Equivalence Principle

Study guides: Exam FAM, Exam ALTAM

The Equivalence Principle sets a premium so that the expected present value of the future loss random variable $L_0$ at issue is zero: the expected present value (EPV) of the premiums equals the EPV of the benefits, plus the EPV of the expenses when a gross premium is being set. It is the standard premium principle of AMLCR Chapter 6 for insurance and annuities on single lives, joint lives and multiple-state models.

$E\left[L_0\right] = 0$

$\text{EPV(premiums)} = \text{EPV(benefits)} + \text{EPV(expenses)}$

$L_0 = S\,v^{K_x+1} - P\,\ddot{a}_{\overline{K_x+1}|}$

$L_0$ is the present value at issue of the insurer's future outgo less future income. The third block is the net loss for a fully discrete whole life policy with sum insured $S$ and annual premium $P$ . Setting $E[L_0]=0$ gives $P = S\,A_x/\ddot{a}_x$ .
Net vs gross: leave expenses out and the result is the net premium; include them and it is the gross premium. Net premiums are also used inside net premium policy values, on the valuation basis.
Variance of the loss: for whole life, $\text{Var}(L_0) = \left(S + P/d\right)^2\left({}^2\!A_x - A_x^2\right)$ . Zero expected loss is not zero risk. The portfolio percentile principle instead chooses $P$ so that the aggregate loss on $N$ policies is negative with a target probability, a percentile of the loss usually found with the normal approximation.
State-dependent and joint-life contracts (ALTAM): the same equation holds. Each EPV is taken over the multiple-state model, or over the joint or last-survivor status, and premiums are valued only while the paying status lasts.

Net Premium and Loss Volatility for Whole Life Using the Standard Ultimate Life Table at $i = 5\%$ ( $A_{50} = 0.18931$ , ${}^2\!A_{50} = 0.05108$ , $\ddot{a}_{50} = 17.0245$ ), find the net annual premium for a fully discrete whole life insurance of $100,000 on $(50)$ . Then find the standard deviation of $L_0$ .
Answer
 $\begin{align*} P &= \frac{100{,}000\,A_{50}}{\ddot{a}_{50}} \\ &= \frac{18{,}931}{17.0245} \\ &= 1{,}111.99 \\ \text{SD}(L_0) &= \left(100{,}000 + \frac{P}{d}\right)\sqrt{{}^2\!A_{50} - A_{50}^2} \\ &= (123{,}351.8)\sqrt{0.015242} \\ &= 15{,}229 \end{align*}$ 
The insurer expects to break even, but the loss on a single policy has a standard deviation of about $15,200.

Gross Premium for a Limited-Pay Policy A whole life policy of $100,000 on $(45)$ is fully discrete, with premiums payable for at most 20 years. Expenses are 50% of the first premium plus $250 at issue, then 5% of each later premium plus $50 at the start of each later premium year. On the SULT at $5\%$ , $A_{45} = 0.15161$ and $\ddot{a}_{45:\overline{20}|} = 12.9391$ . Find the gross premium $G$ .
Answer Equate the EPV of gross premiums to the EPV of benefits plus expenses. Renewal expenses run over $\ddot{a}_{45:\overline{20}|} - 1$ , because the first year carries the initial expense instead.
 $\begin{align*} G\,\ddot{a}_{45:\overline{20}|} &= 100{,}000A_{45} + 0.5G + 250 + (0.05G + 50)(\ddot{a}_{45:\overline{20}|} - 1) \\ G &= \frac{15{,}161 + 250 + 50(11.9391)}{12.9391 - 0.5 - 0.05(11.9391)} \\ &= \frac{16{,}007.96}{11.8421} \\ &= 1{,}351.78 \end{align*}$ 
The net premium is $15{,}161/12.9391 = 1{,}171.72$ , so expenses load the premium by about $180 a year.

Last-Survivor Insurance with Joint-Life Premiums Independent lives $(x)$ and $(y)$ have constant forces of mortality $0.02$ and $0.03$ , and $\delta = 0.05$ . A last-survivor insurance of $100,000 is paid at the moment of the second death. Premiums are payable continuously while both lives survive. Find the annual premium rate.
Answer The joint life has force $0.05$ , so $\bar{A}_{xy} = 0.05/0.10$ and $\bar{a}_{xy} = 1/0.10$ .
 $\begin{align*} \bar{A}_{\overline{xy}} &= \bar{A}_x + \bar{A}_y - \bar{A}_{xy} \\ &= \frac{0.02}{0.07} + \frac{0.03}{0.08} - 0.5 \\ &= 0.160714 \\ P &= \frac{100{,}000\,\bar{A}_{\overline{xy}}}{\bar{a}_{xy}} \\ &= \frac{16{,}071.4}{10} \\ &= 1{,}607.14 \end{align*}$ 
Valuing premiums with the last-survivor annuity $\bar{a}_{\overline{xy}} = 16.786$ would give $957.45. That premium is too small, because no premium is paid after the first death.

How it appears on the exam

FAM gives a life table (usually the SULT) or a survival model, a benefit and an expense basis. It asks for the net or gross annual premium by the equivalence principle, often with limited premium terms or a return-of-premium benefit.

FAM pairs it with the loss random variable: find $E[L_0]$ or $\text{Var}(L_0)$ , or contrast it with the portfolio percentile premium for $N$ policies, or with the premium for a given EPV of profit.

ALTAM applies it to joint-life and last-survivor contracts, and to multiple-state products: disability income, long-term care and CCRCs. The work is building each EPV from transition probabilities, and premiums are payable only in the healthy or both-alive state.

The premium found here is often the first step of a policy-value question; see Reserve.

Exam traps

Wrong premium annuity. Pricing a 20-pay policy with $\ddot{a}_x$ , or a both-alive premium with the last-survivor annuity, understates the premium.

Double-counting first-year expense. If the initial expense replaces the renewal expense at time 0, renewals run over $\ddot{a}-1$ . Using $\ddot{a}$ adds one extra renewal expense.

Net vs gross. A net premium policy value uses the net premium on the valuation basis, not the gross premium actually charged.

Variance factor. $\text{Var}(L_0)$ scales by $(S + P/d)^2$ for annual premiums and $(S + P/\delta)^2$ for continuous ones. Forgetting $P$ gives the variance of the benefit alone, which is too small.