(R^{ei}\right)$ where a natural number K denotes the number of the pricing factors that are used to model the expected excess returns across assets, or the cross-section of expected excess returns. Usually, K is set to be much smaller than N.

Let $f={[f^1,\cdots ,f^K]}^{\top }$ and ${\beta }_i={[{\beta }_{i,1},\cdots ,{\beta }_{i,K}]}^{\top }$ be $K\times 1$ column vectors of the pricing factors and the factor loadings, respectively (We let $A^{\top }$ denote the transpose of a matrix A). If Equations (2) and (3) are satisfied and the ${\beta }_i$ is computed as

$${\beta }_i={\mathrm{\Sigma }}_f^{-1}cov(f,R^{ei})\mathrm{for}i\in \{1,2,\cdots ,N\}$$

(4)

where ${\mathrm{\Sigma }}_f$ is the $K\times K$ covariance matrix of f and $cov(f,R^{ei})$ denotes the $K\times 1$ column vector whose k-th entry is $cov(f^k,R^{ei})$, then we say that there is a multifactor model with factors $f^1,f^2,\cdots ,f^K$ [10]. We refer to the multifactor model described by Equations (2)–(4) as the K factor model. As an illustrative example, we present a specific type of multifactor model in Section 2 in order to clarify the motivation of our work.

1.3. Unsupervised Learning Problem for the Multifactor Models

The main objective of this study is to propose a new method for estimating the pricing factors and the factor loadings of the multifactor model based on a given set of training data $D=\{R_t^{ei}:t=1,2,\cdots ,T,i=1,2,\cdots ,N\}$. The estimator aims to provide the T estimates of the pricing factors ${\widehat{f}}_1,{\widehat{f}}_2,\cdots ,{\widehat{f}}_T\in {\mathbb{R}}^{K\times 1}$ and the N estimates of the factor loadings ${\widehat{\beta }}_1,{\widehat{\beta }}_2,\cdots ,{\widehat{\beta }}_N\in {\mathbb{R}}^{K\times 1}$ that fit well with the conditions in Equations (2)–(4). Note that the estimator studied in this paper has nothing to do with predicting, say, an expected excess return, from unseen data. In other words, the estimation problem studied in this paper is an unsupervised learning problem.

1.4. Notation

We summarize notation used in this paper as follows. Let $N,T$, and K be the number of all assets, the number of time-series observations, and the number of factors, respectively. We assume that $K<N$ and $K<T$ throughout the paper. Let $m,n\in \mathbb{N}$. $I_n$ is an $n\times n$ identity matrix. ${\mathbb{1}}_n$ is a $n\times 1$ column vector of ones. For a matrix $A\in {\mathbb{R}}^{m\times n}$, we let $rank\left(A\right)$ denote the rank of A, $A^{\top }$ its transpose, ${\parallel A\parallel }_F$ its Frobenius norm, and $\mathrm{vec}\left(A\right)\in {\mathbb{R}}^{mn\times 1}$ the vec operation applied to A. For a square matrix S, $tr\left(S\right)$ denotes the trace of S. For matrices A and B, $A\otimes B$ denotes the Kronecker product of A and B. We let ${\mathbb{S}}_{+}^n$ denote the set of all $n\times n$ symmetric positive-semi-definite matrices. For a $V\in {\mathbb{S}}_{+}^n$, we define ${\parallel ·\parallel }_V:{\mathbb{R}}^n\to [0,\infty )$ by ${\parallel x\parallel }_V=\sqrt{x^{\top }Vx}$ for all $x\in {\mathbb{R}}^{n\times 1}$. The set of training data D is compactly represented by a $T\times N$ matrix X whose $(t,i)$-th entry is $R_t^{ei}$. We define a $T\times T$ matrix $P_1={\displaystyle \frac{1}{T}}{\mathbb{1}}_T{\mathbb{1}}_T^{\top }$, which is a projection matrix onto the linear subspace spanned by ${\mathbb{1}}_T$. We define a $T\times T$ matrix $M_1=I_T-P_1$ that annihilates the component that is parallel to the subspace spanned by ${\mathbb{1}}_T$, i.e., ${\mathbb{1}}_T^{\top }\left(M_{1}x\right)=0$ for all $x\in {\mathbb{R}}^{T\times 1}$. We widely use the fact that $P_1$ and $M_1$ are symmetric and idempotent.

2. Motivation

In this section, we elucidate a critical issue that motivates the estimator proposed in this paper, supported by preliminary experiments conducted using real-world data [11]. We consider the three factor model described by the time-series regression

$$R_t^{ei}={\alpha }_i+{\beta }_{i,\mathtt{Mkt}-\mathtt{RF}}{R}_t^{\mathtt{Mkt}-\mathtt{RF}}+{\beta }_{i,1}{f}_t^1+{\beta }_{i,2}{f}_t^2+{ϵ}_t^i$$

(5)

of excess returns $R_t^{ei}$ on the pricing factors $(R_t^{\mathtt{Mkt}-\mathtt{RF}},f_t^1,f_t^2)$, for $t\in \{1,2,\cdots ,T\}$. We fix one of the three factors as the market’s excess return $R_t^{\mathtt{Mkt}-\mathtt{RF}}$ that represents the entire US stock market and choose the remaining two from five candidates, $\{\mathtt{SMB},\mathtt{HML},\mathtt{CMA},\mathtt{RMW},\mathtt{Mom}\}$, which account for 10 combinations. We consider $N=25$ portfolios formed on size and book-to-market equity ratio, often called the $5\times 5$ Size-B/M portfolios, as test assets indexed by $i\in \{1,2,\cdots ,25\}$. It is known that they are well explained when $(f^1,f^2)=\left(\mathtt{SMB},\mathtt{HML}\right)$ [8]. For each i, we run the time-series regression from January 2017 to December 2021 (60 months) and obtain estimates of the factor loadings $({\widehat{\beta }}_{i,\mathtt{Mkt}-\mathtt{RF}},{\widehat{\beta }}_{i,1},{\widehat{\beta }}_{i,2})$, the pricing error ${\widehat{\alpha }}_i$, and the residuals ${\widehat{ϵ}}_t^i$.

