Classifying Factor Velocity with Swarm Intelligence: Market Pricing of Fast- and Slow-Moving Factors

Abstract

Utilizing a dataset of 190 risk factors spanning over three decades, we apply a swarm-based classification model to estimate factor velocity and analyze its implications for asset pricing. Our results show that slower-moving factors generate higher abnormal returns than their faster-moving counterparts, underscoring the critical role of price adjustment speed in market dynamics. Furthermore, our results suggest that trading frictions impede the rapid assimilation of information, contributing to the observed return patterns. This research offers new insights into return predictability and demonstrates the potential of swarm intelligence as a powerful tool for financial modeling.

1. Introduction

Momentum investments have been the centerpiece of investments in stocks for the last two decades. Momentum investing refers to buying stocks that have had high returns in the past and selling those that have had poor returns over the same period. Researchers have identified persistent momentum trends in stock markets as far back as the Victorian era (c. 1830s to 1900). Richard Driehaus is sometimes considered the father of momentum investing. He once said, “far more money is made buying high and selling at even higher prices.” This strategy reportedly delivered compound annual returns of 30% for Driehaus Capital Management in the 12 years after it was set up in the 1980s. This is taken from Richard Driehaus and Momentum Investing. For the original source, please see [1,2,3].

Momentum supporters believe that there is a fundamental reason for the momentum phenomenon to persistently exist, although momentum in stocks totally lacks economic theory. Pioneering academic research on momentum investing can be traced back to [4]. They documented how strategies of buying recent stock winners and selling recent losers generated significantly higher near-term returns than the U.S. market overall from 1965 to 1989. They established the basic time frame for momentum investing success as a 3- to 12-month window on either side. Since then, it has been booming as one of the largest research areas in finance. Recently, they [4] wrote an excellent review piece of the last 30 years of momentum research. Besides momentum, we believe that all of those technical indicators, although not supported by any economic theory, are supported by psychological behaviors of market participants—herding.

While herding itself cannot be easily observed due to the fact that individual trading data are not publicly (or even legally) available, the consequences are. In other words, although we cannot measure how investors herd in buying/selling stocks, we can observe how stock prices (returns) move as a result of herding. When herding exists, certain stocks will be chased (bought) or avoided (sold). These result in winner stocks and loser stocks. As herding behaviors change, winners can become losers and vice versa.

Past studies demonstrate the diverse applications of swarm intelligence algorithms in financial contexts, including portfolio optimization and stock price prediction (see, e.g., [5,6,7,8,9,10]). These studies leverage swarm intelligence in decision making processes by simulating collective agent behaviors observed in nature, such as those of birds, fish, and insects. However, to our knowledge, no existing research has specifically examined how stock market prices influence or interact with the velocity of agents in a swarm system. This study aims to bridge that gap by exploring the market pricing of velocity in financial swarm dynamics.

Velocity is a crucial component of system intelligence in swarm-based models. In natural and artificial swarm systems, velocity reflects the rate of change in an agent’s position, capturing both speed and directional movement. It serves as a fundamental mechanism by which information is propagated and decisions are adjusted dynamically within the swarm. In financial markets, velocity can be viewed as a proxy for the responsiveness of stocks to new information, as well as a key determinant of trend formation and reversals. Stocks that exhibit higher velocity within the swarm framework may represent “leaders,” entities that drive market sentiment, while lower-velocity stocks may act as “followers,” adjusting their movements based on lagged signals from faster-moving assets.

Incorporating velocity into swarm intelligence models of financial markets enhances our understanding of price formation, momentum effects, and return predictability. A system that lacks velocity awareness may fail to capture the hierarchical influence of stocks, where certain assets dictate the movement patterns of others. By analyzing how velocity distributes across financial agents, we can gain deeper insights into price discovery mechanisms and the differentiation between trend-driving and trend-reactive assets.

While the concept of velocity originates from swarm intelligence and captures the rate of directional movement in a multi-agent system, we argue that it also has a meaningful financial interpretation grounded in economic theory. In our setting, factor velocity serves as a proxy for the speed at which return-predictive information is incorporated into asset prices. This notion aligns with the literature on information diffusion and price discovery, where some factors or assets lead in assimilating value-relevant news, while others adjust more slowly [11,12]. High-velocity factors may reflect conditions of greater liquidity or heightened investor attention, leading to rapid but potentially noisy price movements [13,14]. On the other hand, low-velocity factors may be slower to adjust due to frictions, inattention, or underreaction, conditions consistent with behavioral theories that emphasize delayed price discovery and persistent mispricing [15,16]. Thus, velocity within our swarm-based model is not merely a technical artifact but a behaviorally and economically motivated measure of how promptly different factors respond to new information, with implications for trend persistence, herding, and return predictability.

This study therefore introduces a velocity-driven perspective on swarm intelligence in financial markets. By modeling returns of risk factors (also sometimes called factor portfolios in that these factors are weighted averages of stocks and hence analogous to portfolios) formed based on stylized return predictors through a swarm-based system that accounts for velocity differences, we aim to uncover new dimensions of market behavior and pricing dynamics that have previously been overlooked in the literature.

Crucially, we define two related concepts. Swarm velocity (SV) is the empirically estimated parameter derived from our model, representing the factor’s mathematical rate of change in the swarm space. This SV serves as a proxy for the economic concept of factor adjustment speed or information incorporation speed—the rate at which value-relevant information is assimilated into asset prices.

The primary contributions of this research, which bridges the fields of swarm intelligence and asset pricing, are fourfold. First, we introduce and successfully implement a non-parametric swarm intelligence model (derived from particle swarm optimization principles) to estimate a quantitative measure of factor velocity (SV), or the speed at which 190 risk factors adjust their returns relative to a market leader. This represents the first application of this methodology to systematically classify and analyze asset pricing factors.

Second, we find that factors classified as slow-moving (Low SV) generate significantly higher one-month-ahead abnormal returns (alphas) than their faster-moving (High SV) counterparts, even after controlling for established asset pricing models (e.g., FF5, FF6, Q5). The monthly alpha differential is both economically and statistically significant, ranging from 17 to 22 basis points.

Third, our results suggest that the cross-sectional return pattern documented in this study is likely driven by trading frictions and delayed information assimilation. This is evidenced by (1) the significant outperformance of the slow-mover portfolios and (2) the finding that friction-related factors are disproportionately overrepresented in the slow-mover category (more than twice their unconditional share).

Lastly, we demonstrate that the swarm velocity classification is highly persistent, with factors remaining in the same velocity quintiles for up to five years, confirming that SV represents a stable characteristic of the factor rather than temporary noise.

The remainder of the paper is organized as follows. Section 2 introduces the theoretical background of swarm intelligence (SI) and derives our non-optimizing, leader-following model for empirical factor estimation. Section 3 details the empirical methodology, outlining the three-stage pipeline for transforming monthly factor returns into the swarm velocity (SV) metric. Section 4 provides the literature review, contextualizing our SI approach within the broader fields of modern financial AI, herding, and lead–lag effects. Section 5 presents the empirical findings, including the experimental setup, descriptive statistics, return predictability results (alphas), and a detailed economic analysis of factor persistence and composition. Section 6 concludes the paper.

2. Swarm Intelligence

Swarm intelligence is a powerful artificial intelligence tool used to model animal behaviors and solve high-dimensional optimization problems. The basic idea of swarm intelligence is derived from those animals (such as birds, ants, bees, and fish) that rely on group effort to achieve their basic survival needs—seek food and avoid prey. The intelligence behind this collective behavior is how they communicate among one another.

There are two broadly classified categories in swarm intelligence—boids and particle swarm optimization (PSO). While both share the same fundamental theory, they vary in the following manner. Boids is mainly for behaviors of the swarm, and PSO can be used for optimization. Roughly speaking, PSO is used when there is a loss function (i.e., a landscape) and boids without. We should note that although they have been recognized as two separate models, they can be easily combined if needed. For example, although our estimation seems to follow particle swarm optimization, there is no optimization. If needed, we can estimate alignment and cohesion (which reflect another version of swarm speed) along with leader following. We briefly sketch each model in this section.

