Term Insurance Premiums from Temporary Annuities

Why the Premium Uses a Temporary Annuity-Due

A level premium for term insurance is paid only while the insured is alive and the coverage period has not ended. That payment pattern matters: premiums are due at the start of each covered year, while a death benefit is paid at the end of the year of death. To calculate the level net premium, compare the expected present value of those benefits with the expected present value of premiums. The premium present value follows the pattern of a temporary annuity-due, not an annuity paid regardless of survival.

The method in this tutorial is aligned with the SOA Exam FAM (Fundamentals of Actuarial Mathematics) syllabus. It uses the equivalence principle: set the expected present values of benefits and premiums equal, then solve for the level premium. The result is a net premium, with no allowance for expenses, profit, or other loadings.

Definition: Consider an $n$-year discrete term insurance issued at age $x$. It pays a benefit of 1 at the end of the year of death if death occurs within the $n$-year term, and pays nothing if the insured survives to the end of the term. Define ${}_kp_x$ as the probability of surviving $k$ years from age $x$, with ${}_0p_x=1$, and $q_{x+k}$ as the probability of dying in the year after surviving to age $x+k$. The term insurance present value is $$ A^1_{x:\overline{n}|}=\sum_{k=0}^{n-1} v^{k+1}{}_kp_xq_{x+k}. $$ The matching temporary life annuity-due present value is $$ \ddot{a}_{x:\overline{n}|}=\sum_{k=0}^{n-1}v^k{}_kp_x. $$ Here $v=1/(1+i)$, where $i$ is the effective annual interest rate. The annuity pays 1 at times $0,1,\ldots,n-1$, provided the insured is alive at the payment time.

The Level Net Premium Formula

Suppose the benefit is $B$, and a level premium $P$ is due annually at the start of each year while the insured is alive and the term has not ended. The benefit present value scales by $B$, and the premium present value scales by $P$. Equating expected present values gives

B A^1_{x:\overline{n}|}=P\ddot{a}_{x:\overline{n}|}.

Provided $\ddot{a}_{x:\overline{n}|}>0$, solve for $P$:

Theorem: For discrete $n$-year term insurance with benefit $B$ payable at the end of the year of death, and level premiums payable in advance while the insured is alive, the annual level net premium is $$ P=B\frac{A^1_{x:\overline{n}|}}{\ddot{a}_{x:\overline{n}|}}. $$ For a unit benefit, the premium per year is $A^1_{x:\overline{n}|}/\ddot{a}_{x:\overline{n}|}$.

The numerator and denominator must describe the same term and the same mortality and interest assumptions. The insurance value counts only deaths during years 1 through $n$. The annuity value counts premiums at issue and at later anniversaries only if the insured is alive then. There is no premium at time $n$ under this convention: the term has ended.

Why the Quotient Holds

The formula follows by conditioning each cash flow on whether it occurs. In year $k+1$, the probability of a death benefit payment is ${}_kp_xq_{x+k}$; its payment time is $k+1$, so its discount factor is $v^{k+1}$. A premium at time $k$ is paid if the insured has survived $k$ years, an event with probability ${}_kp_x$; that premium is discounted by $v^k$. Summing these expected present values gives the two factors in the equivalence equation.

The derivation can be summarized term by term:

List benefit payment years.
A death in year $k+1$, for $k=0,\ldots,n-1$, occurs with probability ${}_kp_xq_{x+k}$ and produces a benefit at time $k+1$.

Discount expected benefits.
Multiply each payment by its probability and discount factor $v^{k+1}$, then sum. This is $A^1_{x:\overline{n}|}$ for a unit benefit.

List premium payment times.
At time $k$, for $k=0,\ldots,n-1$, a premium is due if the insured is alive. Its probability is ${}_kp_x$, and its discount factor is $v^k$.

Equate and solve.
The expected present value of premiums is $P\ddot{a}_{x:\overline{n}|}$. Equating it to $B A^1_{x:\overline{n}|}$ and dividing by the annuity value yields the level net premium.

This argument also explains when the formula does not apply as written. If premiums are payable at the end of each year, or continue after death, the premium cash-flow pattern changes. If the benefit is paid immediately on death rather than at the end of the year, the benefit present value changes as well. The benefit timing and premium timing must match the contract being valued.

