Funding Constraints and Their Proposed Effect on Conditional Market Betas

Summary

This question presents a pricing equation in which a funding constraint enters the denominator alongside the risk-free rate. The constraint is represented by a Lagrange multiplier, while expected future payoffs, risk aversion, asset covariance, and optimal share holdings also enter the setup. The author seeks to derive how a funding shock changes prices and conditional market betas, with the intuition that a shock affecting all securities could pull betas toward one.

The document supplies related portfolio and pricing equations but does not include a worked derivation or an answer. It therefore offers a theoretical setup and a research question rather than a demonstrated result. The equations as transcribed may also require clarification of notation and assumptions before they can support comparative statics; no empirical evidence or parameter estimates are given.

Key ideas

• The pricing setup includes a funding constraint through a multiplier in the required return denominator.
• The author asks whether common funding shocks can compress conditional betas toward one.
• The document gives portfolio-choice and price equations but no derivation of the proposed beta effect.
• Notation and assumptions would need clarification before drawing theoretical or empirical conclusions.

Tags

# How to derive what effect funding shocks have on conditional market betas?


# How to derive what effect funding shocks have on conditional market betas?












I am unable to derive the correct result eq2 all my answers seem circular, any help would be much appreciated. It should basically end up saying that shocks that affect all securities compress betas towards 1. In this case a funding shock, i.e. increase in borrowing constraints.

Equation 1

$$P_t^s=\frac{E_t (P_{t+1}+\delta_{t+1} )-\gamma \Omega x^*}{1+r^f+\psi_t }$$

Equation 2 $$\frac{\frac{\partial P_t^s}{\partial \psi_t}}{P_t^s}: \frac{-1}{1+r^f+\psi_t} =E_t (P_{t+1}+\delta_{t+1} )-\gamma \Omega x^*$$

Where:

- $\psi$ is the Lagrange multiplier, proxying funding constraints

- $\Omega$ is the covariance matrix

- $\gamma$ represents risk aversion

- $x^*$ vector of shares

- $P_{t+1}+\delta_{t+1}$ is expected future payoff

The price $P_t$ is derived from the equation below:

$Eq3:$ $x^*=1/γ$ $Ω^{-1}$ $(E_t (P_{t+1}+δ_{t+1} )$$-$$(1+r^f+ψ_t ) P_t )$

The optimal portfolio of shares $x^*$ is derived from

$Eq4:$ $max$⁡$〖x'(E_t (P_{t+1}+δ_{t+1} )$$-$$(1+r^f ) P_t)-γ^{i/2}〗$$ x'Ωx$

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