"Risk-Free Consol" in Asset Pricing I: Finite State Models

[oyamad]

While we were following the lecture on Asset Pricing I: Finite State Models, we couldn't really understand the description of the model of A Risk-Free Consol: What is the source of the stochasticity here? What makes m_{t+1} not equal to beta?

[jstac]

m_{t+1} is the SDF process, which is given by eq (11) of that lecture.

So the source of randomness is through the consumption. This affects the valuation of the risk free bond (consol).

Is that OK?

[oyamad]

In the previous model, the consumption c_t = d_t stochastically changes as the dividend d_t stochastically changes. But here with a constant payoff zeta, isn't the consumption constant, equal to zeta?

[jstac]

I'm working off Tom's notes but, if I understand correctly, we take consumption as given and then use the resulting SDF to price a variety of assets. This gives the "fundamental asset pricing equation," which is eq (2) in these notes.

The x_{t+1} in that equation can be the payoff of any asset, including holding a consol for one period (annuity payment plus right to sell).

Does that seem reasonable? Should we add some more discussion of this?

[oyamad]

So is the following the right interpretation?

• There is a consol in addition to the Lucas tree.
• The consumer can take any short or long position.
• We focus on the equilibrium in which the consumer demands a positive amount only for (the claim to the returns to) the Lucas tree, so that c_t = d_t holds in equilibrium and hence the second equality in eq (13) holds.
• In equilibrium the consumer must not have a strict incentive to buy or (short-)sell the consol, so that eq (15) must hold.

[jstac]

The answer to point 1 is yes. For the answers to the other questions, I think this is what's assumed in the link above (Cochrane) and other asset pricing literature.

(Although I'm not sure you can short the asset --- is it necessary?)

[oyamad]

So what is missing in the lecture is a discussion about why we have the second equality in eq (13) in the model of this section (and what is g_{t+1} here).

• Assuming g is the (stochastic) growth factor of the return d_t of the Lucas tree, to get this equality the consumption c_t should be equal to d_t, so that the demand for the consol is zero.
• The FOC for a corner solution would be an inequality. I was trying to reason the equality as the FOC.

[jstac]

Thanks @oyamad , I'll make those changes soon.