Spontaneous Cognition Equilibrium

1. Motivation

This note develops an information-based asset pricing model based on Tirole (2009) where thinking through market contingencies is costly and fear of missing an important detail restrains trading behavior. For example, think about a statistical arbitrageur who decided not to release the throttle on an otherwise profitable trading strategy because she noticed that it had an unexplained industry \beta. Alternatively, consider a value investor who didn’t fully invest in a seemingly undervalued conglomerate because of his unfamiliarity with every single one of its business lines. In both of these examples, traders weren’t necessarily afraid of releasing too much information to the market while building their position; instead, they were afraid of taking on a large position and then being held-up by “Mr. Market” after the fact.

To see how the model works, imagine you’re a market neutral statistical arbitrageur who usually puts together a position of type \mathbf{a} to exploit the momentum anomaly at the monthly horizon. This position has a price of p dollars, usually generates a payout of v > 0 dollars, and costs Mr. Market c_{\mathrm{Transact}} > 0 dollars to put together. You and Mr. Market split the gains to trade with you getting a fraction, \theta_{\mathrm{Trader}}, and Mr. Market getting a fraction, \theta_{\mathrm{Market}}, so that:

(1)   \begin{align*}   1 &= \theta_{\mathrm{Trader}} + \theta_{\mathrm{Market}} \end{align*}

In a more general model the distribution of these bargaining positions would be an equilibrium object, but for now I take them as given.

Portfolio position \mathbf{a} is the best position you could put together using the available information, but you realize that something may well go wrong. The problem is that you simply can’t write out every single possible contingency. There are just too many. With probability \rho > 0, your boiler-plate portfolio position \mathbf{a} will only deliver a value of (v - \delta) dollars where \delta > 0. In this situation, you actually need to put together the position \mathbf{a}' to exploit momentum while staying market neutral. Portfolio \mathbf{a}' is different from \mathbf{a} but impossible to specify ahead of time. Rebalancing your position ex post will cost c_{\mathrm{Rebal}} > 0 dollars. For example, during the Quant Crisis of August 2007 the market suddenly and unexpectedly went sideways for quantitative traders in long-short equity positions. According to Khandani and Lo (2007), some of the most consistently profitable quant funds in the history of the industry reported “month-to-date losses ranging from 5{\scriptstyle \%} to 30{\scriptstyle \%}” of assets under management.

Before trading, you can exert cognitive effort to find out about what may go wrong and how to put together your portfolio accordingly. I assume that once you start trading the position \mathbf{a}' rather than \mathbf{a}, Mr. Market immediately knows that \mathbf{a}' rather than \mathbf{a} is optimal. Put differently, changing your usual behavior is an eye-opener. The entire reason for search for the correct portfolio position is to avoid the ex post rebalancing costs. I denote this cognitive effort by:

(2)   \begin{align*}   \mathrm{Eff}(\pi) &\geq 0 \end{align*}

where \pi denotes the probability that you discover the correct portfolio position \mathbf{a}' conditional on \mathbf{a} not being the right position. Thus, in contrast to the Veldkamp 2006 model, here you invest cognitive effort until your marginal thinking costs equal the change in your expected ex post hold up costs.

Some words...

Model timing.

2. Agents and Assets

Suppose that you are thinking of putting together a portfolio position denoted by the (N \times 1)-dimensional vector, \mathbf{a}. To do this, you have to buy and sell stocks from Mr. Market subject to a fixed transaction cost of c_{\mathrm{Transact}} > 0. For example, imagine you’re a market neutral statistical arbitrageur who usually puts together a position of type \mathbf{a} to exploit the momentum anomaly. Initially, you believe that the price of this portfolio position is p dollars:

(3)   \begin{align*}   \widetilde{\mathrm{E}}[\mathbf{x}^{\top} \mathbf{a}] &= p \end{align*}

where \mathbf{x} denotes the (N \times 1)-dimensional vector of random payouts of the N assets in the market. For example, in the classic Jegadeesh and Titman (1993) setting, you would repurchase 1/6th of your portfolio holdings each month at a total price of p dollars when using a momentum holding period of 6 months.