Worked Calculations

Worked Example: Two-Year Term With Cumulative Survival Probabilities

A hypothetical policy pays 1 at the end of the year of death during a two-year term. The effective annual interest rate is $i=10\%$. The survival probabilities from issue are ${}_1p_x=0.90$ and ${}_2p_x=0.80$. Premiums are paid annually in advance while alive. Find the level net premium per unit of benefit.

Method. Find unconditional death probabilities for each covered year, calculate the term insurance and temporary annuity-due values, and take their quotient. This avoids treating a cumulative survival probability as a conditional one.

The probability of death in year 1 is $1-0.90=0.10$. The probability of death in year 2, measured from issue, is ${}_1p_x-{}_2p_x=0.90-0.80=0.10$. The discount factor is $v=1/1.10=10/11$.

Thus the unit-benefit insurance value is

A^1_{x:\overline{2}|} =0.10v+0.10v^2 =0.10\left(\frac{10}{11}\right)+0.10\left(\frac{100}{121}\right) =\frac{21}{121} =0.1735537\ldots.

The premium payments are at time 0 and at time 1 if alive then. Therefore

\ddot{a}_{x:\overline{2}|} =1+0.90v =1+0.90\left(\frac{10}{11}\right) =\frac{20}{11} =1.8181818\ldots.

The level net premium per unit of benefit is

P=\frac{21/121}{20/11} =\frac{231}{2420} =0.0954545\ldots.

To four decimal places, the answer is 0.0955 per unit of benefit per year. Check: $0.0954545\ldots\times1.8181818\ldots=0.1735537\ldots$, which matches the insurance value.

Worked Example: Constant Annual Mortality Over a Three-Year Term

A hypothetical three-year term policy pays 1 at the end of the year of death. Each year, conditional on being alive at the start of that year, the probability of survival through the year is $p=0.80$ and the probability of death is $q=0.20$. The effective annual interest rate is $i=5\%$. Premiums are due at the start of each covered year while alive. Find the level net premium per unit of benefit.

Method. Build both present values using the same annual survival pattern. Factoring the common sum provides a useful check and avoids dropping a year from either cash-flow stream.

The discount factor is $v=1/1.05=20/21$. The probability of surviving $k$ years is $p^k$, so each premium term is $v^kp^k=(pv)^k$. A death in year $k+1$ has probability $p^kq$, and its discounted benefit term is $v^{k+1}p^kq=qv(pv)^k$. For three years, set $r=pv=(4/5)(20/21)=16/21$. Then

\ddot{a}_{x:\overline{3}|} =1+r+r^2 =1+\frac{16}{21}+\frac{256}{441} =\frac{1033}{441} =2.3424036\ldots.

The insurance value is

A^1_{x:\overline{3}|} =qv(1+r+r^2) =\frac{1}{5}\frac{20}{21}\frac{1033}{441} =\frac{4132}{9261} =0.4461719\ldots.

Taking the quotient cancels the common finite sum:

P=\frac{A^1_{x:\overline{3}|}}{\ddot{a}_{x:\overline{3}|}} =qv =\frac{4}{21} =0.1904761\ldots.

To four decimal places, the level net premium is 0.1905 per unit of benefit per year. Check: $0.1904761\ldots\times2.3424036\ldots=0.4461719\ldots$. The cancellation works here because mortality is constant by year; it should not be assumed when the annual survival probabilities vary.

Worked Example: Varying Survival Probabilities and a Dollar Benefit

A hypothetical three-year policy pays $B=\$20{,}000$ at the end of the year of death. The cumulative survival probabilities are ${}_1p_x=0.95$, ${}_2p_x=0.85$, and ${}_3p_x=0.70$. The effective annual interest rate is $i=5\%$. Premiums are due at the start of each covered year while alive. Find the annual level net premium.

Method. Use differences in cumulative survival to obtain each year's unconditional death probability, then calculate both present values explicitly. This prevents using the wrong conditional mortality rate in a year with changing mortality.

The unconditional probabilities of death in years 1, 2, and 3 are, respectively, $1-0.95=0.05$, $0.95-0.85=0.10$, and $0.85-0.70=0.15$. The discount factor is $v=20/21$, with $v^2=400/441$ and $v^3=8000/9261$. The unit-benefit insurance value is

\begin{aligned} A^1_{x:\overline{3}|} &=0.05v+0.10v^2+0.15v^3\\ &=\frac{1}{21}+\frac{40}{441}+\frac{400}{3087}\\ &=\frac{2481}{9261} =0.2678976\ldots. \end{aligned}

Premiums can be paid at time 0, time 1 if alive, and time 2 if alive. Hence

\begin{aligned} \ddot{a}_{x:\overline{3}|} &=1+0.95v+0.85v^2\\ &=1+\frac{19}{21}+\frac{340}{441}\\ &=\frac{1180}{441} =2.675737\ldots. \end{aligned}

The premium per dollar of benefit is

\frac{2481/9261}{1180/441} =\frac{2481}{24780} =0.1001210\ldots.

