For the second moment of Ax in the continuous case, which formula is correct?

• A. Ax second moment = 1 - (2 delta) * (second moment annuity)
• B. Ax second moment = 1 - delta * (second moment annuity)
• C. Ax second moment = 1 - (2 i) * (second moment annuity)
• D. Ax second moment = 1 - (delta) * (second moment discrete)

Answer: (B)

In continuous-time actuarial math, the discounting uses the force of interest δ, not the nominal rate i. When you look at the second moment of the present value of a unit continuous life annuity, you’re squaring the integral of the discounted payment stream over the lifetime and then taking expectations with respect to survival. Working this out with the double integral and the survival function shows that the effect of discounting enters as a single δ multiplier on the second moment of the corresponding (undiscounted or deterministically structured) payoff.

This leads to a neat relation: the second moment of the continuous Ax equals 1 minus δ times the second moment of the (continuous) annuity payoff. The factor is δ (not 2δ), and i or a discrete-second-moment form does not apply in this continuous framework. Therefore the correct formula uses a single δ multiplying the second moment and matches the continuous-discounting setup.