Risk Modelling and Survival Analysis · Concepts of survival models

Select, Ultimate and Aggregate Mortality Tables Explained

Q: What is the difference between select and aggregate mortality?

Select mortality depends on age at selection and time since selection. Aggregate mortality depends on attained age only and mixes all lives. Use select tables for newly underwritten lives and aggregate for general populations.

Q: How do I read q[x]+t in a select table?

Find the row for the age at selection x. Then go to the column for duration t. If t is at least the select period, use the ultimate column at attained age x + t instead.

Q: What is the select period?

It is the number of years after selection during which mortality still differs from the ultimate rates. It is usually written r. After r years, rates depend on attained age only.

Q: Is the ultimate table the same as an aggregate table?

No. An ultimate table applies to lives whose selection effect has worn off. An aggregate table includes all lives at each age, without regard to when they were selected.

Updated 11 October 2026 · Fact-checked

Select mortality applies to lives recently selected, for example by underwriting, so their death rates depend on age at selection [x] and time since selection t. After the select period, rates depend on attained age only (ultimate). Aggregate mortality ignores selection and uses attained age for all lives. To use a select table, read q[x]+t and switch to the ultimate column once t reaches the select period.

Understand Select, Ultimate and Aggregate Mortality

A mortality rate normally depends on age. But for some groups it also depends on how long ago the life was selected. Selection means the life passed a check at a point in time, such as medical underwriting when buying life insurance, or joining a pension scheme while in active work.

A newly selected life is usually healthier than the general population at the same age. Over time the effect wears off. The period over which it matters is called the select period, usually written r years. Within it, the force of mortality or the rate q depends on two things: the age at selection x, and the time since selection t.

The standard notation is [x]+t. It means a life who is now aged x + t and was selected at age x. So q[x]+t is the probability that this life dies within one year. For example, q[40]+2 is the one-year death probability of a life aged 42 who was selected at age 40. If r = 3, q[40]+2 is generally not equal to q[41]+1 or q42, even though all three lives are aged 42. If r ≤ 2, q[40]+2 = q42.

After the select period, the effect of selection is assumed to have gone. Then q[x]+t = q(x+t) for t ≥ r. These are the ultimate rates, and they depend on attained age only. An aggregate table is different. It is built from all lives of a given age, mixing recently selected and long-standing lives, and does not distinguish them. It depends on attained age only throughout, so it is a single column. A select table is made of a select part (for t < r) followed by the ultimate column.

In tables, rows are usually indexed by age at selection [x]. Columns show t = 0, 1, ..., r-1, then the ultimate rates at age x + r. Reading the table correctly is most of the skill in this topic.

Key rules to remember

Select-period notation: q[x]+t = P(a life aged x + t, selected at age x, dies within 1 year); Valid for t = 0, 1, 2, ... Here t is the duration since selection.
Ultimate rule: q[x]+t = q(x+t) for t ≥ r; r is the select period. The ultimate table depends on attained age only.
Select survival probabilities: p[x]+t = 1 − q[x]+t; l[x]+t+1 = l[x]+t × p[x]+t; Use for any t, but move to the ultimate column when t reaches r.
Multi-year survival from a select age: tp[x] = l[x]+t ÷ l[x] = p[x] × p[x]+1 × ... × p[x]+t−1; After r years, use ultimate lives: for t > r, tp[x] = (l(x+t) ÷ l(x+r)) × (l[x]+r ÷ l[x]), and l[x]+r = l(x+r) only in a consistent table.
Consistency of a select table: l[x]+r = l(x+r); Holds when the table's ultimate l column is defined to begin at the end of the select period.

How to solve Select, Ultimate and Aggregate Mortality questions

Use this method for any question that gives a select and ultimate table or asks you to compare select and aggregate mortality.

1Identify the select period r from the table or the question. Count the select columns.
2Write each life in [x]+t form: age at selection x, duration t, attained age x + t.
3Decide for each year whether t < r (use the select column) or t ≥ r (use the ultimate column at attained age x + t).
4Pick the required quantity: q, p, tp[x] or tq[x]. Write it as a product of one-year probabilities, each in the correct column.
5Use l-values where you can. A ratio of l-values over the select-then-ultimate path avoids many multiplications.
6Compute, check that probabilities lie between 0 and 1, and sanity check: select rates should usually be lower than ultimate rates at the same attained age.
7For conceptual parts, state the cause of selection (underwriting, active-work requirement) and why the effect wears off.

Quickest way: Find the diagonal, then switch columns

When to use it: Numerical questions on survival or death probabilities over several years from a select age.

