Term Insurance Net Level Premiums

Why Term Insurance Premiums Depend on Two Present Values

A term insurance contract covers death only during a specified period. Its annual net level premium must fund the expected present value of benefits payable during that period. But premiums are not necessarily paid for every year of coverage: under the usual annual-in-advance arrangement, a premium is paid at issue and on each anniversary that the insured is alive. The expected number and timing of those payments depend on survival.

The key calculation is therefore a ratio. The numerator is the expected present value of the insured benefit; the denominator is the expected present value of one dollar of premiums. This tutorial uses the equivalence principle to calculate that ratio for discrete term insurance. All contracts and numerical data below are hypothetical.

Definition: An $n$-year discrete term insurance pays a benefit $b$ at the end of the year of death if the insured dies within the next $n$ years, and pays no death benefit if the insured survives the term. Assume annual premiums are payable in advance at times $0,1,\ldots,n-1$, provided the insured is then alive. A net level premium is a constant annual premium calculated using the equivalence principle, without expenses or profit loadings.

The Benefit and Premium Factors

Let $i$ be the effective annual interest rate and $v=1/(1+i)$. For a life aged $x$, write ${}_kp_x$ for the probability of surviving $k$ years and ${}_kq_x$ for the probability of dying within $k$ years. The one-year conditional death probability at age $x+k$ is $q_{x+k}$, so the probability of surviving $k$ years and then dying in the following year is ${}_kp_xq_{x+k}$.

A death in year $k+1$ produces a benefit at time $k+1$. Its discounted expected contribution per dollar of benefit is therefore $v^{k+1}{}_kp_xq_{x+k}$. Summing over the $n$ covered years gives the term insurance factor:

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

The premium paid at time $k$ is made only if the insured has survived to that time. Its expected present value per dollar of annual premium is $v^k{}_kp_x$. The time-0 premium is certain, since the insured is alive at issue. Thus the premium factor is the temporary life annuity-due value:

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

The bar and term indicate that premiums stop after the $n$-year period. In particular, there is no premium at time $n$ under the stated arrangement. A benefit payment at the end of the final covered year is included, while a premium at that time is not.

Theorem: For a level benefit $b$ and annual net premium $P$, the equivalence principle for this $n$-year term insurance gives

P = \frac{bA^1_{x:\overline{n}|}}{\ddot{a}_{x:\overline{n}|}} = \frac{ b\displaystyle\sum_{k=0}^{n-1}v^{k+1}{}_kp_xq_{x+k} }{ \displaystyle\sum_{k=0}^{n-1}v^k{}_kp_x }.

This calculation assumes a fixed annual effective interest rate, the stated mortality probabilities, a benefit payable at the end of the year of death, and premiums payable in advance while alive. If premiums are payable at a different time, or if the benefit is paid at another time, the corresponding present-value factors must change.

Why the Premium Equation Holds

The equivalence principle sets the expected present value of premiums equal to the expected present value of benefits. To see why the factors above appear, let $P$ be the annual premium. In year $k+1$, the probability that a benefit is paid is ${}_kp_xq_{x+k}$, and its payment time is $k+1$. The expected present value of all benefits is therefore $bA^1_{x:\overline{n}|}$.

For the premium stream, the time-$k$ payment occurs only if the insured is alive then. Its expected present value is $Pv^k{}_kp_x$. Summing those payments gives $P\ddot{a}_{x:\overline{n}|}$. Equating the two values and dividing by the positive premium factor yields the net premium formula:

bA^1_{x:\overline{n}|} = P\ddot{a}_{x:\overline{n}|} \quad\Longrightarrow\quad P = \frac{bA^1_{x:\overline{n}|}}{\ddot{a}_{x:\overline{n}|}}.

The annuity factor is generally less than the value for $n$ certain annual payments in advance, because some premiums are not paid after death. The term insurance factor also reflects survival: a later-year death benefit is payable only if the insured first survives to that year.

Worked Calculations

Worked Example: A Three-Year Term With Changing Mortality

A hypothetical policy pays $30,000 at the end of the year of death if death occurs within three years. Annual premiums are paid in advance while the insured is alive. The effective annual interest rate is $i=5\%$. The conditional death probabilities for the three years are $0.10$, $0.20$, and $0.25$, respectively. Calculate the annual net level premium, rounding the final premium to cents.

Method. First calculate the survival probability at the start of each covered year. The survival probabilities for years 1, 2, and 3 are $1$, $0.90$, and $0.90(0.80)=0.72$. The corresponding probabilities of death in each year are $0.10$, $0.90(0.20)=0.18$, and $0.72(0.25)=0.18$.

Here $v=1/1.05$. The term insurance factor is

\begin{aligned} A^1_{x:\overline{3}|} &=0.10v+0.18v^2+0.18v^3\\ &=\frac{0.10}{1.05}+\frac{0.18}{1.05^2}+\frac{0.18}{1.05^3}\\ &=\frac{2}{21}+\frac{8}{49}+\frac{160}{1029}\\ &=\frac{142}{343}\\ &\approx 0.4139942. \end{aligned}

The premium factor uses survival at the premium dates $0,1,2$:

\begin{aligned} \ddot{a}_{x:\overline{3}|} &=1+0.90v+0.72v^2\\ &=1+\frac{0.90}{1.05}+\frac{0.72}{1.05^2}\\ &=1+\frac{6}{7}+\frac{32}{49}\\ &=\frac{123}{49}\\ &\approx 2.5102041. \end{aligned}

The unrounded premium is

P = \frac{30{,}000(142/343)}{123/49} = 30{,}000\left(\frac{142}{861}\right) \approx \$4{,}947.7352.

