Complete Future Lifetime Probabilities from Life Tables

Introduction

A life table usually records survival at integer ages, but a probability about complete future lifetime may have endpoints partway through a year. To find the probability that a life aged \(x\) survives for more than 0.4 years but no more than 1.7 years, for example, the integer-age values alone are not enough: we also need a rule for survival between integer ages.

This tutorial combines integer-age life-table information with a stated fractional-age assumption. The central calculation is to find the probability of surviving to each endpoint and subtract. The interpolation model matters: UDD and constant force can give different fractional-age survival probabilities even when they agree at integer ages.

Definition: The complete future lifetime \(T_x\) is the nonnegative, possibly fractional, time from age \(x\) until death. For \(t\geq 0\), \({}_tp_x=P(T_x>t)\) is the probability of surviving more than \(t\) years from age \(x\). If the life table gives \(\ell_y\) at integer ages, then for integer \(n\), \({}_np_x=\ell_{x+n}/\ell_x\), provided \(\ell_x>0\).

Turning Life-Table Values into Fractional-Age Survival

Suppose \(t=n+r\), where \(n\) is a nonnegative integer and \(0\leq r<1\). First calculate survival through the \(n\) complete years. Then apply the chosen interpolation assumption to the fraction \(r\) of the next year. Define the one-year survival probability \(p_y={}_1p_y=\ell_{y+1}/\ell_y\), and the one-year death probability \(q_y={}_1q_y=1-p_y\).

Under UDD, deaths are distributed uniformly over each year of age. Under constant force, the force of mortality is constant within each year, so fractional-year survival is a power of the annual survival probability. In either case, fractional survival in a year is conditional on having reached the start of that year.

Fractional-age survival: For \(t=n+r\), the life-table survival probability is

{}_tp_x = \frac{\ell_{x+n}}{\ell_x} \begin{cases} 1-rq_{x+n}, & \text{UDD},\\ (p_{x+n})^r, & \text{constant force}. \end{cases}

The factor \(\ell_{x+n}/\ell_x\) accounts for survival through the first \(n\) complete years. The remaining factor accounts only for the fraction \(r\) of the year beginning at age \(x+n\). When \(r=0\), that factor is 1, so the formula returns the ordinary integer-age life-table probability.

If an interval endpoint crosses more than one integer age, the complete-year survival factor includes every intervening year. For example, when \(t=2.4\), the fractional part is applied at age \(x+2\), after survival through ages \(x\) to \(x+2\).

From Survival Probabilities to an Interval Probability

Let \(0\leq a<b\). The event \(a<T_x\leq b\) consists of lives that survive beyond \(a\), but not beyond \(b\). Since the event \(T_x>b\) is contained in \(T_x>a\), subtracting its probability from the probability of surviving beyond \(a\) leaves exactly the probability of dying within the interval.

P(a<T_x\leq b) = P(T_x>a)-P(T_x>b) = {}_ap_x-{}_bp_x.

This argument is a set subtraction, not an assumption that deaths are equally likely throughout the interval. The interpolation assumption is needed to calculate the two survival probabilities; once those are known, the subtraction identity holds. Under UDD or constant force, the lifetime distribution has no point masses within an age year, so changing an endpoint from \(<\) to \(\leq\) does not change the probability.

Write each endpoint as an integer plus a fraction.
For each endpoint \(t\), identify \(n=\lfloor t\rfloor\) and \(r=t-n\).

Calculate survival at each endpoint.
Use the life-table ratio for complete years and the specified interpolation rule for the fractional year.

Subtract in the correct order.
Calculate \({}_ap_x-{}_bp_x\) for \(a<b\); the earlier endpoint has the larger survival probability.

Apply conditioning only if requested.
Given survival beyond \(a\), divide the interval probability by \({}_ap_x\).

Conditional interval probability: For \(0\leq a<b\) and \({}_ap_x>0\),

P(a<T_x\leq b\mid T_x>a) = \frac{{}_ap_x-{}_bp_x}{{}_ap_x} = 1-\frac{{}_bp_x}{{}_ap_x}.

The denominator is survival to the beginning of the interval. Omitting it gives the unconditional probability from age \(x\), not the probability among lives that have already survived beyond \(a\).

