Table-Based Continuous Values Under UDD

Why Annual Life Tables Can Value Continuous Benefits

A life table usually records the probability of death during each year of age, while a continuous life insurance benefit is paid at the moment of death. To value that benefit from annual table entries, we need an assumption about when, within a year, deaths occur. Under the uniform distribution of deaths (UDD), deaths are spread uniformly through each year of age, conditional on dying in that year. This makes it possible to calculate continuous term and whole-life values using annual probabilities.

The calculations below assume a constant effective annual interest rate $i$, with $v=1/(1+i)$ and force of interest $\delta=\ln(1+i)$. A benefit of one is paid immediately at death. If the insured survives the full term of a term policy, no benefit is paid.

Definition: Under UDD, if a life aged $x$ is alive at age $x$, the conditional probability of surviving a fraction $t$ of the year is ${}_tp_x=1-tq_x$, for $0\le t\le1$. Here $q_x$ is the probability of dying between ages $x$ and $x+1$, and $p_x=1-q_x$. Equivalently, conditional on dying in that year, the time of death within the year is uniformly distributed.

UDD is applied separately in every year. Thus the probability of being alive at the start of year $k$ is ${}_kp_x$, obtained from the annual table, and the probability of dying during the next year given survival to its start is $q_{x+k}$. Within that year, the death time has the uniform distribution specified above.

The Value Contribution From One Year

Suppose the insured has survived to age $x+k$. Conditional on dying in the year from $x+k$ to $x+k+1$, write $s$ for the fraction of that year lived before death. Under UDD, $s$ is uniform on $[0,1]$. A payment at that time has discount factor $v^{k+s}=v^k e^{-\delta s}$, since $v=e^{-\delta}$. Averaging over the possible death times gives the within-year discount factor

\int_0^1 e^{-\delta s}\,ds =\frac{1-e^{-\delta}}{\delta} =\frac{1-v}{\delta}.

The probability of reaching the year and then dying in it is ${}_kp_xq_{x+k}$. Consequently, the expected present value contribution from that year is the product of the probability of death in the year, the discount to its start, and the average discount within the year.

Key formula: For $\delta>0$, the contribution to a unit continuous death benefit from deaths in policy year $k$ is

{}_kp_xq_{x+k}\,v^k\frac{1-v}{\delta}.

If $i=0$, then $\delta=0$, and the within-year factor is its limit, $1$: there is no discounting within the year.

For a term policy lasting $n$ years, only deaths in years $0,\ldots,n-1$ produce a benefit. For whole life insurance, contributions continue through all years in the life table. In the finite-table formulas below, $\omega$ denotes the terminal age at which the table has no lives remaining beyond that age.

\bar{A}^{1}_{x:\overline{n}|} =\frac{1-v}{\delta} \sum_{k=0}^{n-1}v^k\,{}_kp_xq_{x+k}, \qquad \bar{A}_x =\frac{1-v}{\delta} \sum_{k=0}^{\omega-x-1}v^k\,{}_kp_xq_{x+k}.

Here $\bar{A}^{1}_{x:\overline{n}|}$ is the present value of a unit benefit paid immediately on death during the $n$-year term, and $\bar{A}_x$ is the present value of a unit continuous whole-life death benefit. For a benefit amount $B$, multiply the appropriate unit value by $B$. The mortality probabilities and the interest rate must refer to the same valuation basis throughout.

Connection to Year-End Insurance Values

The annual discrete term insurance value $A^{1}_{x:\overline{n}|}$ pays at the end of the year of death. Its contribution for year $k$ is $v^{k+1}{}_kp_xq_{x+k}$, rather than the continuous contribution above. Comparing the two year-by-year shows the conversion factor; the same comparison applies to whole-life values.

\frac{v^k(1-v)/\delta}{v^{k+1}} =\frac{1-v}{v\delta} =\frac{i}{\delta}, \qquad \bar{A}^{1}_{x:\overline{n}|} =\frac{i}{\delta}A^{1}_{x:\overline{n}|}, \qquad \bar{A}_x=\frac{i}{\delta}A_x.

