Equivalence Principle for Life Insurance Premiums¶

Definition¶

The equivalence principle chooses a premium so that expected discounted premium income equals expected discounted benefits and, for gross valuation, expenses. Let $Z$ be the benefit present value (PV), $H$ the expense PV and $Y$ the PV of a unit premium stream. For a level premium $P$ , the gross equation is $P\mathbb{E}[Y]=\mathbb{E}[Z+H]$ $H]$ . Net valuation omits $H$ ; the resulting level premium is the net level premium, and "net" always means that expenses and loadings are excluded.

Interpretation¶

This balances expected cash flows at issue under a specified mortality and interest basis. It does not guarantee that every policy or every portfolio has zero realized loss.

Two pairs of bars: the net premium 4 times 10 balances expected benefits of 40, and the gross premium 4.5 times 10 balances expected benefits of 40 plus expenses of 5.

With the numbers from the Examples, the net premium $P=4$ makes $P\,\mathbb{E}[Y]=4\times10$ equal $\mathbb{E}[Z]=40$ $40$ , and the gross premium $P=4.5$ makes $4.5\times10=45$ equal $\mathbb{E}[Z+H]=40+5$ $5$ . The bars balance in expectation only; a realized policy can still produce a profit or a loss.

Move the trial premium $P$ : the orange gap is $\mathbb E[L_0^g]$ , which vanishes where income $P\,\mathbb E[Y]$ meets the outgo. Add a premium-related expense share $r$ and the outgo line tilts, so it must be solved together with $P$ .

Premiums paid annually in advance while the policyholder is alive form a whole life annuity-due. Its actuarial present value (APV) is the expected present value of one unit of premium each year, explaining the denominator $\ddot a_x$ in the whole-life premium formula.

Key Results¶

If $\mathbb{E}[Y]>0$ and expenses do not themselves depend on $P$ , $P=\mathbb{E}[Z+H]/\mathbb{E}[Y]$ $H] / E [Y]$ .
An annual net whole life premium payable in advance while alive is $P=A_x/\ddot a_x$ for unit annual-settlement death benefit.
If premiums are payable for at most $n$ years, the denominator is $\ddot a_{x:\actuarialterm{n}}$ .
Premium-related expenses must remain in the equation and be solved together with $P$ .

Examples¶

If expected benefit PV is 40 and the unit premium-stream APV is 10, the net annual premium is 4. Expenses with APV 5, independent of the premium, raise the gross premium to 4.5.

Prerequisites¶

Sources¶

Life Insurance · Lecture 6

Aliases: net level premium, gross premium equivalence, equivalence principle, net premium, equivalence premium, gross premium, principle of equivalence, office premium, benefit premium Keywords: P = A x / ä x, expected premium income equals expected outgo, E[L 0] = 0, premium calculation, level annual premium

Last updated: 17 October 2026 21:53