The Equivalence Principle for Single-Premium Combinations

What the Equivalence Principle Means

A contract may provide more than one benefit: for example, a life insurance payment on death together with payments while the insured is alive. The equivalence principle gives a basic way to determine the single premium for such a contract. On a specified interest and mortality basis, the premium equals the expected present value of all benefits. The calculation depends on both the benefit amounts and their payment times.

This is a net-pricing principle: it does not include expenses, profit margins, or other loadings. The single premium is assumed payable at issue, at time zero. A premium paid then has present value equal to its own amount. If the contract instead included expenses or a different premium-payment pattern, this simple equality would need to be adjusted.

Definition: Let $Z$ be the present value, at issue, of all benefits under a contract, using the stated interest and mortality assumptions. Under the equivalence principle, the single premium $P$, paid at issue, is $$ P=E[Z]. $$ For benefits whose present values are $Z_1,\ldots,Z_r$, linearity of expectation gives $$ P=E[Z_1+\cdots+Z_r]=E[Z_1]+\cdots+E[Z_r]. $$

The components need not be statistically independent. For the expected value of their sum, linearity alone is sufficient. The important work is to identify each benefit’s amount, payment time, and condition for payment, then calculate its expected present value on a consistent basis.

Payment Timing and the Basic Formula

Use $i$ for the effective annual interest rate and $v=1/(1+i)$ for the annual discount factor. For an insured aged $x$, ${}_kp_x$ is the probability of surviving $k$ years from age $x$, and ${}_kq_x$ is the probability of dying within $k$ years. In a discrete annual model, an insurance benefit on death during year $k+1$ is paid at the end of that year. Its present-value discount factor is $v^{k+1}$.

An annuity-due pays at the beginnings of its payment periods. A payment at time $k$ is made only if the insured is alive at time $k$, so its expected payment is weighted by ${}_kp_x$. In particular, the payment at time zero requires no survival after issue; a payment at time one requires survival to time one.

Key formulas: For whole-life insurance with unit benefit payable at the end of the year of death, $A_x$ is its expected present value. For a whole-life annuity-due with unit payments while alive at times $0,1,2,\ldots$, $\ddot{a}_x$ is its expected present value. Thus, if the contract provides a death benefit $B$ and an annuity-due payment $R$ per year while alive, its single premium is $$ P=BA_x+R\ddot{a}_x. $$

For a finite term of $n$ years, the unit term insurance value $A^1_{x:\overline{n}|}$ sums the discounted death benefits for deaths in the first $n$ years. The $n$-year temporary life annuity-due value $\ddot{a}_{x:\overline{n}|}$ includes payments at times $0,1,\ldots,n-1$, each conditional on being alive at that payment time. A combination with benefit amounts $B$ and $R$ then has single premium $BA^1_{x:\overline{n}|}+R\ddot{a}_{x:\overline{n}|}$.

Why the Components Add

The additivity in the premium formula follows directly from the present value of the benefits. For example, let $K_x$ denote the curtate future lifetime: death occurs in year $K_x+1$. A whole-life death benefit $B$ has present value $Bv^{K_x+1}$. A life annuity-due paying $R$ at each integer time $k\geq 0$ has present value $R\sum_{k=0}^{\infty}v^k I(K_x\geq k)$, where $I(\cdot)$ is one if the condition holds and zero otherwise. Taking expectations gives the two standard factors, since $P(K_x=k)={}_kp_xq_{x+k}$ and $P(K_x\geq k)={}_kp_x$.

E[Z] = B\sum_{k=0}^{\infty}v^{k+1}{}_kp_xq_{x+k} + R\sum_{k=0}^{\infty}v^k{}_kp_x = BA_x+R\ddot{a}_x.

The first sum discounts each death benefit to the end of the year in which death occurs. The second discounts each annuity payment to its scheduled time and includes it only when the insured survives to that time. Since the premium is paid at time zero, equating it to this expectation gives the formula. The same reasoning applies to finite-term sums by restricting the relevant payment years.

Worked Single-Premium Calculations

Worked Example: A Whole-Life Benefit Combination

A hypothetical contract pays $12,000 at the end of the year of death and $600 at the beginning of each year while the insured is alive, starting immediately. Assume an effective annual interest rate of $i=4\%$ and an independent, constant annual survival probability of $p=0.95$ at each year. Find the net single premium.

Method. Calculate the whole-life insurance and annuity-due factors under the stated model, multiply each by its benefit amount, and add. The constant annual survival assumption means $q=1-p=0.05$, and the probability of surviving $k$ years is $p^k$.

First, $v=1/1.04=25/26$, so $vp=(25/26)(0.95)=95/104$. The geometric series for the annuity-due factor is

\ddot{a}_x=\sum_{k=0}^{\infty}(vp)^k =\frac{1}{1-vp} =\frac{1}{1-95/104} =\frac{104}{9}.

For insurance, each year-$(k+1)$ death has probability $p^kq$, and the benefit is discounted by $v^{k+1}$. Therefore,

A_x=\sum_{k=0}^{\infty}v^{k+1}p^kq =\frac{vq}{1-vp} =\frac{(25/26)(0.05)}{9/104} =\frac{5}{9}.

The expected present value of the insurance benefit is $12{,}000(5/9)=\$6{,}666.666\ldots$. The expected present value of the annuity is $600(104/9)=\$6{,}933.333\ldots$. Their unrounded sum is

12{,}000\left(\frac{5}{9}\right)+600\left(\frac{104}{9}\right) =\frac{122{,}400}{9} =\$13{,}600.

The net single premium is $13,600.00. This check uses the exact factors before rounding. The timing trap to avoid is treating the annuity as if its first payment were one year after issue: the stated annuity-due pays $600 immediately, and that payment is included in the factor’s $k=0$ term.