2.1. Boids (Without a Leader)

Ref. [17] was the first to “artificialize” such natural intelligence and create a computer algorithm named boids (for bird-oid object). Reynold’s algorithm is amazingly simple. For any given bird, Reynold devises a set of combined linear equations (vectors) that determine how the bird should fly to its next destination. Reynold created boids in 1986: “Boids is an artificial life program, developed by Craig Reynolds in 1986, which simulates the flocking behaviour of birds.”

The factors that determine how various vectors are combined are separation, alignment, and cohesion. As their names suggest, “separation” is to avoid collision with other birds, “alignment” decides how a particular bird should fly in a direction by referencing its fellow birds, and “cohesion” decides how fast (speed) a particular bird should fly to the center of its fellow birds.

There are countless versions of boids. For example, see Google Scholar: https://scholar.google.com/scholar?q=boids+flocking+algorithm&hl=en&as_sdt=0&as_vis=1&oi=scholart (accessed on 7 March 2025). One can add obstacles. One can add an objective destination (swim to target). One can place boids in a maze. The basic boids, as shown in Figure 1, can be described by the following algorithm.

In the swarm model, let there be $m$ fish (birds) $F=\{f^{(1)},\dots ,f^{(m)}\}$ that move about in an $n$-dimensional space. We then represent their positions at any given time $t$ as $X_t=\{{\stackrel{\rightharpoonup }{x}}_t^{(1)},\dots ,{\stackrel{\rightharpoonup }{x}}_t^{(m)}\}$, in which each vector ${\stackrel{\rightharpoonup }{x}}_t^{(i)}$ ($i=1,\dots ,m$) has $n$ elements $x_{j,t}^{(i)}$ ($j=1,\dots ,n$). In other words, $X_t$ is an $n\times m$ matrix. In our empirical work, each fish is a risk factor (a total of 190 factors), and each dimension is a lagged month (total of 12 lagged months). See the online appendix for the full list of the 190 factors, along with their notations and descriptions.

The velocity of each fish is defined as a weighted average of various forces.

$${\stackrel{\rightharpoonup }{v}}_t^{(i)}=w_A{\stackrel{\rightharpoonup }{v}}_{A,t}^{(i)}+w_C{\stackrel{\rightharpoonup }{v}}_{C,t}^{(i)}+w_S{\stackrel{\rightharpoonup }{v}}_{S,t}^{(i)}$$

(1)

where the weights, $w_A$ for alignment, $w_C$ for cohesion, and $w_S$ for separation, must sum up to one, and each corresponding velocity is defined as follows:

$$\begin{array}{l}{\stackrel{\rightharpoonup }{v}}_{A,t}^{(i)}=\mathrm{avg}\left({\stackrel{\rightharpoonup }{v}}_{t-1}^{(j\ne i)}|f^{(j)}\in F\right)-{\stackrel{\rightharpoonup }{v}}_{t-1}^{(i)}\\ {\stackrel{\rightharpoonup }{v}}_{C,t}^{(i)}=\mathrm{avg}\left({\stackrel{\rightharpoonup }{x}}_{t-1}^{(j\ne i)}|f^{(j)}\in F\right)-{\stackrel{\rightharpoonup }{x}}_{t-1}^{(i)}\\ {\stackrel{\rightharpoonup }{v}}_{S,t}^{(i)}=\max \left\{\left|{\stackrel{\rightharpoonup }{x}}_t^{(i)}-{\stackrel{\rightharpoonup }{x}}_t^{(j\ne i)}\right|,\epsilon \right\}\end{array}$$

(2)

representing alignment, cohesion, and separation, respectively. As easily seen, alignment for each fish involves seeking the average velocity of its fellow fish (excluding itself), cohesion for each fish involves moving toward the center of its fellow fish (excluding itself), and separation is undertaken in order to avoid colliding with its fellow fish.

The update of position is

$${\stackrel{\rightharpoonup }{x}}_t^{(i)}={\stackrel{\rightharpoonup }{x}}_{t-1}^{(i)}+{\stackrel{\rightharpoonup }{v}}_t^{(i)}$$

(3)

In our empirical work, we shall estimate the weights in Equation (1) using data. Our data contain each of 190 risk factors’ lagged returns from $t-1$ to $t-12$. Formally, $x_{j,t}^{(i)}$, defined earlier, can be rewritten as the following matrix (190 fish or risk factors and 12 lagged months or dimensions):

$$\left[\begin{array}{cccc}{x}_{t-1,t}^{(1)}& x_{t-2,t}^{(1)}& \dots & x_{t-12,t}^{(1)}\\ x_{t-1,t}^{(2)}& x_{t-2,t}^{(2)}& \dots & x_{t-12,t}^{(2)}\\ ⋮& ⋮& \ddots & ⋮\\ x_{t-1,t}^{(190)}& x_{t-2,t}^{(190)}& \dots & x_{t-12,t}^{(190)}\end{array}\right]$$

This is a snapshot of the locations of all fish at a given time $t$. From one snapshot to another, it represents a migration (i.e., velocity). Then, this empirically observed velocity is matched (by minimizing the sum of squared errors) with Equation (1) to solve for the weights.

2.2. Particle Swarm Optimization (With a Leader)

Particle swarm optimization (PSO), from its name, is an optimization tool using swarm. See [18,19]. Particles here are the same as fish or birds in the previous sub-section. The change in terms is mainly due to different researchers in the past. To be consistent, we shall use fish for the remainder of this paper.

In a standard PSO, an objective function (aka a fitness function) is given so that all fish can reach the optimal location. The communication mechanism among the fish is the same as among boids, and yet how they move is different.

One can think of PSO as a swarm with a landscape (i.e., the fitness function). Fish will now try to reach either the peak or the bottom (i.e., the global optimum) of the landscape. In this case, they will stop moving once their objective is met. A simple demonstration of a two-dimensional PSO is given in the Appendix A.

In order to achieve convergence in an optimization problem, the velocity of a standard PSO is given as

$${\stackrel{\rightharpoonup }{v}}_t^{(i)}=w_t{\stackrel{\rightharpoonup }{v}}_{t-1}^{(i)}+c_1{r}_1({\stackrel{\rightharpoonup }{p}}_{t-1}^{(i)}-{\stackrel{\rightharpoonup }{x}}_{t-1}^{(i)})+c_2{r}_2({\stackrel{\rightharpoonup }{g}}_{t-1}-{\stackrel{\rightharpoonup }{x}}_{t-1}^{(i)})$$

(4)

where $w_t<1$ is a decaying weight (e.g., we can let $w_t={\delta }^t$ and $\delta <1$), $c_1$ and $c_2$ are two constants, $r_1$ and $r_2$ are two random variables,

$${\stackrel{\rightharpoonup }{p}}_t^{(i)}=\left\{{\stackrel{\rightharpoonup }{x}}_{\tau \le t}^{(i)}|\underset{\tau }{\max }\varphi ({\stackrel{\rightharpoonup }{x}}_{\tau }^{(i)})\right\}$$

(5)

is the personal best position, and

$${\stackrel{\rightharpoonup }{g}}_t=\underset{i}{\max }{\stackrel{\rightharpoonup }{p}}_t^{(i)}$$

(6)

is the global best position, which is determined by maximizing (or minimizing) the fitness function, $\varphi (\cdot )$.

The “personal best” (Equation (5)) represents the best position (i.e., the highest value of the fitness function, as demonstrated in Figure A2) a fish has experienced in its past (hence maximizing over all $\tau \le t$ of the past). Given that ${\stackrel{\rightharpoonup }{p}}_t^{(i)}$ is the personal best of fish $i$, the “global best” (Equation (6)) is simply the best of the bests, as it finds the fish that has highest fitness value among all fish.