However, with probability \rho \in (0,1) your initial ideas about how the market will play out turn out to be wrong, and the portfolio position \mathbf{a} won’t deliver the required payouts. Instead, the position will only be worth (v - \delta) dollars where \delta > 0. For example, perhaps lots of other statistical arbitrageurs are also putting on a similar position. In such a world, your seemingly well-hedged position would be anything but. You could easily lose a large chunk of your assets under management if you didn’t quickly rebalance your portfolio in the event of sudden fire sales as documented in Khandani and Lo (2007). There is a different portfolio \mathbf{a}' that delivers your desired payout, but after you put on the initial position \mathbf{a} it will take an adjustment cost of c_{\mathrm{Rebal}} > 0 dollars to switch over to the portfolio \mathbf{a}'. I assume that it is worth it for you to enter into the market even if you know that you will never discover the correct portfolio position ahead of time:

(4)   \begin{align*}   0 &< (v - \rho \cdot c_{\mathrm{Rebal}}) - c_{\mathrm{Transact}} \end{align*}

3. Information Structure

You and Mr. Market can agree to transact the portfolio position \mathbf{a}. This portfolio position may or may not be what suits your needs as the buyer. If it doesn’t, an initially unknown portfolio position \mathbf{a}' will deliver the desired payouts provided that you can return to the market and rebalance your position. At the initial stage, though, both you and Mr. Market are aware only of \mathbf{a}, although you both know that it may not be the right one. You both may, before contracting, incur a cognitive cost to think about alternatives to \mathbf{a}, and whomever finds out that portfolio position \mathbf{a}' is the right one can decide whether to immediately trade on this information or not. The key idea here is that the discovery of the correct position is an “eye-opener”.

I assume that you have bargaining power \theta_{\mathrm{Trader}} and Mr. Market has bargaining power \theta_{\mathrm{Market}} where \theta_{\mathrm{Trader}} + \theta_{\mathrm{Market}} = 1. For example, if you are the only trader in the market trying to sell the stocks required by portfolio \mathbf{a} and there are lots of other traders lining up to buy them, then your bargaining power, \theta_{\mathrm{Trader}}, will be close to 1. Conversely, if you are one of many traders trying to short a hard to locate stock, then your bargaining power, \theta_{\mathrm{Trader}}, will be close to 0.

Before trading, you can incur thinking costs of \mathrm{Eff}(\pi). Here, \pi denotes the probability that you will discover the correct portfolio position \mathbf{a}' conditional on \mathbf{a} not being the right one. For example, if you wanted to know when lots of other statistical arbitrageurs were also putting on a similar position and thus destroying your market neutrality 50{\scriptstyle \%} of the time, then \pi = 0.50 and you would be \mathrm{Eff}(0.50) dollars in cognitive costs to maintain this level of informativeness. I assume both that:

(5)   \begin{align*}   \mathrm{Eff}(0) &= 0 \quad \text{and} \quad \mathrm{Eff}(1) = \infty \end{align*}

as well as that:

(6)   \begin{align*}   \frac{d^2\mathrm{Eff}}{(d\pi)^2} &> \frac{\rho^2 \cdot (1 - \rho) \cdot \theta_{\mathrm{Market}} \cdot \delta}{(1 - \rho \cdot \pi)^2} \end{align*}

The first assumption reads that learning nothing costs you nothing while it is prohibitively expensive to always know when you need to deviate from the standard position. The second assumption guarantees a unique solution to the cognitive effort optimization problem described in Equation (14) below.

