Two Clocks for the Same Lifetime
A lifetime can be described by the age at which death occurs or by the amount of time remaining from a particular age. These are not different lifetimes: they are two ways to label the same endpoint. The distinction matters whenever a problem gives an age such as 75 but asks about a person who is currently age 52. The relevant future time is not 75 years; it is the number of years between the current age and 75.
Let \(D\) denote age at death. For a person alive at exact age \(x\), let \(T_x\) denote future lifetime, measured in years from age \(x\). In the lifetime model, the individual is known to have survived to age \(x\). Thus \(D=x+T_x\) for that individual, and an event stated in terms of \(D\) can be translated by subtracting \(x\) from its age thresholds.
In standard actuarial notation, \({}_tp_x\) is the probability that a life aged \(x\) survives at least \(t\) years, and \({}_tq_x\) is the probability that the life dies within \(t\) years. Equivalently, these describe future-lifetime events: \({}_tp_x=P(T_x>t)\) and \({}_tq_x=P(T_x\le t)\). These probabilities are conditional on being alive at age \(x\), not unconditional probabilities from birth.
The offset between the two clocks is fixed: a threshold at attained age \(y\), for someone currently aged \(x\), is \(y-x\) years away. If \(y<x\), that threshold is in the past and is not a future-lifetime threshold. If \(y=x\), it is the present time, corresponding to zero years remaining.
| Statement for a life aged \(x\) | Equivalent future-lifetime event | Probability notation |
|---|---|---|
| Survive beyond age \(x+t\) | \(T_x>t\) | \({}_tp_x\) |
| Die by age \(x+t\) | \(T_x\le t\) | \({}_tq_x\) |
| Death after age \(x+a\) and by age \(x+b\), \(a<b\) | \(a<T_x\le b\) | \({}_ap_x-{}_bp_x\) |
Translating Attained-Age Thresholds
To translate a statement, first identify the age at which the life is currently observed. Then subtract that age from each attained-age threshold. Preserve the event's direction: surviving beyond a threshold remains a “greater than” event, while dying by a threshold remains an “at most” event.
Use \(x\) as the time origin. Probabilities for \(T_x\) are conditional on survival to this age.
Convert each attained-age threshold \(y\) into elapsed time \(y-x\).
Use the appropriate survival or death probability, and account for any additional conditioning explicitly.
For an interval, suppose \(0\le a<b\). The event that death occurs after age \(x+a\) and by age \(x+b\) is \(a<T_x\le b\). Its probability can be found by subtracting the probability of surviving beyond \(b\) years from the probability of surviving beyond \(a\) years:
This subtraction works because surviving more than \(b\) years is a subset of surviving more than \(a\) years. The difference consists of those who survive past \(a\) but not past \(b\). In a continuous lifetime model, the probability of death at exactly an endpoint is zero, so endpoint conventions do not change the result. If a model has a point mass at an endpoint, the strict and non-strict inequalities must instead be handled according to that model.
Worked Example: Converting an Age-at-Death Interval
A hypothetical mortality model gives a life aged 40 a probability of 0.82 of surviving 25 years and a probability of 0.71 of surviving 30 years. Find the probability that this life dies after age 65 and by age 70. Assume the model has no probability mass at an exact age.
Method. Convert the attained ages to years from age 40. Age 65 is \(65-40=25\) years away, and age 70 is \(70-40=30\) years away. Thus the requested event is \(25<T_{40}\le30\).
The probability of surviving beyond 25 years is \({}_{25}p_{40}=0.82\). The probability of surviving beyond 30 years is \({}_{30}p_{40}=0.71\). Subtracting the second nested survival probability from the first gives
The probability is 0.11, or 11%. A common mistake is to use 65 and 70 as the time values in \({}_{t}p_{40}\). Those are attained ages, not years remaining from age 40.
Intervals and Conditioning on a Later Age
A problem may ask for an interval probability conditional on surviving to an age later than the current age. In that case, the interval probability from the original age is not yet the answer: it must be divided by the probability of reaching the conditioning age. This follows from the conditional-probability rule \(P(A\mid B)=P(A\text{ and }B)/P(B)\), when the interval event \(A\) is contained in the survival event \(B\).
For example, for a life aged \(x\), the probability of death between attained ages \(x+a\) and \(x+b\), conditional on survival to age \(x+c\), where \(0\le c\le a<b\), is
The numerator is the probability, measured from age \(x\), of reaching the interval and then dying in it. It is contained within the event of surviving \(c\) years. Dividing by \({}_cp_x\) changes the conditioning base to a life known to have reached age \(x+c\). An equivalent calculation can use probabilities measured from age \(x+c\), but the time intervals must then be shortened by \(c\).