In Equation (4), $c_1$ and $c_2$ are learning rates. These are standard parameters in artificial intelligence to attain the most effective learning. First of all, fish tend to move toward the current leader, who is the best among all fish, ${\stackrel{\rightharpoonup }{g}}_{t-1}-{\stackrel{\rightharpoonup }{x}}_{t-1}^{(i)}$, and also tend to move toward its own historical best, ${\stackrel{\rightharpoonup }{p}}_{t-1}^{(i)}-{\stackrel{\rightharpoonup }{x}}_{t-1}^{(i)}$ This is known as “exploitation” in artificial intelligence, as they would like to learn from/exploit given information. Yet more importantly, fish also move randomly based on the two random variables $r_1$ and $r_2$. These random movements are known as “exploration,” meaning that the fish do not only learn from exploitation. This is analogous to mutations in a genetic algorithm. This is key to artificial intelligence to avoid local optima. Note that without such random movements, fish can only move within the initial positions that are arbitrarily assigned.

The update of the position is the same as Equation (3) and repeated here:

$${\stackrel{\rightharpoonup }{x}}_t^{(i)}={\stackrel{\rightharpoonup }{x}}_{t-1}^{(i)}+{\stackrel{\rightharpoonup }{v}}_t^{(i)}$$

(7)

2.3. Combining Boids and PSO

Although boids and PSO are based on the same intelligence (i.e., peer-to-peer communication), their purposes are different. The former is a behavioral model, and yet the latter is an optimization tool. As we can see, the parameters in boids, alignment, cohesion, and separation, describe how fish move around, and every fish is identical (i.e., there is no leader). Ref. [17] argues that with only three parameters followed identically by every fish, amazing group intelligence can be achieved to avoid prey (or barriers) or hunt food. On the other hand, PSO has a different use of intelligence. PSO identifies a leader (i.e., whoever is the global best), and other fish follow. The behavior of each fish is not determined by the three parameters but by following the leader (and its own personal best position), and, in doing so, the global optimum can be reached.

We can combine the two versions of the swarm easily. By combining Equations (1) and (4), the velocity equation can be defined as

$${\stackrel{\rightharpoonup }{v}}_t^{(i)}=w_A{\stackrel{\rightharpoonup }{v}}_{A,t}^{(i)}+w_C{\stackrel{\rightharpoonup }{v}}_{C,t}^{(i)}+w_S{\stackrel{\rightharpoonup }{v}}_{S,t}^{(i)}+{\alpha }_{P,t}^{(i)}({\stackrel{\rightharpoonup }{p}}_{t-1}^{(i)}-{\stackrel{\rightharpoonup }{x}}_{t-1}^{(i)})+{\alpha }_{L,t}^{(i)}({\stackrel{\rightharpoonup }{g}}_{t-1}-{\stackrel{\rightharpoonup }{x}}_{t-1}^{(i)}),$$

(8)

where the random coefficients $c_1{r}_1$ and $c_2{r}_2$ in Equation (4) are replaced by constants ${\alpha }_{P,t}^{(i)}$ and ${\alpha }_{L,t}^{(i)}$, respectively. Here, PSO is used for a behavioral model and no longer used for optimization, as we have no fitness function. In PSO, the leader is the one who has the highest fitness value. In other words, there must exist a fitness function upon which each fish can be evaluated. This fitness function is then maximized (minimized in some cases) once the algorithm converges. In our empirical work, there is no such fitness function. The leader is determined by whoever is being followed the most. This way, the new swarm will possess both behaviors of boids and leader following.

It is apparent that one can modify (by adding, dropping, or editing) the above velocity for any specific need. For example, one can eliminate the personal best if one is only interested in group behavior. Or, one can drop separation, as it is not relevant to evaluating financial assets.

In our empirical work, because we only focus on identifying the leader, we simplify Equation (8) as

$${\stackrel{\rightharpoonup }{v}}_t^{(i)}={\alpha }_{L,t}^{(i)}({\stackrel{\rightharpoonup }{g}}^{t-1}-{\stackrel{\rightharpoonup }{x}}_{t-1}^{(i)})$$

(9)

We drop all of the other terms, as we would like to focus on just one parameter, ${\alpha }_{L,t}^{(i)}$, which represents the speed of lead/lag (which is in turn used in the empirical work of portfolio sorting).

Note that ${\alpha }_{L,t}^{(i)}$ is a “follow-the-leader” parameter that is different for every fish. That is, each fish can choose its own tendency to follow the leader. We could alternatively replace ${\alpha }_L^{(i)}$ by ${\alpha }_L$ (without the superscript). In doing so, every fish must adopt the same speed.

Clearly, the particular version of swarm we adopt in this paper is restricted. A general swarm should contain all of the parameters. By estimating only ${\alpha }_{L,t}^{(i)}$, the effects of other parameters, such as alignment and cohesion, are embedded in ${\alpha }_{L,t}^{(i)}$. Implicitly, we are assuming that these omitted parameters have the same effect for every fish, implying that these effects will not alter the direction of ${\alpha }_{L,t}^{(i)}$, although the magnitudes will change.

3. Empirical Methodology

Our empirical design follows a rigorous three-stage pipeline to establish the relationship between swarm-based dynamics and asset pricing. First, we transform the time-series of monthly factor returns into a sequence of instantaneous positions (${\stackrel{\rightharpoonup }{x}}_t^{(i)}$) in a 12-dimensional swarm space. The change in position from t to t + 1 provides the empirically observed velocity (${\stackrel{\rightharpoonup }{v}}_t^{(i)}$). Second, using the leader-following model (a variant of PSO), we estimate the factor-specific parameter ${\alpha }_{L,t}^{(i)}$ by minimizing the squared error between the model-predicted velocity and the observed empirical velocity. This factor-specific ${\alpha }_{L,t}^{(i)}$ is defined as the swarm velocity (SV) metric. Third, we sort the 190 factors into quintiles each month based on their calculated SV. The core test of predictability involves analyzing the subsequent month’s return differential and risk-adjusted alphas between the slowest- and fastest-moving factor quintiles.

This section provides a detailed explanation of the estimation of the factor-specific swarm velocity metric. We can estimate the parameters of swarm (Equation (1)) using data or factor returns in our study. Specifically, empirical positions (i.e., factor returns) can be labeled as ${\stackrel{\rightharpoonup }{\xi }}_t^{(i)}$ and velocity as ${\stackrel{\rightharpoonup }{\nu }}_t^{(i)}$ (to substitute for ${\stackrel{\rightharpoonup }{x}}_t^{(i)}$ and ${\stackrel{\rightharpoonup }{v}}_t^{(i)}$ in Equation (3)). Hence, similar to Equation (3),

$${\stackrel{\rightharpoonup }{\nu }}_t^{(i)}={\stackrel{\rightharpoonup }{\xi }}_t^{(i)}-{\stackrel{\rightharpoonup }{\xi }}_{t-1}^{(i)}$$

(10)

Our objective function is to minimize the sum of squared errors between ${\stackrel{\rightharpoonup }{\nu }}_t^{(i)}$ and ${\stackrel{\rightharpoonup }{v}}_t^{(i)}$:

$$\underset{\theta }{\min }({\stackrel{\rightharpoonup }{v}}_t^{(i)}-{\stackrel{\rightharpoonup }{\nu }}_t^{(i)}{)}^{\prime }({\stackrel{\rightharpoonup }{v}}_t^{(i)}-{\stackrel{\rightharpoonup }{\nu }}_t^{(i)})=\underset{\theta }{\min }{\displaystyle {\sum }_{j=1}^n}{(v_{j,t}^{(i)}-{\nu }_{j,t}^{(i)})}^2$$

(11)

By taking the partial derivative and setting it to 0,

$${\displaystyle \sum }(v_{j,t}^{(i)}-{\nu }_{j,t}^{(i)}){\displaystyle \frac{\partial v_{j,t}^{(i)}}{\partial {\alpha }_{L,t}^{(i)}}}=0$$

(12)

In this paper, we only focus on leaders and laggards, and hence there is only parameter, ${\alpha }_{L,t}^{(i)}$.

Clearly, Equation (12) in general has no closed-form solution and needs to be solved numerically. However, if we focus on one parameter at a time (i.e., holding other parameters constant), then there is a closed-form solution, which is what we implement in the empirical section.

