the loss-at-issue random variable

When an insurer sells a policy it cannot yet know whether this customer will die next year or live to 100, so it does not yet know whether the deal will gain or lose money. The loss-at-issue random variable is the actuary's name for that unknown outcome: it is the present value, measured at the moment of issue, of the benefits the insurer will pay MINUS the present value of the premiums it will collect, for one particular policyholder, viewed as a gamble whose result depends on how long that person lives.

Precisely, write L for this quantity. If the insured's future lifetime turns out to be such that death occurs in year k, then L equals (present value of the benefit paid at that time) minus (present value of all premiums received up to then). Because the time of death is random, L is random too. The equivalence principle is exactly the statement that the expected value of L is zero, E[L] = 0: on average the deal breaks even. But L still has spread — for some policyholders L is very positive (they die early, the insurer loses), for most it is negative (they live long and pay premiums for years). The variance of L measures how risky one policy is.

This idea is the engine behind premium setting and reserving. Setting E[L] = 0 gives the net premium. The variance of L drives how much capital and margin an insurer needs and underlies the portfolio percentile premium. And the reserve at a later date is simply the expected value of a re-started loss variable looking forward from that date. So although L is abstract, almost every number in life insurance pricing is a fact about its distribution.