## Hedging life insurance with pure endowments.(English)Zbl 1183.91067

Summary: We extend the work of M. A. Milevsky, S. D. Promislow and V. R. Young [“Financial valuation of mortality risk via the instantaneous Sharpe ratio”, preprint; available at http://www.ifid.ca/pdf_workingpapers/WP2005NOV4.pdf] and V. R. Young [“Pricing life insurance under stochastic mortality via the instantaneous Sharpe ratio”, preprint; available at http://ideas.repec.org/p/arx/papers/0705.1297.html] 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.

### MSC:

 91B30 Risk theory, insurance (MSC2010) 35R60 PDEs with randomness, stochastic partial differential equations
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### References:

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