Standard Actuarial Notation for Future Lifetime Distributions

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 $q_x$.

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

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 ${}_{t\vert}q_x$ with ${}_{t}q_x$. ${}_{t}q_x$ = dying within $t$ years. ${}_{t\vert}q_x$ = 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 ($\mathring{}$) = continuous/complete. No circle = discrete/curtate.
• $\mu_x$ is a rate (can exceed 1). $q_x$ is a probability (bounded by 1). Under a positive constant force, $\mu > q$; at zero force both are zero.
• 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 2,203 words, 15 min read) adds 4 worked examples, all 5 common mistakes, a self-check, free in the app.

Learning objectives

• 8c

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