Set Net Single and Level Premiums by Equivalence

Introduction

A premium is not determined by the benefit amount alone. Its value also depends on when the benefit may be paid, how likely each payment time is, when premiums are due, and the interest earned on money held before a payment. The equivalence principle brings these pieces together by comparing expected present values.

This tutorial applies that principle to net premiums: premiums set to cover the expected present value of the insurance benefits, without expenses or profit margins. The central task is to value the benefit cash flows and the premium cash flows on the same mortality and interest basis, with the timing of each payment stated explicitly. This approach is aligned with the SOA Exam FAM (Fundamentals of Actuarial Mathematics) syllabus.

Definition: Under the equivalence principle, the net premium is chosen so that the expected present value of premiums equals the expected present value of benefits, on the stated mortality and interest basis. A net single premium is one premium paid at issue. A level net premium is a constant amount paid at each specified premium date while the policyholder is eligible to pay.

Set the Cash-Flow Timeline First

For a discrete insurance benefit of amount $B$, payable at the end of the year of death, the probability of a payment at the end of year $k+1$ is ${}_kp_x q_{x+k}$. Here ${}_kp_x$ is the probability of surviving $k$ years from age $x$, and $q_{x+k}$ is the probability of dying within the following year. At effective annual interest rate $i$, the discount factor is $v=1/(1+i)$. Thus a payment at time $k+1$ has present value $Bv^{k+1}$.

For a term of $n$ years, with no benefit on survival to the end of the term, the expected present value of benefits is therefore

B A^{1}_{x:\overline{n}|} = B\sum_{k=0}^{n-1} v^{k+1}\,{}_kp_x q_{x+k}.

If level premiums are payable annually in advance, the first premium is due at issue, and later premiums are due at the start of each policy year while the insured is alive. For an $n$-year premium period, their expected present value per dollar of annual premium is

\ddot{a}_{x:\overline{n}|} = \sum_{k=0}^{n-1} v^k\,{}_kp_x.

The term $k=0$ is the premium at issue: ${}_0p_x=1$ and $v^0=1$. A premium at time $k$ is included only if the insured survives to that premium date. A contract with different premium dates or a different rule for when premiums cease requires a different premium-value sum.

Why the Premium Formula Works

Suppose the level annual premium is $P$, payable in advance for at most $n$ years while the insured is alive. Each possible premium payment has amount $P$, and its expected present value is its amount multiplied by the probability of being payable and the discount factor for its payment date. Adding those terms gives $P\ddot{a}_{x:\overline{n}|}$. Applying equivalence to the term benefit gives the following derivation:

Value the benefits.
For a death benefit $B$ paid at the end of the year of death, add $Bv^{k+1}$ weighted by the probability ${}_kp_xq_{x+k}$ for each covered year.

Value the premiums.
For premiums payable annually in advance while alive, add $P v^k {}_kp_x$ for each eligible premium date $k=0,\ldots,n-1$.

Apply equivalence and solve.
Set the two expected present values equal and divide by the premium value per dollar.

B\sum_{k=0}^{n-1}v^{k+1}{}_kp_xq_{x+k} = P\sum_{k=0}^{n-1}v^k{}_kp_x, \qquad P= \frac{B A^{1}_{x:\overline{n}|}} {\ddot{a}_{x:\overline{n}|}}.

The same reasoning gives a single premium: when one premium is paid at issue with certainty, its expected present value is the amount paid. Hence the net single premium is the expected present value of benefits. The ratio formula for $P$ applies only to the premium schedule used to form its denominator; it is not a universal formula independent of payment timing.

Worked Example: Three-Year Term Insurance with Premiums in Advance

A hypothetical policy pays $1,000 at the end of the year of death if death occurs within three years. The annual effective interest rate is 5%. The conditional annual mortality probabilities are $q_x=0.10$, $q_{x+1}=0.20$, and $q_{x+2}=0.25$. Find the net single premium and the level annual net premium payable at issue and at the start of each later year if the insured is alive.

