# Early vs. Late Resolution of Uncertainty

My new paper, ETFs, Learning, and Information in Stock Prices studies the effect of introducing an ETF into a model with endogenous information acquisition. Introducing the ETF has two competing effects on learning: (1) the ETF makes it easier for investors to take aggressive bets on stock-specific information, making it more attractive to learn about stock-specific risks (2) the ETF allows investors to directly trade on systematic risk, which cannot otherwise be diversified away. This makes it more attractive to learn about systematic risk and makes it less attractive to pay a fixed cost and become informed at all.

Which of these forces dominates in equilibrium depends on the economy’s risk bearing capacity, a function of the share of agents who decide to become informed and risk aversion. If risk aversion is sufficiently high, introducing the ETF decreases learning about stock-specific risks, increases learning about systematic risk and decreases the share of agents who become informed. The model provides an explanation for the empirical results in my job market paper, where I link the rise of passive ownership to decreased firm-specific information in stock prices.

My model mainly builds two papers: (1) A Noisy Rational Expectations Equilibrium for Multi-Asset Securities Markets by Anat Admati and (2) A Rational Theory of Mutual Funds’ Attention Allocation
by Kacperczyk, Van Nieuwerburgh and Veldkamp. While working through the second paper, I noticed the following sentence defining the agents’ objective function when deciding how to allocate their limited attention, *“The objective is \(−E[lnE_j[exp(−\rho W_j)]]\).”* For context, the outer expectation is taken with respect to time 0 information, while the inner expectation is taken with respect to time 1 information. Going forward, I will use \(E_t\) to denote the expectation with respect to the time \(t\) information set.

I didn’t understand why the authors put the \(ln\) inside the outer expectation i.e. why the objective function was not \(-E_0[E_1[exp(-\rho W_j)]]\), or standard expected utility. Reading through the appendix of the paper, I saw that putting this \(ln\) inside the outer expectation, *“is a transformation that induces a preference for early resolution of uncertainty.”*

Still confused, I re-read chapter 8 of Information Choice in Macroeconomics and Finance (Veldkamp (2011)) and found this line in section 8.2.4 explaining what the \(ln\) is doing: *“This formulation of utility
is related to Epstein and Zin’s (1989) preference for early resolution of uncertainty.”* This was the last hint I needed to put the pieces together. In this post, I will walk through how to derive this utility function as a special case of Epstein-Zin preferences, why this transformation introduces a preference for an early resolution of uncertainty, and why this matters in models with endogenous information acquisition.

# The Setup

In my model, terminal wealth i.e. wealth at time 2 is defined as: \(w_{2,j}=\left(w_{0,j}-1_{informed,j} c\right) +\mathbf{q_j} (\mathbf{z} - \mathbf{p})\) where \(w_{0,j}\) is initial wealth, \(1_{informed,j}\) is an indicator of whether or not agent \(j\) decides to become informed and \(c\) is the cost (in dollars) of becoming informed. If an agent decides to become informed, they get a signal at time 1 about the asset payoffs, and the precision of this signal depends on how they allocate their limited attention. \(\mathbf{q_j} (\mathbf{z} - \mathbf{p})\) is investor \(j\)’s trading profits: their portfolio \(\mathbf{q_j}\), times the payoff of each asset \(\mathbf{z}\) minus the price \(\mathbf{p}\). The bold typeface denotes a vector, which is needed because there are multiple risky assets.

At time 1, agent \(j\) submits demand \(\mathbf{q_j}\) to maximize expected utility over time two wealth: \(U_{1,j}=E_{1,j}[-exp(-\rho w_{2,j})]\), so investors have Constant Absolute Risk Aversion (CARA) or exponential utility at time 1, with risk aversion \(\rho\).

At time 0, agent \(j\) decides whether or not to pay \(c\) and become informed. If informed, agent \(j\) allocates attention to maximize time 0 expected utility. In line with Kacperczyk et. al. (2016), I define agents’ time 0 objective function as: \(-E_0[ln(-U_{1,j})]/\rho\) which simplifies to: \(U_0 = E_0\left[E_{1,j}[w_{2,j}]-0.5 \rho Var_{1,j}[w_{2,j}] \right]\). This simplification comes from the fact that (1) \(w_{2,j}\) is normally distributed in the model, and (2) \(E[exp(a x)]=exp(a \mu_x + \frac{1}{2}a^2 \sigma_x^2)\) where \(x\) is a normally distributed random variable with mean \(\mu_x\) and standard deviation \(\sigma_x\), and \(a\) is a constant.