In our empirical work, we are only interested in leader following. Hence,

$${\stackrel{\rightharpoonup }{v}}_t^{(i)}={\alpha }_{L,t}^{(i)}({\stackrel{\rightharpoonup }{g}}_{t-1}-{\stackrel{\rightharpoonup }{x}}_{t-1}^{(i)})$$

(13)

The solution (see the Appendix B) is

$${\alpha }_{L,t}^{(i)}={\displaystyle \frac{{\displaystyle {\sum }_{j=1}^n}{\nu }_{j,t}^{(i)}(g_{j,t}-x_{j,t}^{(i)})}{{\displaystyle {\sum }_{j=1}^n}{(g_{j,t}-x_{j,t}^{(i)})}^2}}$$

(14)

While Equation (14) minimizes the empirical sum of squared errors described by Equation (11), it has an interesting interpretation. First of all, we should note that every term in Equation (14) comes from data. ${\nu }_{j,t}^{(i)}$ is the empirical velocity of the $i$-th risk factor in the $j$-th dimension at time $t$. This quantity is obtained by taking a difference of returns of the $i$-th risk factor in two consecutive months. $x_{j,t}^{(i)}$ is the return of the $i$-th risk factor in the $j$-th dimension at time $t$. Later, as we explain our empirical methodology, it will be clear that $x_{j,t}^{(i)}$ is the return of the $i$-th risk factor in the $j$-th lagged month, given that we are using past returns to predict future returns (and hence lagged returns are features/dimensions in the swarm model). Hence, $x_{j,t}^{(i)}$ is defined as $t-j$ return of the risk factor $i$. We use the last 12 returns ($n=12$ in Equation (14)) in our empirical work, and hence there are 12 lagged returns in Equation (14). That is, $x_{1,t}^{(i)}$ is the return of the last month ($t-1$), $x_{2,t}^{(i)}$ is the return of two months ago ($t-2$), …, until the last one $x_{12,t}^{(i)}$ which is the return of the twelfth lagged month.

Finally, $g_{j,t}$ is the “leader,” however it is chosen. As mentioned earlier, when there is a landscape (i.e., fitness function), the leader can be whoever possesses the best fitness value. Or, it can simply be chosen subjectively. In our empirical work, we are seeking to find such a leader. As a result, we loop through each risk factor as the leader and compute the speeds of its followers. If a particular risk factor is not a leader, then the other risk factors will not follow it, and the speeds are therefore small. If this risk factor is indeed the leader, then the other risk factors will follow it closely, and the speeds will be high.

In other words, each risk factor’s position (i.e., the last 12 months of returns) when it is assumed to be the leader is

$$g_{j,t}=x_{j,t}^{(i)}$$

(15)

and, as a result, ${\alpha }_{L,t}^{(i)}=0$.

The average across individual risk factors is

$$\begin{array}{cc}\hfill {\overline{\alpha }}_{L,t}& ={\displaystyle \frac{1}{m-1}}{\displaystyle {\sum }_{i=1}^{m-1}}{\alpha }_{L,t}^{(i)}\hfill \\ & ={\displaystyle \frac{1}{m-1}}{\displaystyle {\sum }_{i=1}^{m-1}}{\displaystyle \frac{{\displaystyle {\sum }_{j=1}^n}{\nu }_{j,t}^{(i)}(g_{j,t}-x_{j,t}^{(i)})}{{\displaystyle {\sum }_{j=1}^n}{(g_{j,t}-x_{j,t}^{(i)})}^2}}\end{array}$$

(16)

It is noteworthy that the method of looping through each risk factor as a potential leader is a core feature that distinguishes our approach from simpler, purely financial lead–lag models. Alternative leader selection mechanisms, such as designating the largest factor by market capitalization or the factor with the highest trading volume, rely on exogenous financial proxies that may not accurately reflect the psychological mechanism of herding. Herding behavior is driven by which stock (or factor) is currently perceived as the “winner” by market participants, which is not necessarily the largest or most liquid factor. Our data-driven approach is superior because the optimal leader is identified endogenously—it is the factor whose past position best explains the collective subsequent movements of the entire swarm, as measured by minimizing the sum of squared errors (SSE) in Equation (11). This method directly captures the empirical leader–follower dynamic that aligns best with the observed market behavior. Consequently, the chosen leader is the one that historically best dictated the direction of factor movements, providing a more direct proxy for herding than simple size or liquidity metrics.

In the next empirical section, we provide a visualization of how estimation is performed. A more generalized optimization can be found in [20,21].

As we can see, in this paper, we apply data to swarm and “back-out” the parameters of the swarm, and we even try to find out who the leader is if the data indeed comply with our assumption that there is swarm in risk factors. This is analogous to computing the implied volatility of the Black–Schole option pricing model in that the model is not used for pricing but for “backing out” the parameters of the model.

4. The Swarm-Related Literature in Finance

Our research intersects two distinct, yet complementary, streams in the financial literature: (1) the broader application of advanced artificial intelligence and machine learning techniques in financial modeling, and (2) the classical behavioral finance topics of herding and lead–lag effects, which provide the economic interpretation for our velocity measure.

4.1. Contextualizing Swarm Intelligence in Modern Financial AI

Our research utilizes swarm intelligence (SI), a metaheuristic rooted in collective agent behavior, positioning it uniquely within the rapidly evolving landscape of quantitative financial modeling. In contemporary finance, the literature has increasingly moved beyond linear factor models to adopt highly flexible, non-linear methodologies, such as deep learning (DL) and reinforcement learning (RL), which are primarily aimed at superior prediction and dynamic strategy execution [22,23]. This evolution recognizes that complex financial systems are best modeled as complex adaptive systems (CAS), where traditional linear relationships often fail to capture market non-linearities and emergent properties [24].

However, while DL and RL prioritize predictive accuracy, they often operate as black box models, making it difficult to extract economic intuition about why a prediction was made. Swarm intelligence and related evolutionary algorithms offer an important middle ground. They were historically a key component of early artificial intelligence in finance, demonstrating strong capabilities in portfolio optimization and parameter estimation in high-dimensional, rugged landscapes where parametric methods fail [10,25].

The SI framework we employ offers a unique advantage over pure prediction models as it provides a powerful, physics-based, and interpretable lens through which to measure a specific behavioral dynamic, the speed and direction of collective factor movement. By focusing on estimating the factor velocity parameter and linking it directly to asset pricing theory (information diffusion and trading frictions), our contribution is positioned squarely in the realm of advanced behavioral and psychological modeling, offering economic insight alongside a modern computational technique.

4.2. Herding

Swarm, although meant for describing low-intelligent animals like ants and bees, has actually been observed in human behaviors. When gathered, humans tend to act collectively and hastily without a thorough decision making process. This is known as herding.

Ref. [26] is believed to be the first author who studied herding in the financial market. He describes investors influenced by their peers in making investment decisions as a social activity. He asserts that when making investment decisions, humans tend to gossip about others’ successes or failures in investing as opposed to making intelligent decisions of their own.

In fact, humans herd in many other activities, as well. When voting for someone politically, people often go with their close friends and relatives as opposed to thinking independently. Nutrition decisions are another example where people tend to listen to their friends and relatives as opposed to actively conducting research. The main reason is that such research usually takes time and people are “lazy.” Another possible reason is that these decisions (voting, picking supplements, and investing) are often gossip topics when people social.

Herding is a significant research topic because researchers find that herding does not generate better returns. Ref. [27] argues that herding results in market disturbances and destabilization, which can move the market away from its fundamentals. As a result, it can be expected that herding is not persistent. It happens sometimes, ceases to happen sometimes, and furthermore contradicts itself sometimes.

Herding, as a result, can be regarded as a reason why technical analyses have been persistent. Financial economists have long frowned on technical analyses due to their lack of economic support. Although they may not have economic support, they have plenty of psychological support. For this reason, Ref. [28] study the determinants of herding and find that when volatility is high (e.g., a crisis), the size of the firm is small, interest rates are high, or the market is opaque, herding is likely to be observed. It is obvious that such situations are not sustainable situations. They happen only temporarily and are naturally not persistent.

Various authors, given data availability and research focus, use different proxies for herding (see, for example, [29,30]). Yet these proxies suffer from various measurement errors, and, so far, there has been no consensus.

