Relationships Between Insurance, Endowment, and Annuity Present Values

Problem Preview: Given ${\ddot{a}}_{45}=16$, ${\ddot{a}}_{65}=10$, ${}_{20}{E}_{45}=0.30$, $d=0.05$. Find the 20-year term insurance. Three identities, 30 seconds: temporary annuity = $16-0.30\times 10=13$. Endowment = $1-0.05\times 13=0.35$. Term = $0.35-0.30=0.05$. Summing 20 individual terms would take five minutes.

HIGH-FREQUENCY: The insurance-annuity identity $A_x=1-d{\ddot{a}}_x$ appears directly or indirectly in the majority of Topics 9 and 10 problems.

From $Y=(1-Z)/d$, taking expectations:

$${\ddot{a}}_x=\frac{1-A_x}{d},A_x=1-d\cdot {\ddot{a}}_x$$

Continuous: ${\bar{A}}_x=1-\delta \cdot {\bar{a}}_x$. The pattern: insurance = 1 minus (discount rate) times (annuity-due).

KEY: The identity $A_x=1-d{\ddot{a}}_x$ appears directly or indirectly in the majority of Topics 9 and 10 problems. Match the discount rate to the payment type: $d$ for due, $\delta$ for continuous, $d^{(m)}$ for $m$-thly.

Common mistakes

• Using $i$ instead of $d$. The identity is $A_x=1-d\cdot {\ddot{a}}_x$, not $1-i\cdot {\ddot{a}}_x$. Using $i$ gives a value that is too low. Trap: a negative insurance value.
• Using the identity with term instead of endowment. $1-d\cdot {\ddot{a}}_{x:\overline{n|}}$ gives the endowment insurance, not term. Subtract ${}_n{E}_x$ to get term. Trap: answer too high by exactly ${}_n{E}_x$.
• Confusing deferred annuity direction. In the splitting identity, the second annuity is at age $x+n$, not age $x$. Using ${\ddot{a}}_x$ instead of ${\ddot{a}}_{x+n}$ creates a circular equation.

Bottom line

• Master identity: $A_x=1-d\cdot {\ddot{a}}_x$ (annual), ${\bar{A}}_x=1-\delta \cdot {\bar{a}}_x$ (continuous)
• Endowment decomposition: $A_{x:\overline{n|}}=A_{x:\overline{n|}}^1+{}_n{E}_x$ (term + pure endowment)
• Whole life splitting: ${\ddot{a}}_x={\ddot{a}}_{x:\overline{n|}}+{}_n{E}_x\cdot {\ddot{a}}_{x+n}$ (temporary + deferred)
• Term shortcut: $A_{x:\overline{n|}}^1=1-d\cdot {\ddot{a}}_{x:\overline{n|}}-{}_n{E}_x$

Exam shortcut

If your approach requires summing more than 5 terms, stop and look for an identity. The exam frequently provides ${\ddot{a}}_x$, ${\ddot{a}}_{x+n}$, and ${}_n{E}_x$, three values that give you the temporary annuity, endowment insurance, and term insurance through simple arithmetic. "1 minus d-a-double-dot", the insurance-annuity identity in one phrase. "Whole = Temp + Deferred." "Endowment = Term + Pure."

The full lesson (about 1,605 words, 11 min read) adds 3 worked examples, all 5 common mistakes, a self-check, free in the app.

Learning objectives

• 9c

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