Inferring Trader Horizons from Trading Volume

1. Motivating Example

This post shows that, if traders face convex transaction costs (i.e., it costs them more per share to buy $2$ shares of stock than to buy $1$ share of stock), then it is possible to infer traders’ investment horizons from trading-volume data. To see why, imagine you are a trader and you’ve just learned some positive news about Apple’s next earnings announcement, which is due out overnight. To take advantage of this revelation, you will need to buy shares of Apple stock at some point today. In order to minimize your transaction costs, you will want to space out your demand for Apple shares as much as possible. So, all else equal, the average demand for Apple’s shares will be slightly higher today than it was yesterday because of your earnings revelation. This same logic applies to information at other horizons. Thus, if a larger fraction of the variation in Apple’s trading volume comes from day-to-day differences, then more of Apple’s traders must be operating at the daily horizon. Whereas, if a larger fraction of the variation in Apple’s trading volume comes from week-to-week differences, then more of Apple’s traders must be operating at the weekly horizon.

In the past, when researchers have studied traders’ investment horizons, they have used data on portfolio positions rather than trading volume. Some have measured trading activity at a couple of investment horizons for a small number of stocks. e.g., Brogaard et al. (2014) use NASDAQ data on a randomly selected sample of $60$ stocks that assigns each trader a typical investment horizon. Others have measured horizon-specific trading activity for a large number of stocks but only at a single horizon. e.g., Cella et al. (2013) sample the portfolio positions of institutional traders at the quarterly frequency using 13F filings. But, collecting data on traders’ portfolio positions is hard. While this approach works, it tends to restrict the analysis to only a handful of stocks (e.g., $60$ randomly selected NASDAQ stocks) or to a single horizon (e.g., the quarterly horizon). Because we can use trading-volume data to infer traders’ investment horizons, we no longer face these data-collection restrictions since trading-volume data is publicly available. Broad cross-sectional studies of traders’ investment horizons are now possible.

2. Traders’ Problem

Let’s begin by outlining the data-generating process and describing the problem faced by traders with an $H$ -period horizon. Suppose that returns are generated by a simple $1$ -factor model,

(1) $\begin{align*} r_{t+1} &= \phantom{-} \beta \cdot f_t + \epsilon_{t+1}, \end{align*}$

where $\epsilon_t \overset{\scriptscriptstyle \mathrm{iid}}{\sim} \mathrm{N}(0,\, \sigma_{\epsilon}^2)$ and the level of the factor evolves according to an $\mathrm{AR}(1)$ model,

(2) $\begin{align*} \Delta f_{t+1} &= - \gamma \cdot f_t + \xi_{t+1}, \end{align*}$

with $\xi_t \overset{\scriptscriptstyle \mathrm{iid}}{\sim} \mathrm{N}(0,\, \sigma_{\xi}^2)$ . Note that this is the exact same factor structure used in Garleanu and Pedersen (2013). The $H$ -period returns in this model are are:

(3) $\begin{align*} r_{t+H} &= \beta \cdot (1 - \gamma)^{H-1} \cdot f_t + \beta \cdot {\textstyle \sum_{h=1}^{H-1}} (1 - \gamma)^{(H-1) - h} \cdot \xi_{t+h} + \epsilon_{t+H}. \end{align*}$

And, conditional on knowing the current level of the factor, $f_t$ , it’s possible to compute the conditional mean and variance of these $H$ -period-ahead returns, $r_{t+H} \mid f_t \sim \mathrm{N}(\mu_{H,t}, \, \sigma_H^2)$ :

(4) $\begin{align*} \mu_{H,t} &= \beta \cdot (1 - \gamma)^{H-1} \cdot f_t \\ \text{and} \quad \sigma_H^2 &= \sigma_{\epsilon}^2 + 1_{\{ H \geq 2 \}} \times \left\{ \, \beta^2 \cdot {\textstyle \sum_{h=1}^{H-1}} (1 - \gamma)^{2 \cdot \{(H-1) - h\}} \cdot \sigma_{\xi}^2 \, \right\}. \end{align*}$

Traders at time $t$ observe the current level of the factor, $f_t$ , and choose how many shares to buy over the course of the next $H$ periods in order to maximize their mean-variance utility. Let $\Delta_H[x_t] = \sum_{h=0}^{H-1} \Delta_1[x_{t-h}]$ denote the change in a trader’s portfolio position over the period from $t$ to $(t+H)$ , and let $\gamma$ denote traders’ risk-aversion parameter. Then, we can write the baseline utility function of a trader with no transaction costs as of time $t$ as:

(5) $\begin{align*} v_t(\Delta_H[x_t]) &= \mu_{H,t} \cdot \Delta_H[x_t] - {\textstyle \frac{\gamma}{2}} \cdot \sigma_H^2 \cdot \Delta_H[x_t]^2. \end{align*}$

The $t$ subscript comes from the fact that traders can observe the level of the factor at time $t$ , $f_t$ , prior to making their investment decision. If we look at the trader’s decision at a different time, $t$ , then he will have a different utility and choose a different portfolio because the level of the factor, $f_t$ , will be different.

