Hedging life insurance with pure endowments

We extend the work of Milevsky et al., Milevsky, M.A., Promislow, S.D., Young, V.R., 2005. Financial valuation of mortality risk via the instantaneous Sharpe ratio (preprint)] and Young, Young, V.R., 2006. Pricing life insurance under stochastic mortality via the instantaneous Sharpe ratio (preprint)] by pricing life insurance and pure endowments together. We assume that the company issuing the life insurance and pure endowment contracts requires compensation for their mortality risk in the form of a pre-specified instantaneous Sharpe ratio. We show that the price P^m,n for m life insurances and n pure endowments is less than the sum of the price P^m,0 for m life insurances and the price P^0,n for n pure endowments. Thereby, pure endowment contracts serve as a hedge against the (stochastic) mortality risk inherent in life insurance, and vice versa.