The identities follow because $1-v=iv$. They hold under UDD and the same interest and mortality assumptions on both sides. When $i=0$, use the limiting conversion factor $i/\delta\to1$. The continuous value is not found by simply multiplying an annual value by the average fraction of a year; the discounting of the actual payment time is what determines the factor.

Worked Example: Two-Year Continuous Term Insurance

A hypothetical policy insures a life aged $x$ for two years, with a unit benefit paid immediately at death. The annual probabilities are $q_x=0.10$ and $q_{x+1}=0.20$. Assume UDD in both years and an effective annual interest rate $i=100\%$. Find the continuous term value.

Method. Calculate the probability-weighted discounted death contributions for the two covered years, then multiply their sum by the UDD within-year factor. The force is $\delta=\ln(2)=0.693147$, and $v=1/2$. Therefore, $\frac{1-v}{\delta}=\frac{0.5}{0.693147}=0.721348$.

In year 0, the probability of death is $q_x=0.10$, so its weighted amount before the within-year factor is $v^0q_x=1(0.10)=0.10$.

The probability of surviving the first year is $p_x=1-0.10=0.90$. In year 1, the probability of death, including survival to the start of that year, is ${}_1p_xq_{x+1}=0.90(0.20)=0.18$. Discounting to the policy start gives $v^1(0.18)=0.50(0.18)=0.09$.

The total before averaging the within-year discount is $0.10+0.09=0.19$. Thus $\bar{A}^{1}_{x:\overline{2}|}=0.721348(0.19)=0.137056$. The unit continuous term value is 0.137056. As a check, the year-end value is $A^{1}_{x:\overline{2}|}=0.50(0.10)+0.25(0.18)=0.095$, and $(i/\delta)A^{1}_{x:\overline{2}|}=(1/0.693147)(0.095)=0.137056$. The common mistake avoided here is using the second-year death probability $0.20$ without first allowing for survival through year 0.

Whole-Life Values From a Finite Life Table

A table-based whole-life calculation follows the same method as the term calculation, but includes every year through the table’s terminal age. A useful way to organize the work is to list, for each year, the annual survival probability, the death probability conditional on reaching that year, and the discount to the start of that year. The table must support the calculation through the final year; omitting a final interval in which death is certain understates the value.

Worked Example: Whole-Life Value With a Terminal Year

For a hypothetical life aged $x$, suppose the annual death probabilities are $q_x=0.10$, $q_{x+1}=0.20$, $q_{x+2}=0.50$, and $q_{x+3}=1$. The table ends at age $x+4$. Assume UDD in each year and $i=100\%$. Calculate the unit continuous whole-life value.

Method. Sum the discounted probability of death in each year, including the final year where $q_{x+3}=1$, and apply the within-year factor $0.721348$ calculated from $\delta=\ln 2$.

The probability of surviving to the start of year 0 is $1$, so the year 0 weighted amount is $v^0(1)(0.10)=0.10$.

Survival to year 1 is $0.90$. Its contribution before the within-year factor is $v^1(0.90)(0.20)=0.50(0.18)=0.09$.

Survival to year 2 is $0.90(0.80)=0.72$. Its contribution is $v^2(0.72)(0.50)=0.25(0.36)=0.09$.

Survival to year 3 is $0.72(0.50)=0.36$. Since everyone alive at the start of this year dies by its end, its contribution is $v^3(0.36)(1)=0.125(0.36)=0.045$.

The total before the within-year factor is $0.10+0.09+0.09+0.045=0.325$. Therefore, $\bar{A}_x=0.721348(0.325)=0.234438$, to six decimal places. The discrete whole-life value is $0.05+0.045+0.045+0.0225=0.1625$; multiplying by $i/\delta=1/0.693147$ gives $0.234438$, confirming the result. The common mistake avoided is stopping after the last year with $q_x<1$: the terminal year still has a positive probability of payment.