A caveat is that the estimation problem described in the previous section does not exactly align with this time-series regression analysis. In the former case, both factors and regression coefficients are estimated, whereas in the latter, predefined factors are used without estimating them. By fixing one element for estimation and utilizing the established knowledge of explanatory power exhibited by various factor combinations, we gain a more intuitive understanding of the essential characteristics that “appropriate” factors and, consequently, factor models necessarily possess. These characteristics form the basis for designing the criterion for the proposed factor model estimator.

Figure 1 (top) illustrates the absolute value of the sample correlation coefficients between ${\widehat{ϵ}}_1^i,\cdots ,{\widehat{ϵ}}_T^i$ and ${\widehat{ϵ}}_1^j,\cdots ,{\widehat{ϵ}}_T^j$ that are identified by the brightness of the $(i,j)$-th grid with the color bar. Figure 1 (bottom) presents the average absolute values of the sample correlation coefficients and the sphericity test statistic of Bartlett [12] for the hypothesis that the correlation matrix equals the identity matrix. The figure reveals that the choice of $(f^1,f^2)=\left(\mathtt{SMB},\mathtt{HML}\right)$ indeed results in the smallest correlation. This is shown in the top-left subfigure of Figure 1 (top), represented by the darkest surface. It is also observed by the smallest values presented by the leftmost two bars in Figure 1 (bottom). The figure demonstrates that excluding either SMB or HML leads to a rise in cross-sectional correlation, while excluding both SMB and HML further increases the cross-sectional correlations.

These observations suggest that “appropriately” chosen pricing factors well regress out covariations in the test assets’ returns, making the unexplained part of the test assets, i.e., ${\widehat{\alpha }}_i+{\widehat{ϵ}}_t^i$, less likely to be correlated across assets. Conversely, when the chosen pricing factors are “inappropriate” for explaining the test assets, it becomes more probable that ${\widehat{\alpha }}_i+{\widehat{ϵ}}_t^i$ is cross-sectionally correlated.

In this context, evaluating an estimated factor model typically involves testing the null hypothesis $H_0:{\alpha }_1=\cdots ={\alpha }_N=0$ using a test statistic of the form:

$$q=c{\widehat{\alpha }}^{\top }{\widehat{W}}^{-1}\widehat{\alpha }$$

(6)

where $\widehat{\alpha }={[{\widehat{\alpha }}_1,\cdots ,{\widehat{\alpha }}_N]}^{\top }$, $\widehat{W}\in {\mathbb{S}}_{+}^N$ is the estimated residual covariance matrix and c is a positive constant independent of $\widehat{\alpha }$ and $\widehat{W}$ [13,14]. Evaluation of the model is conducted by checking whether $\left|q\right|\le \delta$ for a predefined threshold $\delta$, in which case it is concluded that the factor model is correctly estimated. Conversely, if $\left|q\right|>\delta$, it is inferred that the factor model is not correctly estimated. Therefore, it is important to identify a model capable of achieving a small value of $\left|q\right|$ that passes the test for the null $H_0$. However, as shown in Figure 1, $\widehat{W}$ can significantly deviate from $I_N$, especially when “inappropriate” factors are chosen. In such cases, the computed value of q under the assumption of $\widehat{W}=I_N$ may differ unpredictably difference from the value obtained without this assumption. This observation motivates the incorporation of q and $\widehat{W}$ into the estimation criterion such that models with smaller $\left|q\right|$ are preferred.

3. Related Work and Our Contributions

The arbitrage pricing theory of Ross [15] pioneered a line of research wherein statistical factor structure in the covariances of excess returns between assets are considered first, from which it is derived that the cross-section of expected excess returns is explained by the multifactor model (Equations (2)–(4)). The estimation problem of interest to us, wherein both of the pricing factors and the factor loadings are latent and should be estimated, resorts to the principal component analysis (PCA), justified by Chamberlain and Rothschild [16] and Connor and Korajczyk [17], which has been a dominant form of estimator in the literature, evidenced by large number of publications in recent years [18,19,20,21,22,23,24].

The conventional PCA due to Chamberlain and Rothschild [16] and Connor and Korajczyk [17] removes sample mean from the data by using $\tilde{X}=M_{1}X$ instead of X and apply the eigen-decomposition to $\widehat{\mathrm{\Sigma }}={\displaystyle \frac{1}{T}}{X}^{\top }{M}_{1}X={\displaystyle \frac{1}{T}}{\tilde{X}}^{\top }\tilde{X}$ so as to explain as much time-series variation in the de-meaned data $\tilde{X}$ as possible. This is explained in Box A in Figure 2, wherein the objective is to find the factors $\tilde{F}\in {\mathbb{R}}^{T\times K}$ and the loadings $\mathrm{\Lambda }\in {\mathbb{R}}^{N\times K}$ that well approximate $\tilde{X}$. The conventional PCA does not explicitly address pricing error information represented in the Boxes B and C.

