Recursive utility under the ICAPM: Is it relevant?

I show that, under the benchmark homoskedastic framework in Campbell (1993) (C93), the assumption of recursive utility is unnecessary to derive the ICAPM pricing equation. That is, by assuming the standard power utility one obtains the same exact pricing equation. This also holds when deriving the (heteroskedastic) three-factor ICAPM of Campbell, Giglio, Polk, and Turley (2018). The assumption of recursive utility in C93 is no longer innocuous under heteroskedasticity. However, the respective ICAPM is not empirically testable unless one is willing to impose ad-hoc restrictions. Overall, recursive utility is largely irrelevant to obtaining Campbell's ICAPM and its various extensions.