Standard Actuarial Notation (International Actuarial Notation)

Problem Preview: Translate "a 10-year deferred, 15-year temporary life annuity-due for (50) with monthly payments" into notation. Answer: ${}_{10|}{\ddot{a}}_{50:\overline{15|}}^{(12)}$. Five symbols convey what took 14 words. Reading this fluently is the difference between finishing the exam and running out of time.

HIGH-FREQUENCY: The superscript "1" position distinguishes term insurance from pure endowment.

• Bar ($\stackrel{ˉ}{}$): continuous. ${\bar{A}}_x$ = continuous insurance, ${\bar{a}}_x$ = continuous annuity.
• Double dot ($\ddot{}$): annuity-due (beginning of period). ${\ddot{a}}_x$ = whole life annuity-due.
• No accent on $a$: annuity-immediate (end of period).
• No accent on $A$: annual insurance (end of year of death).

TRAP: Bar on $A$ (continuous insurance) and double dot on $a$ (annuity-due) are independent concepts. Bar means benefit at moment of death; double dot means payments at beginning of period. Confusing them changes the formula.

$A_x$ = whole life at age $x$. $A_{x:\overline{n|}}$ = $n$-year endowment insurance.

Common mistakes

• Confusing bar (continuous) with double dot (due). Bar on $A$ = benefit at moment of death. Double dot on $a$ = annuity payments at beginning of period. Independent concepts.
• Misreading the superscript-1 position. 1 over the life status = term (death first). 1 over the term status = pure endowment (survival first). Verify: "Is this death within $n$ or survival to $n$?"
• Forgetting that deferral shifts the age. ${}_{n|}{\ddot{a}}_x={}_n{E}_x\cdot {\ddot{a}}_{x+n}$. The annuity after deferral is at age $x+n$, not $x$.

Bottom line

• Two base symbols: $A$ = insurance, $a$ = annuity.
• Bar = continuous. Double dot = annuity-due. No accent on A = annual. No accent on a = immediate.
• Subscript $x$ = issue age. Angle $\overline{n|}$ = term. Pre-subscript $n|$ = deferred.
• Superscript 1 over x = term insurance. Superscript 1 over n = pure endowment.

Exam shortcut

Spend the first 5 seconds of every question decoding the notation: (1) insurance or annuity, (2) continuous/due/immediate, (3) whole life or term, (4) deferred or not, (5) annual or $m$-thly. Once classified, the formula writes itself. "BAD-SM: Bar = continuous, Accent (dot) = due, Deferral, Superscript = frequency, Moment." "1 over life = term, 1 over term = pure." "E bridges ages."

The full lesson (about 1,449 words, 10 min read) adds 2 worked examples, all 5 common mistakes, a self-check, free in the app.

Learning objectives

• 9e

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