Three Ages and Time Measures in a Select Table
A select table tracks lives by the age at which they are selected and by how long they have been observed since selection. These are not interchangeable with the age a person has reached. A correct lookup therefore begins with the timeline, not with the nearest-looking age label.
For example, a hypothetical person selected at age 40 and observed for two complete years has reached attained age 42. The person belongs to the row for selection age 40, while the relevant duration column is 2. The corresponding mortality entry concerns the year beginning at attained age 42, not a person newly selected at age 42.
In conventional notation, \(q_{[x]+j}\) denotes a one-year mortality probability for a life selected at age \(x\), conditional on being alive at duration \(j\). The square brackets mark the age at selection; the \(+j\) records duration since selection. Thus \(q_{[40]+2}\) is a duration-2 cell for the cohort selected at age 40. Its one-year period is at attained age 42 to 43. By contrast, \(q_{[42]}\) refers to selection at age 42 and duration 0. Those entries can be different because they describe different select cohorts and durations.
A select table may display rows by selection age and columns by duration, or it may place the select age and duration together in a notation such as \(q_{[x]+j}\). Either way, retain both coordinates. Do not replace the selection age with the attained age when choosing the row.
Why the Attained-Age Mapping Works
The mapping is an age-and-time identity. If a life enters at age \(x\), and \(j\) years elapse, the attained age increases by exactly \(j\) years. The duration identifies how far the life has travelled from selection; adding that duration to the selection age gives the age at the start of the corresponding annual interval.
Consequently, a duration-\(j\) one-year cell spans attained ages \(x+j\) to \(x+j+1\). This is a mapping of the table’s coordinates; it does not say that every select-table rate depends only on attained age. In particular, another life at the same attained age may have entered the table at a different age and be at a different duration.
Identify the age \(x\) at which the life entered the select cohort, using the table’s stated age convention.
Determine the duration \(j\) at the start of the one-year period being considered.
Calculate \(x+j\). The year represented by that cell runs from attained age \(x+j\) to \(x+j+1\).
Use selection age \(x\) for the cohort row and duration \(j\) for the duration column. Verify that their sum matches the attained age at the start of the interval.
Worked Select-Table Lookups
Worked Example: Finding the Cell for a Later Policy Year
A hypothetical select table gives the following one-year rates for lives selected at age 38: duration 0, \(0.0041\); duration 1, \(0.0050\); duration 2, \(0.0068\); duration 3, \(0.0082\). A life selected at age 38 has survived two complete years since selection. Identify the duration-2 cell, its attained age, the age interval it covers, and the tabulated rate.
Method. Keep the selection age as the row label and count full years since selection to get the duration. Then add age and duration to identify the attained age.
The selection age is \(x=38\). Two complete years have elapsed, so \(j=2\). The attained age at the start of this duration is \(38+2=40\). The cell therefore represents the one-year interval from age 40 to age 41.
The requested table entry is \(0.0068\), or \(0.68\%\). The age-duration check is \(38+2=40\), matching the interval’s starting age. This is the select-age-38 row at duration 2, not the duration-0 entry for a life selected at age 40.
Common mistake avoided: Treating attained age 40 as the row label would change the selection cohort. The question specifies selection at 38, so the row remains 38.
Worked Example: Translating a Policy-Year Description
A hypothetical table lists the following rates for a cohort selected at age 52: duration 0, \(0.0032\); duration 1, \(0.0040\); duration 2, \(0.0049\). A life was selected exactly at age 52. Determine the table cell and the age interval for the life’s third policy year, and report the rate.
Method. Convert the numbered policy year into elapsed duration at the start of that year. The first policy year starts at duration 0, so the third starts at duration \(3-1=2\).
The selection age is 52, and the duration at the start of the third policy year is 2. The attained age at that point is \(52+2=54\). The year therefore runs from attained age 54 to age 55.
The third-policy-year cell is duration 2, and its rate is \(0.0049\), or \(0.49\%\). The check is that the first policy year begins at age 52, the second at age 53, and the third at age 54.
Common mistake avoided: Using duration 3 for the third policy year counts the year just completed as elapsed before that year begins. At the start of policy year \(k\), the duration is \(k-1\).
Worked Example: Comparing Cohorts at the Same Attained Age
A hypothetical table contains these entries: for selection age 44, the duration-3 rate is \(0.0076\); for selection age 45, the duration-2 rate is \(0.0069\). Identify the attained age and one-year interval represented by each entry. Then explain whether the entries describe the same select-table cell.