# Formulation as Recursive Utility

Too see how the log transformation, \(-E_0[ln(-U_{1,j})]/\rho\), induces a preference for an early resolution of uncertainty relative to expected utility \(E_0[U_{1,j}]\), we can follow the hint in Veldkamp (2011) and cast preferences as recursive utility (Epstein and Zin (1989)).

Start by writing down the formulation of Epstein-Zin preferences on the Wikipedia Page: \(U_t = \left[(1-\beta)c_t^\alpha + \beta \mu_t\left(U_{t+1}\right)^\alpha \right]^{1/\alpha}\) where the elasticity of intertemporal substitution (EIS) is \(1/(1-\alpha)\) and \(\mu_t\) is the certainty equivalent (CE) operator. Note, the Wikipedia page uses \(\rho\) instead of \(\alpha\), but I’ve re-labeled it to avoid confusion with risk aversion.

In my setting, all consumption happens at time 2, so let’s simplify \(U_t\) from the perspective of \(t=0\). To further simplify things, set \(\beta=1\).

Choose the von Neumann-Morgenstern utility index \(u(w)=-exp(-\rho w)\) i.e. the CARA utility at time 1 described above. We can then define the certainty equivalent operator \(\mu_t(U_{t+1})=E_t\left[-ln(-U_{t+1})/\rho\right]\). This \(\mu_t\) is just the inverse function of the von Neumann-Morgenstern utility index. It makes sense to call this a certainty equivalent operator because it returns the amount of dollars for sure that would yield the same utility as the risky investment. Recall that \(U_{1,j}=E_{1,j}[-exp(-\rho w_{2,j})]\) and wealth is normally distributed so \(U_{1,j}=-exp(-\rho E_{1,j}[w_{2,j}]+0.5 \rho^2 Var_{1,j}[w_{2,j}])\)

Starting with setting \(\beta=0\) and \(c_1=0\): \(U_0 = \left[\mu_0\left(U_{1}\right)^\alpha \right]^{1/\alpha}\)

Substituting in the expression for the CE operator: \(U_0 = \left[ E_0\left[-ln(-U_{1})/\rho\right]^\alpha \right]^{1/\alpha}\)

Putting in our expression for \(U_1\): \(U_0 = \left[ E_0\left[-ln(exp(-\rho E_{1,j}[w_{2,j}]+0.5 \rho^2 Var_{1,j}[w_{2,j}]))/\rho\right]^\alpha \right]^{1/\alpha}\)

Simplifying: \(U_0 = \left[ E_0\left[ \left(E_{1,j}[w_{2,j}]-0.5 \rho Var_{1,j}[w_{2,j}]\right) \right]^\alpha \right]^{1/\alpha}\)

Setting \(\alpha=1\) i.e. an infinite EIS: \(U_0 = E_0\left[ \left(E_{1,j}[w_{2,j}]-0.5 \rho Var_{1,j}[w_{2,j}]\right) \right]\)

which matches Equation 6 in Kacperczyk et. al. (2016)! This shows that we can derive their utility function from Epstein-Zin preferences, but does make it totally clear what this transformation has to do with an early vs. late resolution of uncertainty.

# Another Way to View the Recursive Formulation

To make things clearer, let’s work with a more well-known version of Epstein-Zin preferences in Simon Gilchrist’s lecture notes (these were very helpful when I first learned about recursive utility!): \(V_t = \left((1-\beta)c_t^{1-\rho}+\beta[E_t(V_{t+1}^{1-\alpha})]^{(1-\rho)/(1-\alpha)}\right)^{1/(1-\rho)}\)

Setting \(t=0\), \(c_0=0\), \(c_1=0\), \(\beta=1\): \(V_0 = \left([E_0(V_{1}^{1-\alpha})]^{(1-\rho)/(1-\alpha)}\right)^{1/(1-\rho)}\)

