Hong and Stein (1999)

1. Introduction

I replicate main results from Hong and Stein (1999) which constructs an equilibrium model with under-reaction and momentum. First, I give a rough verbal explanation of the model’s results. Then, I outline the basic mathematical framework and work through the equilibrium concept. Finally, I simulate the equilibrium outcomes for different momentum trader horizons and information speeds.

This paper develops an interesting model in which endogenizes the frequency and amplitude of price fluctuations. I work through this paper to better understand the nuts and bolts of this equilibrium concept. Perhaps I might be able to use these statistical wave-like properties to identify and discriminate between different mis-pricing generators.

2. Simple Example

The basic idea behind the model is as follows. Suppose that you have a bunch of traders that receive a demand shock, but only respond slowly. For example, imagine that a bunch of people earn a windfall payment (i.e. win the lottery or find out about a long lost rich uncle) and decide to buy new houses. It would take them a while to search for the appropriate house that fits their exact needs. For instance, perhaps one family needs to be in a nice school district, another needs to be near the airport for frequent trips, and so on… These guys represent slow moving information or demand. However, though it would take time for each of the people to purchase their new home, and anyone who knew about the windfall payments would know that the demand for expensive houses was going to jump up in the future.

Now, suppose that no one knows that the windfall payments have already occurred; but there is, instead, a group of traders that know a windfall payment might occur at anytime. It could have been today. It could have been yesterday. It might actually happen in a week. Yet, while this group doesn’t know when the payment has been made, each of the agents can infer the likelihood from the price movements. If the price drifts up, then it is more likely that helicopters have dumped the cash. A trader who acts on these price movements is a momentum trader.

There are 2 additional quirks: 1) The informed traders don’t realize that other people have also recieved a windfall payments. 2) Momentum traders enter sequentially and are very simple minded. They don’t know how many of their own kind there are. They don’t meet anyone for lunch to discuss what are the best ways to back out whether or not there has been a windfall payment. All they do is make their best guess based on the price growth over the past 6 months. That’s it.

What happens now? The first couple of momentum traders that walk into the market will see a price jump after the windfall actually occurs and trade into it. However, the next few momentum traders will see the price growth induced by the earlier momentum traders and get very excited and trade into the asset even further. These momentum traders are responding to a price movement that was solely generated by other momentum traders. This pattern repeats itself until the bottom falls out. So, it is as if the later momentum traders pay a tax for being late. Early price movements accelerate, then over shoot their fundamental value and collapse.

3. Economic Framework

First, let’s consider a world with only a unit mass of naive, but informed traders. Agents live in a discrete time $T$ period world where $T$ is large. There is a riskless asset with a $0$ return as well as a risky asset in positive net supply $q$ which pays out a dividend $d_T$ at time $T$ . The asset has an expected dividend $d_t$ and price $p_t$ at each interim period $t$ . These traders all start out with the same information and the same endowment of the riskless asset $m_0$ and risky asset $q_0$ . Each trader $i$ has CARA utility over consumption at time $T$ :

(1) $\begin{equation*} U(C^i) \ = \ - \mathbb{E} \left[ \ e^{-\alpha C^i} \ \right] \end{equation*}$

The traders are informed because they receive a series of signals $\varepsilon_t$ about the size of the dividend $d_T$ at each point in time $t$ . However, these signals move slowly throughout the population. Specifically, suppose that there are $z$ different flavors of traders of equal size. Each flavor of traders sees a different, independent component of the signal $\varepsilon_t$ at each point in time. So, for example, at time $t$ , traders of type $z=1$ see the component $\varepsilon^1_{t+z-1}$ of the shock $\varepsilon_{t+z-1}$ . At time $t+1$ , type $z=1$ traders see the second component of the shock $\varepsilon_{t+z-1}$ as well as the first component of the shock $\varepsilon_{t+z}$ . Likewise, at time $t$ , traders of type $z=2$ see the component $\varepsilon^2_{t+z-1}$ of the shock $\varepsilon_{t+z-1}$ as so on.

Thus, the traders of each flavor rotate which component of the shock they see until they have seen all $z$ independent components and know the full shock. This information rotation structure means that, after $\tau$ periods since time $t$ , agents have seen a fraction $\tau z$ of the total signals available for the shock $\varepsilon_{t+z-1}$ .

Information rotation structure in Hong and Stein (1999).

Traders are naive because they do not condition on the observed price when they formulate their expectations. Traders see their components of the shock at each point in time, update their beliefs about the future value of the dividend, and then place their order believing that they will adopt a buy and hold until date $T$ strategy, but they do not impound the market clearing price into their information set.

Equilibrium Concept: Walrasian equilibrium with private valuations.

