variance of the insurance present value

The actuarial present value A_x tells you the average value of a death benefit, but an average hides the spread. For any one policy the actual present value is a random variable — call it Z — because it depends on exactly when death occurs. If death comes next year, Z is large (a lightly discounted payment); if it comes in fifty years, Z is tiny (a heavily discounted one). The variance of the insurance present value measures how widely Z swings around its mean A_x: it is the insurer's risk on a single policy, the thing that averages away across a large pool but bites hard on any one contract.

There is a beautifully simple way to compute it. The present value of a unit benefit paid at the end of the year of death is v raised to the time-of-death; its square is just v at double the force of interest. So the second moment of Z — written with a 2 in front, as in 2A_x — is exactly the same insurance function but computed at an interest rate doubled in the force-of-interest sense (replace v by v squared). The variance is then the standard 'mean of the square minus the square of the mean': Var(Z) = 2A_x - (A_x)^2. For instance, if A_x = 0.30 and 2A_x = 0.12, the variance is 0.12 - 0.09 = 0.03, giving a standard deviation of about 0.17 — large relative to the mean of 0.30.

This variance is what justifies pooling and capital. A single death-benefit contract is volatile; but the law of large numbers says that the variance of the average over n independent policies shrinks like 1/n, so a big, well-diversified book is far more predictable than one policy. Actuaries use this same machinery to set percentile premiums, size safety loadings, and estimate the capital needed to stay solvent at a chosen confidence level. The key caveat: this clean formula assumes the lives are independent and the interest rate is fixed — correlated mortality (a pandemic) or random interest rates make the real variance larger than 2A_x - (A_x)^2 suggests.