UDD Conditional Mortality Within a Year

Why Conditioning Matters Within the Year

A probability of surviving from exact age $x$ to a fractional age answers a different question from the probability of surviving a further fraction of a year given survival to an intermediate fractional age. Under the uniform distribution of deaths assumption (UDD), both probabilities can be found from the same linear interpolation, but conditional survival requires a denominator for the group still alive at the starting fractional age.

This tutorial focuses on an interval that begins at age $x+r$ and ends at age $x+r+s$, with both ages in the same year of age. It is aligned with the SOA Exam FAM (Fundamentals of Actuarial Mathematics) syllabus. There is no interest rate or cash flow in these probability calculations; interest is not applicable.

Definition: Let $q_x={}_1q_x$ be the probability that a life aged $x$ dies before age $x+1$. Under UDD, deaths are spread uniformly through that year, so the proportion of the age-$x$ cohort surviving to age $x+t$ is $1-tq_x$, for $0\leq t\leq1$. For $0\leq r<1$, $0\leq s\leq1-r$, and $1-rq_x>0$, conditional survival from age $x+r$ to age $x+r+s$ is $$ {}_s p_{x+r} =P(T_x>r+s\mid T_x>r) =\frac{1-(r+s)q_x}{1-rq_x}. $$

Here $T_x$ denotes the future lifetime of a life aged $x$. The condition $1-rq_x>0$ ensures that there is a positive probability of being alive at the conditioning age. The endpoint restriction $r+s\leq1$ keeps the entire interval within the same year of age.

Deriving the Conditional Survival Formula

The derivation is a direct application of conditional probability. Under UDD, the proportion of the original cohort alive at age $x+r$ is $1-rq_x$, and the proportion alive at age $x+r+s$ is $1-(r+s)q_x$. A life alive at the later age must also have been alive at the earlier age. Therefore, the probability of being alive at the later age conditional on being alive at the earlier age is the later survivor proportion divided by the earlier one.

\begin{aligned} {}_s p_{x+r} &=P(T_x>r+s\mid T_x>r)\\ &=\frac{P(T_x>r+s)}{P(T_x>r)}\\ &=\frac{1-(r+s)q_x}{1-rq_x}. \end{aligned}

The same reasoning gives the conditional probability of dying during this interval. Conditional on being alive at $x+r$, death before $x+r+s$ is the complement of survival:

{}_s q_{x+r} =1-{}_s p_{x+r} =\frac{s q_x}{1-rq_x}.

The numerator $s q_x$ represents the fraction of the original age-$x$ cohort that dies during the interval from $x+r$ to $x+r+s$: the interval has length $s$ within a full year in which the cohort’s total death probability is $q_x$. The denominator converts that fraction into a probability among those who actually reached age $x+r$.

Interpretation: Under UDD, the unconditional probability of dying in the interval $(x+r,x+r+s]$, measured from age $x$, is $s q_x$. The conditional probability among lives who reach age $x+r$ is larger when $r>0$, because it divides by the smaller survivor proportion $1-rq_x$.

Applying the Formula Carefully

A reliable calculation starts by identifying the beginning of the year of age, the annual probability $q_x$, and the two fractional positions. In particular, $s$ is the length of the interval, while $r+s$ is the endpoint’s position within the year. The formula uses both.

Identify the annual mortality input.
Use $q_x$ for the integer age $x$ at the start of the year containing both fractional ages.

Locate the starting and ending ages.
Write the start as $x+r$, the end as $x+r+s$, and check that $0\leq r<1$ and $0\leq s\leq1-r$.

Form the conditional ratio.
Divide the survivor proportion at the endpoint, $1-(r+s)q_x$, by the survivor proportion at the starting age, $1-rq_x$.

Check the result.
The conditional survival probability must be between 0 and 1. Its complement is the conditional death probability over the interval.