4. Asset Pricing

In this world, how much effort should you expend trying to figure out if \mathbf{a} is the correct portfolio position? At what point is it worth it to just trade and then clean up any mess after the fact? Well, first let’s consider the case where you don’t discover the correct portfolio position ahead of time. Conditional on not finding an alternative portfolio, \mathbf{a}', the posterior probability that \mathbf{a} is not correct conditional on searching with intensity \pi is given by:

(7)   \begin{align*}   \hat{\rho}(\pi) &= \frac{\rho \cdot (1 - \pi)}{1 - \rho \cdot \pi} \end{align*}

The numerator is probability that \mathbf{a}' is the correct portfolio, \rho, times the probability that you didn’t discover this fact during your search, (1 - \pi). The denominator is the probability that you didn’t find an alternative portfolio, 1 - \rho \cdot pi. If \mathbf{a}' is the appropriate portfolio then Mr. Market captures a fraction \theta_{\mathrm{Market}} of the renegotiation gain creating a hold-up:

(8)   \begin{align*}   h &= \theta_{\mathrm{Market}} \cdot (\delta - c_{\mathrm{Rebal}}) \end{align*}

where (\delta - c_{\mathrm{Rebal}}) dollars is the surplus available to be split after realizing ex post that \mathbf{a}' is the correct portfolio. Let \pi^* denote your equilibrium level of search. The ex ante price p(\pi^*) for portfolio \mathbf{a} accounts for the possible hold-up so that:

(9)   \begin{align*}   p(\pi^*) - \left\{ c_{\mathrm{Transact}} - \hat{\rho}(\pi^*) \cdot h \right\} &= \theta_{\mathrm{Market}} \cdot \left\{ (v - c_{\mathrm{Transact}}) - \hat{\rho}(\pi^*) \cdot c_{\mathrm{Rebal}} \right\} \end{align*}

This equation reads that the price you are willing to pay for the portfolio \mathbf{a} given your search equilibrium intensity \pi^* minus the transaction cost and expected hold up costs must equal Mr. Markets share of the gains to trade. Rewriting this equation to isolate the price function yields:

(10)   \begin{align*}   p(\pi^*) &= c_{\mathrm{Transact}} + \theta_{\mathrm{Market}} \cdot \left\{ (v - c_{\mathrm{Transact}}) - \hat{\rho}(\pi^*) \cdot \delta \right\} \end{align*}

Next, let’s consider the case where you find out that \mathbf{a}' is the correct portfolio after expending some cognitive effort. You now how 2 choices:

  1. You can trade portfolio \mathbf{a}' and disclose its correctness to Mr. Market.
  2. You can still trade portfolio \mathbf{a} and then rebalance your position ex post.

By disclosing \mathbf{a}', you will realize a fraction \theta_{\mathrm{Trader}} of the gains to trade, \theta_{\mathrm{Trader}} \cdot (v - c_{\mathrm{Transact}}). If you conceal \mathbf{a}' and continue to trade position \mathbf{a} anyways, you will get v - (c_{\mathrm{Rebal}} + h) - p(\pi^*) where the middle term comes from the fact that you know you will have to rebalance your portfolio and the price is given by Equation (10) above. Combining these two expressions and simplifying then says that revealing the correct portfolio position yields an efficiency gain of:

(11)   \begin{align*}   \Delta U_{\mathrm{Trader}} &= \theta_{\mathrm{Trader}} \cdot c_{\mathrm{Rebal}} + \left\{ 1 - \hat{\rho}(\pi^*) \right\} \cdot \theta_{\mathrm{Market}} \cdot \delta > 0 \end{align*}

The first term in this equation says that you avoid paying your share of the rebalancing costs. The second term in this equation says that you capture Mr. Market’s expected share of the gains to rebalancing. After all, if you start trading portfolio \mathbf{a}' immediately, then the price no longer has to account for the fact that Mr. Market might hold you up later for his share of the gains to rebalancing, \delta. The important thing about Equation (11) is that it’s always positive. Thus, you always want to trade on \mathbf{a}' when you find out that this is the right portfolio.

