This thesis aims to investigate the effect oflongevity risk in the context of life annuities. It develops different tools and frameworks to measure this risk as a step to facilitate the risk management oflongevity risk. Particular attention is directed to stochastic modelling which allows the uncertainty of future projections to be incorporated. Hence, simulation methods are used to consider the distribution of the annuity cost, as well as the more often quoted point estimates. A theoretical extension of the use of the entropy measure applied in population biology by Demetrius (1976) has been developed to measure the effect of a proportionate change in the force of mortality on the cost of life annuity. The properties of the corresponding entropy measure have been then investigated using the Gompertz and the Sithole et al (2000) mortality projection models. Numerical results suggest that, at very high or low levels of mortality, the effect of mortality changes on the value of life annuity is of reduced importance. A full Bayesian model has been developed which incorporates the estimation of the parameters of both the Sithole et al (2000) and the Lee - Carter (1992) mortality projection models within the simulation of the annuity cost. This has been extended to an environment in which the future rates of interest are stochastic. The effect of parameter uncertainty of the Sithole et al (2000) mortality projection model has been considered and shown to be less important than the associated model uncertainty.