Sacrificing Noise Traders

1. Introduction

One way to look at the stock market is as an information aggregation technology. For instance, imagine that you are the CEO of a pencil making company, and have to decide whether or not to stick with making old-fashioned wood pencils or to switch over to making mechanical pencils. If equity shares in lumber companies are publicly traded, you can pop open your laptop and look at their valuation online. Suppose you see that all lumber companies have low valuations and few other customers. In this world, you should really consider updating your product line. Note that it would be much harder to make this inference if all lumber company equity was privately held. No private equity shop is going to answer the phone and tell you that one of their investments is in the toilet. What’s more, (1) more analysts and (2) better informed analysts will study each lumber company’s business operations if there is publicly traded equity and no one has to know who these better informed analysts are ahead of time either. As I think Kevin Costner once said: “If there’s profits, they will come.”

The big question, though, is: Where do these profits come from? What entices informed traders to enter the market and push their information into prices? Asset pricing models such as Grossman and Stiglitz (1980) and Kyle (1985) give us the answer. Informed traders’ profits come directly from the stupidity of noise traders. These profits are transfer payments from the pockets of one group of citizens to the pockets of another. For a social planner, having prices that tell people about the fundamental values of important companies is a good thing. However, noise traders are people too, and sacrificing too many of them to get accurate prices is bad.

In this post, I use a simple, one period, Kyle (1985)-type model to ask the question: How many noise traders do you need to throw to the dogs in order to get accurate prices? Specifically, I think about a world with an asset that pays out v \overset{\scriptscriptstyle \mathrm{iid}}{\sim} \mathrm{N}(0,\sigma_v^2) and has price p. If you are the social planner, you then try to maximize the benefits of having informative prices, \mathrm{Cov}[p,v], minus the costs of wasting lots of noise traders who could be doing other productive things, c(\text{noise traders}), subject to the constraint that it has to be worth it for informed traders to enter into the market, \mathrm{E}[\text{informed trader profit}] \geq \bar{\pi}:

(1)   \begin{align*} \max_{\text{noise traders} \geq 0} \left\{ \mathrm{Cov}[p,v] - c(\text{noise traders}) \right\} \quad &\text{subject to} \quad \mathrm{E}[\text{informed trader profit}] \geq \bar{\pi} \end{align*}

The crazy thing about setting the problem up this way is that the number of noise traders in the market doesn’t affect how informative the prices are:

(2)   \begin{align*} \mathrm{Cov}[p,v] &= \text{constant} \times \sigma_v^2 \end{align*}

Put differently, as the social planner, you need to sacrifice enough noise traders so that informed traders actually like being traders and won’t change careers. If there aren’t enough noise traders, informed traders won’t make their reservation wage, \bar{\pi}, and will switch over to being butchers or bakers. However, pumping more and more noise traders into the market won’t make prices any more informative.

2. Economic Model

How does the model work? Imagine that Alice decides to be an informed trader rather than a butcher. For all of her time studying the markets, she get rewarded with a signal, s, about the fundamental value of a lumber company, Logs Inc:

(3)   \begin{align*} s &= v + \epsilon \quad \text{where} \quad \epsilon \overset{\scriptscriptstyle \mathrm{iid}}{\sim} \mathrm{N}(0,\sigma_\epsilon^2) \end{align*}

Since everything in the model is nice and normally distributed, her posterior beliefs about the fundamental value of Logs Inc will be normally distributed with variance \mathrm{Var}[v|s] = (1/\sigma_v^2 + 1/\sigma_\epsilon^2)^{-1} and mean \mathrm{E}[v|s] = \mathrm{SNR} \cdot s where \mathrm{SNR} denotes Alice’s signal to noise ratio:

(4)   \begin{align*} \mathrm{SNR} &= \frac{\sigma_v^2}{\sigma_v^2 + \sigma_\epsilon^2} \end{align*}

For instance, if Alice just saw the fundamental value, v, directly then \sigma_\epsilon = 0 and here signal to noise ratio would be \mathrm{SNR} = 1. Conversely, as \sigma_\epsilon \to \infty her signal becomes meaningless and her signal to noise ratio tends to \mathrm{SNR} \to 0.

