Notes on Information Aversion

1. Motivation

In spite of how they are modeled in Merton (1971), traders don’t pay attention to their portfolio every second of every day. What’s more, this lumpy rebalancing behavior has important asset-pricing implications. If traders aren’t continuously adjusting their portfolio, then they have to bear both payout risk and allocation-error risk, which lowers the amount they are willing to pay for risky assets in equilibrium. Researchers have modeled this portfolio inattention in a variety of ways. For example, Sims (2003) points out that it’s hard to process that much information. Alternatively, Abel, Eberly, and Panageas (2013) study agents who have to pay a transactions cost every time they checking in on their portfolio.

This post discusses Andries and Haddad (2015) which proposes a different reason for portfolio inattention: information aversion. If traders subscribe to prospect theory à la Kahneman and Tversky (1979)—that is, if they value payouts relative to a reference point and are loss averse—then traders will prefer to check in on their portfolio less often to save themselves from the possibility of experiencing painful temporary losses.

2. Prospect Theory

When computing the certainty-equivalent value of a lottery, \mathrm{CE}(x), a trader adhering to prospect theory values the payouts relative to a reference point and places extra weight on bad outcomes. Gul (1991) gives axiomatic formulation of this idea by defining the reference point at the lottery’s certainty-equivalent value,

(1)   \begin{align*} \mathrm{CE}(x) &= \frac{1}{\mathrm{Z}} \cdot \left( \, \int_{\{ x : \mathrm{CE}(x) \leq x \}} x \cdot d\mathrm{F}(x) + (1 + \theta) \cdot \int_{\{ x : \mathrm{CE}(x) > x \}} x \cdot d\mathrm{F}(x) \, \right), \end{align*}

where \mathrm{Z} = 1 + \theta \cdot \int_{\{ x : \mathrm{CE}(x) > x\}} d\mathrm{F}(x).

Let’s plug in some numbers to get a better sense of what this definition means. Let’s think about a lottery with a pair of equally likely outcomes, \mathrm{Pr}(x_1 = 2) = \sfrac{1}{2} and \mathrm{Pr}(x_1 = 0) = \sfrac{1}{2}, as pictured below: plot--static-info-aversion--23jun2015 A trader with disappointment-aversion parameter \theta = \sfrac{1}{5} would then assign this lottery a certainty-equivalent value:

(2)   \begin{align*} \mathrm{CE}(x_1) &= \frac{2 \cdot \sfrac{1}{2} + (1 + \sfrac{1}{5}) \cdot 0 \cdot \sfrac{1}{2}}{1 + \sfrac{1}{5} \cdot \sfrac{1}{2}} = \frac{10}{11}. \end{align*}

Because the trader places extra weight on the bad outcome of x_1 = 0, his certainty-equivalent value for the lottery is less than the expected value of the lottery, \mathrm{CE}(x_1) = \sfrac{10}{11} < 1 = \mathrm{E}(x_1).

3. Dynamic Reformulation

If we want to think about the optimal time between portfolio rebalancing decisions, then we need a dynamic version of prospect theory. Andries and Haddad (2015) use the recursive dynamic extension below,

(3)   \begin{align*} \mathrm{CE}_{t-1}(\mathrm{CE}_t(x_{t+1}))  &=  \frac{1}{\mathrm{Z}_{t-1}}  \cdot  \left( \, \int_{\{x_t:\mathrm{CE}_{t-1}(x_t) \leq \mathrm{CE}_t(x_{t+1})\}} \mathrm{CE}_t(x_{t+1}) \cdot d\mathrm{F}_{t-1}(x_t)  \right. \\ &\qquad \quad \left. + \,  (1 + \theta) \cdot \int_{\{x_t:\mathrm{CE}_{t-1}(x_t) > \mathrm{CE}_t(x_{t+1})\}} \mathrm{CE}_t(x_{t+1}) \cdot d\mathrm{F}_{t-1}(x_t) \, \right), \end{align*}

where \mathrm{Z}_{t-1} = 1 + \theta \cdot \int_{\{x_t:\mathrm{CE}_{t-1}(x_t) > \mathrm{CE}_t(x_{t+1})\}} d\mathrm{F}_{t-1}(x_t).

