Why Table Labels Matter
A select-and-ultimate mortality table presents rates with more than one kind of label. A select rate is indexed by the age at which a life was selected and by the duration since selection. An ultimate rate is indexed by an age label on the ultimate basis. Since the labels have different meanings, a correct numerical entry can still be the wrong answer if it came from the wrong row or column.
This tutorial focuses on reading table entries, not on using them to calculate multi-year survival probabilities. The central habit is simple: identify what each row and column represents before looking up a number, then report the rate with its full label and units.
Brackets are part of the select-rate label: in \(q_{[x]+j}\), the bracketed \(x\) is the select age, and \(j\) is the duration index. The ultimate-rate label \(q_y\) uses an age index \(y\). Do not assume that a row label has the same role in every section of a table. Read the table’s headings and notes; they define the meaning of its labels.
A Table with Separate Select and Ultimate Sections
The following hypothetical table gives select rates for durations 0, 1, and 2, followed by a separate ultimate table. All rates are shown as decimals. Thus, \(0.0058\) represents \(0.58\%\), or a probability of \(0.0058\).
| Select age \(x\) | Duration 0: \(q_{[x]}\) | Duration 1: \(q_{[x]+1}\) | Duration 2: \(q_{[x]+2}\) |
|---|---|---|---|
| 40 | 0.0048 | 0.0054 | 0.0062 |
| 41 | 0.0051 | 0.0058 | 0.0067 |
| 42 | 0.0055 | 0.0063 | 0.0072 |
| Ultimate age \(y\) | Ultimate rate \(q_y\) |
|---|---|
| 40 | 0.0060 |
| 41 | 0.0065 |
| 42 | 0.0071 |
| 43 | 0.0078 |
| 44 | 0.0086 |
| 45 | 0.0095 |
| 46 | 0.0105 |
In the first table, locate a select rate by using both its select-age row and duration column. For example, the entry in the age-41 row and duration-1 column is \(q_{[41]+1}=0.0058\). In the second table, locate an ultimate rate by its ultimate-age label: the entry for ultimate age 44 is \(q_{44}=0.0086\).
The table’s layout can vary. Durations may be labeled by words such as “year 1” or by the indices 0, 1, and 2; an ultimate section might be beside or below the select section. The headings, not the page layout, determine which rate an entry represents.
A Reliable Lookup Method
A lookup is safest when it is treated as a sequence of decisions rather than a quick scan for a familiar age. First determine whether the requested rate is select or ultimate. Then match the request to the corresponding age label and, for a select rate, the duration label. Finally, read the intersecting cell and preserve the table’s stated units.
Decide whether the question asks for a select rate \(q_{[x]+j}\) or an ultimate rate \(q_y\). Use the section heading and notation.
For a select rate, find the row for the age at selection \(x\). For an ultimate rate, find the row or entry labeled with ultimate age \(y\).
For a select rate, locate the column for duration \(j\). Do not substitute a duration column for an age label.
Read the intersecting value, state its complete rate label, and convert from decimal to percentage only if useful. Confirm the table’s units before reporting the result.
Why the Intersection Gives the Requested Rate
A table is an organized set of labeled values. In its select section, each row fixes the select age, and each column fixes the select duration. Consequently, one cell corresponds to the pair \((x,j)\). The notation for that cell is \(q_{[x]+j}\), so the cell at the intersection of row \(x\) and duration column \(j\) is the rate specified by those two indices. In an ultimate section, an age label \(y\) identifies the indexed rate \(q_y\).
This is not an extra mortality assumption: it is how the table’s indexing works. If a table has a note that defines a different layout or a different starting index for duration columns, follow that note. A heading such as “first select year” may label the same period differently from a heading “duration 0”; the table legend must resolve the convention.
Worked Example: Finding a Select Rate and Reporting It
A hypothetical insurer’s table is the select table shown above. A question asks for the one-year select mortality rate for a life selected at age 41 at duration 1. Find the table entry and report the rate as both a decimal and a percentage.
Method. The word “select” and the specified duration identify a select-table entry, not an ultimate entry. The select age is \(x=41\), so use the row labeled 41. The duration is \(j=1\), so use the column labeled “Duration 1: \(q_{[x]+1}\).”
