Yield to maturity of long-duration Gertler-Karadi private claim


I am trying to understand how to derive the yield to maturity (in Dynare-consistent way) of a long duration (infinite duration) private bond (private capital claim) with cash flows equal to the MPK net of depreciation. I am following Gertler-Karadi, 2013, IJCB, equation 47, with the difference that my security is a perpetuity, and also I abstract from the “slightly different payoff structure” that they impose.

It seems that this equation (eq 47, in GK 2013) contains some typos, which makes it hard to understand how it should be properly written (after assuming infinite duration and abstracting from their “different payoff structure”), and thus try to impose a recursive structure (infinite sum).

Happy if anyone could shed some light on this.

If I remember correctly, there is a typo in equation 47. Due to time limitations, I try to come back to your problem by the end of this week.

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Equation (47) in Gertler & Karadi (2013, p. 36) is given by
\begin{align} P_tQ_t &= E_t\sum_{s=1}^{40} \frac{(Z_{t+s}-\delta_{t+s})P_{t+s}}{(1+i_{kt}^*)^s}\end{align} +\frac{P_{t+40} Q_{ss}}{(1+i_{kt}^*)^{40} }
but according to the payoff structure of the perfectly state-contingent claim to capital (see page 29 for details) the numerator of the first term, (Z_{t+s}-\delta_{t+s})P_{t+s}, should be (1-\delta)^{s-1}\left(\prod_{l=1}^s\xi_{t+l}\right)Z_{t+s}. Their formula in the numerator might make sense, if you assume that replacement costs of depreciated capital are fixed at 1 (in real terms) and abstract from “capital quality” shocks (or alternatively, replace Z_{t+s} with \tilde{Z}_{t+s}=\xi_{t+s}Z_{t+s}). However, at a glance, I cannot see if the authors make this assumption for replacement cost.

Thank you @Max1 for your attempt.

I guess what I am after is the following. Imagine we have a government bond which is a perpetuity (consol) promising to yield a constant coupon payment normalised to 1 unit of the final good every period. Then, we can price the asset according to this equation
Screenshot 2024-06-17 at 10.50.24 AM

and solving with respect to the gross yield to maturity, RL_{t}^{b}, we end up with this
Screenshot 2024-06-17 at 10.50.35 AM

Now, imagine I want to do the same for the private capital claim, which is assumed to be a perpetuity (consol) with payoff equal to the MPK less depreciation. We can price the private asset following
Screenshot 2024-06-17 at 10.44.40 AM

My problem is how to solve wrt RL_{t}^{k}. My attempt yields something like that
Screenshot 2024-06-17 at 10.55.44 AM
but I doubt this is correct.

Your formula looks like the period to period return not the yield to maturity. I think you can solve your equation number 3 only numerically, if you assume an upper bound for the sum. @jpfeifer Is this correct?

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Yes, this looks like the computation of an internal rate of return. It generally requires solving a polynomial equation with a numerical solver.

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