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#1




Covariance of Life Annuity and Life Insurance?
So this question:
Z is the present value random variable for a unit whole life insurance on (x) with benefits paid at the end of the year of death. Y is the present value random variable for a unit whole life annuitydue on (x). With the given values, Ax = 0.3, 2Ax = 0.16, d = 0.04 and calculate: Cov(Z,Y) I did it by doing Cov(Y,Z) = E(YZ)  E(Y)E(Z) = E(Y*(1dY))E(Y)E(Z) = E(Y)dE(Y^2)  E(Y)E(Z) = (1.3)/.04.04*(1.16)/.04^2  (1.3)/.04*.3 = 3.5  5.25 = 8.75 I used the fact that Z = 1dY. However, this doesn't work out in the problem. The solution used Y = (1Z)/d, which changes E(YZ) to E(Z(1Z)/d) which is positive 3.5 instead of negative 3.5 Am I missing something? This is driving me crazy Last edited by erichuang1996; 11092019 at 06:10 PM.. 
#2




Your E[Y^2] is incorrect. You can't calculate it that way because E[Y^2] is NOT equal to the EPV of annuity evaluated at twice of force of interest.
To get E[Y^2], the quickest way is to calculate Var[Y] first, which is (0.160.3^2)/(0.04^2) = 43.75. Then, E[Y^2] = Var[Y] + (E[Y])^2 = 43.75 + [(10.3)/0.04]^2 = 350. Thus, E[YZ] = (10.3)/0.04  0.04*350 = +3.5. 
#4




There is a much easier way to calculate Cov (Y,Z). Substitute Y in terms of Z and use the fact that Cov(Z,Z) is the variance. The results is Var(Z)/d.
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