Benefit-Premium Identities for Net Premiums

Why Insurance and Annuity Values Are Connected

A net level premium equates the expected present value of premiums with the expected present value of benefits. In many discrete life insurance calculations, the two present values are linked: the same survival and death probabilities appear in both. Using that relationship can simplify a calculation and provide an independent check on its result.

The identity depends on the contract’s timing. In this tutorial, premiums are payable annually in advance while the insured is alive, and a death benefit is payable at the end of the year of death. Interest is at an effective annual rate $i$, with $v=1/(1+i)$ and $d=1-v$. These are discrete, annual-payment identities; they should not be applied unchanged when benefit or premium timing differs.

Definition: For a life aged $x$, ${}_kp_x$ is the probability of surviving $k$ years, and ${}_kq_x=1-{}_kp_x$ is the probability of dying within $k$ years. The temporary life annuity-due factor for $n$ years is $$ \ddot{a}_{x:\overline{n}|}=\sum_{k=0}^{n-1}v^k\,{}_kp_x. $$ The $n$-year term insurance factor, for a unit benefit at the end of the year of death, is $$ A^{1}_{x:\overline{n}|}=\sum_{k=0}^{n-1}v^{k+1}\,{}_kp_xq_{x+k}. $$ The $n$-year endowment insurance factor adds a unit benefit at time $n$ if the insured survives to that time: $$ A_{x:\overline{n}|}=A^{1}_{x:\overline{n}|}+v^n{}_np_x. $$

The annuity-due factor includes a payment at time 0, then one at the beginning of each later year in which the insured is alive, up to $n$ possible payments. The term insurance factor includes a death benefit only for deaths in the first $n$ years. The pure endowment component $v^n{}_np_x$ is the expected present value of the maturity benefit, not an additional annuity payment.

The Temporary Insurance–Annuity Identity

For a term contract, the key relationship is

Theorem: Under the discrete timing assumptions above, $$ A^{1}_{x:\overline{n}|}+d\ddot{a}_{x:\overline{n}|}=1-v^n{}_np_x. $$ Equivalently, the right side is the probability-weighted present value of receiving one unit at time 0 unless the insured survives to time $n$.

The identity follows by combining the death benefit and annuity terms year by year. In year $k+1$, the probability of death in that year is ${}_kp_xq_{x+k}={}_kp_x-{}_{k+1}p_x$. Also $d=1-v$. Therefore,

\begin{aligned} A^{1}_{x:\overline{n}|} &=\sum_{k=0}^{n-1}v^{k+1}\bigl({}_kp_x-{}_{k+1}p_x\bigr),\\ d\ddot{a}_{x:\overline{n}|} &=\sum_{k=0}^{n-1}(v^k-v^{k+1}){}_kp_x. \end{aligned}

When these sums are added, the terms involving $v^{k+1}{}_kp_x$ cancel successively. The initial term is ${}_0p_x=1$, and the final uncancelled term is $-v^n{}_np_x$. Thus the sum is $1-v^n{}_np_x$, as stated. The surviving term is why the finite-term identity is not generally equal to 1.

For a whole-life contract, take the limit as $n$ increases. Since $0<v<1$ and ${}_np_x\leq 1$, $v^n{}_np_x$ tends to zero. The corresponding identity is

A_x+d\ddot{a}_x=1.

Using the Identities for Net Level Premiums

The equivalence principle sets the expected present value of premiums equal to the expected present value of benefits. With a level annual premium $P$ payable in advance while the insured is alive, the premium present value is $P\ddot{a}$, using the annuity factor for the premium payment period. If the benefit factor is $A$, then $P=A/\ddot{a}$.

