Why Two Measures of Future Lifetime Are Used
A person who is currently age 40 and dies at age 44.6 has lived 4.6 more years. That elapsed time is not an integer. Yet some actuarial questions count only the whole years completed before death: in this example, four. These are two different measurements of the same future lifetime, and keeping them separate prevents errors in survival calculations and in later life-contingency applications.
The distinction matters even when the two measurements are closely related. Two people can each complete two years after a given age, while one dies shortly after the second anniversary and the other shortly before the third. Their curtate future lifetimes agree, but their complete future lifetimes do not.
The subscript \(x\) identifies the exact age at which future lifetime is being measured. It is not itself a duration. For example, if the starting age is \(x=40.4\) and death occurs at age 44.6, the future lifetime is measured from age 40.4, not from birth and not from the preceding integer age.
The floor convention assigns a death exactly \(k\) years after age \(x\) to \(K_x=k\). In a continuous lifetime model, the probability of death at any one exact time is zero, so this boundary convention does not affect probabilities. If a problem explicitly uses a discrete convention for deaths at anniversaries, follow that convention consistently. In the examples below, the floor convention is used.
From Age at Death to the Two Lifetimes
When the starting age and age at death are both given, first find elapsed time by subtraction. Then count the completed anniversaries from the starting age. These are not necessarily the same operation: subtraction gives \(T_x\), while counting completed years gives \(K_x\).
Use the exact attained age \(x\) from which the future lifetime is measured.
Subtract the starting age from the age at death. This gives \(T_x\), in years.
Count the anniversaries of age \(x\) reached by the time of death. Equivalently, apply the stated integer-year convention to \(T_x\) to obtain \(K_x\).
\(T_x\) is a duration and can include a fractional year; \(K_x\) is a nonnegative integer count.
Worked Example: A Death Between Two Anniversaries
A hypothetical person is observed at exact age 38.4. The recorded age at death is 43.9. Find the complete and curtate future lifetimes, and check the result using the anniversary ages. Treat the two ages as given observations; no mortality distribution or interest assumption is needed.
Method. Subtract ages to find elapsed time, then identify the number of whole years completed. This avoids treating an age at death as though it were a future-lifetime count.
Complete future lifetime. The elapsed time from age 38.4 to age 43.9 is
Curtate future lifetime. Five years after age 38.4 is age \(38.4+5=43.4\). Six years after age 38.4 is age \(38.4+6=44.4\). Death at age 43.9 occurs after the fifth anniversary but before the sixth, so five whole years have been completed:
The check is consistent: \(5.5\) years consists of five complete years plus another \(0.5\) year. The death age, 43.9, is not the value of either future-lifetime measure.
Why the Integer Count Is the Floor
The relationship between the two measures follows from dividing elapsed time into consecutive one-year intervals. If death occurs between the second and third anniversaries after age \(x\), exactly two whole years have been completed. More generally, for a nonnegative elapsed time \(T_x\), there is an integer \(k\) such that \(k\leq T_x<k+1\). The number of whole years completed is then \(k\), which is precisely the greatest integer not exceeding \(T_x\).
This is a definition-based justification, not a statement that the fractional part of lifetime has any particular distribution. Knowing \(K_x\) locates death within a one-year interval, but it does not tell us where within that interval death occurs. For instance, a curtate lifetime of two years is compatible with many complete lifetimes, from two years up to (but not including) three years under the floor convention.
Worked Example: Different Elapsed Lifetimes, Same Curtate Lifetime
Two hypothetical lives are each measured from exact age 52.6. Life A dies at age 54.85, and Life B dies at age 55.40. Find \(T_{52.6}\) and \(K_{52.6}\) for each life, and compare their results. The death ages are stipulated observations; no mortality model, distribution parameters, or interest rate is assumed.
Method. Calculate each elapsed lifetime from the common starting age, then locate each death relative to the second and third anniversaries. This prevents an assumption that equal curtate lifetimes imply equal elapsed lifetimes.