Worked Example: Conditioning on Survival to an Attained Age
For a hypothetical life aged 52, suppose the probabilities of surviving 8, 15, and 20 years are respectively 0.94, 0.86, and 0.75. Find the probability that the life dies after age 67 and by age 72, given survival to age 60. Assume a continuous lifetime model.
Method. The interval from age 67 to age 72 corresponds, from age 52, to \(15<T_{52}\le20\). The conditioning event, survival to age 60, corresponds to \(T_{52}>8\).
The probability of being in the death interval, measured from age 52, is
The probability of surviving to the conditioning age 60 is \({}_{8}p_{52}=0.94\). Since the death interval is contained in survival to age 60, divide the interval probability by 0.94:
To four decimal places, the answer is 0.1170. The denominator is survival to age 60, not survival to age 52: the question specifies that the life is already known to have reached age 60.
Working Directly With an Age-at-Death Model
Sometimes a problem specifies a model for age at death rather than giving actuarial survival probabilities. The same translation still applies, but conditioning on being alive at the current age is essential. For a life aged \(x\), only outcomes with \(D>x\) belong to the future-lifetime model. A probability calculated for \(D\) without this conditioning may describe the original population, not the life currently under consideration.
Worked Example: Finding a Residual-Lifetime Probability From a Uniform Model
In a hypothetical population, age at death \(D\) is uniformly distributed from age 60 to age 100. A person is known to be alive at age 70. Find the probability that the person dies after age 80 and by age 90. Assume the uniform model assigns no probability to any single exact age.
Method. Given survival to age 70, the possible ages at death now range from 70 to 100, a span of \(100-70=30\) years. The requested interval, after age 80 and by age 90, has length \(90-80=10\) years.
Because the conditional distribution remains uniform over the remaining 30-year range, the desired probability is the interval length divided by the remaining range:
The answer is \(1/3\), or 0.3333 to four decimal places. In future-lifetime terms, the same event is \(10<T_{70}\le20\). A common error is to divide by the original 40-year range from age 60 to 100. That would ignore the information that the person has already survived to age 70.
Common Traps and a Reliable Check
The most frequent error is confusing an attained age with a duration. A person aged 48 who is being considered for survival to age 73 has 25 years of future lifetime to that threshold, not 73 years. A second error is to compute an interval probability but overlook that the question conditions on survival to a later age. In that situation, the numerator and denominator must refer to compatible events.
- Wrong time origin: putting an attained age directly into a future-lifetime duration. Correct it by subtracting the current age.
- Wrong conditioning base: using a probability conditional on being alive at the original age when the problem says the life has survived to a later age. Divide by the probability of reaching that later age, or restart the calculation at that age.
- Wrong interval subtraction: subtracting survival probabilities in the wrong order. For \(a<b\), \({}_ap_x\ge{}_bp_x\), so the interval probability is \({}_ap_x-{}_bp_x\), which cannot be negative.
A useful final check is to ask whether the computed duration is plausible and whether the probability refers to the stated population of lives. If the requested age is later than the current age, the corresponding duration should be positive. If a conditional probability is requested, verify that the condition appears in the denominator or is built into the chosen starting age.
Check Your Understanding
For each problem, identify the future-lifetime time interval before calculating. Assume continuous lifetime models unless stated otherwise.
- A hypothetical life aged 33 is considered for survival to age 58. How many years of future lifetime does this threshold represent? (Answer: 25 years.)
- For a life aged 46, suppose \({}_{15}p_{46}=0.88\) and \({}_{20}p_{46}=0.79\). Find the probability of death after age 61 and by age 66. (Answer: 0.09.)
- A hypothetical life aged 55 has probability 0.14 of dying within 13 years. What is the probability of dying by age 68? (Answer: 0.14.)
- For a life aged 52, suppose \({}_{8}p_{52}=0.94\), \({}_{15}p_{52}=0.86\), and \({}_{20}p_{52}=0.75\). Find the probability of death after age 67 and by age 72, given survival to age 60. (Answer: 0.1170 to four decimal places.)
- For a hypothetical life aged 41, suppose \({}_{24}p_{41}=0.79\) and \({}_{30}p_{41}=0.68\). Find the probability of death after age 65 and by age 71. (Answer: 0.11.)