Again, to get a better sense of what this definition means, let’s plug in some numbers. Specifically, let’s look at a 2-period version of the binomial model above where every period the payout is equally likely to either decrease or increase by 1. plot--dynamic-info-aversion--23jun2015 Starting at time 1, the certainty-equivalent value of the payout at time t is either

(4)   \begin{align*} \mathrm{CE}(x_2|x_1 = 2) &= \frac{3 \cdot \sfrac{1}{2} + (1 + \sfrac{1}{5}) \cdot 1 \cdot \sfrac{1}{2}}{1 + \sfrac{1}{5} \cdot \sfrac{1}{2}} = \frac{21}{11}, \quad \text{or} \\ \mathrm{CE}(x_2|x_1 = 0) &= \frac{1 \cdot \sfrac{1}{2} + (1 + \sfrac{1}{5}) \cdot -1 \cdot \sfrac{1}{2}}{1 + \sfrac{1}{5} \cdot \sfrac{1}{2}} = -\frac{1}{11}. \end{align*}

Rolling back the clock to time 0, we can then write the certainty-equivalent value of the time 2 payout as:

(5)   \begin{align*} \mathrm{CE}_0(\mathrm{CE}_1(x_2)) &= \frac{\sfrac{21}{11} \cdot \sfrac{1}{2} + (1 + \sfrac{1}{5}) \cdot -\sfrac{1}{11} \cdot \sfrac{1}{2}}{1 + \sfrac{1}{5} \cdot \sfrac{1}{2}} = \frac{9}{11}. \end{align*}

Because traders subscribe to prospect theory, they value a lottery with an expected value of \mathrm{E}_0(x_2) = 1 at the certainty-equivalent value of \mathrm{CE}_0(x_2) = \sfrac{9}{11}.

4. Information Aversion

Here’s where things get interesting. If traders adhere to prospect theory, then it matters how often they check the certainty-equivalent value of the lottery. For instance, suppose that you decided to only evaluate the 2-period lottery at time 0, then you’d give it a certainty-equivalent value:

(6)   \begin{align*} \mathrm{CE}_0(x_2) &= \frac{3 \cdot \sfrac{1}{4} + 1 \cdot \sfrac{1}{2} + (1 + \sfrac{1}{5}) \cdot -1 \cdot \sfrac{1}{4}}{1 + \sfrac{1}{5} \cdot \sfrac{1}{4}} = \frac{19}{21}. \end{align*}

But, this is greater than the certainty-equivalent value of the lottery when you check on it every period:

(7)   \begin{align*} \mathrm{CE}_0(x_2) > \mathrm{CE}_0(\mathrm{CE}_1(x_2)). \end{align*}

Checking in on the process more frequently means that traders are more likely to experience painful temporary losses that won’t matter in the long run. This simple insight is quite general, and it marks the jumping off point for the analysis in Andries and Haddad (2015).

In particular, they show that traders always assign a lower certainty-equivalent value to a lottery when they are given additional interim signals under prospect theory. For instance, consider a baseline case where a trader values an uncertain future payout, \mathrm{CE}(x|x_0), to case where the trader also gets an arbitrary intermediate signal, s \in \mathcal{S}, from some distribution \mathrm{G}(s), which moves his beliefs:

(8)   \begin{align*} \mathrm{F}_0(x) \to \mathrm{F}_0(x|s). \end{align*}

The authors show that, for all \mathrm{F}_0(x) and \{ \mathrm{F}_0(x|s), \mathrm{G}(s) \}_{s \in \mathcal{S}} such that

(9)   \begin{align*} \mathrm{F}_0(x) = \int_{s \in \mathcal{S}} \mathrm{F}_0(x|s) \cdot d\mathrm{G}(s), \end{align*}

we have that \mathrm{CE}_0(\mathrm{CE}_s(x)) \leq \mathrm{CE}_0(x). Under prospect theory, traders don’t like to be given intermediate updates about their portfolio.

5. Certainty-Equivalent Rate

Let’s now extend this idea to a continuous-time setting where traders get a terminal payout x_T from the geometric-Brownian-motion process below:

(10)   \begin{align*} \frac{dx_t}{x_t} &= \mu \cdot dt + \sigma \cdot dz_t. \end{align*}

A risk-neutral trader would obviously value this payout at \mathrm{V}(x_T) = x_0 \cdot e^{\mu \cdot T}. But, how would a trader value the payout if he were information averse and only looked at the process every \ell > 0 minutes, \mathrm{V}_{\ell}(x_T)? So, for example, if the terminal payout is T = 4 minutes away and he checks the process every minute, \ell = 1, then

(11)   \begin{align*} \mathrm{V}_1(x_4) = \mathrm{CE}_0(\mathrm{CE}_1(\mathrm{CE}_2( \mathrm{CE}_3(x_4) ) ) ). \end{align*}

Alternatively, if he checked the process every other minute, \ell = 2, then \mathrm{V}_2(x_4) = \mathrm{CE}_0( \mathrm{CE}_2( x_4) ).

