Dear all,

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?