UDD Conditional Survival After a Fractional Age

Introduction

A life-table probability such as $q_x$ describes mortality over a full year of age, but a survival question may begin or end between birthdays. Under the uniform distribution of deaths assumption (UDD), the number of deaths is spread uniformly across each integer-age year. This gives a linear interpolation for survival within that year. For conditional survival across several birthdays, the key is to apply that interpolation separately to each year segment.

The calculation is conditional: a person who has already survived to a fractional age is not exposed to the full annual mortality rate for that year. The remaining survival probability within the year must be divided by the probability of having survived to the starting fractional age.

Definition: Under UDD, for an integer age $k$ and $0\leq u\leq 1$, the number alive at exact age $k+u$ is interpolated linearly between the life-table values at ages $k$ and $k+1$: $$ \ell_{k+u}=\ell_k-u d_k=\ell_k(1-uq_k), $$ where $d_k=\ell_k-\ell_{k+1}$ and $q_k=d_k/\ell_k$. Thus $1-uq_k$ is the probability, relative to a life aged $k$, of surviving to age $k+u$.

Conditional Survival Within One Year

Let $0\leq r\leq s\leq1$. A life aged $k+r$ has already survived the fraction $r$ of the year of age $k$. The conditional probability of surviving from age $k+r$ to age $k+s$ is the number alive at $k+s$ divided by the number alive at $k+r$. The common starting factor $\ell_k$ cancels.

{}_{s-r}p_{k+r} =\frac{\ell_{k+s}}{\ell_{k+r}} =\frac{1-sq_k}{1-rq_k}.

The numerator is the fraction of the original age-$k$ cohort alive at $k+s$; the denominator is the fraction already alive at the conditional starting age. This ratio is why simply subtracting two interpolated survival fractions, or applying a fraction of $q_k$ directly to the conditional life, does not give the correct answer.

Worked Example: Survival Between Two Fractional Ages in One Year

A hypothetical life table has $q_{60}=0.12$. Under UDD, find the probability that a life aged 60.25 survives to age 60.80. State the method and avoid treating $0.12$ as the mortality probability for the remaining 0.55 years.

Method. Both ages are in the year of age 60, so use the within-year conditional survival ratio with $r=0.25$, $s=0.80$, and $q_{60}=0.12$.

The fraction of the age-60 cohort alive at age 60.80 is $1-0.80(0.12)=1-0.096=0.904$. The fraction alive at age 60.25 is $1-0.25(0.12)=1-0.030=0.970$. Therefore,

{}_{0.55}p_{60.25} =\frac{0.904}{0.970} =0.931958\ldots \approx 0.9320.

The check is that the conditional probability is less than 1, as the later age is older, and greater than the unconditional probability $0.904$ of surviving from age 60 to 60.80, because the life is known to have survived to 60.25. The common mistake avoided here is using $1-(0.80-0.25)q_{60}=0.9340$, which incorrectly treats mortality as a fraction of the annual rate conditional on reaching age 60.25.

Survival Across Integer Birthdays

When the interval crosses one or more integer ages, divide it into three possible pieces: the remaining part of the first year, any complete years between the endpoints, and the elapsed part of the final year. The first and last pieces use conditional UDD factors. Each complete intervening year uses the ordinary one-year survival probability $p_j=1-q_j$.

Theorem: Suppose a life starts at age $k+r$ and the later age is $m+s$, where $k\leq m$, $0\leq r<1$, and $0\leq s<1$. If $m>k$, and the relevant life-table survivals are positive, then under UDD $$ {}_{(m+s)-(k+r)}p_{k+r} = \frac{1-q_k}{1-rq_k} \left(\prod_{j=k+1}^{m-1}(1-q_j)\right) (1-sq_m). $$ If $m=k$, use the same-year formula $(1-sq_k)/(1-rq_k)$. An empty product, as occurs when there are no complete intervening years, equals 1.

This expression also handles integer endpoints. For example, if the target age is the integer age $m$, take $s=0$; the final factor is then 1. If the starting age is an integer, take $r=0$, so the first factor becomes $1-q_k$, the full-year survival probability. In practice, writing the endpoints as age plus fractional part helps prevent an annual mortality rate from being assigned to the wrong age interval.

Identify the age-year segments.
Write the starting and ending ages as integer ages plus fractional parts. Mark each full year that lies between them.

Calculate the first partial-year factor.
For a start at $k+r$, survival to the next birthday is $(1-q_k)/(1-rq_k)$.

Include complete years.
For each complete year of age $j$, multiply by $1-q_j$.

Calculate the final partial-year factor.
For a target $m+s$, multiply by $1-sq_m$, the survival from age $m$ to $m+s$ relative to age $m$.

The product is a conditional probability from the original fractional starting age, not a probability from the nearest integer birthday. Each factor describes survival conditional on reaching the beginning of its own segment, and multiplying these successive conditional probabilities gives the probability of surviving the whole interval.

Worked Example: Crossing Two Integer Birthdays

A hypothetical life table has $q_{70}=0.15$, $q_{71}=0.20$, and $q_{72}=0.10$. Under UDD, find the probability that a life aged 70.8 survives to age 72.5. State the method and avoid treating all three mortality rates as if they applied to the entire interval.

Method. Split the interval into survival from age 70.8 to 71, a complete year from 71 to 72, and half a year from 72 to 72.5.

For the first segment, the fraction of the age-70 cohort alive at age 70.8 is $1-0.8(0.15)=0.88$, while the fraction alive at 71 is $1-0.15=0.85$. Hence the conditional probability of reaching 71 from 70.8 is $0.85/0.88=0.965909\ldots$.

For the complete year from age 71 to 72, survival is $1-q_{71}=1-0.20=0.80$. For the final half-year, survival from age 72 to 72.5 is $1-0.5(0.10)=0.95$.