Method and timing. Calculate the probability of death in each covered year for the benefit value. For the level premium, use the advance-payment dates $0,1,2$. The premiums are not payable after the three-year term ends.

Survival and death probabilities. The probability of being alive at the start of years 1, 2, and 3 is $1$, $0.90$, and $0.90(0.80)=0.72$. The probabilities of death in the three years are $0.10$, $0.90(0.20)=0.18$, and $0.72(0.25)=0.18$.

Discount factors. $v=1/1.05=0.952381$, $v^2=0.907029$, and $v^3=0.863838$, shown to six decimal places for the calculation.

Net single premium. Weight each end-of-year benefit by the probability of death in that year and its discount factor:

1000(0.10v+0.18v^2+0.18v^3) = 1000(0.095238+0.163265+0.155491) = \$413.99.

Level premium value. The premium dates are $0,1,2$, and the probabilities of being alive on those dates are $1,0.90,0.72$. Therefore, the expected present value per dollar of annual premium is

$$ 1+0.90v+0.72v^2 = 1+0.857143+0.653061 = 2.510204. $$

The annual premium is $413.994\div2.510204=164.924$, so the net level premium is $164.92 to the nearest cent. As a check, $164.924(2.510204)$ is approximately $413.99, matching the benefit value before rounding. A common mistake is to put the first premium at time 1; that would omit the certain premium due at issue and price a different payment schedule.

When Benefits Include Survival

An endowment benefit pays on death during the term or on survival to the end of the term. Its expected present value includes both kinds of payment. For an $n$-year endowment with benefit $B$, death benefits paid at the end of the year of death, and a survival benefit paid at time $n$, the value is

B\left( \sum_{k=0}^{n-1}v^{k+1}{}_kp_xq_{x+k} + v^n{}_np_x \right).

This expression weights each possible payment by its probability and discounts it to issue. If premiums are annual in advance for at most $n$ years while alive, their value per dollar is still $\sum_{k=0}^{n-1}v^k{}_kp_x$. Equivalence then gives the level premium as the endowment benefit value divided by that premium value. The insurance benefit and the premium schedule must each be valued on their own timing rules.

Worked Example: Two-Year Endowment with a Two-Payment Premium Schedule

A hypothetical contract pays $1,000 at the end of the year of death if death occurs within two years, or $1,000 at time 2 if the insured survives two years. The annual effective interest rate is 4%, with $q_x=0.10$ and $q_{x+1}=0.20$. Find the net single premium and the annual net premium payable at issue and at time 1 if alive.

Method. Separate the death outcomes from survival to time 2, then value each payment. The annual premium is payable only at times 0 and 1, so its value per dollar is $1+{}_1p_xv$.

The probability of death in year 1 is $0.10$. The probability of death in year 2 is $0.90(0.20)=0.18$, and the probability of survival to time 2 is $0.90(0.80)=0.72$. The discount factors are $v=1/1.04=0.961538$ and $v^2=0.924556$.

The net single premium is

1000(0.10v+0.18v^2+0.72v^2) = 1000(0.096154+0.166420+0.665680) = \$928.25.

The expected present value per dollar of annual premium is

$$ 1+0.90v = 1+0.865385 = 1.865385. $$

Thus the annual level premium is $928.254\div1.865385=497.621$, or $497.62 to the nearest cent. Multiplying the unrounded premium by $1.865385$ recovers approximately $928.25. The year-2 survival benefit belongs in the benefit value, but there is no premium at time 2 under the stated schedule.

Whole-Life Level Premiums

The same equivalence principle applies to whole-life insurance. For a benefit $B$ paid at the end of the year of death, its expected present value is $B A_x$, where the discrete whole-life insurance value is the sum over all possible years of death. If annual premiums are payable in advance while the insured is alive, continuing for life, their value per dollar is $\ddot{a}_x$. Therefore $P=B A_x/\ddot{a}_x$, provided both values use the same mortality and interest basis and the stated payment timing.