The conventional PCA dicussed above assumes that the mean of the data matrix X is equal to zero. The risk-premium PCA (RP-PCA) is proposed by removing this assumption based on the observation that this assumption might be restrictive if the means have information about the factor structure. The RP-PCA differs from regularized PCA estimators employed in other applications, e.g., low-rank matrix approximation [25,26,27] and matrix completion [28,29,30], in the sense that it adds an economically motivated regularization term tailored to account for cross-sectional pricing errors.

The RP-PCA explicitly takes into account Box B in Figure 2, wherein the risk premia of the assets and the factor risk premia are compared. In other words, Box B aims to find $\overline{F}\in {\mathbb{R}}^{K\times 1}$ and $\mathrm{\Lambda }\in {\mathbb{R}}^{N\times K}$ in such a way that it minimizes the pricing errors, which are estimated by the difference between $\overline{X}$ and $\overline{F}{\mathrm{\Lambda }}^{\top }$, measured by $\parallel \overline{X}-\overline{F}{\mathrm{\Lambda }}^{\top }{\parallel }_{I_N}$. By simultaneously considering the time-series variations (Box A) and the cross-section of pricing errors (Box B) in the framework of a regularized minimization problem, Lettau and Pelger [22,31] showed that their estimator can find pricing factors that cannot be detected by the conventional PCA and that it can estimate factors more efficiently than the conventional PCA in the presence of “weak” factors.

Our factor model estimator is developed to address the weaknesses inherent in the RP-PCA. Specifically, a crucial problem of the RP-PCA is its lack of explicit consideration for the real-world scenario, characterized by correlated pricing errors observed in Section 2. This circumstance can result in substantial differences between $\parallel \overline{X}-\overline{F}{\mathrm{\Lambda }}^{\top }{\parallel }_{I_N}$ and $\parallel \overline{X}-\overline{F}{\mathrm{\Lambda }}^{\top }{\parallel }_V$, with the later being the more suitable distance function for pricing model estimation when V equals the precision matrix of pricing errors, cf., the definition of the test statistic q in Equation (6). Despite its significance in handling real financial data, however, the incorporation of ${\parallel ·\parallel }_V$ for a general V into pricing model estimation has not been explored in the literature.

A significant challenge in utilizing the precision matrix arises from the inherent estimation errors associated with the covariance structure matrix V, which must be estimated and thus introduces inaccuracies. To tackle this issue, we propose a method that integrates all potential matrices V within predefined ranges into the estimation process. This is accomplished by formulating a novel min-max optimization problem designed to address the estimation errors linked to V, ultimately leading to more robust estimates in the factor model estimation framework.

Our Contributions

First, we propose a new estimator for pricing factors and the associated factor loadings which are defined in the framework of adversarial machine learning. To this end, we restate the estimation problem for asset pricing model as a min-max optimization problem. It is important to mention that the factor model examined in this paper is static, meaning that factor loadings remain constant over time. This static nature simplifies the relationships between factors and asset returns, allowing for a more straightforward analysis on the underlying structure of the data without the need for dynamic adjustments. Then, the estimates of pricing factors and factor loadings will aim to closely approximate the time-series fluctuations in the training set of excess returns, while ensuring that the cross-section of pricing errors remains jointly small, even in the presence of correlated pricing errors across assets. Specifically, we extend an existing PCA-based factor model estimator by allowing for the distance to be defined by a seminorm ${\parallel ·\parallel }_V$ for any arbitrary $V\in {\mathbb{S}}_{+}^n$, which entails the consideration of cross-sectional correlations.

Second, we provide an optimization method that approximately solves the proposed min-max problem. We employ the alternating optimization procedure, which solves minimization and maximization problems, iteratively. In particular, we explain that it is challenging to solve the minimization part of the problem of our interest and prove that the proposed algorithm converges and generates well-defined iterates. By doing so, we introduce a novel computational framework for examining factor pricing model estimators defined to cover estimation errors inherent in the covariance structure matrix V that allows for a broader and more general measurement of pricing errors.

4. The Proposed Estimator of the K Factor Models

In this section, we present our proposed estimator of the K factor models, which is defined by an min-max optimization problem. We explain the objective function of the proposed optimization problem and relate it to the existing factor model estimators.

4.1. The Proposed Min-Max Problem

Based on the preliminary empirical findings in Section 2, it appears that correlations between pricing errors of distinct assets may not be negligible if inappropriate pricing factors are employed to explain asset excess returns. Drawing on these preliminary results and the concept outlined in Figure 2, we propose a novel estimator of the K factor model. The estimates of F and $\mathrm{\Lambda }$ are determined as solutions to the following min-max problem.

$$\underset{\begin{array}{c}\mathrm{\Lambda }\in {\mathbb{R}}^{N\times K}\\ F\in {\mathbb{R}}^{T\times K}\end{array}}{min}\underset{V\in \mathcal{V}}{max}\psi (\mathrm{\Lambda },F,V)\triangleq {\displaystyle \frac{1}{NT}}{∥\tilde{X}-\tilde{F}{\mathrm{\Lambda }}^{\top }∥}_F^2+{\displaystyle \frac{\eta }{N}}{∥\overline{X}-\mathrm{\Lambda }\overline{F}∥}_V^2$$

(7)

The objective function $\psi :{\mathbb{R}}^{N\times K}\times {\mathbb{R}}^{T\times K}\times \mathcal{V}\to [0,\infty )$ is specified by a non-negative real number $\eta$ that we call a regularization parameter and the set $\mathcal{V}$ of symmetric positive-semi-definite matrices defined as

$$\mathcal{V}=\left\{V\in {\mathbb{R}}^{N\times N}:V\in {\mathbb{S}}_{+}^N,V^l\le V\le V^u\right\}.$$

(8)

Here, the inequalities are evaluated element-wise and $V^l$ and $V^u$ are pre-specified matrices used to handle estimation errors in V.