The literature also associates herding with momentum (see [31] for the original momentum research). Ref. [32] define herding as how a group of investors moves in and out of the market simultaneously, which is then connected to momentum trading. Recently, Ref. [33] studied the influence of industry herding on momentum returns. Ref. [34] also studied the impact of herding behavior on momentum. Ref. [35] found that uninformed country-level herding is highly related to momentum. Ref. [28] discovered strong evidence of international herding. Ref. [36] estimated swarm parameters using a proprietary broker/dealer dataset and found mild herding among brokers and dealers. Ref. [37] provides an overview of the recent theoretical and empirical research on herd behavior in financial markets.

4.3. Lead–Lag

Asset returns that move faster, referred to as leaders in our study, could be driven by several factors. One possibility is that they exhibit higher liquidity, allowing investors to incorporate value-relevant information more quickly [14,38]. Another potential explanation is that these assets attract heightened investor attention, often due to media coverage or prevailing market sentiment [13,39]. When assets gain excessive attention, their prices may deviate from fundamentals, leading to a temporary overvaluation that eventually corrects over time. If investor-driven overreaction propels fast-moving assets, we would expect them to subsequently underperform, as measured by alphas relative to standard asset pricing models, such as the Fama–French factor models [40,41] or the Hou–Xue–Zhang factor models [42]. In other words, the overvaluation fueled by heightened investor attention should lead to lower subsequent alphas as prices revert to their intrinsic values. However, if the primary driver of rapid price movement is superior liquidity conditions, then these assets would likely incorporate value-relevant information efficiently, leaving no room for abnormal performance and ensuring fair pricing.

Conversely, assets that move slower, referred to as laggards, may do so for several reasons. One possibility is that they operate in more mature sectors where the information flow is relatively stable, leading to fewer price-adjusting events [40]. As a result, the absence of significant informational updates causes slow price movements. Another potential explanation is that these assets face higher trading frictions, such as lower liquidity, wider bid–ask spreads, and higher transaction costs [43]. In this case, market participants take longer to process and incorporate new information, leading to a slower diffusion of price-relevant signals. If slow movement results from operating in a stable industry with predictable fundamentals, there may be no systematic impact on future returns. However, if frictions and impediments to trading hinder the incorporation of information, the direction of future abnormal returns depends on whether the market predominantly underreacts to good news or bad news. If slow-moving assets experience an underreaction to positive information, they could generate positive alphas, whereas an underreaction to negative information could lead to negative alphas. This remains an empirical question, and we rely on data-driven analysis to uncover the relationship between asset velocity and return predictability.

5. Empirical Findings

In this section, we briefly introduce the data we use and how a swarm model is applied in order to estimate the speed of following the leader, and we investigate the market pricing of slow- and fast-moving risk factors.

5.1. Data

The test assets are monthly Hou–Xue–Zhang 190 risk factors over 23 years obtained from the Q-factor data library, https://global-q.org/index.html (accessed on 7 March 2025). It contains portfolios in six categories with different periods:

• Frictions (February 1990~December 2022), 10 factors;
• Intangibles (January 1990~December 2022), 31 factors;
• Investments (January 1973~December 2022), 32 factors;
• Momentum (July 1979~December 2022), 42 factors;
• Profitability (January 1985~December 2022), 44 factors;
• Value vs. growth (January 1985~December 2022), 31 factors.

In order to run our swarm model, we adopt the common period from February 1990 to December 2022, a total of 395 months.

We use lag for 12 months as features. That is, the universe of the swarm is a 12-dimensional space. In other words, on each month, a risk factor (i.e., a fish) is positioned in a 12-dimensional space according to its last 12 months of returns. As each month passes by, the risk factor moves to the next 12 months of returns.

However, to help visualize how we fit a swarm model via data, we only use lag for 2 months as a demonstration in Figure 2. Figure 2 plots frictions’ 10 risk factors for 4 consecutive months, April through July of 1990. Each panel contains positions (i.e., returns) of the last two months. For example, Panel (A) plots 10 risk factors’ February returns on the x-axis and March returns on the y-axis.

We pick two risk factors as an example; beta_1 (the market beta estimated using daily returns within a month) is marked in red, and tv_1 (total volatility computed from daily returns over the same period) is marked in yellow. From Figure 2, we can clearly observe how they move from month to month. As explained earlier, on each month, the position of a risk factor is its last two months of returns. The returns of the two risk factors are given below.

In

Table 1. This table presents the monthly returns of the beta_1 and tv_1 factors from February 1990 to June 1990 (Panel A), their corresponding mappings to position space (Panel B), and the computation of their swarm-based velocities (Panel C).

Panel A. Monthly return
Data		beta_1				tv_1
2/1990		3.5051				1.4045
3/1990		7.4907				1.6897
4/1990		0.3085				−1.5450
5/1990		8.5011				2.3310
6/1990		−0.9011				−0.7728
Panel B. Position space
Position	beta_1				tv_1
	X		y		X		Y
4/1990	3.5051		7.4907		1.4045		1.6897
5/1990	7.4907		0.3085		1.6897		−1.5450
6/1990	0.3085		8.5011		−1.5450		2.3310
7/1990	8.5011		−0.9011		2.3310		−0.7728
Panel C. Monthly velocity
Velocity	beta_1				tv_1
	X			y	X		Y
5/1990	3.9856			−7.1822	0.2852		−3.2347
6/1990	−7.1822			8.1926	−3.2347		3.8760
7/1990	8.1926			−9.4022	3.8760		−3.1038

, Panel (A), beta_1’s position is (3.5051, 7.4907), and tv_1’s position is (1.4045, 1.6897). In Panel (B), beta_1’s position is (7.4907, 0.3085), and tv_1’s position is (1.6897, −1.5450). From month to month, we can now see the migration of the two risk factors. We then use these positions to estimate the swarm speed. Then, Panel C illustrates how the velocity is calculated, which is measured as the difference between two consecutive positions.

These numerical results are then fed into Equation (16) to calculate the speed parameter. In our empirical work, 12 lags are used. We also tried different numbers of lags, and the results are qualitatively similar. They are available upon request. In terms of the number of risk factors, we follow [42] and use 190 risk factors. Hence, we use $m=190$ and $n=12$ in our swarm model.

To visualize the movements of the risk factors (i.e., fish), we use beta_1 and tv_1 as an example and plot their movements in Figure 3.

The movement of a risk factor is represented by the velocity. As described in Equation (10), the actual velocity from data can be calculated by taking the difference of the positions of two consecutive months. Panel (A) of Figure 2 plots the positions of chosen risk factors in April 1990, and Panel (B) plots the positions of chosen risk factors in May 1990. Taking beta_1 as an example, Figure 3 plots its positions in these two consecutive months. Its velocity is calculated (as the difference of positions) as (3.98, –7.18), which is the difference in two consecutive positions, (3.5051, 7.4907) and (7.4907, 0.3085), in April and May of 1990, respectively. Similarly, the velocity for tv_1 (orange) is (0.28, –3.23). The result is plotted in Panel (A) of Figure 3. Similarly, the transitions from May to June and June to July of 1990 are plotted in Panels (B) and (C).

In estimating the parameters, such as ${\alpha }_{L,t}^{(i)}$ in Equation (14), we simply iterate the parameter value until the model’s velocity matches the data’s velocity as closely as possible (i.e., minimized sum of squared errors).

5.2. Experimental Design and Procedure

This section clarifies the experimental environment, the fixed parameter settings of our adapted swarm model, and the step-by-step procedure used to conduct the monthly asset pricing tests.

The core empirical methodology is a time-series analysis built on monthly cross-sectional sorting. The fixed parameters of our estimation environment are determined based on the input data and the structural design of the swarm space. The agents, or “particles,” in the swarm are the 190 cross-sectional risk factors sourced from the Q-factor data library. The dimension of the swarm space is fixed at D = 12, representing the preceding 12 months of returns for each factor. This D = 12 choice serves as the input features that define the factor’s position vector (${\stackrel{\rightharpoonup }{x}}_t^{(i)}$) in the swarm space. The core estimation period spans T = 395 months, from February 1990 to December 2022.

