Modelling Extra Risk

Problem Preview: Standard mortality has constant force 0.02, interest force is 0.05, and the extra mortality force is 0.03. At the shifted interest force 0.08, standard insurance is 0.20 and the annuity is 10. Including extra deaths gives substandard insurance of 0.50 and a premium of 0.05 per unit benefit. Omitting the extra-death term understates the premium.

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.

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

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

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

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: ${}_{t}p^*_x = {}_{t}p_x \cdot e^{-ct}$.

$${}_{t}p^*_x = {}_{t}p_x \cdot e^{-ct}$$

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: $q^* = q + c$ instead of computing $\mu + c$ then converting back.
• Forgetting the $c \cdot \bar{a}$ term in the insurance. $\bar{A}^* = \bar{A}^{@\delta+c} + c \cdot \bar{a}^{@\delta+c}$, 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: $\bar{P}^* = \bar{P}^{@\delta+c} + c$. Insurance: $\bar{A}^* = \bar{A}^{@\delta+c} + c\bar{a}^{@\delta+c}$. 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: $\mu^* = \mu + c$ gives ${}_{t}p^*_x = {}_{t}p_x \cdot e^{-ct}$, turning every substandard calculation into a standard one at interest rate $\delta+c$.
• Under constant addition: annuity = standard at $\delta+c$; insurance $\bar{A}^* = \bar{A}^{@\delta+c} + c\bar{a}^{@\delta+c}$; premium $\bar{P}^* = \bar{P}^{@\delta+c} + c$.
• Constant multiple: $q^* = kq$, no closed-form shortcut, compute numerically; the method fails when $kq > 1$.

Exam shortcut

When you see "constant addition to the force," immediately write ${}_{t}p^*_x = {}_{t}p_x e^{-ct}$ and $\bar{a}^* = \bar{a}^{@\delta+c}$. 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 2,158 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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