Modelling Extra Risk

Problem Preview: Standard values at $\delta =0.05$: ${\bar{A}}_{40}=0.18$, ${\bar{a}}_{40}=16.4$. At $\delta =0.08$: ${\bar{A}}_{40}=0.28$, ${\bar{a}}_{40}=12.0$. A substandard life has constant extra force $c=0.03$. The substandard premium is $0.28/12.0+0.03=0.05333$, nearly 5x the standard premium of 0.01098. Half the candidates forget the $c\cdot \bar{a}$ term in the insurance formula.

HIGH-FREQUENCY: The constant addition to the force of mortality and its effect on insurance and annuity values.

Treat a life aged $x$ as if aged $x+c$. All formulas use standard table values at the shifted age.

$$P^{\text{rated}}=P_{x+c},{}_t{V}^{\text{rated}}={}_t{V}_{x+c}$$

Simple but inflexible, assumes extra risk follows the standard mortality shape.

$${\mu }_{x+t}^{\ast }={\mu }_{x+t}+c$$

The key result:

KEY: The $e^{-ct}$ factor is the most powerful tool in this topic. It converts every substandard survival probability into a standard one: ${}_t{p}_x^{\ast }={}_t{p}_x\cdot e^{-ct}$.

Common mistakes

• Adding $c$ to $q$ instead of $\mu$. The constant addition goes to the force of mortality, not the one-year rate. Adding to $q$ does not produce the $e^{-ct}$ survival factor. Trap: $q^{\ast }=q+c$ instead of computing $\mu +c$ then converting back.
• Forgetting the $c\cdot \bar{a}$ term in the insurance. ${\bar{A}}^{\ast }={\bar{A}}^{@\delta +c}+c\cdot {\bar{a}}^{@\delta +c}$, not just ${\bar{A}}^{@\delta +c}$. The second term accounts for extra deaths. Trap: a premium that is too low.
• Confusing "add c to premium" with "add c to insurance." Premium shortcut: ${\bar{P}}^{\ast }={\bar{P}}^{@\delta +c}+c$. Insurance: ${\bar{A}}^{\ast }={\bar{A}}^{@\delta +c}+c{\bar{a}}^{@\delta +c}$. These are different additions.

Bottom line

• Age rating: use age $x+c$ in all standard formulas. Simplest method, no new formulas; the extra risk follows the standard mortality curve shape.
• Constant addition to force: ${\mu }^{\ast }=\mu +c$ gives ${}_t{p}_x^{\ast }={}_t{p}_x\cdot e^{-ct}$, turning every substandard calculation into a standard one at interest rate $\delta +c$.
• Under constant addition: annuity = standard at $\delta +c$; insurance ${\bar{A}}^{\ast }={\bar{A}}^{@\delta +c}+c{\bar{a}}^{@\delta +c}$; premium ${\bar{P}}^{\ast }={\bar{P}}^{@\delta +c}+c$.
• Constant multiple: $q^{\ast }=kq$, no closed-form shortcut, compute numerically; the method fails when $kq>1$.

Exam shortcut

When you see "constant addition to the force," immediately write ${}_t{p}_x^{\ast }={}_t{p}_x{e}^{-ct}$ and ${\bar{a}}^{\ast }={\bar{a}}^{@\delta +c}$. If the problem gives standard values at both $\delta$ and $\delta +c$, you have everything. Do not re-derive, apply directly. "Shift, Add, Multiply" (the three methods in order of complexity. "CAF: Combine forces, Annuity shifts, Full insurance adds $c\bar{a}$." "Premium plus c") the fastest formula in this topic.

The full lesson (about 2,025 words, 14 min read) adds 3 worked examples, all 5 common mistakes, a self-check, free in the app.

Learning objectives

• 10e

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