For term insurance, the temporary identity gives an alternate expression for the benefit factor. For endowment insurance, adding the pure endowment to the term benefit gives a particularly simple identity. For whole-life insurance, the limiting identity applies.

\begin{aligned} \text{Term:}\quad P&=\frac{A^{1}_{x:\overline{n}|}}{\ddot{a}_{x:\overline{n}|}} =\frac{1-v^n{}_np_x}{\ddot{a}_{x:\overline{n}|}}-d,\\[4pt] \text{Endowment:}\quad A_{x:\overline{n}|}+d\ddot{a}_{x:\overline{n}|}&=1, \qquad P=\frac{1}{\ddot{a}_{x:\overline{n}|}}-d,\\[4pt] \text{Whole life:}\quad A_x+d\ddot{a}_x&=1, \qquad P=\frac{1}{\ddot{a}_x}-d. \end{aligned}

Each premium formula assumes that the premium payment period matches the annuity factor shown. If premiums stop before the benefit period ends, or if premium payments are in arrears, the premium present value must be changed before applying equivalence.

Worked Example: Whole-Life Insurance with Constant Annual Mortality

A hypothetical whole-life policy pays 1 at the end of the year of death and charges level premiums annually in advance while the insured is alive. The effective annual interest rate is $i=5\%$. Assume independent year-to-year mortality with the same conditional survival probability $p=0.8$ each year, so $q=0.2$. Find the net level annual premium per unit of benefit.

Method. Calculate the whole-life insurance and annuity factors as geometric series, then check them with the whole-life identity.

First, $v=1/1.05=20/21$, so $vp=(20/21)(0.8)=16/21$. The annuity-due factor is

\ddot{a}_x=\sum_{k=0}^{\infty}(vp)^k =\frac{1}{1-16/21} =\frac{21}{5} =4.2.

The death benefit is paid at time $k+1$ if the insured survives $k$ years and dies in the next year. Thus

A_x=\sum_{k=0}^{\infty}v^{k+1}p^kq =\frac{vq}{1-vp} =\frac{(20/21)(0.2)}{5/21} =0.8.

The discount rate is $d=1-v=1/21$. Check the identity: $d\ddot{a}_x=(1/21)(21/5)=0.2$, so $A_x+d\ddot{a}_x=0.8+0.2=1$. The net premium is

P=\frac{A_x}{\ddot{a}_x} =\frac{0.8}{4.2} =\frac{4}{21} \approx 0.1905.

The answer is $0.1905$ per unit of benefit, rounded to four decimal places. The identity check confirms the computed factors before the premium is calculated.

Worked Example: Three-Year Term Insurance

A hypothetical policy pays 1 at the end of the year of death if death occurs within three years. Premiums are paid annually in advance while the insured is alive, for at most three years. The effective annual interest rate is $i=10\%$, and the conditional probability of surviving each year is $p=0.9$, independently from year to year. Find the net level annual premium per unit of benefit.

Method. Use the temporary insurance–annuity identity, retaining the pure endowment term on its right-hand side.

Here $v=10/11$, $d=1/11$, and $vp=(10/11)(0.9)=9/11$. The three-year annuity-due factor is

\ddot{a}_{x:\overline{3}|} =1+\frac{9}{11}+\left(\frac{9}{11}\right)^2 =\frac{121+99+81}{121} =\frac{301}{121}.

The probability of surviving three years is $0.9^3=0.729$, so the pure endowment factor is

v^3{}_3p_x =\left(\frac{10}{11}\right)^3(0.9)^3 =\left(\frac{9}{11}\right)^3 =\frac{729}{1331}.

Apply the identity:

A^{1}_{x:\overline{3}|} =1-\frac{729}{1331}-\frac{1}{11}\left(\frac{301}{121}\right) =\frac{1331-729-301}{1331} =\frac{301}{1331}.

Therefore,

P=\frac{301/1331}{301/121} =\frac{1}{11} \approx 0.0909.

The net premium is $0.0909$ per unit of benefit, rounded to four decimal places. As a check, the benefit factor plus $d$ times the annuity factor is $301/1331+301/1331=602/1331=1-729/1331$, as required.

Worked Example: Two-Year Endowment Insurance

A hypothetical two-year endowment policy pays 1 at the end of the year of death if death occurs within two years, or pays 1 at the end of year 2 if the insured survives. The net level premium is paid annually in advance while the insured is alive. The effective annual interest rate is $i=5\%$; assume a conditional survival probability of $0.8$ in each year. Find the net premium per unit of benefit.