This min-max optimization problem can be regarded as a zero-sum game, wherein one player, representing the minimization part, aims to uncover the factors and loadings that provide the most accurate explanation for the time-series variation in asset returns (represented by the first term in Equation (7)) as well as the cross-sectional variation in their mean transformed by V (the second term in Equation (7)). For instance, when a fixed matrix V is decomposed using the Cholesky decomposition as $V=Q^{\top }Q$, the cross-section of pricing errors to be minimized can be computed as $\parallel \overline{X}-\mathrm{\Lambda }\overline{F}{\parallel }_V^2$, which is equivalent to $\parallel Q(\overline{X}-\mathrm{\Lambda }\overline{F}){\parallel }^2$.

Concurrently, the other player, representing the maximization part, aims to construct a hypothesis test that poses the greatest challenge for its counterpart to generate a good estimate of the pair $(\widehat{\mathrm{\Lambda }},\widehat{F})$. This is accomplished by trying to find V that maximizes $\parallel \overline{X}-\mathrm{\Lambda }\overline{F}{\parallel }_V$ in Equation (7) as much as possible, i.e., maximizing the pricing errors generated by the factor model currently estimated by its counterpart. This maximization part allows the minimization part to uncover the pair $(\mathrm{\Lambda },F)$ that “works” across a wide range of adverse environments. Therefore, the interaction between the two players drives an iterative process, fostering the discovery of an improved and robust estimator similar to the “generator” and the “discriminator” in the celebrated generative adversarial network (GAN) [32].

The algorithm to find the estimates, which are defined as a solution to the min-max problem (7), is based on an alternating procedure between the minimization and maximization, similar to the methodology employed in adversarial deep learning approaches, e.g., [32,33]. The reason for employing this alternating method, which yields an approximate solution, is to address the numerical instability highlighted by [34]. By utilizing the alternating procedure, we aim to mitigate the numerical challenges associated with solving such optimization problems. Algorithm 1 presents our proposed estimator. The minimization step, for a fixed V, involves finding the estimates $\widehat{F}$ and $\widehat{\mathrm{\Lambda }}$ by employing the alternating least squares method. This method updates F and $\mathrm{\Lambda }$ iteratively, with one fixed while the other is updated, as explained in the next subsection. The maximization step, for fixed $\mathrm{\Lambda }$ and F, can be reformulated as follows:

$$\begin{array}{cc}\underset{V}{max}& tr\left((\overline{X}-\mathrm{\Lambda }\overline{F}){(\overline{X}-\mathrm{\Lambda }\overline{F})}^{\top }V\right)\hfill \\ \mathrm{s}.\mathrm{t}.& V\in {\mathbb{S}}_{+}^N,V^l\le V\le V^u.\hfill \end{array}$$

(9)

This problem can be solved numerically using a semi-definite programming solver, such as the one provided by [35].

Algorithm 1: Proposed factor model estimator

4.2. The Minimization Part

Unlike the maximization part of (7), which is a convex optimization problem, its minimization counterpart is non-convex. This necessitates the development of an algorithm specifically designed to address the challenges of non-convex optimization. We propose a method based on the alternating least squares method as described in lines 5–9 of Algorithm 1. Specifically, let us consider the minimization part of Problem (7) for a fixed V:

$$\underset{\mathrm{\Lambda },F}{min}\varphi (\mathrm{\Lambda },F)\triangleq {\displaystyle \frac{1}{NT}}{∥\tilde{X}-\tilde{F}{\mathrm{\Lambda }}^{\top }∥}_F^2+{\displaystyle \frac{\eta }{N}}{∥\overline{X}-\mathrm{\Lambda }\overline{F}∥}_V^2$$

(10)

The following reformulation of the objective function $\varphi :{\mathbb{R}}^{N\times K}\times {\mathbb{R}}^{T\times K}\to [0,\infty )$

$$\varphi (\mathrm{\Lambda },F)={\displaystyle \frac{1}{NT}}tr\left(M_1(X-F{\mathrm{\Lambda }}^{\top }){(X-F{\mathrm{\Lambda }}^{\top })}^{\top }{M}_1+\eta P_1(X-F{\mathrm{\Lambda }}^{\top })V{(X-F{\mathrm{\Lambda }}^{\top })}^{\top }{P}_1\right)$$

(11)

is useful, which is derived in Appendix B.

If $\varphi$ in Equation (11) only has the first term, i.e., $\eta =0$, then Problem (10) reduces to the conventional PCA estimator where the pricing factors and factor loadings are estimated by computing the eigen-decomposition of $X^{\top }{M}_{1}X$. If we replace $\eta$ with $1+\gamma$ and V with $I_N$, then our proposed objective function $\varphi$ becomes a function ${\varphi }_{RP}:{\mathbb{R}}^{N\times K}\times {\mathbb{R}}^{T\times K}\to [0,\infty )$ defined by

$${\varphi }_{RP}(\mathrm{\Lambda },F)={\displaystyle \frac{1}{NT}}tr\left(M_1(X-F{\mathrm{\Lambda }}^{\top }){(X-F{\mathrm{\Lambda }}^{\top })}^{\top }{M}_1+(1+\gamma )P_1(X-F{\mathrm{\Lambda }}^{\top }){(X-F{\mathrm{\Lambda }}^{\top })}^{\top }{P}_1\right)$$

(12)

for $\gamma \in [-1,\infty )$, which is exactly the same as the objective function that defines the RP-PCA [22,31]. Consequently, our estimator defined by Problem (10), more generally by (7), subsumes the conventional PCA and RP-PCA as special cases.