Worked Example: UDD Within One Age Year

A hypothetical life table has \(\ell_{60}=100{,}000\), \(\ell_{61}=96{,}000\), and \(\ell_{62}=91{,}200\). Assume UDD in each year. Calculate \(P(0.4<T_{60}\leq1.7)\).

Method. Calculate survival at 0.4 years and 1.7 years from age 60, then subtract. The annual death probabilities needed are \(q_{60}=1-96{,}000/100{,}000=0.04\) and \(q_{61}=1-91{,}200/96{,}000=0.05\).

At \(t=0.4\), there are no complete years and \(r=0.4\). UDD gives \({}_{0.4}p_{60}=1-0.4(0.04)=0.984\).

At \(t=1.7\), survival through the first year is \(\ell_{61}/\ell_{60}=96{,}000/100{,}000=0.96\). The remaining 0.7 year is within age 61, so its conditional survival factor is \(1-0.7(0.05)=0.965\). Thus \({}_{1.7}p_{60}=0.96(0.965)=0.9264\).

Subtract in earlier-endpoint minus later-endpoint order: \(P(0.4<T_{60}\leq1.7)=0.984-0.9264=0.0576\). The probability is \(0.0576\), or 5.76%. As a check, survival to 0.4 years, 0.984, exceeds survival to 1.7 years, 0.9264, and their difference is positive. A common mistake is to apply \(q_{60}\) to the 0.7 fraction after age 61; that year uses \(q_{61}=0.05\), not \(q_{60}=0.04\).

Why the Interpolation Formula Works

The two interpolation rules follow from different within-year assumptions. Under UDD, conditional on reaching age \(y\), a proportion \(r\) of the deaths expected during the year from \(y\) to \(y+1\) have occurred by age \(y+r\). The fraction of the original cohort still alive is therefore \(1-rq_y\).

\ell_{y+r} = \ell_y-r(\ell_y-\ell_{y+1}) = \ell_y(1-rq_y), \qquad 0\leq r\leq1.

Dividing by \(\ell_y\) gives the UDD conditional survival factor. Under constant force, if the force during the year is \(\mu\), survival for \(r\) years is \(e^{-\mu r}\), while annual survival is \(p_y=e^{-\mu}\). Hence \(e^{-\mu r}=(p_y)^r\). These models agree at \(r=0\) and \(r=1\), but generally not between those endpoints.

The distinction is consequential: a life table supplies the survival at integer ages, but does not by itself identify the distribution of deaths inside each year. A fractional-age probability is therefore not determined by the integer-age entries until the interpolation assumption is specified.

Worked Example: Constant Force Across an Age Boundary

A hypothetical life table has \(\ell_{70}=100{,}000\), \(\ell_{71}=81{,}000\), and \(\ell_{72}=51{,}840\). Assume constant force within each year. Find \(P(0.5<T_{70}\leq1.5)\), and then find the probability conditional on surviving beyond 0.5 years.

Method. Use the annual life-table ratios to obtain the fractional-year survival factors under constant force, subtract to get the unconditional interval probability, and divide by survival to 0.5 years for the conditional probability.

The annual survival probabilities are \(p_{70}=81{,}000/100{,}000=0.81\) and \(p_{71}=51{,}840/81{,}000=0.64\).

For \(t=0.5\), the constant-force factor is \((p_{70})^{0.5}=\sqrt{0.81}=0.9\), so \({}_{0.5}p_{70}=0.9\).

For \(t=1.5\), first survive the full year to age 71, with probability 0.81. Then survive half of the next year, with conditional probability \((p_{71})^{0.5}=\sqrt{0.64}=0.8\). Thus \({}_{1.5}p_{70}=0.81(0.8)=0.648\).

The unconditional interval probability is \(0.9-0.648=0.252\). Conditional on survival beyond 0.5 years, it is \(0.252/0.9=0.28\). The answers are 0.252 and 0.280, respectively. Checking the calculation, the later survival probability, 0.648, is smaller than 0.9. A common mistake is to use \(p_{70}\) for the half-year after age 71; that fractional year uses \(p_{71}=0.64\).

Worked Example: UDD Over Several Age Years

A hypothetical life table gives \(\ell_{50}=200{,}000\), \(\ell_{51}=190{,}000\), \(\ell_{52}=171{,}000\), \(\ell_{53}=153{,}900\), and \(\ell_{54}=123{,}120\). Assume UDD in each year. Calculate \(P(1.25<T_{50}\leq3.6)\).