Notice that \(c^{1-\alpha}\) is a version of Constant Relative Risk Aversion (CRRA) utility. CRRA utility simplifies to log utility if relative risk aversion is equal to 1. So, with this in mind, set \(\alpha=1\): \(V_0 = \left(exp[E_0(ln[V_1])]^{(1-\rho)}\right)^{1/(1-\rho)}\)

Set \(\rho=0\) (i.e. infinite EIS as we did above): \(V_0=exp[E_0(ln[V_1])]\)

This is equivalent to maximizing: \(\quad V_0=E_0(ln[V_1])\) because \(exp(x)\) is a monotone function.

In my setting: \(\quad V_1=E_1[exp(-\rho w)]\) i.e. time 1 utility times -1

So the final maximization problem is: \(\quad V_0=-E_0(ln[-V_1])\)

With Epstein-Zin, there is a preference for an early resolution of uncertainty if \(\alpha>(1/EIS)\). As set up here, \(\alpha=1\) and \(1/EIS=0\), so agents have a preference for early resolution of uncertainty. For expected utility, we would set \(\alpha=0\), and then there would be no preference for early resolution of uncertainty.

# Why this Matters

As I said above, \(U_0 = E_0\left[ \left(E_{1,j}[w_{2,j}]-0.5 \rho Var_{1,j}[w_{2,j}]\right) \right]\) introduces a preference for the early resolution of uncertainty (see e.g. Veldkamp, 2011). There are two types of uncertainty in the model: (1) uncertainty about payoffs at \(t=2\), conditional on signals at \(t=1\) (2) uncertainty about portfolio you will hold at \(t=1\) from the perspective of \(t=0\). With these preferences, agents are not averse to uncertainty resolved before time two i.e. are not averse to the uncertainty about which portfolio they will hold.

An intuitive way to see this is that increases in expected variance, \(E_0\left[ Var_{1,j}[w_{2,j}]\right)\), linearly decrease utility. With expected utility, \(-E_0[E_1[exp(-\rho w)]]\), simplifies to \(-E_0[exp\left(-\rho E_{1,j}[w_{2,j}]+0.5 \rho^2 Var_{1,j}[w_{2,j}]\right)]\). Because variance is always positive, utility is decreasing faster than linearly in expected variance.

A more nuanced argument requires a discussion of why learning about particular risks is useful. Expected excess portfolio return achieved through learning depends on the covariance between your portfolio \(q\) and asset payoffs \(f-p\), \(cov(q,f-p)\). Specializing in learning about one asset leads to a high covariance between payoffs and holdings of that asset. The actual portfolio you end up holding, however, can deviate substantially from the time 0 expected portfolio. Learning a little about every risk leads to smaller deviations between the realized and time 0 expected portfolio, but also lowers \(cov(q,f-p)\).

With expected utility, investors are averse to time 1 portfolio uncertainty (i.e. risk that signals will lead them to take aggressive bets), so do not like portfolios that deviate substantially from \(E_0\left[q\right]\) The utility cost of higher uncertainty from specialization offsets the utility benefit of higher portfolio returns, removing the “planning benefit” experienced by the mean-variance specification.

Recursive utility investors are not averse to risks resolved before time 2, so specialization is a low-risk strategy. Lowers time 2 portfolio risk by loading portfolio heavily on an asset whose payoff risk will be reduced by learning.

This also shows why it is desirable to introduce a preference for an early resolution of uncertainty in endogenous learning models. Think about an investor who wants to learn about AAPL. They do this so they can hold a lot of Apple (AAPL) when it does well, and hold little AAPL when it does poorly. An expected utility investor would be hesitant to learn too much about AAPL, because the fact that their portfolio will vary substantially depending on the signal they get seems risky to them.

# Wrap Up

In this post, I showed (1) why adding a \(ln\) inside the outer expectation induces a preference for an early resolution of uncertainty and (2) why this is useful in models with endogenous learning. For those interested in these topics, I recommend reading Costis Skiadas’ Asset Pricing Theory (2009) textbook for a discussion of recursive utility and Laura Veldkamp’s Information Choice in Macroeconomics and Finance (2011) for discussion of endogenous learning models.