It is difficult to solve Problem (10) due to the existence of the matrix V that accepts an arbitrary matrix in ${\mathbb{S}}_{+}^N$. On the other hand, the conventional PCA (or our estimator with $\eta =0$) and RP-PCA estimators can find an exact solution simply by applying the eigen-decomposition to the matrix in the form of $X^{\top }(I_T+{\displaystyle \frac{\gamma }{T}}\mathbb{1}{\mathbb{1}}^{\top })X$ because the first-order optimality conditions of their corresponding minimization problems imply that F in Equation (12) can be removed using the relation $F=X{\mathrm{\Lambda }}^{\top }{\left({\mathrm{\Lambda }}^{\top }\mathrm{\Lambda }\right)}^{-1}$. This substitution is impossible for Problem (10) due to the existence of V in the second term in Equation (11). Appendix B offers a detailed explanation for this issue.

In order to find an approximate solution to Problem (10), we employ the alternating least squares method where $\varphi (\mathrm{\Lambda },F)$ is minimized for one variable at a time with the other variable fixed. In the update step for pricing factors, we find the minimum of $\varphi ({\mathrm{\Lambda }}_{*},F)$ over F with a fixed ${\mathrm{\Lambda }}_{*}$ by solving

$$\begin{array}{cc}& ({\mathrm{\Lambda }}_{*}^{\top }\otimes I_T)\left[I_N\otimes M_1+\eta (V\otimes P_1)\right]\mathrm{vec}\left(X\right)\hfill \\ & =({\mathrm{\Lambda }}_{*}^{\top }\otimes I_T)\left[I_N\otimes M_1+\eta (V\otimes P_1)\right]({\mathrm{\Lambda }}_{*}\otimes I_T)\mathrm{vec}\left(F\right)\hfill \end{array}$$

(13)

for F, and in the update step for factor loadings, we find the minimum of $\varphi (\mathrm{\Lambda },F_{*})$ over $\mathrm{\Lambda }$ with a fixed $F_{*}$ by solving

$$\begin{array}{cc}& (I_N\otimes F_{*}^{\top })\left[I_N\otimes M_1+\eta (V\otimes P_1)\right]\mathrm{vec}\left(\mathrm{X}\right)\hfill \\ & =(I_N\otimes F_{*}^{\top })\left[I_N\otimes M_1+\eta (V\otimes P_1)\right](I_N\otimes F_{*})\mathrm{vec}\left({\mathrm{\Lambda }}^{\top }\right)\hfill \end{array}$$

(14)

for $\mathrm{\Lambda }$. Sufficient conditions for existence and uniqueness of solutions to Equations (13) and (14) are given in the following proposition, which is proved in Appendix C.

Proposition 1.

Suppose that $V\in {\mathbb{S}}_{+}^N$. Then, there exist solutions to Equations (13) and (14). If it is additionally assumed that V is positive-definite, $\eta >0$ and ${\mathrm{\Lambda }}_{*}$ and $F_{*}$ have full column rank, i.e., $rank\left({\mathrm{\Lambda }}_{*}\right)=rank\left(F_{*}\right)=K$, then the solutions are unique.

Algorithm 2 summarizes the alternating least squares method that defines the estimates as $\widehat{\mathrm{\Lambda }}={\mathrm{\Lambda }}_{n_{alt}}$ and $\widehat{F}=F_{n_{alt}}$. For iterates $({\mathrm{\Lambda }}_n,F_n)$ generated by the algorithm for any $V\in {\mathbb{S}}_{+}^N$ and $\eta \in [0,\infty )$, it is true that the sequence $\left\{\varphi ({\mathrm{\Lambda }}_n,F_n)\right\}$ is monotonically decreasing and is therefore convergent. Indeed, it is clear that the function $F\mapsto \varphi ({\mathrm{\Lambda }}_{*},F)$ is convex for any ${\mathrm{\Lambda }}_{*}$. Thus, $F_n$ that satisfies the first-order optimality condition (Equation (15)) is the global minimum of the convex function $F\mapsto \varphi ({\mathrm{\Lambda }}_{n-1},F)$. It similarly holds for the function $\mathrm{\Lambda }\mapsto \varphi (\mathrm{\Lambda },F_{*})$.

Algorithm 2: Alternating minimization

We empirically demonstrate the convergence of Algorithm 2 through experiments conducted on real-world data. As in Section 2, we use the monthly excess returns of $5\times 5$ Size-B/M portfolios, i.e., $N=25$, over a period of $T=60$ months, from January 2017 to December 2021, represented by a $T\times N$ matrix X. We select the regularization parameters $\eta =10$ in Equation (11) and fix $K=4$ as in the simulation study of Lettau and Pelger [22], while varying the input $V\in {\mathbb{S}}_{+}^n$ to handle cross-sectional correlation in the pricing errors.