Method. Calculate the temporary annuity factor, then use the endowment identity to obtain the benefit factor and premium.

With $v=20/21$, the two-year annuity-due factor is

\ddot{a}_{x:\overline{2}|} =1+v(0.8) =1+\frac{16}{21} =\frac{37}{21}.

The first-year death benefit factor is $v(0.2)=4/21=84/441$. The second-year death benefit factor is $v^2(0.8)(0.2)=64/441$. Thus

A^{1}_{x:\overline{2}|} =\frac{84}{441}+\frac{64}{441} =\frac{148}{441}.

The maturity benefit factor is $v^2(0.8)^2=256/441$, so the endowment benefit factor is $A_{x:\overline{2}|}=(148+256)/441=404/441$. The identity check gives

A_{x:\overline{2}|}+d\ddot{a}_{x:\overline{2}|} =\frac{404}{441}+\frac{1}{21}\frac{37}{21} =\frac{404+37}{441} =1.

The premium is

P=\frac{404/441}{37/21} =\frac{404}{777} \approx 0.5199.

The net premium is $0.5199$ per unit of benefit, rounded to four decimal places. The endowment identity provides a check on the combined death and maturity benefits, not on the term benefit alone.

Common Mistake: Dropping the Survival Term

A frequent error is to use $A^{1}_{x:\overline{n}|}+d\ddot{a}_{x:\overline{n}|}=1$ for finite-term insurance. That is the whole-life identity, not the term identity. In the three-year example, applying it incorrectly would give

A^{1}_{x:\overline{3}|} \overset{\text{wrong}}{=}1-\frac{301}{1331} =\frac{1030}{1331},

instead of the correct $301/1331$. The error is the omitted survival term $v^3{}_3p_x=729/1331$. The correction is to use $A^{1}_{x:\overline{n}|}+d\ddot{a}_{x:\overline{n}|}=1-v^n{}_np_x$ for term insurance, and to add that pure endowment factor to the term benefit before using the endowment identity.

These identities matter because they expose errors in timing, benefit type, or payment period. Before using one, identify whether the contract pays only on death during a finite term, pays a survival benefit as well, or covers the whole of life. Then confirm that the annuity factor reflects the actual premium dates and survival-contingent payment conditions. This approach is aligned with the SOA Exam FAM (Fundamentals of Actuarial Mathematics) syllabus.

Check Your Understanding

For each problem, use the identity appropriate to the stated benefit and show your own intermediate calculations before checking the final answer.

A hypothetical whole-life policy pays 1 at the end of the year of death and charges annual premiums in advance while the insured is alive. The effective annual interest rate is $25\%$, and the conditional survival probability is $0.75$ each year. Find the net level premium per unit of benefit. (Answer: $0.2000$)
A hypothetical two-year term policy pays 1 at the end of the year of death, with annual premiums in advance while the insured is alive. The effective annual interest rate is $25\%$, and the conditional survival probability is $0.75$ each year. Find the term insurance factor and net level premium. (Answer: $A^{1}_{x:\overline{2}|}=0.3200$; $P=0.2000$)
A hypothetical two-year endowment policy pays 1 at death during the term or at the end of year 2 on survival. Premiums are annual in advance while alive. The effective annual interest rate is $10\%$; the first-year conditional survival probability is $0.8$, and the second-year conditional survival probability is $0.5$. Find the endowment benefit factor and net level premium. (Answer: $A_{x:\overline{2}|}=102/121$; $P=102/209\approx0.4880$)
A hypothetical four-year term policy has a valid mortality-table annuity-due factor of $3.6$ and a pure endowment factor $v^4{}_4p_x=0.7$. The effective annual interest rate is $5\%$, and premiums are annual in advance for at most four years while alive. Use the identity to find the term insurance factor and net level premium. (Answer: $A^{1}_{x:\overline{4}|}=0.1286$; $P\approx0.0357$)

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