The generation time is commonly defined as the mean age of mothers at birth. In matrix population models, a general formula is available to compute this quantity. However, it is complex and hard to interpret. Here, we present a new approach where the generation time is envisioned as a return time in an appropriate Markov chain. This yields surprisingly simple results, such as the fact that the generation time is the inverse of the sum of the elasticities of the growth rate to changes in the fertilities. This result sheds new light on the interpretation of elasticities (which as we show correspond to the frequency of events in the ancestral lineage of the population), and we use it to generalize a result known as Lebreton's formula. Finally, we also show that the generation time can be seen as a random variable, and we give a general expression for its distribution.