Question

For a whole life insurance of $1 on (41) with death benefit payable at the end of year of death, you are given:

(i) i = 0.05
(ii) p40 = 0.9972
(iii) A41 - A40 = 0.00822
(iv) 2A41 - 2A40 = 0.00433
(v) Z is the present value random variable for this insurance.
Calculate Var(Z).

Answer 1

To calculate the variance of a present value random variable (Z) for a whole life insurance of $1 on (41) with death benefit payable at the end of the year of death, we can follow these steps:

Step-by-step explanation:

To calculate the variance of a present value random variable (Z) for a whole life insurance of $1 on (41) with death benefit payable at the end of the year of death:

1. First, we need to find the present value of the insurance, A41. We are given that A41 - A40 = 0.00822, which implies A41 = A40 + 0.00822. Using the formula for present value of a life insurance, we have A41 = 1 - 2A41 + 0.00822. Solving for A41 gives us A41 = 0.00411.
2. Next, we need to find the expected value of Z, which is the present value of the death benefit. For whole life insurance, the expected present value of the death benefit is equal to the sum of the expected present values of each individual death benefit. Using the given information, we have E(Z) = A41 * 1 + (1 - A41) * 0. This simplifies to E(Z) = A41 = 0.00411.
3. Finally, to calculate Var(Z), we can use the formula Var(Z) = E(Z^2) - [E(Z)]^2. Since E(Z) = 0.00411, we can substitute this value into the formula and calculate Var(Z).