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:

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

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. $1 - d \cdot \ddot{a}_{x:\overline{n\vert}}$ 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: $A_{x:\overline{n\vert}} = A^{1}_{x:\overline{n\vert}} + {}_{n}E_x$ (term + pure endowment)
• Whole life splitting: $\ddot{a}_x = \ddot{a}_{x:\overline{n\vert}} + {}_{n}E_x \cdot \ddot{a}_{x+n}$ (temporary + deferred)
• Term shortcut: $A^{1}_{x:\overline{n\vert}} = 1 - d \cdot \ddot{a}_{x:\overline{n\vert}} - {}_{n}E_x$

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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