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. $A_{40:\overline{15\vert}} = 0.42$, $d = 0.04$. 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 condition balances expected premium income and expected benefits. It does not minimize the variance of the loss.

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: $P^{1}_{x:\overline{n\vert}} = A^{1}_{x:\overline{n\vert}}/\ddot{a}_{x:\overline{n\vert}}$. For endowment: $P_{x:\overline{n\vert}} = A_{x:\overline{n\vert}}/\ddot{a}_{x:\overline{n\vert}} = 1/\ddot{a}_{x:\overline{n\vert}} - d$.

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 $P = A/\ddot{a}$, balancing expected income and benefits.
• Alternative form: $P = 1/\ddot{a} - d$ (whole life and endowment), enabled by the identity $A = 1 - d\ddot{a}$
• Portfolio percentile: solve with the loss standard deviation at the candidate premium; the safety loading falls as the number of independent policies grows.
• 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 2,480 words, 17 min read) adds 4 worked examples, all 5 common mistakes, a self-check, free in the app.

Learning objectives

• 10b

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