Here, it’s useful to define an object called the certainty-equivalent rate:

(12)   \begin{align*} \mu_{\ell} = \frac{1}{\ell} \cdot \log \mathrm{CE}_0(\sfrac{x_{\ell}}{x_0}). \end{align*}

Under prospect theory, if a trader checks in on the payout process every \ell minutes, then he runs the risk of experiencing painful temporary losses that he wouldn’t otherwise notice. As a result, even though the actual valuation is growing at a rate of \mu per minute, the trader’s effective valuation is only growing at a rate of \mu_{\ell} per minute after accounting for the additional anguish he feels. Thus, the value of the lottery at time T to a trader who checks in on it every \ell minutes is given by:

(13)   \begin{align*} \mathrm{V}_{\ell}(x_T) &= x_0 \cdot e^{\mu_{\ell} \cdot T}. \end{align*}

Consistent with the idea that the wedge between \mu_{\ell} and \mu comes from loss-averse traders checking in on the lottery too often, the authors show that \lim_{\ell \to \infty}[\mu_{\ell}] = \mu and \frac{\partial}{\partial \theta}[\mu_{\ell}] < 0. That is, if a trader never checks in on the lottery, then his valuation is the same as the risk-neutral valuation.

6. Portfolio Problem

Once we have this certainty-equivalent rate—an analogue of the risk-neutral rate in standard models—we can start doing some asset pricing. Consider the problem of a representative agent with value function,

(14)   \begin{align*} v_0^{1 - \alpha}  &=  \int_0^{\ell} e^{- \rho \cdot t} \cdot c_t^{1-\alpha} \cdot dt + e^{-\rho \cdot \ell} \cdot \mathrm{CE}_0(v_{\ell})^{1-\alpha}, \end{align*}

who chooses 3 things: how much to consume, c_t; what fraction of his wealth to invest in the risky asset, s_t; and, how often to check in on the his portfolio’s performance, \ell. Because the agent only checks in on his portfolio every \ell minutes, let’s look at the case where he consumes deterministically in the interim. In this setting, the agent’s budget constraint is given by,

(15)   \begin{align*} dw_t &= - c \cdot dt + s \cdot w_t \cdot dx_t + r \cdot w_t \cdot (1 - s \cdot x_t) \cdot dt, \end{align*}

where his initial wealth is w_0 = w, he can’t be insolvent w_t \geq 0, \alpha > 0 is his intertemporal elasticity of substitution, \rho > 0 is his discount rate, and r > 0 is the risk-free rate.

Let’s define m as the amount that the agent has to save in order to finance his deterministic consumption from time 0 to time \ell:

(16)   \begin{align*} m &= \int_0^{\ell} e^{-r \cdot t} \cdot c \cdot dt. \end{align*}

The authors show that the agent’s optimal policy depends on the relationship between the certainty-equivalent rate and the risk-free rate:

(17)   \begin{align*} m &= \begin{cases} 1 - e^{\ell \cdot \left\{ - \frac{\rho}{\alpha} + \frac{1 - \alpha}{\alpha} \cdot \mu_{\ell} \right\}} &\text{if } \mu_{\ell} > r \\ 1 - e^{\ell \cdot \left\{ - \frac{\rho}{\alpha} + \frac{1 - \alpha}{\alpha} \cdot r \right\}} &\text{if } \mu_{\ell} \leq r \end{cases} \qquad \text{and} \qquad s = \begin{cases} 1 - m_{\ell} &\text{if } \mu_{\ell} > r \\ 0 &\text{if } \mu_{\ell} \leq r \end{cases}. \end{align*}

In other words, if prospect theory lowers the agent’s certainty-equivalent rate below the risk-free rate, then the he should only save in the safe asset. However, if the certainty-equivalent rate is high enough, then the agent should invest a fraction s = e^{\ell \cdot \left\{ - \frac{\rho}{\alpha} + \frac{1 - \alpha}{\alpha} \cdot \mu_{\ell} \right\}} of his wealth in the risky asset.

Given this portfolio allocation, we can now ask: what’s the optimal length of time the agent should wait, \ell, before he checks in on his portfolio? He faces a trade off. Checking in more often means that he can keep his allocation closer to the optimal level, but it also means lowering the certainty-equivalent rate of the risky asset. Andries and Haddad (2015) show that the optimal inattention strategy is given by the solution to the differential equation below,

(18)   \begin{align*} \frac{\partial}{\partial \log(\ell)}[\mu_{\ell^\star}] &= \left( \, \frac{\rho}{1 - \alpha} - \mu_{\ell^\star} \, \right) \cdot \left( \, 1 - \frac{f(\sfrac{\rho}{(1-\alpha)} - r,\ell^\star)}{f(\sfrac{\rho}{(1-\alpha)} - \mu_{\ell^\star},\ell^\star)} \, \right), \end{align*}

where f(x,\ell) = \sfrac{x}{(\exp\{\sfrac{(1 - \alpha)}{\alpha} \cdot x \cdot \ell\} - 1)}. They find that the agent is less attentive when he has a bigger disappointment aversion parameter, \frac{\partial \ell^\star}{\partial \theta^{\phantom{\star}}} > 0, and when the risky asset’s payout is more volatile, \frac{\partial \ell^\star}{\partial \sigma^{\phantom{\star}}} > 0.