Worked Example: Whole-Life Insurance with Constant Annual Mortality

A hypothetical whole-life policy pays $10,000 at the end of the year of death. Assume the conditional probability of death is $q=0.20$ in every year, so $p=0.80$, and the effective annual interest rate is 25%, so $v=0.80$. Premiums are paid annually in advance while the insured is alive. Find the net single premium and the level annual net premium.

Method. Use the geometric series for survival-weighted benefit values and for advance premiums. Under the stated assumptions, the probability of surviving $k$ years is $p^k$.

A death benefit is paid at the end of year $k+1$ with probability $p^kq$. Its value per dollar of benefit is therefore

A_x=\sum_{k=0}^{\infty}v^{k+1}p^kq =\frac{vq}{1-vp} =\frac{(0.80)(0.20)}{1-(0.80)(0.80)} =\frac{0.16}{0.36} =\frac{4}{9}.

The net single premium is $10{,}000(4/9)=\$4{,}444.44$. For advance premiums, the value per dollar is

\ddot{a}_x=\sum_{k=0}^{\infty}(vp)^k =\frac{1}{1-vp} =\frac{1}{0.36} =\frac{25}{9}.

Consequently, $P=10{,}000(4/9)/(25/9)=\$1{,}600.00$ per year. The expected present value of these premiums is $1{,}600(25/9)=\$4{,}444.44$, which equals the benefit value. A common error is to use $v^k$ for a death benefit paid at the end of year $k+1$; that discounts the benefit as though it were paid one year earlier.

Timing Checks and Common Traps

Before finalizing a premium, write down every possible payment date and the condition for payment. Then check that the discount exponent matches that date and that each survival probability describes the insured's status at that date. These checks often catch errors more quickly than redoing the arithmetic.

Confusing the first premium date. An annual premium in advance includes a payment at issue. An annual premium in arrears does not. Their expected present values differ.
Discounting death benefits to the wrong date. A benefit at the end of year $k+1$ is discounted by $v^{k+1}$, not $v^k$.
Using the wrong survival condition. A premium at time $k$ is payable only if the insured survives to time $k$; a death benefit for year $k+1$ requires survival to its start and death in that year.
Including premiums beyond the premium period. A limited premium schedule stops when specified, even if coverage continues. Conversely, a premium due at issue is not omitted just because the policyholder has not yet survived a year.
Mixing benefit and premium assumptions. The numerator and denominator must use the same stated mortality and interest basis, while still reflecting their separate cash-flow timing.

A useful final check is to multiply the proposed level premium by the expected present value of its payment stream. Before rounding, that product should equal the expected present value of benefits under the equivalence principle. The resulting net premium covers expected benefits on the chosen basis; it is not a gross premium that includes expenses.

Check Your Understanding

For each problem, identify the payment timing before calculating. Amounts in the answers are rounded to the nearest cent where appropriate.

A hypothetical two-year term policy pays $5,000 at the end of the year of death. Let $i=0$, $q_x=0.10$, and $q_{x+1}=0.20$. Premiums are annual in advance at times 0 and 1 while alive. Find the level net premium. (Answer: $736.84)
A one-year hypothetical policy pays $12,000 at the end of the year if death occurs during that year. Let $i=0.04$ and $q_x=0.05$. Find the net single premium. (Answer: $576.92)
A hypothetical three-year term policy pays $2,000 at the end of the year of death. Let $i=0$, $q_x=0.05$, $q_{x+1}=0.10$, and $q_{x+2}=0.20$. Find the level net premium payable at times 0, 1, and 2 while alive. (Answer: $225.31)
A hypothetical two-year endowment pays $10,000 at the end of the year of death or at time 2 on survival. Let $i=0$, $q_x=q_{x+1}=0.10$, and let premiums be payable at times 0 and 1 while alive. Find the level net premium. (Answer: $5,263.16)
A hypothetical whole-life policy pays $16,000 at the end of the year of death. Assume $q=0.25$ every year, $p=0.75$, and $i=1/3$, so $v=0.75$. Find the net single premium and the annual net premium payable in advance while alive. (Answer: Net single premium $6,857.14; annual premium $3,000.00)

What You'll Learn