Worked Example: Survival Over a Half-Year Interval

A hypothetical life table assigns an annual death probability $q_{52}=0.20$ to lives aged 52. Under UDD, find the probability that a life aged 52 years and 3 months survives to age 52 years and 9 months, given that the life is alive at 52 years and 3 months. No interest rate applies because this is a mortality probability, not a present-value calculation.

Method. Use conditional survival under UDD. The distribution assumption is UDD within the year of age 52, with annual parameter $q_{52}=0.20$. The starting fraction is $r=3/12=0.25$. The ending fraction is $9/12=0.75$, so the interval length is $s=0.75-0.25=0.50$.

The survivor proportion at the starting age, measured from age 52, is $1-rq_{52}=1-(0.25)(0.20)=1-0.05=0.95$. The survivor proportion at the ending age is $1-(r+s)q_{52}=1-(0.75)(0.20)=1-0.15=0.85$.

Therefore,

{}_s p_{52+r} =\frac{0.85}{0.95} =\frac{17}{19} =0.8947.

The conditional probability of dying in the six-month interval is $1-0.8947=0.1053$. As a check, the equivalent death calculation is $(s q_{52})/(1-rq_{52})=(0.50)(0.20)/0.95=0.10/0.95=0.1053$. The common mistake this avoids is using the unconditional interval death probability $s q_{52}=0.10$ as though it were conditional on reaching age 52.25.

What Changes When the Starting Age Changes?

The annual value $q_x$ describes mortality for the cohort at exact age $x$, not a new full-year probability beginning at every fractional age. Under UDD, the deaths are distributed across the original year. A life who has already survived part of that year belongs to a selected group: some lives in the original cohort have died, while this life has not. The conditional denominator accounts for that selection.

For a fixed interval length $s$, increasing $r$ leaves the unconditional death probability $s q_x$ for that slice of the original cohort unchanged. But the proportion of the original cohort still alive, $1-rq_x$, decreases as $r$ increases when $q_x>0$. Consequently, the conditional death probability $s q_x/(1-rq_x)$ increases. This is a consequence of conditioning, not a change in the assumed annual death probability.

Worked Example: Conditional Death Probability Near Year-End

For a hypothetical population aged 68, suppose $q_{68}=0.12$. Under UDD, a benefit analyst needs the probability of death between ages 68.40 and 68.75, conditional on survival to age 68.40. Find both conditional survival and conditional death probabilities. No interest rate applies. The mortality model is UDD for the year of age 68, with annual parameter $q_{68}=0.12$.

The starting fraction is $r=0.40$. The interval ends at fraction $0.75$, so its length is $s=0.75-0.40=0.35$. The endpoint is within the same year because $r+s=0.75\leq1$.

The starting survivor proportion is $1-rq_{68}=1-(0.40)(0.12)=1-0.048=0.952$. The endpoint survivor proportion is $1-(r+s)q_{68}=1-(0.75)(0.12)=1-0.09=0.91$. Thus conditional survival is

{}_{0.35}p_{68.40} =\frac{0.91}{0.952} =\frac{455}{476} =0.9559.

The conditional death probability is the complement: $1-0.9559=0.0441$. Directly, it is $(0.35)(0.12)/0.952=0.042/0.952=0.0441$. The two probabilities add to 1 before rounding because they describe the exhaustive outcomes of surviving or dying over this interval, conditional on being alive at its start. The common mistake avoided here is dividing by 1 rather than by 0.952, which would give the unconditional interval death probability 0.042.

Solving for an Annual Death Probability

The conditional formula can also be rearranged when an interval probability is given and the annual mortality input is unknown. This is useful when a problem supplies fractional-age information rather than $q_x$ directly. Before solving, confirm that the given interval has a valid length and that the resulting annual probability lies between 0 and 1.

Worked Example: Recovering the Annual Probability from Conditional Survival

A hypothetical mortality study assumes UDD during the year of age 41. For a life alive at age 41.20, the probability of surviving to age 41.70 is reported as 0.90. Find the implied annual death probability $q_{41}$, then find the conditional probability of death over that interval. No interest rate applies; the distribution is UDD and the unknown parameter is $q_{41}$.

