The variance of a whole life annuity is given by which expression?

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Multiple Choice

The variance of a whole life annuity is given by which expression?

The quantity being tested is how to express the spread (variance) of the present value of a whole life annuity. If you let X be the present value of the stream of payments for life aged x, then X can be written as a sum of discounted survival indicators: X = sum_{k≥1} v^k I(T ≥ k), where I(T ≥ k) is 1 if the person is alive at the end of year k and 0 otherwise, and v is the discount factor. The expected value of X is the actuarial present value of the annuity, denoted Ax. The variance comes from the second moment E[X^2] minus the square of the mean. When you work through the double sum for E[X^2] using the survival probabilities, the cross-term structure permits a tidy simplification in terms of Ax and the discount rate d (with d = i/(1+i)). The resulting variance is (2Ax − Ax^2) / d^2. So the expression (2Ax − Ax^2) / d^2 is the variance of the present value of a whole life annuity. The other forms differ by a factor of d or d^2 and do not match the correct second-moment calculation.

The quantity being tested is how to express the spread (variance) of the present value of a whole life annuity. If you let X be the present value of the stream of payments for life aged x, then X can be written as a sum of discounted survival indicators: X = sum_{k≥1} v^k I(T ≥ k), where I(T ≥ k) is 1 if the person is alive at the end of year k and 0 otherwise, and v is the discount factor.

The expected value of X is the actuarial present value of the annuity, denoted Ax. The variance comes from the second moment E[X^2] minus the square of the mean. When you work through the double sum for E[X^2] using the survival probabilities, the cross-term structure permits a tidy simplification in terms of Ax and the discount rate d (with d = i/(1+i)). The resulting variance is (2Ax − Ax^2) / d^2.

So the expression (2Ax − Ax^2) / d^2 is the variance of the present value of a whole life annuity. The other forms differ by a factor of d or d^2 and do not match the correct second-moment calculation.

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