The experiment is executed via a monthly rolling procedure, which consists of three distinct phases. The first phase involves the observation and modeling of factor dynamics. We first compute the position vector (${\stackrel{\rightharpoonup }{x}}_t^{(i)}$) for all 190 factors at the end of each month t using their returns from t − 11 to t. The empirically observed velocity (${\stackrel{\rightharpoonup }{v}}_t^{(i)}$) is then calculated as the change in position from the previous month: ${\stackrel{\rightharpoonup }{v}}_t^{(i)}={\stackrel{\rightharpoonup }{x}}_t^{(i)}-{\stackrel{\rightharpoonup }{x}}_t^{(i)}$. Subsequently, the leader position (${\stackrel{\rightharpoonup }{x}}_t^{(\mathrm{leader})}$) is identified as the factor exhibiting the highest cross-sectional return over the preceding 12-month period, t − 11 to t. Finally, using the time-series of $v_t^{\left(i\right)}$ and the position difference (${\stackrel{\rightharpoonup }{x}}_t^{(\mathrm{leader})}-{\stackrel{\rightharpoonup }{x}}_t^{(i)}$), we estimate the factor-specific swarm velocity (SV), ${\alpha }_{L,t}^{(i)}$, for all 190 factors by minimizing the error via the closed-form solution (Equation (14)).

The second phase establishes the link between dynamics and returns. Specifically, at the end of month t, we sort the 190 factors into five equal-weight quintile portfolios based on their estimated SV. The Low SV portfolio comprises the slow-moving factors, and the High SV portfolio comprises the fast-moving factors. We then calculate the equal-weighted return for each of the five portfolios over the subsequent month, t + 1. This step establishes the core link between factor dynamics (measured at t) and subsequent return predictability (observed at t + 1).

The third phase involves the formal statistical evaluation. We calculate the time-series average of the return differential between the High SV and Low SV portfolios (High−Low). To test whether the predictability constitutes an unmodeled risk premium, the time-series returns of all five portfolios are regressed against various established factor models (FF5, FF6, Q, Q5). The intercept terms (α) are the primary metric for assessing abnormal performance. Finally, we examine the long-term return predictability by analyzing portfolio returns when the portfolios formed at month t are held and tested over horizons extending up to 12 months ahead (as summarized in Table 3). This comprehensive procedure is repeated for every month within the sample period, generating a time-series of monthly portfolio returns used for all subsequent statistical inference.

5.3. Speed Result

The speed results are plotted in Figure 4 and Figure 5. Figure 4 presents a distribution of average speeds across risk factors. Each observation in this distribution is an average speed of a risk factor over the entire sample period. It can be seen that (1) all speeds are positive, demonstrating a definite swarm behavior (somewhat surprising!), and (2) the mode of the distribution is slightly less than 0.5, with the next highest peak slightly higher than 0.5.

Figure 5 presents the time-series of average speed across risk factors. As we can see, the variability of speed over time is not noteworthy, and the mean and the median are very close to each other, demonstrating no significant skewness. Figure 5 also plots the 25th and 75th percentiles, and they indicate a roughly symmetrical distribution across risk factors.

5.4. Market Pricing of Factor Velocity

To investigate how the market prices factors with different velocity measures derived from swarm intelligence, we conduct a portfolio-level analysis. Each month, we sort 190 factors from the Q-factor data library into quintiles based on their swarm velocity (SV). We then compute the equal-weighted returns for each quintile and compare the return differences between the top and bottom quintiles. Our portfolio-level analysis is conducted in an out-of-sample fashion in the time-series sense. Specifically, for each month, factor velocities are computed using only return data from the prior 12 months, and these estimates are then used to sort factors and evaluate one-month-ahead returns. This rolling structure ensures that no future information is used in forming the velocity measure, thereby preserving the out-of-sample integrity of the return predictions. To further address concerns about predictive validity across time, we perform a subsample analysis by splitting the sample into two equal subperiods and examining the return predictability of factor velocity separately within each. The results (available upon request) remain directionally consistent with the full-sample findings, with slow-moving factors continuing to earn higher average returns than fast-moving ones. These findings reinforce the robustness and generalizability of swarm velocity as a predictive signal.

The primary metrics used to evaluate the performance of portfolios sorted by swarm velocity (SV) are the modified Jensen’s Alpha (α), serving as the gold standard for measuring risk-adjusted abnormal returns in empirical asset pricing (e.g., [40,41,44,45]); among many others). Specifically, α is calculated as the intercept term in a time-series regression of the monthly returns of each factor portfolio ($R_t^p$) on established market risk factors ($F_t^k$) using the general K-factor model, $R_t^p={\alpha }^p+\sum _{k=1}^K{{\beta }^{k}F}_t^k+{ϵ}_t^p$. A positive alpha (α > 0) indicates superior performance, returns exceeding those predicted by the model given its risk, suggesting an abnormally high return after accounting for portfolio p’s exposure to market risk factors. Conversely, a negative alpha indicates underperformance. The statistical significance of alpha is determined by its t-statistic (Newey–West adjusted to account for serial correlation), where an absolute value typically exceeding 1.96 confirms that the abnormal return is statistically robust across the alternative factor models tested. Following prior studies, we estimate alphas for each factor portfolio and the return difference between the top and the bottom SV quintiles relative to the following established models: the Fama–French Five-Factor Model (FF5, including excess market return (MKT−RF), size factor (SMB), book-to-market factor (HML), investment growth factor (CMA), and operating profitability factor (RMW)); the Fama–French Six-Factor Model (FF6, extending FF5 with a momentum factor (UMD)); the augmented FF6 model (FF6PS, including the Pastor–Stambaugh illiquidity factor (PS)); the Hou, Xue, and Zhang Four-Factor Model (Q, including excess market return (R_MKT), size factor (R_ME), investment growth factor (R_IA), and operating profitability factor (R_ROE)); and the extended Q-Factor Model (Q5, including the growth-related factor (R_EG)).

Table 2. Univariate portfolio sorts on swarm velocity. Each month, we sort the 190 factors into quintiles based on their swarm velocity (SV). Column 1 presents the average characteristics of swarm velocity (SV), the sorting variable. Column 2 reports the mean equal-weighted returns (Mean) across factors within each quintile. Columns 3 to 8 present the alphas for each quintile portfolio relative to various factor models, including the Fama–French Five-Factor Model (FF5), the Fama–French–Carhart Six-Factor Model (FF6), the Fama–French–Carhart Six-Factor Model augmented with the Pastor–Stambaugh liquidity factor (FF6PS), the Hou–Xue–Zhang Four-Factor Model (Q), and the Hou–Mo–Xue–Zhang Five-Factor Model (Q5). Rows 1 to 5 present the results for each SV-based quintile, from the lowest (Low) to the highest (High). The last row (High–Low) reports the differences in mean returns and alphas between the top and bottom quintiles. Newey–West-adjusted t-statistics with six lags are reported in parentheses. The sample period spans from March 1991 to December 2022. Numbers in boldface indicate statistical significance at the 5% level or better.

Quintile	SV	Mean	FF5	FF6	FF6PS	Q	Q5
Low	0.21	0.43	0.41	0.25	0.25	0.31	0.24
		(4.85)	(3.94)	(3.81)	(3.71)	(3.52)	(2.57)
2	0.37	0.23	0.16	0.13	0.14	0.12	0.06
		(4.43)	(3.18)	(2.52)	(2.52)	(2.36)	(1.20)
3	0.47	0.07	0.06	0.06	0.07	0.04	0.00
		(1.48)	(1.27)	(1.15)	(1.33)	(0.79)	(−0.04)
4	0.58	0.20	0.17	0.10	0.10	0.11	0.07
		(3.46)	(3.27)	(2.40)	(2.38)	(2.37)	(1.55)
High	0.73	0.17	0.16	0.07	0.07	0.09	0.01
		(2.99)	(3.26)	(1.58)	(1.60)	(1.76)	(0.11)
High–Low	0.52	−0.26	−0.25	−0.18	−0.17	−0.22	−0.21
		(−3.39)	(−2.86)	(−2.52)	(−2.42)	(−2.61)	(−2.63)

presents the results. Column 1 reports the average characteristics of the sorting variable, Column 2 provides the mean equal-weighted returns within each quintile, and Columns 3 to 8 display alphas relative to various factor models. These models include (1) the Fama–French Five-Factor Model (FF5), which includes market (MKT), size (SMB), book-to-market (HML), investment (CMA), and profitability (RMW) factors [40], with the resulting intercept term labeled FF5 alpha; (2) the Fama–French Six-Factor Model (FF6), which adds the momentum (UMD) factor to FF5, producing FF6 alpha [40,44]; (3) the Fama–French–Carhart Six-Factor Model augmented with the Pastor–Stambaugh liquidity factor [14], generating FF6PS alpha; (4) the Hou–Xue–Zhang Four-Factor Model (Q-factor model), which includes R_MKT, R_ME, investment (R_IA), and profitability (R_ROE) [42], with the corresponding Q alpha; and (5) the Hou–Mo–Xue–Zhang Five-Factor Model [46], which extends the Q-factor model with the growth (R_EG) factor, producing Q5 alpha. The last row of Table 2 reports the differences in mean returns and alphas between the top and bottom quintiles, with Newey–West-adjusted t-statistics) accounting for serial correlation. Our sample period spans March 1991 to December 2022 to ensure all 190 factors are available each month.