First, we consider the case when $V=I_N$ where Problem (10) is reduced to the RP-PCA, and so the exact solution can be found. Let us denote the exact solution by $({\mathrm{\Lambda }}_{*},F_{*})$ and the minimum value of the objective function by ${\varphi }_{*}=\varphi ({\mathrm{\Lambda }}_{*},F_{*})$. For a set of iterates $\{({\mathrm{\Lambda }}_n,F_n):n=0,1,\cdots ,n_{alt}\}$ created by Algorithm 2, let us define ${\varphi }_n=\varphi ({\mathrm{\Lambda }}_n,F_n)$. We run the algorithm five times with five different random initializations $({\mathrm{\Lambda }}_0,F_0)$, and exhibit the suboptimality measured by ${\varphi }_n-{\varphi }_{*}$ and the distance between the iterates and the exact solution $\parallel F_n{\mathrm{\Lambda }}_n^{\top }-F_{*}{\mathrm{\Lambda }}_{*}^{\top }{\parallel }_F$ in Figure 3a,b where we can clearly see that the suboptimality and the distance both converge to zero for every random initialization.

Next, we consider a $V\in {\mathbb{S}}_{+}^N$ computed as follows. We run the time-series regressions (Equation (5)) of X on $\{\mathtt{Mkt}-\mathtt{RF},\mathtt{SMB},\mathtt{HML}\}$ and $\{\mathtt{Mkt}-\mathtt{RF},\mathtt{RMW},\mathtt{CMA}\}$, obtain the estimated residuals and compute their sample covariance matrices denoted by ${\mathrm{\Sigma }}_1$ and ${\mathrm{\Sigma }}_2\in {\mathbb{S}}_{+}^N$. Then, we normalize ${\mathrm{\Sigma }}_1$ and ${\mathrm{\Sigma }}_2$ by dividing them by $tr\left({\mathrm{\Sigma }}_1\right)/N$ and $tr\left({\mathrm{\Sigma }}_2\right)/N$, respectively, in order for the normalized covariance matrices to have $tr\left({\mathrm{\Sigma }}_1\right)=tr\left({\mathrm{\Sigma }}_2\right)=N$, which equals the trace of $I_N$. We run Algorithm 2 for $V\in \{{\mathrm{\Sigma }}_1^{-1},{\mathrm{\Sigma }}_2^{-1}\}$ and plot the objective function values ${\varphi }_n$ for $V={\mathrm{\Sigma }}_1^{-1}$ and $V={\mathrm{\Sigma }}_2^{-1}$ in Figure 3c,d, respectively.We can observe that only five iteration steps are enough for the objective function values to converge for every random initialization.

5. Experiments

We evaluate the performance of the proposed estimator and other related estimators using a set of empirical data which comprises monthly returns of portfolios of stocks listed in the Center for Research in Security Prices (CRSP). The portfolios of stocks are divided into deciles based on 37 characteristics that were also considered in [31,36], and we also use the first and the tenth decile portfolios. Each of the decile portfolios was constructed as a value-weighted, long-only portfolio comprising US stocks within the corresponding decile. Consequently, our dataset consists of $N=74$ portfolios in the cross-section. The dataset covers a sample period extending from November 1963 to December 2019, totaling 674 months. We consider the Fama–French three-factor (FF3) and five-factor (FF5) models [8,9], as well as the conventional principal component analysis (PCA) [37,38] and RP-PCA [22], as benchmarks for our evaluation. We assess their performance across three criteria, both in-sample and out-of-sample, as investigated in [31].

For the in-sample analysis, we use the entire dataset from November 1963–December 2019. For the out-of-sample (OOS) analysis, we employ a rolling estimation method with a 240-month estimation period, i.e., $T=240$, and a 1-month prediction period, moving the estimation period forward by 1 month at a time. Our first estimation period spans from November 1963 to October 1983, and the first OOS prediction is made for November 1983. As a result, we have a total of 434 OOS observations.

The first criterion is the maximum Sharpe ratio that is obtained by linearly combining factors based on the weights $w\in {\mathbb{R}}^K$ calculated by $w={\widehat{\mathrm{\Sigma }}}_F^{-1}{\widehat{\mu }}_F.$ Here, ${\widehat{\mathrm{\Sigma }}}_F\in {\mathbb{R}}^{K\times K}$ and ${\widehat{\mu }}_F\in {\mathbb{R}}^K$ are, respectively, the sample covariance matrix and the sample mean of the estimated factors ${\widehat{F}}_{t-239},\cdots ,{\widehat{F}}_t\in {\mathbb{R}}^K$. As a result, $w\in {\mathbb{R}}^K$ represents the weights that maximize the Sharpe ratio of the portfolio composed of the estimated factors. The OOS returns of the maximum Sharpe ratio portfolio, corresponding to the weights w, are computed by $r_{t+1}^{oos}=w^{\top }{F}_{t+1}$ and are gathered for each prediction period $t+1$ to compute the OOS maximum Sharpe ratio (OOS-SR). Second, we consider the root mean squared (RMS) pricing error across N test assets ${\mathrm{RMS}}_{\alpha }=\sqrt{{\widehat{\alpha }}^{\top }\widehat{\alpha }/N}$ where $\widehat{\alpha }$ is the vector of ordinary least squares (OLS) intercepts estimated by regressing the returns of the test assets on the estimated factors. Third, we evaluate the average unexplained variance across N test assets ${\overline{\sigma }}_e^2={\displaystyle \frac{1}{N}}{\sum }_{i=1}^N{\sigma }_{{\widehat{e}}_i}^2/{\sigma }_{R_i}^2$ where ${\widehat{e}}_i\in {\mathbb{R}}^{T\times 1}$ represents the residual estimated by regressing the returns of asset i on the estimated factors. ${\sigma }_{{\widehat{e}}_i}^2$ and ${\sigma }_{R_i}^2$ represent the variances of the residual and return of asset i, respectively.