5. Cognitive Effort

Note that even before getting into any mathematical details, it’s clear that a social planner would ask you to look for new contingencies until the marginal cost of looking for the next important detail would just offset of the expected rebalancing costs, \pi^{**}:

(12)   \begin{align*}   \left. \frac{d \mathrm{Eff}}{d\pi} \right|_{\pi^{**}} &= \rho \cdot c_{\mathrm{Rebal}} \end{align*}

Moreover, in the absence of any rebalancing costs, c_{\mathrm{Rebal}} = 0, any investment in cognition is purely rent-seeking!

So what happens in the model? You choose your level of cognitive effort by solving the optimization problem:

(13)   \begin{align*}   U_{\mathrm{Trader}} &= \max_{\pi \in [0,1]} \Big\{  \rho \cdot \pi \cdot \theta_{\mathrm{Trader}} \cdot (v - c_{\mathrm{Transact}})     \\   &\qquad \qquad \qquad + \ \rho \cdot (1 - \pi) \cdot \left\{ v - (c_{\mathrm{Rebal}} + h) - p(\pi^*)\right\}   \\   &\qquad \qquad \qquad \qquad + \ (1 - \rho) \cdot \left\{ v - p(\pi^*) \right\}   \\   &\qquad \qquad \qquad \qquad \qquad - \ \mathrm{Eff}(\pi) \Big\} \end{align*}

What are the terms in this equation? Well, there are 3 possible outcomes: \mathbf{a}' could be the right portfolio and you could discover it right away, \mathbf{a}' could be the right portfolio and you might not discover it until it’s too late, and \mathbf{a} could be the right portfolio all along. The first 3 terms represent your payouts in each of these states weighted by the probabilities that they occur. The final term is just your cognitive costs.

Differentiating with respect to your cognitive effort level, \pi, and observing that in equilibrium is has to be that \pi = \pi^* yields:

(14)   \begin{align*}   \left.\frac{d\mathrm{Eff}}{d\pi} \right|_{\pi^*} &= \rho \cdot c_{\mathrm{Rebal}} + \rho \cdot h - \rho \cdot \left( \frac{\rho \cdot (1 - \pi^*)}{1 - \rho \cdot \pi^*} \right) \cdot \theta_{\mathrm{Market}} \cdot \delta \end{align*}

This equation is pretty interesting. It says that you will search until your marginal cost of mulling over your portfolio equals your expected rebalancing costs plus a pair of additional terms. The first extra term says that you will increase your cognitive efforts in order to avoid being held up by Mr. Market after the fact. The second additional term says that you won’t fully account for this hold up problem since you can adjust the price ex ante. The sum of these additional terms will always be greater than 0; thus, you will always expend too much cognitive effort. Put differently, this model is populated by Woody Allen traders who neurotically search for negative contingencies.

6. Discussion

This model delivers a couple of interesting implications. First, spontaneous cognition is a natural source of noise in financial markets. This is an attractive feature since noise traders are akin to theoretical dark matter. Without them, informed traders would be unable to exploit their informational advantage in existing noisy rational expectations models. Nevertheless, these traders are inherently difficult to identify in the data and make welfare analysis problematic. How should the social planner weight noise trader utility?

Second, traders spend too much time trying to identify the perfect portfolio position. If you are a statistical arbitrageur, you obviously can’t sit on the sidelines until you have a sure fire strategy. You have to trade even when you are not completely certain about your strategy’s payouts. No one will pay you fees to sit on their money. Interestingly, I find that traders might well choose to expend too much cognitive effort looking for holes in your strategy in order not to be held-up by Mr. Market. I have never seen another model predict this.

Third and finally, traders specialize in identifying bad news for themselves. A trader has little motivation to look for confirming evidence that position \mathbf{a} is correct. After all, this would be his portfolio position if he exerted no effort whatsoever and spent the morning in sweatpants on his couch. In fact, the adverse selection problem can be severe enough that traders will not go through with the trade unless they find something wrong with the boiler-plate portfolio position.