Survival After The Select Period

Survival Once the Ultimate Period Has Begun

A select mortality table distinguishes lives by how recently they were selected; after the select period ends, the table uses ultimate rates indexed by attained age. When a survival calculation begins after that transition, each year of the calculation uses the ultimate rate for the age at the start of that year. The central task is therefore to track the age interval correctly before multiplying.

This tutorial considers survival calculations whose starting point is already in the ultimate period. It does not calculate survival through both select and ultimate periods. If a life selected at age \(x\) has a select period of \(r\) years, the ultimate period begins at attained age \(x+r\). A calculation beginning at or after that point uses ultimate rates.

Definition: In the ultimate period, \(q_y\) is the probability that a life alive at exact age \(y\) dies before reaching age \(y+1\), and \(p_y=1-q_y\) is the probability that the life survives that year. The \(n\)-year survival probability from age \(y\) is denoted \({}_np_y\). It is the probability that a life alive at age \(y\) survives to age \(y+n\).

The starting age determines the first rate, not the age at selection or the age at which a policy was issued. For example, survival from age 52 to age 55 uses \(p_{52}\), \(p_{53}\), and \(p_{54}\). The last included interval is age 54 to 55; a rate at age 55 is not needed.

The Product of Annual Survival Probabilities

For an ultimate mortality table, the general calculation is

{}_np_y=\prod_{j=0}^{n-1}p_{y+j} =\prod_{j=0}^{n-1}(1-q_{y+j}).

Every factor corresponds to one complete year, and the attained age increases by one from one factor to the next. The rates do not have to be equal. In fact, unless the problem states that they are equal, use the rate for each specific attained age.

The justification is the multiplication rule for conditional probabilities. To survive \(n\) years, the life must survive the first year, then the second year given survival through the first, and so on. Each conditional one-year survival probability is the applicable \(p\) at that year’s starting age.

\begin{aligned} {}_np_y &=P(\text{survive to }y+n\mid\text{alive at }y)\\ &=P(\text{survive to }y+1\mid\text{alive at }y)\\ &\quad{}\times P(\text{survive to }y+2\mid\text{alive at }y+1)\\ &\quad{}\times\cdots\times P(\text{survive to }y+n\mid\text{alive at }y+n-1)\\ &=p_y p_{y+1}\cdots p_{y+n-1}. \end{aligned}

This argument uses conditional probabilities and does not require an assumption that the annual survival events are independent. Each factor already conditions on being alive at the start of its year. The corresponding probability of death before age \(y+n\), for a life alive at age \(y\), is the complement:

{}_nq_y=1-{}_np_y.

Here \({}_nq_y\) means death within the \(n\)-year interval, not the one-year rate \(q_y\). Keeping the notation distinct helps prevent a multi-year probability from being mistaken for a single annual rate.

A Reliable Calculation Process

Locate the starting age.
Confirm that the life is already in the ultimate period. Record the attained age at the start of the requested interval.

List the annual intervals.
For survival from age \(y\) to age \(y+n\), list \(y\) through \(y+n-1\). These are the ages whose annual rates are needed.

Convert each mortality rate.
For every listed age, calculate \(p_{y+j}=1-q_{y+j}\). Do not substitute one age’s rate for another.

Multiply and check.
Multiply the annual survival probabilities. If the question also asks for death during the interval, subtract the survival probability from 1.

Worked Example: Three Years of Ultimate Survival

A hypothetical ultimate mortality table gives \(q_{44}=0.015\), \(q_{45}=0.018\), and \(q_{46}=0.020\). A life is alive at age 44, already in the ultimate period. Calculate the probability of surviving to age 47 and the probability of dying before age 47.

Method. Use the three rates for the intervals age 44 to 45, age 45 to 46, and age 46 to 47. This avoids accidentally including a rate at age 47, which would apply to the next interval.

First convert each mortality rate to an annual survival probability:

p_{44}=1-0.015=0.985,\qquad p_{45}=1-0.018=0.982,\qquad p_{46}=1-0.020=0.980.

Multiply the three annual survival probabilities:

{}_3p_{44}=0.985(0.982)(0.980) =0.967270(0.980) =0.9479246.

Thus the probability of survival to age 47 is \(0.9479\), to four decimal places. The probability of death before age 47 is

{}_3q_{44}=1-0.9479246=0.0520754\approx 0.0521.

The check is that survival and death probabilities sum to \(0.9479246+0.0520754=1\). No interest assumption is needed because this is a probability calculation, not a present-value calculation.

Indexing Rates After the Select Period

A select label identifies the age at selection, but ultimate rates are indexed by attained age. Once the select period has ended, the calculation begins with the ultimate rate at the attained age at that transition—or at the later age where the requested interval begins. The selection age itself is not automatically the starting age for an ultimate-period calculation.

