How to derive what effect funding shocks have on conditional market betas? [closed]

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$

closed as unclear what you're asking by SRKXNov 3 '15 at 7:57

Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

• I'm not a specialist of these models, but this seems super unclear to me. I tried to clean up a bit you're formatting, maybe you should finish up if you want to get any answers on this. And please correct the typo in your question's title and make it a question. – SRKX Nov 2 '15 at 3:11
• What exactly is the optimization problem here? Throwing a up a bunch of resulting equations without showing the original question is not helpful at all. – user32416 Nov 2 '15 at 4:43
• Ok this is still not clear and the formatting is still not right. I'm closing this until you've cleaned up the formatting and made clear what you're looking to do. – SRKX Nov 3 '15 at 7:57