Table 2 reveals that factors in the lowest SV quintile earn significantly higher abnormal returns than those in the highest SV quintile, with an alpha difference ranging from 17 to 22 basis points per month. Importantly, this return differential is primarily driven by the superior performance of slower-moving factors. The time period robustness test shows that the predictive power of the SV factor holds across both the pre-Global Financial Crisis or GFC (March 1993–December 2008) and post-GFC (January 2009–December 2022) periods, with slow-moving factors outperforming fast-moving factors at the one-month-ahead horizon, suggesting that its predictive strength is not confined to a specific economic cycle or market regime.

We further examine whether swarm speed exhibits return predictability beyond a one-month horizon. To test this, we relate the swarm velocity measure to future returns of the previously formed quintile factor values over horizons extending from 2 months to 12 months ahead.

Table 3. Long-term return predictability. This table presents the Hou–Mo–Xue–Zhang Five-Factor alphas for the top and bottom quintile portfolios sorted by swarm velocity (SV), along with their alpha differentials across horizons ranging from 2 (t + 2) to 12 months (t + 12) ahead. Rows 1 and 2 report the results for the lowest (Low) and highest (High) SV-based quintiles. The last row (High–Low) reports the differences in mean returns and alphas between the top and bottom quintiles. Newey–West-adjusted t-statistics with six lags are reported in parentheses. The sample period spans from March 1991 to December 2022.

Quintile	t + 2	t + 3	t + 4	t + 5	t + 6	t + 7	t + 8	t + 9	t + 10	t + 11	t + 12
Low	0.18	0.17	0.18	0.16	0.21	0.21	0.18	0.17	0.18	0.16	0.16
	(2.05)	(1.88)	(1.90)	(1.53)	(2.27)	(2.28)	(1.82)	(1.78)	(1.99)	(1.81)	(1.60)
High	0.01	0.06	0.05	0.04	0.05	0.03	0.03	0.05	0.01	0.04	0.08
	(0.20)	(1.10)	(0.87)	(0.80)	(0.87)	(0.46)	(0.51)	(0.91)	(0.21)	(0.75)	(1.51)
High–Low	−0.17	−0.11	−0.10	−0.11	−0.11	−0.13	−0.11	−0.09	−0.13	−0.12	−0.05
	(−2.18)	(−1.38)	(−1.29)	(−1.08)	(−1.32)	(−1.64)	(−1.25)	(−1.15)	(−1.58)	(−1.41)	(−0.67)

presents the Hou–Mo–Xue–Zhang Five-Factor alphas (Q5) for these horizons. The results show that the return spread between the top and bottom SV quintile factor values remains negative across all horizons, ranging from 5 to 17 basis points per month. However, only the two-month-ahead alpha spread is statistically significant, driven by the continued outperformance of slow-moving factors.

Next, we explore the characteristics of fast-moving versus slow-moving factors. Panel A of

Table 4. Characteristics of slow-mover and fast-mover portfolios. Panel A presents the top 10 slow-moving factors (left four columns) and the top 10 fast-moving factors (right four columns), based on swarm velocity (SV). For each factor, the table reports the average SV (column labeled “SV”) and the probabilities of inclusion in the slow-mover portfolio (bottom SV quintile, column labeled “Slow mover”) and the fast-mover portfolio (top SV quintile, column labeled “Fast mover”). All values are mean-adjusted to facilitate comparison. Panel B summarizes factor representation by category. The “Factor presence” column indicates the percentage of factors in each of the six categories relative to the total universe of 190 factors. The “Slow mover” and “Fast mover” columns report the proportion of each category represented in the slow- and fast-mover portfolios, respectively. The complete list of factors, including notations and category-specific descriptions, is provided in the online Appendix C. Numbers in boldface indicate statistical significance at the 5% level or better.

Panel A. Top slow-mover and fast-mover factors.
Top 10 Slow-Movers						Top 10 Fast-Movers
Characteristics	SV		Slow Mover	Fast Mover		Characteristics	SV		Slow Mover	Fast Mover
beta_1	−0.28		0.69	−0.20		cm_12	0.37		−0.20	0.78
tv_1	−0.29		0.68	−0.20		sim_12	0.33		−0.20	0.77
r11_1	−0.31		0.67	−0.20		abr_12	0.33		−0.20	0.76
ivff_1	−0.24		0.61	−0.20		ilr_12	0.32		−0.20	0.73
r1n	−0.29		0.57	−0.20		sm_12	0.34		−0.20	0.72
r11_6	−0.25		0.57	−0.20		droe_12	0.28		−0.20	0.70
p52w_6	−0.26		0.56	−0.20		cm_6	0.27		−0.19	0.67
r6_1	−0.26		0.54	−0.19		resid6_12	0.25		−0.20	0.64
ivq_1	−0.22		0.50	−0.20		droe_6	0.23		−0.20	0.57
fp_6	−0.20		0.46	−0.20		tbiq_12	0.25		−0.20	0.55
Panel B. Presence in slow-mover and fast-mover portfolios by category.
Category		Factor Presence			Slow Mover			Fast Mover
Frictions		5.26			11.55			4.64
Intangibles		16.32			17.48			7.71
Investments		16.84			8.31			20.32
Momentum		22.11			25.37			36.2
Profitability		23.16			17.03			24.36
Value vs. Growth		16.32			20.26			6.77
Sum (%)		100			100			100

lists the top 10 slowest-moving factors (left panel) and the top 10 fastest-moving factors (right panel). For comparison, values in the table are demeaned. The slowest-moving factors exhibit an average velocity below the mean of all 190 factors, with a probability exceeding 50% of falling into the slow-moving quintile and a 20% lower likelihood of appearing in the fast-moving quintile. Conversely, the fastest-moving factors exhibit significantly higher velocity scores, with probabilities more than 50% above the mean for appearing in the fast-mover quintile and approximately 20% lower for falling into the slow-mover quintile.

To improve interpretability and identify underlying drivers of return differentials, we follow [42,46] and categorize factor-level results by their respective themes, including frictions, intangibles, investments, momentum, profitability, and value vs. growth. Panel B of Table 4 shows that friction-related factors are disproportionately overrepresented in the slow-mover risk factor, appearing at more than twice their unconditional share. In contrast, investment-related factors are significantly underrepresented, comprising less than 50% of their unconditional share. Within the fast-mover risk factor, momentum-related factors dominate, accounting for 36.20% of the risk factor versus their unconditional share of 25.37%, while intangibles and value vs. growth factors are notably underrepresented. The non-random factor categorization bias in Table 4, Panel B, is formally confirmed using a Chi-Squared (${\chi }^2$) goodness-of-fit test. The test compares the observed distribution of factor categories within the extreme quintiles against the unconditional distribution and finds the bias statistically significant at the 1% level or better for both slow- and faster-mover portfolios, validating that the factor bias is non-random.

