I have a question regarding the derivation of the bond duration formula for the Woodford (2001) type bonds with geometric coupon as these bonds are often used in the literature, e.g. Sims & Wu (2021), Carlstrom et al. (2017), Chen et al. (2012).
Sims & Wu (2021, p.145) use the formula from Carlstrom et al. (2017, p. 209).
Carlstrom et al. (2017, p. 209) write: “The duration and (gross) yield to maturity on these bonds are defined as: duration= {(1 − \kappa)} ^{−1} , gross yield to maturity = Q_t ^{− 1} + \kappa .”
On the other hand, within their appendix Chen et al. (2012) offer a different formula for the bond duration of the same type of bonds, namely \frac{R_{L,t}}{R_{L,t}-\kappa}.
According to my calculations, the later formula should correspond to the Macaulay duration.
Up to this point it is not clear to me how to derive the formula of Carlstrom et al. (2017, p. 209).
Also, the original formulas from Woodford (2001) look a little bit different.
Therefore, my question: Are Carlstrom et al. (2017, p. 209) using a different version of the bond duration, which I missed or is it just a typo?
We know the price of such a bond with a coupon decaying at rate \rho and starting with \rho^0=1 next period and using discount factor \beta=1/(1+r) is given by
P = \rho^{-1} \left[ {{{\left( {\beta \rho } \right)}^1} + {{\left( {\beta \rho } \right)}^2} + ...} \right] = \rho^{-1} \frac{{\beta \rho }}{{1 - \beta \rho }} = \rho^{-1} \frac{{\frac{\rho }{{1 + r}}}}{{1 - \frac{\rho }{{1 + r}}}} = \rho^{-1} \frac{{\frac{\rho }{{1 + r}}}}{{\frac{{1 + r}}{{1 + r}} - \frac{\rho }{{1 + r}}}} = \rho^{-1} \frac{{\frac{\rho }{{1 + r}}}}{{\frac{{1 + r - \rho }}{{1 + r}}}} = \frac{{{1}}}{{1 + r - \rho }}
From this follows \frac{{\partial P}}{{\partial r}} = \frac{{ - 1}}{{{{\left( {1 + r - \rho } \right)}^2}}} =\frac{{ - 1}}{{\left( {1 + r - \rho } \right)}}P
and therefore the duration is D = - \frac{{\frac{{\partial P}}{{\partial r}}}}{P} = \frac{1}{{\left( {1 + r - \rho } \right)}}
We are interested in the Macaulay duration D_M, which is related via D = \frac{{{D_M}}}{{1 + r}}
Thus: {D_M} = D\left( {1 + r} \right) = \frac{{1 + r}}{{1 + r - \rho }} = \frac{1}{{\frac{{1 + r - \rho }}{{1 + r}}}} = \frac{1}{{1 - \frac{\rho }{{1 + r}}}} = \frac{1}{{1 - \rho \beta }}=(1-\rho \beta )^{-1}
The free parameter here is \rho. For any given value of the discount factor (which need not be the \beta from the preferences), you can always chose \rho to get the desired \kappa=\rho \beta .
However, most of the literature seems to employ \kappa=\rho and therefore duration \frac{1}{1-\kappa}. That implies ignoring the steady state yield to maturity R^L\neq 1 (but it typically is close to 1, although this is not a guarantee for a decent-sized approximation error).