Rounded to cents, the annual net level premium is $4,947.74. As a check, the expected present value of benefits is $30{,}000(142/343)\approx \$12{,}419.83$. Using the unrounded premium, the expected present value of premiums is also approximately $\$12{,}419.83$. The common mistake avoided here is using the death probabilities as though they were unconditional: the second- and third-year death probabilities must include survival to those years.

Worked Example: Zero Interest and Four Premium Dates

A hypothetical four-year term policy pays $20,000 at the end of the year of death. The conditional death probability is $0.10$ in each of the four years. The effective annual interest rate is $i=0\%$. Premiums are payable in advance while alive. Find the annual net level premium.

Method. At zero interest, $v=1$. Survival to the start of years 1 through 4 is $1$, $0.90$, $0.81$, and $0.729$. The probabilities of death in the four years are therefore $0.10$, $0.09$, $0.081$, and $0.0729$.

\begin{aligned} A^1_{x:\overline{4}|} &=0.10+0.09+0.081+0.0729\\ &=0.3439, \\[4pt] \ddot{a}_{x:\overline{4}|} &=1+0.90+0.81+0.729\\ &=3.439. \end{aligned}

The premium is

P = \frac{20{,}000(0.3439)}{3.439} = \frac{6{,}878}{3.439} = \$2{,}000.

The expected present value of benefits is $6,878, and the expected present value of premiums is $2{,}000(3.439)=\$6{,}878$, so the result checks exactly. The common mistake avoided is treating the premium factor as four certain payments. In fact, its four terms are weighted by survival to each premium date.

Worked Example: Comparing the Coverage and Premium Periods

A hypothetical two-year term policy pays $25,000 at the end of the year of death. The conditional death probabilities are $0.20$ in the first year and $0.30$ in the second year. The effective annual interest rate is $i=10\%$. Annual premiums are payable in advance while alive. Calculate the annual net level premium, rounding the final amount to cents.

Method. Survival to the second year is $0.80$, so the probability of death in year 2 is $0.80(0.30)=0.24$. With $v=1/1.10=10/11$, the term insurance factor is

\begin{aligned} A^1_{x:\overline{2}|} &=0.20v+0.24v^2\\ &=0.20\left(\frac{10}{11}\right) +0.24\left(\frac{100}{121}\right)\\ &=\frac{22}{121}+\frac{24}{121}\\ &=\frac{46}{121}. \end{aligned}

There are two possible premium dates, at issue and one year later if alive. Thus

\ddot{a}_{x:\overline{2}|} = 1+0.80v = 1+\frac{8}{11} = \frac{19}{11} = \frac{209}{121}.

The annual premium is

P = \frac{25{,}000(46/121)}{209/121} = \frac{1{,}150{,}000}{209} \approx \$5{,}502.3923.

Rounded to cents, the premium is $5,502.39. The benefit present value is $25{,}000(46/121)\approx \$9{,}504.13$; using the unrounded premium gives the same expected present value for premiums. The common mistake avoided is charging for two certain premiums: the second premium is conditional on survival to the second premium date.

Common Exam Trap: Counting Years Instead of Valuing Premiums

A term lasting $n$ years does not imply that its premium factor is $n$. That value would apply only to $n$ certain payments made in advance with no discounting. Under the assumptions here, each premium after issue is contingent on survival, and at positive interest it is also discounted. The correct factor is $\ddot{a}_{x:\overline{n}|}$, not $n$.

For example, in the four-year zero-interest calculation above, replacing the correct premium factor $3.439$ with $4$ would give the incorrect premium $6{,}878/4=\$1{,}719.50$. The correct factor is $1+0.90+0.81+0.729=3.439$, so the correct premium is $6{,}878/3.439=\$2{,}000$. The error understates the premium because it incorrectly treats all four premiums as certain.

A second check is to keep the benefit and premium periods straight. The final covered death benefit is discounted to time $n$, but there is no premium at time $n$ under this annual-in-advance contract. Extending the premium sum to $k=n$ adds a payment that the contract does not require.

Key takeaway: For annual net level premiums on discrete term insurance, divide the expected present value of covered death benefits by the expected present value of one dollar of premiums. Include each covered year’s survival-weighted death benefit and each premium date’s survival-weighted payment; do not substitute the term length for the premium factor.

Check Your Understanding

For each problem, assume the stated annual premiums are payable in advance while the insured is alive, and the death benefit is paid at the end of the year of death. Round monetary answers to cents where needed.

A two-year term has $i=0\%$, conditional death probabilities $0.20$ and $0.25$, and a benefit of $10,000. Find the annual net level premium. (Answer: $2,222.22)
A three-year term has $i=5\%$, conditional death probability $0.10$ in each year, and a benefit of $20,000. Find the annual net level premium. (Answer: $1,904.76)
A one-year term has $i=4\%$, a first-year death probability of $0.06$, and a benefit of $50,000. Find the annual net level premium. (Answer: $2,884.62)
A three-year term has $i=0\%$, conditional death probabilities $0.15$, $0.20$, and $0.10$, and a benefit of $25,300. Find the annual net level premium. (Answer: $3,880.00)
A three-year term has $i=0\%$, conditional death probability $0.20$ in each year, and a benefit of $12,000. Find the correct annual net level premium, and identify the incorrect result obtained by dividing the expected present value of benefits by three certain premiums. (Answer: Correct premium $2,400.00; incorrect premium $1,952.00)

What You'll Learn