6. Discussion

Table 1. In-sample performance. The top-performing estimator is highlighted in bold for each $(K,\eta )$.

		SR	RMSα	${\overline{\mathit{\sigma }}}_{\mathit{e}}^2$ (%)
Panel A: $K=3$
FF3		0.193	0.31	17.48
PCA	$\eta =1$	0.216	3.18	13.71
	$\eta =10$	0.216	3.18	13.71
	$\eta =20$	0.216	3.18	13.71
RP-PCA	$\eta =1$	0.257	2.89	13.72
	$\eta =10$	0.361	2.75	13.84
	$\eta =20$	0.397	2.80	13.91
Ours	$\eta =1$	0.421	3.05	13.87
	$\eta =10$	0.442	2.89	14.01
	$\eta =20$	0.445	2.89	14.02
Panel B: $K=5$
FF5		0.317	0.26	16.02
PCA	$\eta =1$	0.317	2.52	10.31
	$\eta =10$	0.317	2.52	10.31
	$\eta =20$	0.317	2.52	10.31
RP-PCA	$\eta =1$	0.383	2.34	10.32
	$\eta =10$	0.575	1.91	10.44
	$\eta =20$	0.601	1.86	10.46
Ours	$\eta =1$	0.603	1.92	10.45
	$\eta =10$	0.630	2.00	10.45
	$\eta =20$	0.629	1.90	10.47
Panel C: $K=7$
PCA	$\eta =1$	0.381	2.22	8.89
	$\eta =10$	0.381	2.22	8.89
	$\eta =20$	0.381	2.22	8.89
RP-PCA	$\eta =1$	0.449	2.03	8.90
	$\eta =10$	0.596	1.78	8.96
	$\eta =20$	0.618	1.74	8.98
Ours	$\eta =1$	0.618	1.79	8.97
	$\eta =10$	0.646	1.83	8.98
	$\eta =20$	0.644	1.76	8.98
Panel D: $K=10$
PCA	$\eta =1$	0.410	2.07	7.25
	$\eta =10$	0.410	2.07	7.25
	$\eta =20$	0.410	2.07	7.25
RP-PCA	$\eta =1$	0.461	1.97	7.25
	$\eta =10$	0.602	1.75	7.31
	$\eta =20$	0.625	1.71	7.32
Ours	$\eta =1$	0.697	2.35	7.30
	$\eta =10$	0.651	1.79	7.33
	$\eta =20$	0.650	1.73	7.33
Panel E: $K=15$
PCA	$\eta =1$	0.507	1.70	5.42
	$\eta =10$	0.507	1.70	5.42
	$\eta =20$	0.507	1.70	5.42
RP-PCA	$\eta =1$	0.590	1.51	5.43
	$\eta =10$	0.714	1.19	5.46
	$\eta =20$	0.726	1.15	5.46
Ours	$\eta =1$	0.733	1.20	5.46
	$\eta =10$	0.740	1.14	5.47
	$\eta =20$	0.740	1.12	5.47

and

Table 2. Out-of-sample performance. The top-performing estimator is highlighted in bold for each $(K,\eta )$.

		SR	RMSα	${\overline{\mathit{\sigma }}}_{\mathit{e}}^2$ (%)
Panel A: $K=3$
FF3		0.150	0.25	16.47
PCA	$\eta =1$	0.107	3.04	15.78
	$\eta =10$	0.107	3.04	15.78
	$\eta =20$	0.107	3.04	15.78
RP-PCA	$\eta =1$	0.133	2.96	15.70
	$\eta =10$	0.296	2.53	15.35
	$\eta =20$	0.325	2.45	15.32
Ours	$\eta =1$	0.302	2.52	15.39
	$\eta =10$	0.299	2.61	15.42
	$\eta =20$	0.327	2.45	15.39
Panel B: $K=5$
FF5		0.302	0.19	13.85
PCA	$\eta =1$	0.235	2.21	11.98
	$\eta =10$	0.235	2.21	11.98
	$\eta =20$	0.235	2.21	11.98
RP-PCA	$\eta =1$	0.270	2.12	11.97
	$\eta =10$	0.486	1.75	12.04
	$\eta =20$	0.498	1.70	12.06
Ours	$\eta =1$	0.422	1.78	11.98
	$\eta =10$	0.491	1.72	12.05
	$\eta =20$	0.500	1.69	12.06
Panel C: $K=7$
PCA	$\eta =1$	0.298	2.19	10.62
	$\eta =10$	0.298	2.19	10.62
	$\eta =20$	0.298	2.19	10.62
RP-PCA	$\eta =1$	0.368	2.07	10.64
	$\eta =10$	0.482	1.76	10.76
	$\eta =20$	0.489	1.74	10.77
Ours	$\eta =1$	0.459	1.75	10.73
	$\eta =10$	0.490	1.74	10.76
	$\eta =20$	0.493	1.72	10.76
Panel D: $K=10$
PCA	$\eta =1$	0.346	1.84	8.97
	$\eta =10$	0.346	1.84	8.97
	$\eta =20$	0.346	1.84	8.97
RP-PCA	$\eta =1$	0.433	1.73	8.96
	$\eta =10$	0.502	1.55	9.00
	$\eta =20$	0.506	1.53	9.01
Ours	$\eta =1$	0.476	1.59	9.01
	$\eta =10$	0.507	1.53	9.01
	$\eta =20$	0.508	1.53	9.01
Panel E: $K=15$
PCA	$\eta =1$	0.372	1.61	6.98
	$\eta =10$	0.372	1.61	6.98
	$\eta =20$	0.372	1.61	6.98
RP-PCA	$\eta =1$	0.488	1.41	6.97
	$\eta =10$	0.549	1.23	6.97
	$\eta =20$	0.552	1.21	6.97
Ours	$\eta =1$	0.525	1.26	6.97
	$\eta =10$	0.552	1.21	6.97
	$\eta =20$	0.553	1.21	6.97