Method. Find each endpoint survival by combining all complete years with the appropriate fractional-year factor, then subtract. The annual survival probabilities are \(p_{50}=0.95\), \(p_{51}=0.90\), \(p_{52}=0.90\), and \(p_{53}=0.80\); equivalently, \(q_{51}=0.10\), \(q_{52}=0.10\), and \(q_{53}=0.20\).

For \(t=1.25\), survival through the first year is \(\ell_{51}/\ell_{50}=190{,}000/200{,}000=0.95\). The additional 0.25 year is within age 51. Under UDD its survival factor is \(1-0.25(0.10)=0.975\). Therefore, \({}_{1.25}p_{50}=0.95(0.975)=0.92625\).

For \(t=3.6\), survive the first three complete years with probability \(\ell_{53}/\ell_{50}=153{,}900/200{,}000=0.7695\). The remaining 0.6 year is within age 53, where the UDD survival factor is \(1-0.6(0.20)=0.88\). Therefore, \({}_{3.6}p_{50}=0.7695(0.88)=0.67716\).

Subtracting gives \(P(1.25<T_{50}\leq3.6)=0.92625-0.67716=0.24909\). The probability is 0.2491 to four decimal places. The check is that survival decreases across the interval, and the difference is between 0 and 1. The main trap is to interpolate 3.6 years using only the first and last life-table values; the fractional part belongs to the year beginning at age 53, after the three complete years have been accounted for.

Common Errors and Practical Checks

Fractional-age calculations are compact, but a small indexing error can change the result. Before subtracting, check the endpoint decomposition and the age attached to each annual probability.

Using the wrong year’s mortality. For \(t=n+r\), use \(q_{x+n}\) under UDD or \(p_{x+n}\) under constant force for the fractional part.
Mixing interpolation assumptions. Do not use \(1-rq_y\) for UDD in one endpoint and \((p_y)^r\) for constant force in another unless the problem explicitly assigns different assumptions.
Reversing the subtraction. If \(a<b\), then \({}_ap_x\geq{}_bp_x\), so the interval probability is \({}_ap_x-{}_bp_x\).
Confusing unconditional and conditional probabilities. The unconditional interval probability is a difference. The conditional probability given survival to the interval’s start is that difference divided by \({}_ap_x\).
Assuming life-table values determine fractional survival by themselves. Integer-age values do not specify death timing within a year; use the interpolation assumption stated in the problem.

A useful numerical check is that survival probabilities cannot increase with time. Thus, for \(a<b\), a valid calculation must satisfy \(0\leq{}_bp_x\leq{}_ap_x\leq1\), and consequently \(0\leq P(a<T_x\leq b)\leq{}_ap_x\). If the interval probability is negative or larger than the probability of reaching its start, revisit the endpoint ages, interpolation factors, and subtraction order.

Key takeaway: To find the probability that complete future lifetime falls between two fractional endpoints, calculate survival to each endpoint under one consistent interpolation assumption and subtract later survival from earlier survival. For a probability conditional on reaching the interval’s start, divide that difference by survival to the start.

Check Your Understanding

For each problem, use the stated interpolation assumption and give probabilities to four decimal places.

A hypothetical table has \(q_{40}=0.08\) and \(q_{41}=0.10\). Under UDD, calculate \(P(0.25<T_{40}\leq1.5)\). (Answer: 0.1060)
A hypothetical table has \(\ell_{60}=10{,}000\), \(\ell_{61}=9{,}000\), and \(\ell_{62}=8{,}100\). Under constant force, calculate \(P(0.5<T_{60}\leq1.5)\). (Answer: 0.0949)
A hypothetical table has \(q_{50}=0.04\) and \(q_{51}=0.12\). Under UDD, calculate \(P(0.6<T_{50}\leq1.2)\). (Answer: 0.0390)
A hypothetical table has \(\ell_{70}=50{,}000\), \(\ell_{71}=45{,}000\), and \(\ell_{72}=36{,}000\). Under constant force, calculate \(P(0.25<T_{70}\leq1.5)\). (Answer: 0.1690)
A hypothetical table has \(q_{30}=0.05\) and \(q_{31}=0.15\). Under UDD, calculate \(P(0.5<T_{30}\leq1.5\mid T_{30}>0.5)\). (Answer: 0.0987)

What You'll Learn