One-Year Term Insurance From Life-Table Counts

From Life-Table Counts to a One-Year Insurance Value

A life table records how many members of a hypothetical cohort survive to each age. Two consecutive entries therefore contain the information needed to calculate the probability that a person aged $x$ dies before reaching age $x+1$. For a one-year term policy, that conditional probability determines whether a benefit is paid; the interest rate determines its present value.

The key is to interpret the counts conditionally. The number who die between the two ages is found by subtracting the later survivor count from the earlier one. To obtain the probability for an individual who is alive at age $x$, divide that number by the number alive at age $x$—not by the original cohort size unless that size is $\ell_x$.

Definition: In a life table, $\ell_x$ is the number of members of the cohort surviving to exact age $x$. The number who die between ages $x$ and $x+1$ is $\ell_x-\ell_{x+1}$. Provided $\ell_x>0$, the probability of death within that year, conditional on being alive at age $x$, is $$ {}_1q_x=\frac{\ell_x-\ell_{x+1}}{\ell_x}=1-\frac{\ell_{x+1}}{\ell_x}. $$

This probability is also commonly written $q_x$. It is conditional on survival to age $x$: it is not the proportion of the original cohort that dies during the year. The life-table entries can be actual counts or values based on a chosen cohort size, often called a radix. The ratio gives the same probability if both entries use the same scale.

Valuing the One-Year Death Benefit

Consider a one-year term policy issued to a person aged $x$. Assume the policy pays a level death benefit $B$ at the end of the year if the insured dies between ages $x$ and $x+1$. It pays nothing if the insured survives to age $x+1$. Let $i$ be the effective annual interest rate, and let $v=1/(1+i)$ be the discount factor for one year.

Let $I$ be the indicator of death during the year: $I=1$ if death occurs during the year and $I=0$ otherwise. Under the life-table mortality assumption, $I$ has a Bernoulli distribution with parameter ${}_1q_x$. The present value of the benefit is $BvI$, so its expected present value is

\operatorname{EPV}=E[BvI]=BvE[I]=Bv\,{}_1q_x =\frac{B}{1+i}\frac{\ell_x-\ell_{x+1}}{\ell_x}.

This is a value per policy issued to someone alive at age $x$. It is not the total expected benefit for the entire life-table cohort unless the per-policy value is multiplied by the number of policies in that cohort. The formula also depends on the stated timing: it discounts the benefit for one full year because the benefit is paid at the end of the year, regardless of when death occurs during that year.

Why the Formula Holds

The life-table calculation supplies the probability of each outcome for one insured. The present value is $Bv$ on death and zero on survival. Weighting those two possible values by their probabilities gives the expected present value:

\begin{aligned} \operatorname{EPV} &=(Bv)P(\text{death during the year})+0\cdot P(\text{survival to }x+1)\\ &=(Bv)\frac{\ell_x-\ell_{x+1}}{\ell_x}+0\\ &=\frac{B}{1+i}\frac{\ell_x-\ell_{x+1}}{\ell_x}. \end{aligned}

The two probabilities sum to one: death during the year has probability $(\ell_x-\ell_{x+1})/\ell_x$, and survival to age $x+1$ has probability $\ell_{x+1}/\ell_x$. The result does not require an assumption about the exact time of death within the year, because the payment is specified to occur at year-end in either case. If a benefit were paid immediately at death, its discount period would depend on the timing of death, and this calculation would not apply as written.

Worked Example: A One-Year Benefit From a Cohort Table

A hypothetical life table has $\ell_{52}=12{,}000$ and $\ell_{53}=11{,}880$. A one-year term policy issued at age 52 pays $B=\$100{,}000$ at the end of the year if the insured dies during the year. Assume a 5% effective annual interest rate and that the insured’s one-year mortality follows the table. Find the expected present value per policy.

Method. First calculate the number of deaths represented by the two counts, then divide by the age-52 survivor count to obtain the conditional probability. The death indicator is Bernoulli with that probability.

Deaths in the year: $12{,}000-11{,}880=120$.

Conditional death probability: ${}_1q_{52}=120/12{,}000=0.01$.

Discount factor: $v=1/(1+0.05)=1/1.05=0.95238095\ldots$.

Expected present value: the present value is $\$100{,}000v$ on death and zero on survival, so

\operatorname{EPV} =\$100{,}000(0.95238095\ldots)(0.01) =\$952.38095\ldots

Rounded to the nearest cent, the expected present value is $\$952.38$. Check: a 1% death probability applied to the $\$100{,}000$ benefit gives an undiscounted expected benefit of $\$1{,}000$, and discounting that amount by one year at 5% gives $\$1{,}000/1.05=\$952.38$. A common mistake is to use the $\$1{,}000$ undiscounted amount as the policy value.

Worked Example: Counts From a Smaller Life-Table Cohort

For a hypothetical population, $\ell_{68}=8{,}400$ and $\ell_{69}=8{,}190$. A one-year term policy pays $\$80{,}000$ at year-end if the insured, aged 68 at issue, dies during the year. Assume the insured follows the life-table mortality and the effective annual interest rate is 4%. Calculate the expected present value per policy.

Method. Use the difference between successive survivor counts for the deaths, and use $\ell_{68}$ as the denominator because the policy is issued to someone alive at age 68.

Deaths in the year: $8{,}400-8{,}190=210$.

