Effect of Changes in Underlying Assumptions

Problem Preview: An actuary prices a whole life policy at 6% interest, then the yield assumption drops to 5%. Under constant force $\mu = 0.03$: $\bar{A}_x$ jumps from $0.03/0.09 = 0.333$ to $0.03/0.08 = 0.375$, a 12.5% increase. One percentage point of interest rate change, millions of dollars in reserve impact across a portfolio.

HIGH-FREQUENCY: The recursion $A_x = vq_x + vp_x A_{x+1}$ is the primary tool for analyzing single-age assumption changes.

$$A_x = vq_x + vp_x \cdot A_{x+1}, \quad \ddot{a}_x = 1 + vp_x \cdot \ddot{a}_{x+1}$$

Higher $i$ → lower $v$ → every discounted payment is worth less → both $A_x$ and $\ddot{a}_x$ decrease.

KEY: Holding nonnegative cash flows and mortality fixed, higher interest cannot increase their present value. The decrease is strict when a positive payment can occur after time zero; a payment made only at issue is unchanged.

Common mistakes

• Getting the direction wrong for mortality on annuities. Higher mortality = shorter lifetimes = fewer annuity payments = lower annuity values. Not "more risk = higher value."
• Applying a change at every age when only one is affected. If $q_{70}$ changes but $q_{71}$ does not, then $A_{71}$ stays the same. Use the recursion at age 70 only.
• Forgetting that ${}^{2}A_x$ changes when interest changes. The second moment uses $2\delta$. If $\delta$ changes, $2\delta$ changes too.

Bottom line

• Higher interest cannot increase the value of fixed nonnegative cash flows; the decrease is strict when a positive payment can occur after issue.
• At positive interest, uniformly higher future mortality raises $A_x$ but lowers $\ddot{a}_x$; whole life insurance and annuities move in opposite directions.
• At positive interest, uniformly lower future mortality lowers $A_x$ and raises $\ddot{a}_x$.
• The recursion $A_x = vq_x + vp_x A_{x+1}$ isolates single-age changes: $\Delta A_x = v\Delta q_x(1-A_{x+1})$

Exam shortcut

For directional questions, the constant force model gives instant answers: $\mu/(\mu+\delta)$ is increasing in $\mu$ and decreasing in $\delta$. For quantitative single-age changes, use the recursion shortcut: $\Delta A_x = v\Delta q_x(1-A_{x+1})$. If the approach requires recomputing entire tables, stop and look for the recursion. "Insurance and annuity: mortality opposites." "Interest affects everything the same way, down." "Recursion: v-q plus v-p-A."

The full lesson (about 1,769 words, 12 min read) adds 3 worked examples, all 5 common mistakes, a self-check, free in the app.

Learning objectives

• 9d

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