The concentration of specific factor categories in the extreme velocity quintiles strongly supports our behavioral interpretation. The heavy overrepresentation of friction factors (e.g., beta1, tv1, ivff1) within the slow-mover portfolio directly confirms the hypothesis that the low swarm velocity is driven by trading impediments and delayed information diffusion. These factors, which are often proxies for low liquidity or high idiosyncratic risk, inherently lead to slow price adjustments, as institutional investors may be slow to trade or incorporate new information due to high transaction costs. Conversely, the significant domination of the fast-mover portfolio by momentum factors (e.g., cm12, sim12, abr12) is consistent with the literature on investor attention and herding. Momentum factors are highly sensitive to recent price action and news, leading to rapid collective movements. These factors act as the trend-setters or “leaders” within the swarm system, and their high swarm velocity confirms their role as the fastest responders to market sentiment. This empirical factor alignment validates that the estimated swarm velocity is a direct, data-driven measure of a factor’s speed of price adjustment.

Next, we examine the persistence of swarm velocity (SV) using a portfolio transition matrix, which reports the probability that a factor in the $i$-th SV quintile at month t remains in the $j$-th SV quintile at month $t+n$, where n ranges from 1 to 60 months (i.e., from 1 month ahead to 5 years ahead). Under the null hypothesis of random portfolio assignment, the probability of a factor remaining in the same quintile over time would be 20%. Panel A of

Table 5. Transition matrix for swarm-velocity-sorted portfolios. This table presents the transition matrix for quintile portfolios sorted by swarm velocity (SV). At the end of each month t, all factors are sorted into ascending SV quintiles. For each t-month SV quintile, the table reports the time-series average of the percentage of factors that transition into each SV quintile at future horizons of t + 1, t + 3, t + 6, t + 12, t + 24, t + 36, t + 48, and t + 60 months, respectively. Numbers in boldface indicate statistical significance at the 5% level or better.

Panel A, month t + 1
	Quintile in Month t
Quintile in Month t + 1	Low	2	3	4	High
Low	79	11	4	4	2
2	10	68	14	4	3
3	4	14	64	14	3
4	4	4	14	68	11
High	2	3	3	11	81
Sum (%)	100	100	100	100	100
Panel B, month t + 3
	Quintile in Month t
Quintile in Month t + 3	Low	2	3	4	High
Low	70	18	6	4	3
2	17	50	22	7	3
3	6	22	45	22	5
4	4	6	22	50	18
High	3	3	5	18	71
Sum (%)	100	100	100	100	100
Panel C, month t + 6
	Quintile in Month t
Quintile in Month t + 6	Low	2	3	4	High
Low	60	23	10	6	3
2	21	37	24	11	5
3	10	24	33	24	8
4	6	11	24	37	22
High	3	5	8	22	62
Sum (%)	100	100	100	100	100
Panel D, month t + 12
	Quintile in Month t
Quintile in Month t + 12	Low	2	3	4	High
Low	45	25	16	10	5
2	24	26	23	17	9
3	16	23	24	23	14
4	10	17	22	27	23
High	5	8	14	24	50
Sum (%)	100	100	100	100	100
Panel E, month t + 24
	Quintile in Month t
Quintile in Month t + 24	Low	2	3	4	High
Low	42	24	18	11	5
2	24	27	22	18	9
3	17	22	24	22	14
4	12	18	22	25	23
High	6	9	14	24	49
Sum (%)	100	100	100	100	100
Panel F, month t + 36
	Quintile in Month t
Quintile in Month t + 36	Low	2	3	4	High
Low	40	24	19	13	5
2	24	26	22	19	9
3	18	23	23	21	15
4	12	18	22	24	23
High	6	9	15	23	48
Sum (%)	100	100	100	100	100
Panel G, month t + 48
	Quintile in Month t
Quintile in Month t + 48	Low	2	3	4	High
Low	40	25	18	12	6
2	24	25	22	19	9
3	17	23	23	21	14
4	13	18	22	25	22
High	6	10	15	23	48
Sum (%)	100	100	100	100	100
Panel H, month t + 60
	Quintile in Month t
Quintile in Month t + 60	Low	2	3	4	High
Low	41	25	18	12	5
2	24	26	23	18	10
3	17	22	23	23	15
4	11	17	22	25	24
High	7	11	15	22	46
Sum (%)	100	100	100	100	100

shows that for the one-month-ahead transition matrix, 79% (81%) of factors in the highest (lowest) SV quintile at month t remain in the same quintile at month $t+1$. Moreover, for all SV quintiles, the probability of remaining in the same quintile at $t+1$ exceeds the random benchmark of 20%, indicating meaningful short-term persistence. Panels B through H of Table 5 present the SV transition matrices at 3-, 6-, 12-, 24-, 36-, 48-, and 60-month horizons. The results reveal strong and sustained persistence in SV. For example, 70%, 60%, 45%, 42%, 40%, and 41% of factors in the high-SV portfolio remain in the high-SV quintile after 3, 6, 12, 24, 36, 48, and 60 months, respectively. Similarly, the corresponding persistence rates for factors in the low-SV portfolio are 71%, 62%, 50%, 49%, 48%, and 46%. These probabilities are substantially higher than the 20% expected under random assignment and provide compelling evidence that swarm velocity is highly persistent over time. The persistence of swarm velocity (SV) is formally confirmed by testing the factor retention rates against the 20% random benchmark (null hypothesis). The retention rates in the extreme portfolios (Low SV and High SV) are statistically significant at the 1% level across all horizons tested (up to 60 months).

These findings, combined with our risk factor return results, support the hypothesis that trading frictions impede the swift incorporation of value-relevant information, leading to subsequent outperformance of slow-moving factors. On the other hand, the statistically insignificant alphas of future returns of fast movers are inconsistent with the idea that heightened investor attention results in temporary overvaluation of fast movers, leading to subsequent underperformance.

While the persistence of the negative alpha differential across all major asset pricing models (FF5, FF6, FF6PS, Q, Q5) strongly suggests that the abnormal return is not compensation for known, priced risk, we acknowledge that it could theoretically be attributed to an unpriced macroeconomic risk premium. However, we find the behavioral interpretation to be more compelling and coherent; the high concentration of friction factors in the slow-moving quintile (Table 4, Panel B) provides empirical support for the mechanism of delayed price discovery. In this view, the alpha is not a risk premium but rather a premium earned for exploiting market inefficiency caused by trading impediments and slow information assimilation, which is the direct economic counterpart to a factor’s low swarm velocity.

6. Conclusions

This study presents a novel application of swarm intelligence to classify factor velocity and investigate its role in asset pricing. Swarm is an ideal methodology to model the herding behavior of people trading in the investment world. Given that individuals’ trading data are not available, it is common to follow risk factors (or portfolios) as a proxy of investors’ herding.

By leveraging a swarm-based approach, we systematically identify risk factors that act as market leaders and followers based on their return movement speed. Our results indicate that slow-moving factors generate higher one-month-ahead returns than fast-moving ones, a finding that holds across multiple asset pricing models. This return differential is likely due to trading frictions or delayed information processing by market participants.

These findings open new avenues for understanding return predictability and market efficiency, emphasizing how velocity classification can be integrated into existing asset pricing frameworks. Future research could enhance these models by incorporating alternative swarm intelligence techniques and exploring different test assets. Our study underscores the potential of artificial intelligence in finance, offering a fresh perspective on how collective behaviors shape stock returns.

The velocity-based classification of risk factors provides a compelling economic proof of concept regarding the role of price adjustment speed in market behavior. The finding that factors in the slowest quintile earn significantly higher risk-adjusted returns strongly suggests that these returns are compensation for exploiting an information anomaly (delayed incorporation or underreaction) rather than a known risk factor. However, this study’s primary contribution is focused on factor discovery and economic interpretation. Because our analysis relies on aggregate monthly factor portfolio returns and does not incorporate high-frequency data for modeling transaction costs, liquidity impact, or portfolio turnover, we must caution against drawing immediate conclusions for real-world trading strategies. The observed alpha serves as a robust academic measure of predictive power; however, future research is essential for establishing the full empirical robustness of the factor. This includes extending the framework to individual stocks or other asset classes and using granular trading data to determine exploitability after accounting for implementation costs.