Worked Example: Starting at the Ultimate Age

A hypothetical select table has a two-year select period for a life selected at age 50. The life has survived to the end of that period and is now age 52, where the ultimate period begins. The ultimate rates are \(q_{52}=0.010\), \(q_{53}=0.012\), and \(q_{54}=0.014\). Calculate the probability of surviving from age 52 to age 55 and, for a hypothetical group of 10,000 lives alive at age 52, the expected number who reach age 55.

Method. Begin with attained age 52 and use the ultimate rates for ages 52, 53, and 54. The select age of 50 identifies the earlier selection but is not an ultimate rate index for this calculation.

The annual survival probabilities are

p_{52}=1-0.010=0.990,\qquad p_{53}=1-0.012=0.988,\qquad p_{54}=1-0.014=0.986.

Therefore,

{}_3p_{52}=0.990(0.988)(0.986) =0.978120(0.986) =0.96442632.

The probability is \(0.9644\), to four decimal places. The expected number reaching age 55 is

$$ 10{,}000(0.96442632)=9{,}644.2632. $$

Rounding the expected count to the nearest whole life gives 9,644. The complementary probability of death before age 55 is \(1-0.96442632=0.03557368\); multiplying this by 10,000 gives 355.7368 expected deaths. The expected survivors and deaths total 10,000 before rounding, which checks the calculation.

A useful way to verify the list of rates is to count intervals, not ages named in the question. From age 52 to age 55 there are three intervals, so exactly three annual survival factors are required. Their starting ages are 52, 53, and 54.

Worked Example: Survival Through Two Years, Then Death

A hypothetical ultimate table gives \(q_{63}=0.022\), \(q_{64}=0.026\), and \(q_{65}=0.031\). A life is alive at age 63. Calculate the probability that the life survives to age 65 and then dies before reaching age 66. Also calculate the probability of surviving all the way to age 66.

Method. The first event requires survival in the age-63 and age-64 intervals, followed by death in the age-65 interval. The second event requires survival in all three intervals.

Convert the annual rates to survival probabilities:

p_{63}=0.978,\qquad p_{64}=0.974,\qquad p_{65}=0.969.

The probability of reaching age 65 is

{}_2p_{63}=0.978(0.974)=0.952572.

Given survival to age 65, the probability of dying before age 66 is \(q_{65}=0.031\). Thus the probability of the specified sequence—survive to 65, then die before 66—is

0.952572(0.031)=0.029529732\approx 0.0295.

For survival to age 66, use \(p_{65}\) instead of \(q_{65}\):

{}_3p_{63}=0.978(0.974)(0.969) =0.952572(0.969) =0.923042268\approx 0.9230.

The two outcomes after reaching age 65 sum to the probability of reaching age 65: \(0.029529732+0.923042268=0.952572\). This checks that the final interval has been handled correctly.

Common Mistakes and Checks

The most common error is repeating the first year’s rate across the entire interval. That silently assumes a constant annual mortality rate, which is not given merely because the calculation uses one ultimate table. In the last example, using \(p_{63}\) three times would give

$$ (0.978)^3=0.935441352, $$

not the correct probability \(0.923042268\). The rates at ages 64 and 65 differ from the rate at age 63, so their corresponding survival probabilities must be used. A repeated-rate calculation would be appropriate only if the problem explicitly specified a constant rate over the relevant years.

Another error is using the wrong set of ages. From age 63 to age 66, use ages 63, 64, and 65—not 63, 64, 65, and 66. The latter list includes an extra interval, age 66 to 67. A final check is to calculate the number of factors: an interval lasting \(n\) complete years must have exactly \(n\) annual survival factors.

Key takeaway: For survival beginning in the ultimate period, identify the attained age at the start, list each year’s starting age, and multiply \(1-q\) for those ages. Use the complement \(1-{}_np_y\) for death during the full interval, and check that the survival and death probabilities sum to 1.

Check Your Understanding

For each question, assume the stated rates are hypothetical ultimate one-year mortality probabilities. Give probabilities to four decimal places where needed.

A life is alive at age 70. Given \(q_{70}=0.020\) and \(q_{71}=0.030\), find the probability of surviving to age 72. (Answer: 0.9506)
A life is alive at age 45 in the ultimate period. Given \(q_{45}=0.012\), \(q_{46}=0.016\), and \(q_{47}=0.020\), calculate the probability of surviving to age 48. (Answer: 0.9527)
A hypothetical select period for a life selected at age 40 lasts two years. At age 42 the life enters the ultimate period, with \(q_{42}=0.010\) and \(q_{43}=0.015\). Find the probability of surviving from age 42 to age 44. (Answer: 0.9752)
For a life alive at age 60, the ultimate rates are \(q_{60}=0.025\), \(q_{61}=0.030\), and \(q_{62}=0.040\). Find the probability of surviving to age 62 and then dying before age 63. (Answer: 0.0378)
A hypothetical group of 25,000 lives is alive at age 80 in the ultimate period. Given \(q_{80}=0.050\), \(q_{81}=0.060\), and \(q_{82}=0.070\), calculate the expected number surviving to age 83, rounded to the nearest whole life. (Answer: 20,762)

What You'll Learn