There is a competitive market maker for Logs Inc stock, Bob, who observes aggregate demand, y, and sets the price equal to his conditional expectation of its fundamental value:

(5)   \begin{align*} y &= x + z \quad \text{where} \quad z \overset{\scriptscriptstyle \mathrm{iid}}{\sim} \mathrm{N}(0,\sigma_z^2) \\ p &= \mathrm{E}[v|y] \end{align*}

Here x denotes Alice’s informed demand for Logs Inc stock in units of shares and z denotes noise trader demand for Logs Inc stock. When I say that Bob only observes aggregate demand, I mean that if Bob sees a buy order of 10 shares, he has no idea whether (a) Alice wants to buy 20 shares and the noise traders want to sell 10 shares or (b) Alice wants to sell 10 shares and the noise traders want to buy 20 shares. The assumption of perfect competition for Bob means that he has to set the price equal to his conditional expectation. If he tried to hedge his bets and deviate from p = \mathrm{E}[v|y], someone else would step in and scoop his business.

If Alice is trying to maximize her expected profit, \pi_x:

(6)   \begin{align*} \pi_x &= (v - p) \cdot x \end{align*}

then an equilibrium would be a choice of demand for Alice, x = \beta \cdot s, and a pricing rule for Bob, p = \lambda \cdot y, such that:

  1. Given Bob’s pricing rule, p = \lambda \cdot y, Alices demand maximizes her expected profit.
  2. Given Alice’s demand rule, x = \beta \cdot s, Bob’s price equals his conditional expectation of Logs Inc’s fundamental value.

Taking Bob’s pricing rule as given and maximizing Alice’s expected profit, \mathrm{E}[\pi_x|v], yields an equation for her optimal demand given her realized signal, s:

(7)   \begin{align*} x &= \left( \frac{\mathrm{SNR}}{2 \cdot \lambda} \right) \cdot s \end{align*}

Substituting this demand rule into Bob’s conditional expectation then characterizes the equilibrium parameters \beta and \lambda that govern Alice and Bob’s demand and pricing rules:

(8)   \begin{align*} \lambda = \frac{\mathrm{Cov}[v,y]}{\mathrm{Var}[y]} &= \frac{\sqrt{\mathrm{SNR}}}{2} \cdot \frac{\sigma_v}{\sigma_z} \\ \beta = \frac{\mathrm{SNR}}{2 \cdot \lambda} &= \sqrt{\mathrm{SNR}} \cdot \frac{\sigma_z}{\sigma_v} \end{align*}

In words, \lambda \propto \sigma_v/\sigma_z means that the price of Logs Inc stock will be more responsive to aggregate demand shocks when there is more information to be revealed or when there are few noise traders to mask the information. Conversely, Alice’s demand will be more responsive to a strong private signal when there is more noise trading for her to hid behind or when she wasn’t expecting to discover much in the first place.

3. Unconditional Moments

With the model in place, we can now get to the interesting part of the analysis. Namely, the number of noise traders in the market doesn’t affect how informative prices are. To see this, note that prices have the following functional form:

(9)   \begin{align*} p &= \frac{\mathrm{SNR}}{2} \cdot (v+\epsilon) + \frac{\sqrt{\mathrm{SNR}}}{2} \cdot \frac{\sigma_v}{\sigma_z} \cdot z \end{align*}

The key piece of this equation is that the coefficient on the fundamental value of Logs Inc, \mathrm{SNR}/2, doesn’t have any dependence on the number of noise traders in the market. i.e., if the fundamental value of Logs Inc goes up by \mathdollar 1, then the price of Logs Inc will go up by \mathdollar \mathrm{SNR}/2 on average, and this relationship will be true whether the volatility of noise trader demand is 1{\scriptstyle \mathrm{mil}} shares per period or 1 share per period. More precisely, we have that the covariance of Logs Inc’s price with its fundamental value is:

(10)   \begin{align*} \mathrm{Cov}[p,v] &= \frac{\mathrm{SNR}}{2} \cdot \sigma_v^2 \end{align*}

No matter how much noise trader demand there is, the price is always equally informative about the fundamental value. This is a very strong prediction!

4. Planner’s Problem

OK. So, we’ve written down a really simple model, and this model says that the number of noise traders doesn’t impact how informative prices are about asset fundamentals. What does this say about the original question? How should you, the social planner, decide the number of noise traders to sacrifice so that everyone in the economy can use the resulting price signals?