3. Convex Transaction Costs

The key assumption is that, on top of this baseline utility function, traders also face convex transaction costs. i.e., when a trader with an $H$ -period horizon changes his position over $H$ the course of periods, he pays a transaction cost,

(6) $\begin{align*} \mathit{tc}(\Delta_H[x_t]) &= \min_{\{\Delta_1[x_{t-h}] \}_{h=0}^{H-1}} \, \left\{ \, {\textstyle \frac{\kappa}{2} \cdot \sum_{h=0}^{H-1}} \Delta_1[x_{t-h}]^2 \, \middle| \, \Delta_H[x_t] \, \right\}, \end{align*}$

where $\kappa > 0$ is a positive constant that captures the severity of these transaction costs. Thus, traders with an $H$ -period investment horizon maximize the following objective function:

(7) $\begin{align*} \max_{\Delta_H[x_t]} \, \left\{ \, v_t(\Delta_H[x_t]) - \mathit{tc}(\Delta_H[x_t]) \, \right\}. \end{align*}$

They choose the $H$ -period change in portfolio positions that maximizes their mean-variance utility and they implement this change in a way that minimizes their transactions costs.

The convex transaction costs imply that traders smooth out their trading across periods over their $H$ -period horizon. It’s easiest to see why this is the case when $H = 2$ , since $\Delta_1[x_{t+1}] = \Delta_2[x_t] - \Delta_1[x_t]$ . Suppose that traders know optimal final position $\Delta_2[x_t]$ . Then, traders’ optimization problem from Equation (7) becomes:

(8) $\begin{align*} \max_{\Delta_1[x_t]} \left\{ \, \mu_{2,t} \cdot \Delta_2[x_t] - {\textstyle \frac{\gamma}{2}} \cdot \sigma_H^2 \cdot \Delta_2[x_t] - \sfrac{\kappa}{2} \cdot \left\{ \, \Delta_1[x_t]^2 + (\Delta_2 [x_t] - \Delta_1 [x_t])^2 \, \right\} \, \right\}. \end{align*}$

Taking the first-order condition with respect to $\Delta_1[x_t]$ ,

(9) $\begin{align*} 0 &= - \kappa \cdot \left[ \, \Delta_1 [x_t] - (\Delta_2[x_t] - \Delta_1[x_t]) \, \right], \end{align*}$

then implies that $\Delta_1[x_t] = \sfrac{\Delta_2[x_t]}{2}$ . This simple exercise verifies that, when there are convex transaction costs, traders will want to split their orders up evenly across their investment horizon. In general, if traders have an $H$ -period horizon, then traders will choose $\Delta_1[x_{t+h}] = \sfrac{\Delta_H[x_t]}{H}$ . Traders with an $H$ -period investment horizon will have trading volume that is characterized by smooth $H$ -period long intervals, like the ones described in the figure below.

4. Fluctuations in Volume

If traders at horizon $H$ are characterized by trading that’s smoothed over $H$ periods, then we should be able to use inference tools like the wavelet-variance estimator to infer traders’ investment horizons. In a nutshell, this estimator computes the fraction of variation in a time series that comes from comparing successive $H$ -period long intervals. See Percival and Walden (2000) for more information.

I run a pair of numerical experiments to show that this intuition is correct (code). First, using a data-generating process with $\beta = 0.90$ , $\gamma = 0.75$ , and $\sigma_\epsilon = \sigma_\xi = 1$ , I simulate a long return time series. From Equation (7), it’s possible to compute the optimal portfolio position of trader with horizon $H$ ,

(10) $\begin{align*} x_{H,t} &= \frac{\mu_{H,t}}{\gamma \cdot \sigma_H^2 + \sfrac{\kappa}{H}}. \end{align*}$

For $5$ different horizons, $H^\star \in \{ \, 1, \, 4, \, 16, \, 64, \, 256 \, \}$ , I then simulate the trading-volume time series that would occur if all traders operated at horizon $H^\star$ . The figure below shows the fraction of the trading-volume variance occurring at each horizon according to the wavelet-variance estimator. Just as you’d expect, there is a spike the fraction of trading-volume variance at the true horizon, $H^\star$ … whatever that $H^\star$ happens to be. i.e., there is a spike in the dashed green line, which corresponds to the trading-volume data where traders have a horizon of $H^\star = 16$ , precisely at $\log_2(H) = 4$ .

In addition to this single-horizon experiment, I also run a numerical experiment with traders operating at $2$ different horizons. Specifically, using the same baseline parameters, I simulate a trading-volume time series where half of the volume comes from traders operating at the $H^\star = 4$ -period horizon and half of the volume comes from traders operating at the $H^\star = 64$ -period horizon. The figure below shows the fraction of the trading-volume variance occurring at each horizon according to the wavelet-variance estimator. Again, just as you’d expect, there’s is a spike the fraction of trading-volume variance at both the $H^\star = 4$ – and $H^\star = 64$ -period horizons.