Premium Calculation Principles

Problem Preview: A 15-year endowment of 50,000 on (40), with expenses of 60% first-year premium, 8% renewal premium, $100/year per policy. $A_{40:\overline{15|}}=0.42$, $d=0.04$. The net premium is $1,448. But the gross premium is $1,751, and half the candidates forget the $(1-c)$ factor in the denominator.

HIGH-FREQUENCY: The equivalence principle formula $P=A/\ddot{a}$ is the single most-tested premium calculation on FAM.

Set $E[L_0]=0$:

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

This premium simultaneously minimizes $\text{Var}(L_0)$, a testable result.

KEY: The equivalence principle premium $P=A/\ddot{a}$ is the most tested formula in Topic 10. The alternative form $P=1/\ddot{a}-d$ is often faster when $A$ is not given directly.

For $n$-year term: $P_{x:\overline{n|}}^1=A_{x:\overline{n|}}^1/{\ddot{a}}_{x:\overline{n|}}$. For endowment: $P_{x:\overline{n|}}=A_{x:\overline{n|}}/{\ddot{a}}_{x:\overline{n|}}=1/{\ddot{a}}_{x:\overline{n|}}-d$.

NOTE: Premium ordering: $P_{x:\overline{n|}}^1<P_x<P_{x:\overline{n|}}$ (term is cheapest, endowment is most expensive). Endowment always pays, so it costs the most.

Common mistakes

• Forgetting to subtract 1 from the annuity for renewal expenses. Renewal (years 2, n) uses $\ddot{a}-1$, not $\ddot{a}$. Including year 1 in renewal overstates the expense.
• Omitting the $(1-c)$ factor. Percentage-of-premium expenses put $G$ on both sides. The clean formula has $(1-c)$ in the denominator. Forgetting it makes the equation unsolvable.
• Using the wrong $z$ value. Portfolio percentile at $\alpha =0.05$: use $z=1.645$, not 1.96 (which is two-sided 95%).

Bottom line

• Equivalence principle: $E[L_0]=0$ gives $P=A/\ddot{a}$, the most tested formula in Topic 10; this premium also minimizes $\text{Var}(L_0)$
• Alternative form: $P=1/\ddot{a}-d$ (whole life and endowment), enabled by the identity $A=1-d\ddot{a}$
• Portfolio percentile: adds safety loading $z_{\alpha }\cdot \text{SD}(L_0)/\sqrt{n}$; loading vanishes as $n\to \infty$
• Gross premium: $G=(\text{benefit EPV}+\text{non-premium expenses})/((1-c)\ddot{a})$, percentage-of-premium creates the $(1-c)$ factor

Exam shortcut

For gross premium problems, set up three columns: benefit EPV, expense EPV (by type), premium income EPV. Write $G\cdot \ddot{a}=\text{benefit}+\text{expenses}$ and collect all $G$ terms on the left before solving. For portfolio percentile, compute the equivalence premium first, then add loading, your answer must exceed the equivalence premium. "A over a-double-dot", the fundamental premium formula.

The full lesson (about 2,151 words, 14 min read) adds 4 worked examples, all 5 common mistakes, a self-check, free in the app.

Learning objectives

• 10b

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