Suppose that the force of mortality μ is a constant 0.04 and the force of interest is a constant 0.06.
(a) An insurer sells to one individual a single-premium, whole-life policy, with 100 payable at the moment of death, and also sells, to an independent life, a life annuity with benefits payable continuously at the rate of c per year. If W denotes the present value of the benefits on the two contracts combined, find Var(W) as a function of c.
(b) Suppose now that the two contracts in (a) are sold to the same individual. What is the variance of W in this case? Is it less than, greater than or equal to the variance in (a)? Explain why. For what values of c, if any, will the variance be equal to 0? Explain.