present findings that offer several observations for 15 factor model specifications with $\eta \in \{1,10,20\}$ and $K\in \{3,5,7,10,15\}$. These hyperparameter settings were selected based on the findings from [31], where they demonstrated superior performance within this specific range. We begin our discussion with the OOS findings presented in Table 2: First, increasing the number of factors K leads to improvements in the Sharpe ratio, pricing errors RMS_α and idiosyncratic unexplained variance ${\overline{\sigma }}_e^2$ across all of the three PCA-based estimators. This result suggests that a larger number of factors provides a better approximation of the stochastic discount factor (SDF), enhances the ability to explain pricing information, and captures variations in asset returns. These findings align with the observations drawn in [31,36], providing additional support for the notion that a higher value of K contributes to an enhanced factor model estimation.

Second, the results consistently demonstrate the superior performance of our method compared to other factor models in the OOS analysis, particularly in terms of OOS-SR. Our method consistently achieves the highest SR values among all of the factor model configurations, indicating its ability to better approximate the SDF. Indeed, a notable observation is that, when the number of factors K takes smaller values such as $K=3$ or 5, the superiority of our method in terms of the SR becomes more evident. In comparison to FF3/FF5, the conventional PCA and RP-PCA, our method consistently outperforms them by a substantial margin. As an example, when considering the case wherein $K=3$ in Panel A of Table 2, the RP-PCA shows some advantage over the conventional PCA in terms of SR. However, our proposed method outperforms both estimators by a significant margin, indicating its superior performance in capturing and exploiting the underlying factors driving asset returns. This further underscores the strength and effectiveness of our approach, especially when confronted with a smaller number of factors. This highlights the robustness and reliability of our approach in capturing relevant information, in cases wherein the explanatory power of the factor models is inherently constrained.

Third, we observe that increasing the value of $\eta$ results in higher OOS-SR for both the RP-PCA and our proposed method. However, it is important to note that the magnitude of incremental changes differs between the two approaches. Our proposed estimator consistently achieves higher SR even at lower values of $\eta$ for all K and exhibits stable changes in OOS-SR with respect to increases in the value of $\eta$ in contrast to the unpredictable changes observed for the RP-PCA. For instance, in Panel B of Table 2, when $K=5$, our method shows an increase in the OOS-SR by 16.4% from 0.422 to 0.491, while the RP-PCA exhibits an increase of 80% from 0.270 to 0.486 as $\eta$ varies from 1 to 10. This observation can be attributed to the unique design of our estimator, particularly the utilization of a min-max optimization framework. The maximization part incorporates the specification of $\eta$, leading to smaller changes in the outcomes with respect to changes in the value of $\eta$. Conversely, the increment in SR for RP-PCA is more pronounced, suggesting a greater sensitivity to changes in $\eta$.

Furthermore, our proposed estimator consistently demonstrates a comparative advantage in terms of OOS-RMS_α among the PCA-based factor models. Specifically, our method consistently achieves smaller RMS_α values compared to RP-PCA. For $K\ge 10$ and $\eta \ge 10$, we can observe that RP-PCA and our method yield identical values up to the first decimal point. Additionally, we observe that both our method and RP-PCA outperform conventional PCA, indicating that regularization of pricing error information to obtain factor model estimates is indeed effective. In contrast, for ${\overline{\sigma }}_e^2$ values, we observe that PCA outperforms the other two methods in 7 out of 15 configurations, which can be attributed to the fact that the criterion of PCA solely takes into account time-series variation.

Finally, the in-sample performance, as presented in Table 1, demonstrates improvement in all three performance metrics for all estimators as the number K of factors increases. Specifically, all metrics consistently improve with increasing K in all but one case. In terms of SR, our proposed method outperforms other estimators in all configurations of $(K,\eta )$, indicating its ability to approximate the SDF better than other methods within the sample. Both the RP-PCA and our method outperform PCA in terms of RMS_α, except for one case. This can be attributed to the fact that RP-PCA and our method take into account the first-order information in the estimation process whereas PCA measures the time-series variation only by construction, consequently performing the best in terms of in-sample ${\overline{\sigma }}_e^2$ for all cases.

7. Conclusions

In this paper, we introduce a novel estimator for factor pricing models by presenting a min-max optimization problem that effectively incorporates both the time-series variations of realized excess returns and the cross-sectional pricing errors. We also present an algorithm designed to approximately solve this optimization problem, which utilizes an iterative method widely employed in adversarial machine learning.

Through extensive empirical experiments using real-world data, we demonstrate that our proposed estimator consistently outperforms existing static factor model estimators. Specifically, the portfolios implied by the factors estimated through our method exhibit larger in-sample and out-of-sample Sharpe ratios compared to those of portfolios implied by other related estimators. This highlights the superior risk-adjusted returns achievable via our approach.

Furthermore, our estimator shows comparable performance in terms of out-of-sample pricing errors and unexplained variations, indicating its effectiveness in maintaining accuracy while addressing the complexities inherent in financial modeling. Overall, our findings suggest that this novel estimator represents a significant advancement in the field of factor pricing models.