Standard Actuarial Notation for Future Lifetime Distributions | Free Exam FAM Lesson

Problem Preview: A select-and-ultimate table with 2-year select period gives $q_{[30]} = 0.005$ , $q_{[30] + 1} = 0.008$ , $q_{32} = 0.012$ . You need $_{4} p_{[30]}$ . For years 3 and 4 you need ultimate rates at attained ages 32 and 33. The rate $q_{33}$ comes from the $x = 31$ row, not the $x = 33$ row. Using the wrong row gives 0.95614 instead of 0.96050.

HIGH-FREQUENCY: Correct reading and application of IAN notation appears in virtually every FAM mortality problem.

$_{t} p_{x}$ , probability that a life aged $x$ survives at least $t$ more years.

$_{t} q_{x}$ , probability that a life aged $x$ dies within $t$ years. When $t = 1$ , drop the subscript: $p_{x}$ and $q_{x}$ .

$_{t ∣ u} q_{x} =_{t} p_{x} \cdot_{u} q_{x + t}$ , survive $t$ years, then die within the following $u$ years. When $u = 1$ : $_{t ∣} q_{x} =_{t} p_{x} \cdot q_{x + t}$ .

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Common mistakes

Reading ultimate rates from the wrong row. After the select period, the ultimate rate $q_{33}$ is found where $x + d = 33$ . In a 2-year select table, that is the $x = 31$ row. Trap: using $q_{[33]}$ (select rate for a newly selected 33-year-old) instead of the ultimate rate.
Dropping the brackets too early. Brackets drop only after the select period ends. If the select period is 2 years, $q_{[x] + 0}$ and $q_{[x] + 1}$ use brackets; $q_{x + 2}$ does not. Trap: applying ultimate rates one year too early.
Confusing $_{t ∣} q_{x}$ with $_{t} q_{x}$ . $_{t} q_{x}$ = dying within $t$ years. $_{t ∣} q_{x}$ = surviving $t$ years then dying in the next year. Very different quantities.

Bottom line

$p$ = survive, $q$ = die. Left subscript = duration, right subscript = current age, vertical bar = deferral.
Circle accent ( $˚$ ) = continuous/complete. No circle = discrete/curtate.
$μ_{x}$ is a rate (can exceed 1). $q_{x}$ is a probability (bounded by 1). Under constant force, $μ > q$ .
Select notation: $[x]$ = age at selection, $[x] + t$ = duration $t$ since selection. Brackets drop after the select period ends.

Exam shortcut

Before computing anything in a select-and-ultimate problem, draw a timeline. Mark the selection age, current duration, select period boundary, and target age. Label each year with the correct rate, bracketed for select years, unbracketed for ultimate. This 30-second investment prevents the most common error: pulling the wrong rate. "p survives, q kills." "Brackets = recently screened." "Bar = wait then die." "Ultimate reads by attained age."

The full lesson (about 1,702 words, 11 min read) adds 2 worked examples, all 5 common mistakes, a self-check, free in the app.

Learning objectives

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