How Limited Asset Awareness Affects Equilibrium Prices

Summary

The document poses a two-risky-asset equilibrium problem in which some investors are unaware that the second asset exists. It gives the benchmark equilibrium price vector for the case where all investors know both assets, then asks how equilibrium prices change when a subset can trade only the first asset. The proposed result keeps the first asset’s price at its benchmark level and lowers the second asset’s price by an adjustment involving the unaware investors’ share, aggregate supply, and the second asset’s variance not explained by its covariance with the first.

It also asks for a relationship between expected returns and market covariance, including an additional term for the second asset. The document supplies no proof or solution, so the pricing claims are posed as exercises rather than established results. The setup assumes normally distributed returns and constant absolute risk aversion, and the question’s notation for the distribution parameters appears inconsistent. Any interpretation should therefore check the model assumptions and derivation before treating the formulas as general asset-pricing conclusions.

Key ideas

• The setup compares equilibrium prices when all investors know both risky assets with prices when some investors are unaware of one asset.
• The stated result predicts no price change for the first asset and a lower price for the asset that some investors cannot see.
• The proposed price adjustment depends on the unaware investors’ fraction, total asset supply, and residual variance of the second asset.
• The problem also asks how limited awareness changes a covariance-based expected-return relationship.
• The document gives questions and claimed formulas but no proof, and its distribution notation should be checked.

Tags

# Equilibrium with H agents when some of them are not aware of some assets


# Equilibrium with H agents when some of them are not aware of some assets












Assume there are H agents with constant absolute risk aversion $\alpha$. There is a risk-free asset, and two risky assets with distribution $S1$ ~ $N(\mu; \Sigma)$, where $\mu \in \mathbb{R}^2$ and $\mu \in \mathbb{R}^{2\times2}$. Assume $H_U$ agents are not aware of the existence of the second risky asset. If $H_U = 0$, then the equilibrium prices are $S_0^* = \frac{1}{R_f}\mu - \frac{\alpha}{HR_f}\Sigma\theta_{tot}$ where $\theta_{tot}$ is the total supply of risky assets.

Assume now $0 < H_U < H$

a) Prove that in equilibrium $(S_0)_1 = (S_0^*)_1$ and

$(S_0)_2 = (S_0^*)_2 - \frac{\alpha}{HR_f}\frac{H_U}{H-H_U}(Var((S_1)_2) - \frac{cov((S_1)_1,(S_1)_2)^2}{Var((S_1)_1)})(\theta_{tot})_2$

and hence $(S_0)_2 < (S_0^*)_2$

b) Let $R_1$ and $R_2$ be the returns of the two risky assets. Prove that there exist $A > 0$ and $\lambda$ such that

$E[R_1] = R_f +\lambda\frac{cov(R_1,R_m)}{Var(R_m)}$

$E[R_1] = A + R_f +\lambda\frac{cov(R_2,R_m)}{Var(R_m)}$

$\lambda = E[R_m] - R_f - A\frac{(S_0)_2(\theta_{tot})_2}{S_0^T\theta_{tot}}$

where $R_m$ is the return of the market portfolio.

My doubt comes early in the problem: how should I reflect the condition on $H_U$ mathematically? I understand the formula for $S_0^*$ given in the introduction, but dont know how to introduce that $H_U$ information. Thank you in advance.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.