Conditional death probability: ${}_1q_{68}=210/8{,}400=0.025$.

Discount factor: $v=1/(1+0.04)=1/1.04=0.96153846\ldots$.

Expected present value:

\begin{aligned} \operatorname{EPV} &=\$80{,}000(0.96153846\ldots)(0.025)\\ &=\$2{,}000(0.96153846\ldots)\\ &=\$1{,}923.07692\ldots \end{aligned}

Rounded to the nearest cent, the value is $\$1{,}923.08$. Check: the undiscounted expected benefit is $0.025(\$80{,}000)=\$2{,}000$; dividing by $1.04$ gives $\$1{,}923.08$. Using 210 as the probability would be a units error: 210 is a count, not a probability. The count must be divided by the number alive at age 68.

Worked Example: Scaling the Life Table Does Not Change the Probability

A hypothetical life table is reported on a radix of 3,650, with $\ell_{40}=3{,}650$ and $\ell_{41}=3{,}577$. A one-year term policy issued at age 40 pays $\$50{,}000$ at the end of the year if the insured dies during the year. Assume the insured follows the table and the effective annual interest rate is 2.5%. Find the expected present value, to the nearest cent.

Method. Treat the table values as cohort counts on the stated radix. Their difference gives the deaths represented on that radix; dividing by the number alive at age 40 produces the per-insured probability.

Deaths in the year: $3{,}650-3{,}577=73$.

Conditional death probability: ${}_1q_{40}=73/3{,}650=0.02$, since $3{,}650(0.02)=73$.

Discount factor: $v=1/(1+0.025)=1/1.025=0.975609756\ldots$.

Expected present value:

\begin{aligned} \operatorname{EPV} &=\$50{,}000(0.975609756\ldots)(0.02)\\ &=\$1{,}000(0.975609756\ldots)\\ &=\$975.609756\ldots \end{aligned}

Rounded to the nearest cent, the value is $\$975.61$. Check: the cohort size affects the counts, but the ratio $73/3{,}650$ is the probability for one life. Multiplying the benefit by 73, or treating all 3,650 table members as insured under one policy, would not give the value per policy.

Common Errors and Practical Checks

A reliable calculation separates three quantities: the life-table count of deaths, the conditional probability of death, and the discounted benefit. Keeping these distinct helps prevent errors in both arithmetic and interpretation.

Using the wrong denominator. The denominator is $\ell_x$, the number alive at the start of the insured year. Dividing by a cohort’s original size instead calculates a proportion of that original cohort, which is not the probability conditional on being alive at age $x$.
Confusing a count and a probability. The difference $\ell_x-\ell_{x+1}$ is a count on the table’s scale. Divide it by $\ell_x$ before using it as a probability.
Forgetting the discount. The expected benefit without discounting is $B\,{}_1q_x$. For a payment one year from issue, the expected present value is $B\,{}_1q_x/(1+i)$.
Discounting for the wrong period. Under the stated end-of-year payment rule, use one year of discounting even if death occurs earlier in the year. A benefit paid at the moment of death would require a different timing model.
Changing the table scale inconsistently. Scaling both $\ell_x$ and $\ell_{x+1}$ by the same factor leaves their ratio unchanged. Scaling only one entry does not represent the same life table.

Two quick checks are useful. First, $0\leq{}_1q_x\leq1$ when the table has nonnegative counts and $\ell_{x+1}\leq\ell_x$. Second, for a positive benefit and nonnegative interest, the expected present value cannot exceed the undiscounted expected benefit $B\,{}_1q_x$. A result that violates either check signals a likely error.

These calculations connect a life-table summary to the value of a simple hypothetical insurance contract. The counts describe cohort survival; their ratio gives the probability relevant to an insured already alive at age $x$; and the contract’s payment rule determines how that probability translates into a present value. The same distinction between cohort counts and conditional probabilities is essential whenever life-table data are used to value coverage.

Check Your Understanding

For each hypothetical policy, assume the insured follows the stated life table and the death benefit is paid at the end of the year. Give the expected present value per policy, rounded to the nearest cent.

A table has $\ell_{35}=20{,}000$ and $\ell_{36}=19{,}700$. The benefit is $\$100{,}000$ and the effective annual interest rate is 3%. (Answer: $\$1{,}456.31$)
A table has $\ell_{57}=9{,}600$ and $\ell_{58}=9{,}456$. The benefit is $\$40{,}000$ and the effective annual interest rate is 5%. (Answer: $\$571.43$)
A table has $\ell_{71}=7{,}500$ and $\ell_{72}=7{,}350$. The benefit is $\$120{,}000$ and the effective annual interest rate is 2%. (Answer: $\$2{,}352.94$)
A table on a radix of 4,800 has $\ell_{44}=4{,}800$ and $\ell_{45}=4{,}728$. The benefit is $\$75{,}000$ and the effective annual interest rate is 4%. (Answer: $\$1{,}081.73$)
A table has $\ell_{62}=32{,}000$ and $\ell_{63}=31{,}520$. A hypothetical insurer issues 40 identical one-year policies, each with a $\$125{,}000$ benefit paid at year-end on death. Assume every insured follows this mortality and the effective annual interest rate is 6%. What is the expected present value of the total benefits for the 40 policies? (Answer: $\$70{,}754.72$)

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