The stochastic nature of future mortality arises from both period (time-related) and cohort (year-of-birth-related) effects. Existing index-based longevity hedging strategies mitigate the risk associated with period effects, but often overlook cohort effects. The negligence of cohort effects may lead to sub-optimal hedge effectiveness, if the liability being hedged is a deferred pension or annuity which involves cohorts that are not covered by the data sample. In this paper, we propose a new hedging strategy that incorporates both period and cohort effects. The resulting longevity hedge is a value hedge, reducing the uncertainty surrounding the τ-year ahead value of the liability being hedged. The proposed method is illustrated with data from the male population of England and Wales. It is found that the benefit of incorporating cohort effects into a longevity hedging strategy depends heavily on the persistence of cohort effects and the choice of q-forwards.