Worked Example: Solving for the Insurance Amount

A hypothetical insurer charges a net single premium of $10,000 for a contract that pays $500 at the beginning of each year while the insured is alive and pays an unknown whole-life death benefit at the end of the year of death. Assume $i=5\%$ and a constant annual survival probability $p=0.90$. Find the death benefit.

Method. Find both factors using the same mortality and interest assumptions, subtract the annuity’s expected present value from the total premium, and divide the remainder by the insurance factor.

Here $q=0.10$, $v=1/1.05=20/21$, and $vp=(20/21)(0.90)=6/7$. Thus

\ddot{a}_x=\frac{1}{1-vp}=\frac{1}{1-6/7}=7, \qquad A_x=\frac{vq}{1-vp} =\frac{(20/21)(0.10)}{1/7} =\frac{2}{3}.

The annuity component has expected present value $500(7)=\$3,500$. The amount of premium allocated to the death benefit is $10,000-3,500=\$6,500$. If the death benefit is $B$, then

B\left(\frac{2}{3}\right)=6{,}500, \qquad B=6{,}500\left(\frac{3}{2}\right)=\$9{,}750.

The death benefit is $9,750. Check: $9,750(2/3)=\$6,500$, and $6,500+3,500=\$10,000$. A common mistake is to divide the entire premium by $A_x$, which would incorrectly assign the annuity’s cost to the insurance benefit as well.

Worked Example: A Three-Year Term and Temporary Annuity

A hypothetical three-year contract pays $5,000 at the end of the year of death if death occurs within the three-year term. It also pays $1,000 at times $0$, $1$, and $2$, provided the insured is alive at each payment time. Assume $i=25\%$, and assume a constant annual survival probability of $p=0.80$. Find the net single premium.

Method. Use finite sums for the three-year term insurance and temporary annuity-due. Here $q=0.20$ and $v=1/1.25=0.80$, so $vp=0.64$. The temporary annuity factor includes the payment at time zero and the later payments only if alive at those times:

\ddot{a}_{x:\overline{3}|} =1+vp+(vp)^2 =1+0.64+0.4096 =2.0496.

The term insurance factor includes death in years one, two, or three. The respective discounted probabilities are $vq=0.16$, $v^2pq=0.16(0.64)$, and $v^3p^2q=0.16(0.64)^2$. Hence

A^1_{x:\overline{3}|} =vq\{1+vp+(vp)^2\} =0.16(2.0496) =0.327936.

The insurance component is $5,000(0.327936)=\$1,639.68$. The annuity component is $1,000(2.0496)=\$2,049.60$. Therefore,

P=1{,}639.68+2{,}049.60=\$3{,}689.28.

The net single premium is $3,689.28. The timing trap is to omit the time-zero annuity payment and use only $0.64+0.4096$ for its factor. The correction is to include the certain payment at issue, represented by the initial $1$ in the annuity-due sum.

Checks and Common Traps

A reliable calculation makes the contract’s timing explicit before using a factor. In particular, list when each benefit can be paid and what event makes it payable. Then apply the interest and mortality assumptions consistently to every component.

Mixing benefit timing. A death benefit payable at the end of the year of death uses a discount exponent one greater than the number of full years survived. An annuity-due payment at time $k$ is discounted by $v^k$.
Using the wrong survival condition. A payment at time $k$ requires survival to time $k$, so its probability is ${}_kp_x$. The time-zero annuity-due payment has probability one.
Combining inconsistent factors. Insurance and annuity values in one equivalence calculation must use the same interest rate, mortality assumptions, and contract timing.
Rounding components too early. Keep exact fractions or unrounded decimals through the sum, then round the final premium or benefit amount to cents.
Forgetting what the principle excludes. The equality gives a net single premium on the stated basis; it does not automatically allow for expenses, profit, or a risk margin.

A final arithmetic check is to substitute a solved amount back into the original equation. For an unknown insurance benefit, the insurance expected present value plus the other benefit values should equal the stated premium. For an unknown annuity amount, the same equality should hold after substituting the solved payment.

Key takeaway: For a single premium paid at issue, the equivalence principle sets the premium equal to the sum of the expected present values of all benefits. Correctly identifying each benefit’s payment time and survival condition is essential to obtaining the right factors.

Check Your Understanding

For each problem, use the stated model and timing. Round monetary answers to the nearest cent where needed.

A hypothetical whole-life contract pays $4,000 at the end of the year of death and $200 at the beginning of each year while alive, starting immediately. Assume $i=25\%$ and constant annual survival probability $p=0.80$. Find its net single premium. (Answer: $2,333.33)
A hypothetical contract has a $9,000 net single premium, a whole-life annuity-due of $400 per year while alive, and a whole-life death benefit $B$ payable at the end of the year of death. Assume $i=4\%$ and constant annual survival probability $p=0.96$. Find $B$. (Answer: $7,600)
A hypothetical two-year contract pays $8,000 at the end of the year of death if death occurs within two years, and $300 at times zero and one if alive at each payment time. Assume $i=25\%$ and constant annual survival probability $p=0.80$. Find the net single premium. (Answer: $2,591.20)
A whole-life contract has a net single premium of $13,000. It pays $12,000 at the end of the year of death and an unknown amount $R$ at the beginning of each year while alive, starting immediately. Assume $i=5\%$ and constant annual survival probability $p=0.90$. Find $R$, rounded to the nearest cent. (Answer: $714.29)
A hypothetical three-year contract pays $4,000 at the end of the year of death if death occurs within three years, and $600 at times zero, one, and two if alive at each payment time. Assume $i=25\%$ and constant annual survival probability $p=0.80$. Find the net single premium. (Answer: $2,541.50)

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