Default probability and bond price

Hi, I am trying to solve a model in which there is a government sector issuing defaultable sovereign bonds. The government bond rate of return depends on the bond price and default probability. The asset pricing equation for the bond is:

exp(R_b_govt(1))= (1-exp(pt(1)))*(delta_m+(1-delta_m)*(iota+exp(Qg(1))))/exp(Qg);

where R_b_govt is the government bond rate of return, Qg is the bond price, pt is the default probability, delta_m is the portion of the mature bond, iota is the coupon

Now I want to look at the IRFs to a shock of increasing default probability. It’s very strange that the IRFs are different when pt is persistent or not.

When I set pt as an AR(1), so:
p_t = \lambda p_{t-1} + (1-\lambda) p_{ss} + std_d e_d
The IRFs of pt, Qg and R_b_govt are:

If pt is just a shock, so:
p_t = p_ss + e_d
The IRFs are:

The p_t only impact on the equilibria through the asset pricing equation so I am very puzzled about why the results are different. Anyone can give some suggestions?

The default probability only enters via the asset price equation, which itself is forward-looking. If there is no persistence, the shock will have died out next period, and as it is mean 0, it’s unpredictable. Hence, there is no effect on future asset prices.

Thanks professor Pfeifer. I changed the asset pricing equation a little bit, so now the realized rate of return depends on the realized default probability at time t.
exp(R_b_govt)= (1-exp(pt))(delta_m+(1-delta_m)(iota+exp(Qg)))/exp(Qg(-1));
Now the IRFs look much more reasonable:

I am adding defaultable government bond to a Gertler and Karadi (2011) framework but in a two-country environment (bank borrows from one country and lend to the other country).

I am wondering what do you think of the asset pricing setup right now, since both rate of return and default probability are ex-post and realized at time t, which is not commonly seen in the sovereign default literature.

I don’t think this makes sense. Why should today’s default affect the price of new bonds if it is mean 0 (at least up to first order)?

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Hi professor Pfeifer:

Thanks a lot for your feedback! It’s very helpful!

The bond is issued in t-1, and carries to time t. At time t, the government is possible to default and if default happens, it will lower the rate of return for the sovereign bonds because of the haircut.

The default probability it not mean 0, and it’s endogenously determined by the debt to GDP ratio. A similar setup is the paper by Bi,Leeper and Leith (2014)(I attach the paper to this post
Bi,Leeper and Leith(2014).pdf (346.4 KB)

Is there anything I understand wrong? Really appreciate for your question and it is
really helpful for me to deepen my understanding.

Or do you think I can use the initial setup? Which asset pricing equation is forward-looking but the shock to default probability is persistent.

I am not sure I understand the point. If you have the default probability depend on the debt level, then the default probability will usually be persistent, because debt is persistent. That would give you effects in the forward-looking asset pricing equation.