Method. Apply the age-duration mapping separately to each selection cohort. Compare the resulting attained ages only after both calculations.
For the life selected at age 44, duration 3 gives attained age \(44+3=47\). Thus \(q_{[44]+3}=0.0076\) covers ages 47 to 48.
For the life selected at age 45, duration 2 gives attained age \(45+2=47\). Thus \(q_{[45]+2}=0.0069\) also covers ages 47 to 48.
Both entries refer to the same attained-age interval, 47 to 48, but they are not the same select-table cell: one belongs to the age-44 selection cohort at duration 3, and the other to the age-45 selection cohort at duration 2. Their rates differ by \(0.0076-0.0069=0.0007\), or \(0.07\) percentage points.
Common mistake avoided: Concluding that equal attained ages require equal select-table entries discards the selection-age and duration coordinates. The table’s cells are indexed by both.
Worked Example: Auditing an Age-Duration Label
A hypothetical table includes a rate of \(0.0091\) in the row for selection age 60, duration 4. An analyst labels it as the rate for the year from attained age 63 to 64. Check the label. If it is incorrect, give the correct attained age, interval, and table notation.
Method. Reconstruct attained age from the printed selection age and duration, then compare it with the analyst’s interval.
The table coordinates are \(x=60\) and \(j=4\), so the attained age at the start of the year is \(60+4=64\). The duration-4 cell covers ages 64 to 65. Therefore the analyst’s label, ages 63 to 64, is incorrect.
The corrected label keeps the same rate, \(0.0091\), but assigns it to the year beginning at attained age 64. The check is \(64-60=4\), which recovers the printed duration.
Common mistake avoided: Reading duration 4 as the fourth policy year would shift the interval back one year. Duration 4 means four full years have elapsed; the corresponding year is the fifth policy year.
Common Indexing Errors
Select-table errors often arise from moving between several valid descriptions of the same life: age when selected, current attained age, elapsed duration, or numbered policy year. Before using a table entry, write down which description each number represents.
- Using attained age as the selection-age row. A life selected at 40 and now aged 42 remains in the selection-age-40 row.
- Counting the current year as already elapsed. The first policy year starts at duration 0; the third starts at duration 2.
- Forgetting the interval endpoint. A duration-\(j\) one-year cell starts at attained age \(x+j\) and ends at age \(x+j+1\).
- Assuming equal attained age means equal cell. Different selection ages can reach the same attained age at different durations.
- Ignoring the table’s age convention. Use the selection-age definition specified for the table. Do not substitute a calendar date, issue age, or rounded age unless the table’s convention calls for it.
The mapping \(x+j\) is useful whenever a select-table description is written in words but the table is organized by selection age and duration. It also provides a quick consistency check: if the row, duration, and stated attained age do not satisfy the equation, at least one has been misread. This tutorial concerns identifying the correct coordinates and age interval; the rates themselves must be taken from the table applicable to the stated select cohort and duration.
Check Your Understanding
For each problem, identify the selection age and duration before stating the attained age or table cell.
- A hypothetical select table gives duration-1 rate \(0.0057\) for lives selected at age 36. What attained age and one-year interval does this cell represent? (Answer: \(q_{[36]+1}=0.0057\); attained age 37; interval ages 37 to 38.)
- A life is selected at age 49. Which duration corresponds to the start of its fourth policy year, and what is its attained age then? (Answer: duration 3; attained age 52; the interval is ages 52 to 53.)
- A table entry is \(q_{[57]+2}=0.0083\). State the selection age, duration, attained age at the start of the interval, and age interval. (Answer: selection age 57; duration 2; attained age 59; interval ages 59 to 60.)
- Two hypothetical cells are \(q_{[42]+4}=0.0102\) and \(q_{[44]+2}=0.0096\). Do they refer to the same attained-age interval? Explain using the age-duration mapping. (Answer: Yes. Both start at attained age 46 and cover ages 46 to 47, but they are different select-table cells.)
- A rate in the selection-age-63 row and duration-5 column is labelled “ages 67 to 68.” Is the label consistent? Give the correct interval and state how many complete years have elapsed since selection at its start. (Answer: No. \(63+5=68\), so the correct interval is ages 68 to 69; five complete years have elapsed.)