At the intersection of that row and column, the table gives \(0.0058\). The complete label is \(q_{[41]+1}\). To express this decimal as a percentage, multiply by 100:
Answer: \(q_{[41]+1}=0.0058=0.58\%\). The common mistake to avoid is taking the age-41 ultimate entry, \(q_{41}=0.0065\), which is a different rate on a different basis.
Worked Example: Reading an Ultimate-Age Entry
A hypothetical benefits analyst needs the ultimate rate labeled age 44 in the table. The analyst also wants to report its value in percentage form and explain why it is not a select rate. Find the entry.
Method. The request explicitly says “ultimate rate” and gives ultimate age \(y=44\). Use the ultimate section and find the entry labeled \(q_{44}\), rather than searching the select table for a row with the number 44.
The ultimate table lists \(0.0086\) at age 44. Convert to a percentage:
The answer is \(q_{44}=0.0086=0.86\%\). It is an ultimate entry because its label is \(q_y\) in the ultimate section; it is not \(q_{[44]+j}\), which would require a select-age row and a duration.
Check. The requested age and the table’s ultimate-age label are both 44, and the rate is reported as a probability and as a percentage without changing its meaning.
Worked Example: Resolving Competing Age Labels
A hypothetical report displays the select table and ultimate table above. A reviewer asks for two distinct values: the select rate for select age 42 at duration 0, and the ultimate rate at age 42. Identify both entries, state their labels, and determine which is larger.
First lookup: select rate. Select age 42 directs us to the row labeled 42 in the select table. Duration 0 directs us to the first duration column, labeled \(q_{[x]}\). The intersection is \(0.0055\), so the requested select rate is \(q_{[42]}=0.0055\).
Second lookup: ultimate rate. Ultimate age 42 directs us to the ultimate table entry labeled \(q_{42}\), which is \(0.0071\).
Comparison. Both values are decimals on the same probability scale, so compare them directly: \(0.0071-0.0055=0.0016\). The ultimate entry is larger by \(0.0016\), or 0.16 percentage points. This comparison describes only the two requested entries; it does not establish a general relationship between select and ultimate rates.
Answer: The select rate is \(q_{[42]}=0.0055\), while the ultimate rate is \(q_{42}=0.0071\); the ultimate entry is larger. The common mistake is to see the same age number and treat the entries as interchangeable.
Common Lookup Errors
Many table-reading errors are indexing errors rather than arithmetic errors. A disciplined answer names the rate being read and shows how its label matches the table.
- Using the wrong table section. An age appearing in both sections does not make \(q_{[x]}\) and \(q_x\) the same entry. Check whether the request is select or ultimate.
- Swapping age and duration. In a select entry, the row gives select age and the column gives duration. A duration label is not an age.
- Ignoring brackets. The brackets in \(q_{[x]+j}\) mark the select-age index. Keep them in the reported label when they are part of the table’s notation.
- Assuming a percentage is a decimal. A value printed as \(0.58\%\) equals \(0.0058\), not \(0.58\). Check whether the table prints decimals, percentages, or another scale.
- Reporting only a number. Writing “0.0058” without naming \(q_{[41]+1}\) makes it difficult to verify that the intended row and column were used.
Check Your Understanding
Use the two hypothetical tables above. Give each requested entry with its full label; treat the printed values as decimals.
- What is the select rate for select age 40 at duration 2, and what is its percentage equivalent? (Answer: \(q_{[40]+2}=0.0062=0.62\%\).)
- What is the ultimate rate at age 45? (Answer: \(q_{45}=0.0095=0.95\%\).)
- A reviewer asks for the select rate at select age 41 and duration 0. Which row and column should be used, and what is the entry? (Answer: Select-age row 41, duration-0 column; \(q_{[41]}=0.0051\).)
- Find both the select rate for select age 42 at duration 2 and the ultimate rate at age 42. Which is larger, and by how much? (Answer: \(q_{[42]+2}=0.0072\), \(q_{42}=0.0071\); the select entry is larger by \(0.0001\), or 0.01 percentage points.)
- A table lists an ultimate entry \(q_{44}=0.0086\) and a select entry \(q_{[40]+1}=0.0054\). A report calls both “the rate for age 44.” Identify the labeling error and state the correct labels and values. (Answer: Only \(q_{44}=0.0086\) is the ultimate rate at age 44; \(q_{[40]+1}=0.0054\) is the select rate for select age 40 at duration 1.)