Multiplying by the benefit gives $P=20{,}000(0.1001210\ldots)=\$2{,}002.42$, rounded to the nearest cent. Check: using the unrounded premium, $P\ddot{a}_{x:\overline{3}|}=\$5{,}357.95\ldots$, which equals $20{,}000A^1_{x:\overline{3}|}$ before rounding. The displayed dollar premium is rounded only at the final step.

Common Timing and Calculation Traps

The premium formula is simple, but its two present-value factors represent different events and different payment times. A calculation can have correct mortality inputs and still be wrong if it treats the two streams as having the same timing.

Using an ordinary annuity instead of an annuity-due. Premiums are paid at the start of a year under the stated contract. The first premium is at time 0, so its discount factor is 1. Moving the first payment to time 1 changes the premium stream.
Including a premium at the end of the term. For an $n$-year term with premiums due at the start of each covered year, the final possible premium is at time $n-1$, conditional on survival to then. There is no payment at time $n$.
Discounting the benefit to the wrong time. Under discrete end-of-year-of-death benefit timing, death in year $k+1$ leads to a payment at time $k+1$, discounted by $v^{k+1}$, not $v^k$.
Confusing conditional and unconditional death probabilities. The probability of death in year $k+1$ from issue is ${}_kp_xq_{x+k}$. If cumulative survival probabilities are supplied instead, the unconditional probability of death in that year is the difference between survival to the start and survival to the end of that year.
Using a unit-benefit premium for a different benefit amount. The quotient is per unit of benefit. For benefit $B$, multiply that quotient by $B$.

For example, with ${}_1p_x=0.90$, ${}_2p_x=0.80$, and $i=10\%$, a common wrong calculation is to use $0.20$ as the conditional probability of death in year 2. That is the cumulative probability of death by the end of year 2, not the conditional year-2 mortality rate. The correct unconditional death probabilities are $0.10$ in year 1 and $0.90-0.80=0.10$ in year 2. Thus the insurance value is $0.10v+0.10v^2$, and the matching premium value is $1+0.90v$, giving $0.0955$ per unit of benefit, to four decimal places. Using $0.20$ as the second-year probability overstates the benefit value.

Key takeaway: For discrete term insurance with annual premiums in advance while alive, calculate the term benefit value using end-of-year death payments and divide it by the temporary annuity-due value over the identical term. State the timing assumptions, keep cumulative and conditional probabilities distinct, and check that the expected present values balance.

Check Your Understanding

For each problem, use the stated end-of-year benefit and beginning-of-year premium timing. Round premium rates to four decimal places and dollar premiums to the nearest cent.

A two-year unit-benefit term insurance has constant annual survival probability $p=0.80$ and effective annual interest rate $i=0\%$. Find the level net premium per unit of benefit. (Answer: 0.2000 per year)
A three-year unit-benefit term insurance has cumulative survival probabilities ${}_1p_x=0.90$, ${}_2p_x=0.75$, and ${}_3p_x=0.60$, with effective annual interest rate $i=5\%$. Find the level net premium per unit of benefit. (Answer: 0.1422 per year)
A three-year term has constant annual survival probability $p=0.75$, and the effective annual interest rate is $i=8\%$. Find the level net premium per unit of benefit. (Answer: 0.2315 per year)
A three-year unit-benefit term has cumulative survival probabilities ${}_1p_x=0.96$, ${}_2p_x=0.88$, and ${}_3p_x=0.76$, with effective annual interest rate $i=4\%$. Calculate the level net premium per unit of benefit. (Answer: 0.0801 per year)
A two-year term pays $\$20{,}000$ at the end of the year of death. Cumulative survival probabilities are ${}_1p_x=0.95$ and ${}_2p_x=0.90$, and the effective annual interest rate is $i=5\%$. Premiums are annual and payable in advance while alive. Find the level annual net premium in dollars. (Answer: $\$976.19$)

What You'll Learn