the equivalence principle

When an insurer sells a policy, two streams of money flow in opposite directions: the policyholder pays premiums in, and the insurer pays benefits out, both possibly stretched over many years and both contingent on whether the person lives or dies. How should the insurer decide what premium is fair? The equivalence principle gives the simplest honest answer: set the premium so that, at the moment the policy is issued, what comes in is worth exactly as much as what goes out.

Precisely, it says: choose the premium so that the actuarial present value of future premiums equals the actuarial present value of future benefits. Both sides are EPVs — discounted for interest and weighted by the probabilities of survival and death. Set them equal and solve for the premium. For example, if a single lump-sum premium is being charged for a death benefit, the equivalence principle simply sets that premium equal to the actuarial present value of the death benefit; for a benefit worth 0.30 in EPV, the fair single premium is 0.30. For level annual premiums, you divide the benefit's EPV by the EPV of an annuity of 1 per year over the premium-paying period.

This is called the net premium or benefit premium: it covers the promised benefits and nothing else — no expenses, no profit margin, no cushion for bad luck. That is the catch. A premium set purely by equivalence has, by construction, a roughly 50/50 chance of falling short once randomness plays out, and it ignores the insurer's running costs and the need for a safety buffer. Real-world pricing starts from the equivalence principle and then loads on expenses, margins and prudence. But the principle is the clean conceptual anchor from which net premiums, reserves and most of life actuarial mathematics are derived.