Here $r=0.20$, $s=0.70-0.20=0.50$, and $r+s=0.70$. Let $q=q_{41}$. The given conditional survival probability gives

\frac{1-(0.70)q}{1-(0.20)q}=0.90.

Multiply both sides by the denominator: $1-0.70q=0.90(1-0.20q)=0.90-0.18q$. Rearrange: $1-0.90=0.70q-0.18q$, so $0.10=0.52q$. Therefore,

q_{41}=\frac{0.10}{0.52} =\frac{5}{26} =0.1923.

The conditional death probability is the complement of the supplied survival probability, $1-0.90=0.10$. Check with the death formula: $(0.50)(5/26)/(1-(0.20)(5/26))=(5/52)/(25/26)=0.10$. The result is a valid annual probability, and the conditional survival input is reproduced. The common mistake avoided is treating 0.90 as an unconditional survivor proportion from age 41; it is conditional on being alive at age 41.20.

Boundary Checks and Exam Traps

The assumptions and interval limits are part of the formula, not optional details. For a valid within-year interval, $s=0$ gives conditional survival 1 and conditional death 0. If $r+s=1$, the endpoint is the next integer age. When $q_x<1$, the conditional probability of surviving to that boundary is $(1-q_x)/(1-rq_x)$. If $q_x=0$, there are no deaths during the year under this model, so conditional survival is 1 for every permitted interval. If $q_x=1$, the formula is valid only when there is a positive probability of being alive at the starting age; in particular, a start at $r=1$ cannot be conditioned on being alive.

A frequent error is to use $1-sq_x$ as the conditional survival probability. That expression treats $s q_x$ as if it were the conditional death probability. Under UDD, $s q_x$ is the unconditional probability of dying in this slice of the original cohort. The conditional probability must account for having reached $x+r$:

Common trap: For $r>0$, do not replace $\frac{1-(r+s)q_x}{1-rq_x}$ with $1-sq_x$. The correct conditional death probability is $\frac{s q_x}{1-rq_x}$, not $s q_x$. Also do not use $r+s$ as the interval length: the length is $s$, while $r+s$ locates the endpoint within the year.

The formula depends on UDD across the relevant year. It should not be carried unchanged into an interval spanning more than one year: in that case, the appropriate annual mortality inputs and the survival across each year must be handled. Nor does this formula describe a different within-year mortality assumption. Always match the interpolation model to the one stated in the problem.

Key takeaway: Under UDD, conditional survival from $x+r$ to $x+r+s$ is the endpoint survivor proportion divided by the starting survivor proportion. Keep the interval length $s$, endpoint position $r+s$, and conditioning denominator $1-rq_x$ distinct.

Check Your Understanding

For each problem, use UDD only over the specified year of age. Round probabilities to four decimal places unless the answer is an annual death probability requested to four decimal places.

A hypothetical life table has $q_{60}=0.25$. Find the probability that a life alive at age 60.20 survives to age 60.50. (Answer: 0.9211)
For a hypothetical group with $q_{73}=0.16$, calculate the conditional probability of surviving from age 73.50 to age 73.75. State the conditional death probability as a check. (Answer: survival $0.9565$; death $0.0435$)
Under UDD, a hypothetical cohort has $q_{48}=0.30$. Find the probability of death between ages 48.40 and 48.90, conditional on survival to age 48.40. (Answer: 0.1705)
For a hypothetical cohort under UDD, conditional death probability from age $x+0.25$ to age $x+0.75$ is 0.12. Find the implied annual probability $q_x$. (Answer: 0.2264)
Under UDD, conditional survival from age $x+0.30$ to age $x+0.80$ is 0.95. Find the implied annual probability $q_x$. (Answer: 0.0971)

What You'll Learn