Write the life as [x] and list the years: durations 0, 1, 2, ...
Mark where duration reaches r. From that year the age alone matters.
Move along the row for the select years, then jump to the ultimate column at the matching attained age.
Multiply the p values, or use l[x]+t ÷ l[x] while inside the select period.
Check one number against the table: l at attained age x + r should match across row and ultimate column.

Common mistakes in Select, Ultimate and Aggregate Mortality

Treating q[x]+t as q(x+t) inside the select period.
Both have the same attained age, so they look the same.
Fix: Remember that the select rate also depends on age at selection. Use the select row for x and column t.
Reading the table by attained age as the row label.
Ultimate tables use attained age as the row, so students assume the same for select tables.
Fix: In a select table the row is age at selection [x]. Compute the attained age as x + t only afterwards.
Staying in the select columns after the select period ends.
Students forget that the effect of selection is assumed to have vanished.
Fix: When t ≥ r, use the ultimate column at attained age x + t.
Confusing aggregate with ultimate mortality.
Both depend on attained age only.
Fix: Ultimate rates are what select lives experience after the select period. Aggregate rates mix all lives at an age, selected recently or not, and ignore duration.
Using the wrong age at selection when a policy was bought earlier.
Students use the current age as the selection age.
Fix: If a life aged 45 bought a policy 2 years ago, write [43]+2, not [45].

Worked examples

Example 1

A select table has a select period of 2 years. For a life selected at age 50: q[50] = 0.0010, q[50]+1 = 0.0016, and the ultimate rates are q52 = 0.0022 and q53 = 0.0026. Find the probability that the life, selected at age 50, survives 4 years.

Show the solution

Write the path by duration: t = 0 uses q[50], t = 1 uses q[50]+1.
Since r = 2, from t = 2 onwards use ultimate rates at attained age 52 and 53.
p[50] = 1 − 0.0010 = 0.9990.
p[50]+1 = 1 − 0.0016 = 0.9984.
p52 = 1 − 0.0022 = 0.9978.
p53 = 1 − 0.0026 = 0.9974.
4p[50] = 0.9990 × 0.9984 × 0.9978 × 0.9974.
0.9990 × 0.9984 = 0.99740160.
0.99740160 × 0.9978 = 0.99520738 (approx).
0.99520738 × 0.9974 = 0.99261973 (approx).

Answer: 4p[50] ≈ 0.9926

Example 2

A select and ultimate table has a select period of 2 years. It gives l[40] = 9,800, l[40]+1 = 9,760, and l42 = 9,700. Find (a) q[40]+1 and (b) the probability that a life selected at age 40 survives from duration 1 to attained age 43, given l43 = 9,640. Assume l42 is the ultimate l at age 42 and l[40]+2 = l42.

Show the solution

(a) q[40]+1 = 1 − l[40]+2 ÷ l[40]+1.
Since l[40]+2 = l42 = 9,700, q[40]+1 = 1 − 9,700 ÷ 9,760.
9,700 ÷ 9,760 = 0.993852, so q[40]+1 = 0.006148.
(b) The life is at duration 1 (age 41). Survival to age 43 is 2 years: one select year then one ultimate year.
First year: p[40]+1 = 9,700 ÷ 9,760 = 0.993852.
Second year is in the ultimate part: p42 = l43 ÷ l42 = 9,640 ÷ 9,700 = 0.993814.
Product = (9,700 ÷ 9,760) × (9,640 ÷ 9,700) = 9,640 ÷ 9,760 = 0.987705.

Answer: (a) q[40]+1 ≈ 0.00615; (b) 2p[40]+1 = 9,640 ÷ 9,760 ≈ 0.9877

Exam tips

Write [x]+t for every life before you look at the table. It prevents most reading errors.
In MCQs, check whether the question gives the select period. If it does not, look at the table layout.
For written answers, explain why select rates are lower, such as underwriting or active-work conditions, and why the effect fades.
When l-values are given, use ratios of l. They save time and reduce rounding errors.
State your assumption that selection effects vanish after the select period.

Practice questions from Concepts of survival models

Select, Ultimate and Aggregate Mortality: frequently asked questions

What is the difference between select and aggregate mortality?

Select mortality depends on age at selection and time since selection. Aggregate mortality depends on attained age only and mixes all lives. Use select tables for newly underwritten lives and aggregate for general populations.

How do I read q[x]+t in a select table?

Find the row for the age at selection x. Then go to the column for duration t. If t is at least the select period, use the ultimate column at attained age x + t instead.

What is the select period?

It is the number of years after selection during which mortality still differs from the ultimate rates. It is usually written r. After r years, rates depend on attained age only.

Is the ultimate table the same as an aggregate table?

No. An ultimate table applies to lives whose selection effect has worn off. An aggregate table includes all lives at each age, without regard to when they were selected.