Well, the first thing to see is that all of Alice’s profits from being an informed trader rather than a butcher come at the expense of noise traders:

(11)   \begin{align*} \mathrm{E}[\pi_x] &= - \mathrm{E}[\pi_z] = \frac{\sqrt{\mathrm{SNR}}}{2} \cdot \sigma_z \cdot \sigma_v \end{align*}

Essentially, the rest of the economy is subsidizing the financial market by an amount \sqrt{\mathrm{SNR}} \cdot \sigma_z \cdot \sigma_v/2. It might be worth it if it’s really helpful for everyone in the economy to see an accurate valuation of Logs Inc. There are lots of transfer payments which people are happy to make (e.g., welfare, social security, etc…), but the key observation is that it’s a transfer payment. What’s more, since adding more noise traders doesn’t affect price informativeness, you are going to want to sacrifice the the minimum number of noise trader required to make sure that Alice is a trader and not a butcher:

(12)   \begin{align*} \frac{\sqrt{\mathrm{SNR}}}{2} \cdot \sigma_z \cdot \sigma_v &\geq \bar{\pi} \end{align*}

To get an answer in numbers of noise traders, suppose that each noise trader that you anoint contributes demand variance, \hbar, so that total noise trader demand variance is given by \sigma_z^2 = N \cdot \hbar.

In this world, you need to sacrifice N noise traders to make sure Alice becomes an informed trader:

(13)   \begin{align*} N &= 4 \cdot \frac{\bar{\pi}^2}{\hbar} \cdot \left( \frac{\sigma_v^2 + \sigma_\epsilon^2}{\sigma_v^4} \right) \end{align*}

This equation says that you need to sacrifice (a) more noise traders when Alice can make more money being a butcher, (b) fewer noise traders when each of them is willing to trade more wildly, (c) fewer noise traders when there is more information about Logs Inc to be discovered, and (d) more noise traders when Alice’s signal about Logs Inc is more noisy.

5. Conclusion

Stock prices are useful signals that we pay for with noise trader demand. This post then used a Kyle (1985)-type model to answer a simple question: As a social planner, how many noise traders should you sacrifice? The interesting fact that pops out of this model is that noise trader demand volatility doesn’t affect price informativeness. It only affects informed trader profits. So as a social planner, you want to have just enough noise trader demand volatility in the market to get Alice to figure out the value of Logs Inc.

A natural question to conclude with is: How general is this result? Surely there are times when noise trader demand shocks affect price informativeness in the real world. In the model, these 2 aspects of the economy are completely divorced due to the fact that the equilibrium price impact coefficient, \lambda, and the equilibrium demand coefficient, \beta, in some sense undo one another:

(14)   \begin{align*} \lambda \times \beta &= \text{constant} \end{align*}

Put differently, and increase in noise trader demand will make Alice trade more aggressively since it will be harder for Bob to figure out whether or not changes in aggregate demand are due to Alice or the noise traders. However, as a result, Bob will respond by moving the price around less in response to equal sized changes in aggregate demand.

To see how delicate this canceling out actually is, imagine the Bob has beliefs about the volatility of the underlying asset that are off by (100 \times \eta)\%. e.g., if \eta = 0.05 then he would believe that fundamental volatility was \mathdollar 1.05 when it was in fact \mathdollar 1.00. When Alice gets a really strong signal abou the fundamental value, \mathrm{SNR} = 1, this canceling out seems to be quite robust:

(15)   \begin{align*} \mathrm{Cov}[p,v] &= \frac{1}{2} \cdot \sigma_v^2 \cdot \left\{ 1 + \eta + \eta^2 + \cdots \right\} \end{align*}

Small errors in Bob’s beliefs decay pretty quickly. However, when \mathrm{SNR} \searrow 0, problems can occur and the delicate balance between \lambda and \beta can break down:

(16)   \begin{align*} \mathrm{Cov}[p,v] &= \frac{\mathrm{SNR}}{2} \cdot \sigma_v^2 \cdot \left\{ 1 + \eta \cdot (2 - \mathrm{SNR}) + \cdots \right\} \end{align*}

In percentage terms, small errors in Bob’s beliefs could really add up in situations where the \mathrm{SNR} is low. Since \mathrm{SNR} \leq 1, the factor multiplying \eta is greater than unity. Thus, when Alice gets really weak signals about the fundamental value of Logs Inc, minor errors in Bob’s understanding of the market can lead to wildly incorrect pricing.