Worked Example: Deriving Annual Probabilities From Life Table Counts

A hypothetical life table gives $\ell_{60}=10{,}000$, $\ell_{61}=9{,}500$, $\ell_{62}=8{,}500$, and $\ell_{63}=6{,}000$. A two-year term policy on a life aged 60 pays $B=\$10{,}000$ immediately at death. Assume UDD and $i=100\%$. Find the expected present value of the benefit, rounded to the nearest cent.

Method. First convert the life table counts to annual death probabilities using $q_y=(\ell_y-\ell_{y+1})/\ell_y$. Then use only the first two years, because the policy expires at age 62.

For age 60, $q_{60}=(10{,}000-9{,}500)/10{,}000=500/10{,}000=0.05$, so $p_{60}=0.95$. For age 61, $q_{61}=(9{,}500-8{,}500)/9{,}500=1{,}000/9{,}500=2/19$.

The weighted amount before the within-year factor is $v^0q_{60}+v^1p_{60}q_{61} =0.05+0.50(0.95)(2/19) =0.05+0.05=0.10$. With $v=0.5$ and $\delta=\ln2$, the continuous unit value is $0.721348(0.10)=0.0721348$. Multiplying by the benefit gives $\$10{,}000(0.0721348)=\$721.35$, rounded to the nearest cent. This is a two-year term value, not a whole-life value: the age-62-to-63 mortality interval is outside the coverage and must not be included.

Common Errors and Practical Checks

Using the wrong conversion factor. The factor from a year-end discrete value to a continuous value is $i/\delta$, not $(1-v)/\delta$. The latter is the within-year factor applied to the sum discounted to the start of each year.
Forgetting survival to a later year. A year-$k$ contribution includes ${}_kp_x$, not just $q_{x+k}$. A life must be alive at the start of that year to generate its death benefit.
Including deaths outside the term. A term value with duration $n$ includes only $k=0,\ldots,n-1$. Do not add a benefit for death after coverage expires.
Applying UDD across more than one year as though it were a single uniform distribution. UDD is applied within each annual age interval. Annual survival between intervals still comes from the table.
Using the formula at zero interest without taking a limit. The expression $(1-v)/\delta$ is written as a ratio, but at $i=0$ both numerator and denominator are zero. Its limit is 1, consistent with no discounting.

A final check is to compare the continuous value with the corresponding year-end value under the same basis. For positive interest, $i/\delta>1$, so the continuous value is larger: a death benefit paid during the year is paid earlier than one paid at year-end. Also check that each year’s death probability is multiplied by survival to that year and that the sum covers exactly the insured period. These checks help distinguish errors in mortality indexing, coverage duration, and payment timing.

Check Your Understanding

Assume UDD within every year and the stated effective annual interest rate. Work from the annual probabilities given; answers are rounded to four decimal places unless money is requested.

A two-year continuous term policy has $q_x=0.08$ and $q_{x+1}=0.15$, with $i=100\%$. Find its unit value. (Answer: 0.1075)
A whole-life table has $q_x=0.20$, $q_{x+1}=0.25$, and $q_{x+2}=1$, with no lives remaining beyond age $x+3$. At $i=100\%$, calculate the unit continuous whole-life value, including the terminal year. (Answer: 0.3246)
A life table gives $\ell_x=5{,}000$, $\ell_{x+1}=4{,}500$, and $\ell_{x+2}=3{,}600$. A two-year unit continuous term benefit is valued at $i=0$. Derive the two annual death probabilities and find the value. (Answer: 0.2800)
A two-year continuous term policy has $q_x=0.10$, $q_{x+1}=0.20$, and benefit $B=\$30{,}000$. At $i=100\%$, calculate the expected present value of the benefit to the nearest cent. (Answer: $4,111.68)

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