{}_{1.7}p_{70.8} =\frac{0.85}{0.88}(0.80)(0.95) =\frac{0.646}{0.88} =0.7340909\ldots \approx 0.7341.

As a check, relative to $\ell_{70}$, the numbers alive at ages 70.8 and 72.5 are $0.88\ell_{70}$ and $(0.85)(0.80)(0.95)\ell_{70}=0.646\ell_{70}$, respectively. Their ratio is $0.646/0.88=0.7341$ to four decimals. The common mistake avoided is dividing by the wrong starting fraction or forgetting that survival from age 70.8 is conditional on already being alive then.

Why the Product Formula Holds

The multi-year formula follows directly by writing each segment as a ratio of life-table survivals. The life-table counts at integer ages connect the UDD interpolations, so the adjacent factors telescope to the ratio of the count at the endpoint to the count at the starting age.

Start with the conditional ratio.
For endpoint age $m+s$ and starting age $k+r$, conditional survival is $\ell_{m+s}/\ell_{k+r}$.

Apply UDD at each endpoint.
Use $\ell_{k+r}=\ell_k(1-rq_k)$ and $\ell_{m+s}=\ell_m(1-sq_m)$.

Connect integer ages.
For each complete year $j$, $\ell_{j+1}/\ell_j=1-q_j$. Thus $\ell_m/\ell_k=\prod_{j=k}^{m-1}(1-q_j)$.

Combine and cancel.
Substitution gives $(1-sq_m)\prod_{j=k}^{m-1}(1-q_j)/(1-rq_k)$. Separating the $j=k$ factor yields the stated first-partial, complete-year, and final-partial product.

\frac{\ell_{m+s}}{\ell_{k+r}} = \frac{\ell_m(1-sq_m)}{\ell_k(1-rq_k)} = \frac{(1-sq_m)\prod_{j=k}^{m-1}(1-q_j)}{1-rq_k}.

This justification also provides a useful calculation check: multiply all segment factors and compare the result with $\ell_{m+s}/\ell_{k+r}$, if the interpolated life-table counts are available. The two methods must agree.

Worked Example: Using Life-Table Counts Across Several Years

A hypothetical table has $\ell_{64}=10{,}000$, with annual death probabilities $q_{64}=0.10$, $q_{65}=0.20$, $q_{66}=0.25$, and $q_{67}=0.08$. Under UDD, find the probability that a life aged 64.4 survives to age 67.25. State the method and avoid applying $q_{64}$ to the entire time from age 64.4.

Method. Use a remaining fraction of the year of age 64, complete years 65 and 66, and a final quarter-year of age 67.

The first factor is survival from 64.4 to 65. The fraction of the age-64 cohort alive at 64.4 is $1-0.4(0.10)=0.96$, and at 65 it is $1-0.10=0.90$. Thus this factor is $0.90/0.96=0.9375$.

Survival from 65 to 66 is $1-0.20=0.80$, and from 66 to 67 is $1-0.25=0.75$. The final factor, from 67 to 67.25, is $1-0.25(0.08)=0.98$.

{}_{2.85}p_{64.4} =\frac{0.90}{0.96}(0.80)(0.75)(0.98) =(0.9375)(0.60)(0.98) =0.55125 \approx 0.5513.

For a life-table check, the same factors give the ratio of the interpolated survivor count at 67.25 to that at 64.4; the starting count $\ell_{64}$ cancels. The result is below each individual segment survival factor, as expected for survival over several segments. The common mistake avoided is using the final factor $1-0.25q_{66}$: the final partial year is at age 67, so it must use $q_{67}$.

Common Mistakes and Boundary Checks

Fractional-age questions often turn on which integer-age mortality rate belongs to a segment and whether a calculation is conditional on survival to its starting age. Before multiplying, check each segment against its age interval.

Using an unconditional fraction as a conditional probability. From age $k+r$, the probability of reaching $k+s$ is $(1-sq_k)/(1-rq_k)$, not $1-(s-r)q_k$.
Using the wrong year's $q$. The last partial-year factor uses the rate for the integer age at the start of that year. A target at age $m+s$ uses $q_m$.
Omitting complete years. Every fully traversed year contributes its own factor $1-q_j$. If the interval crosses multiple birthdays, one annual factor is not enough.
Including a year twice. If the target is exactly an integer age, its final partial-year factor is 1. The full year ending at that age has already been included.
Ignoring the conditional denominator. The first partial-year factor divides by $1-rq_k$, accounting for survival to the fractional starting age.

Key takeaway: Under UDD, conditional survival from a fractional age to a later age is the ratio of the interpolated survivor counts at those ages. For a multi-year interval, calculate the first partial year conditionally, multiply the intervening complete-year survivals, and then apply the final partial-year factor.

Check Your Understanding

For each hypothetical life table, assume UDD within every year of age and round probabilities to four decimal places.

If $q_{80}=0.10$, find the probability that a life aged 80.2 survives to age 80.7. (Answer: $0.9490$)
If $q_{50}=0.08$ and $q_{51}=0.12$, find the probability that a life aged 50.5 survives to age 51.4. (Answer: $0.9123$)
If $q_{40}=0.10$, $q_{41}=0.20$, and $q_{42}=0.10$, find the probability that a life aged 40.75 survives to age 42.6. (Answer: $0.7317$)
If $q_{33}=0.15$, $q_{34}=0.05$, and $q_{35}=0.25$, find the probability that a life aged 33.3 survives to age 35.2. (Answer: $0.8033$)
If $q_{60}=0.20$ and $q_{61}=0.10$, find the probability that a life aged 60.4 survives to age 61.5. Identify the partial and complete year segments in your calculation. (Answer: $0.8261$)

What You'll Learn