4. No Momentum Traders

To solve the model, I follow the same general strategy as in a Grossman and Stiglitz (1983) equilibrium, but I give the traders naive rather than rational expectations. First, I write out the optimization problem for each naive, informed trader $i$ . I assume that $z=2$ and solve a $1$ period problem, but the solutions below easily generalize to $z>2$ and multiple periods.

Each trader maximizes his consumption utility by choosing his asset holdings subject to a budget constraint where $m^i$ represents his riskless asset holdings and $q^i$ represents his risky asset holdings:

(2) $\begin{align*} V^i &= \max_{m^iq^i} \left\{ -\mathbb{E} \left[ e^{-\rho w^i} \right] \right\} \\ &\textit{subject to} \\ m^i + \tilde{p} q^i &\leq m_0 + \tilde{p}_0 q_0 \end{align*}$

I assume that each trader has a unit mass of wealth. After substituting in the budget constraint, I get:

(3) $\begin{align*} V^i &= \max_{q^i} \left\{ -\mathbb{E}^i \left[ e^{-\rho \left( (d-\tilde{p}) q^i \right)} \right] \right\} \end{align*}$

The first order condition with respect to $q^i$ characterizes the risky holdings as follows:

(4) $\begin{align*} q^i &= \frac{\mathbb{E}^i\left[ d-\tilde{p} \right]}{\rho\mathbb{V}^i\left[ d-\tilde{p} \right]} \end{align*}$

Next, I guess that price is linear in the public information, the private signal about tomorrows information and the total quantity. $\varepsilon_0$ denotes the public signal available to traders of both flavors. $\varepsilon_1$ denotes the sum of the private signals for each type of trader .

(5) $\begin{align*} \tilde{p} &= \alpha + \beta \varepsilon_0 + \gamma \varepsilon_1 - \delta Q \end{align*}$

I solve for $\tilde{p}$ by substituting the function for $q^i$ into the budget constraint. Since both flavors of agents are symmetric and ignorant of the information in the prices themselves, the price functional simplifies to:

(6) $\begin{align*} \tilde{p} &= \left( \varepsilon_0 + \dfrac{1}{2} \varepsilon_1 \right) - \delta Q \end{align*}$

$\delta$ is a function of the risk aversion parameter $\rho$ as well as the variance $\sigma_\varepsilon^2$ . I pick the risk aversion parameter in order to set $\delta=1$ for simplicity.

Price, dividend, return and signal series from Hong and Stein (1999) with no momentum traders.

5. With Momentum Traders

Now, I add in momentum traders. In order to do this, I allow the naive, informed traders to believe that the risky asset supply is a random variable. Thus, they remain blissfully unaware that there are momentum traders at all. Momentum traders also have CARA utility but, rather than living until date $T$ , these traders have shorter term horizons and die out at date $t+j$ if they enter at date $t$ . Momentum traders earn their name because, rather than observing the sequence of dividend shocks like the naive informed traders above, momentum traders update their beliefs solely on past price movements: $\Delta_k p_{t-1}=p_{t-1} - p_{t-k-1}$ .

For simplicity, I pick $k=1$ below. I conjecture that moment traders demand is a linear function of past price growth:

(7) $\begin{align*} f_t &= \theta + \phi \Delta p_{t-1} \end{align*}$

I denote price in the momentum regime as $p_t$ rather than $\tilde{p}_t$ . Informed traders solve the exact same problem as before, since they see the supply shock as a random variable rather than an informative signal. Now, momentum traders affect the quantity available. I can rewrite the pricing equation from above as:

(8) $\begin{align*} \begin{split} p_t &= \dfrac{1}{z}\left[z\varepsilon_t+(z-1)\varepsilon_{t+1}+ (z-2)\varepsilon_{t+2}+...+\varepsilon_{t + z-1}\right] \\ &\qquad \qquad - \left(Q-\left[\theta j + \phi \sum_{i=1}^j \Delta p_{t-i} \right] \right) \end{split} \end{align*}$

So that the price today reflects both the current knowledge of all of the naive informed traders as well as the myopic response of the momentum traders where the summation over $j$ comes into play since their are $j$ generations of momentum traders in the market at any given time. As is standard in models with CARA agents, the momentum traders choose $\phi$ according to the rule below where $\rho$ represents the moment trader’s risk aversion parameter:

(9) $\begin{align*} \phi &= \rho \left( \frac{\mathbb{C}\left[ \Delta_j p_{t+j}, \Delta p_{t-1} \right]}{\mathbb{V}\left[\Delta p_{t-1}\right] \mathbb{V}\left[ \Delta_j p_{t+j} \right]} \right) \end{align*}$

An equilibrium is a price $p$ and a quantity demanded by the momentum traders $\phi$ such that both the pricing and mean variance equations above are satisfied. I solve for the equilibrium numerically in R.

Price, dividend, return and signal series from Hong and Stein (1999) with momentum traders at the 20 period horizon.

6. Code

Click HERE to view the code used to create these plots.