For Life A,
The second anniversary is age \(52.6+2=54.6\), and the third is age \(52.6+3=55.6\). Death at age 54.85 is between these anniversaries, so
For Life B,
This death is also after age 54.6 and before age 55.6, so
Both lives have the same curtate future lifetime, two years, but their complete future lifetimes differ by \(2.80-2.25=0.55\) years. As a final check, each elapsed time is at least two years and less than three years, as required for a curtate lifetime of two.
Starting at a Fractional Age
The starting age need not be an integer. If someone is observed at age 47.75, the relevant anniversaries for future-lifetime measurement are ages 48.75, 49.75, 50.75, and so on. Counting from age 47.00 instead would answer a different question, because it would use a different starting point.
Worked Example: A Fractional Starting Age and a Later Death
A hypothetical life is observed at exact age 47.75 and dies at age 50.10. Determine the complete and curtate future lifetimes, identify the two anniversaries that bracket the death, and state the error made by using the integer age 47 as the starting age. Treat the ages as exact; no mortality distribution or interest assumption is needed.
Method. Measure elapsed time from the stated exact starting age, then check the anniversary interval. This avoids shifting the origin of measurement to the last birthday.
The elapsed time is
The second anniversary after age 47.75 is \(47.75+2=49.75\), and the third is \(47.75+3=50.75\). Since \(49.75<50.10<50.75\), death occurs after two complete years but before three:
If age 47 were incorrectly used as the starting point, the subtraction would give \(50.10-47=3.10\) years. That is not the future lifetime from age 47.75; it includes the \(0.75\) year before the stated observation age. The correct result is checked by the anniversary interval: \(2\leq2.35<3\).
Common Exam Traps
The most frequent error is to confuse a duration with a count. The following incorrect calculation shows the issue.
Wrong calculation: A life observed at age 60.2 dies at age 63.7. The age at death is 63.7, so the curtate future lifetime is 63 years.
Correction: The starting age is 60.2, so elapsed time is \(63.7-60.2=3.5\) years. Three anniversaries have been completed, and the fourth has not; therefore \(T_{60.2}=3.5\) and \(K_{60.2}=3\). The age at death is an age, not a duration or a count of years lived since age 60.2.
Other checks help catch related errors:
- Do not round the complete lifetime to get the curtate lifetime. A value such as 2.8 years corresponds to two completed years, not three.
- Do not discard a fractional starting age. The anniversaries are counted from the exact age \(x\), not automatically from the nearest birthday.
- Keep the measures distinct in words and notation. \(T_x\) is elapsed time; \(K_x\) is an integer count.
- Check the boundary convention. Under the floor convention, a death exactly \(k\) years after age \(x\) gives \(K_x=k\).
Check Your Understanding
For each problem, use the exact stated starting age and the floor convention for \(K_x\). Each case gives observed ages, so no interest rate or mortality distribution is assumed.
- A hypothetical life is observed at age 29.2 and dies at age 34.6. Find the complete and curtate future lifetimes. (Answer: \(T_{29.2}=5.4\) years and \(K_{29.2}=5\).)
- A life is observed at exact age 61.75 and dies at age 62.00. How much time elapses, and how many full years are completed? (Answer: \(T_{61.75}=0.25\) years and \(K_{61.75}=0\).)
- A hypothetical life is observed at age 40.5 and dies exactly at age 44.5. Under the stated floor convention, find \(T_{40.5}\) and \(K_{40.5}\). (Answer: \(T_{40.5}=4.0\) years and \(K_{40.5}=4\).)
- Two lives are both measured from age 33.4. One dies after 1.1 years and the other after 1.9 years. Find each curtate lifetime and the difference between their complete future lifetimes. (Answer: Both curtate lifetimes are 1 year; their complete future lifetimes differ by 0.8 years.)
- A hypothetical life is observed at age 55.8 and dies at age 60.1. Find \(T_{55.8}\), \(K_{55.8}\), and the age at the fourth anniversary after observation. (Answer: \(T_{55.8}=4.3\) years, \(K_{55.8}=4\), and the fourth anniversary is age 59.8.)