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 $A_x=v{q}_x+v{p}_x{A}_{x+1}$ is the primary tool for analyzing single-age assumption changes.

$$A_x=v{q}_x+v{p}_x\cdot A_{x+1},{\ddot{a}}_x=1+v{p}_x\cdot {\ddot{a}}_{x+1}$$

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

KEY: Higher interest decreases all present values, both insurance and annuity. No exceptions. This is the one directional rule with no ambiguity.

Under constant force: ${\bar{A}}_x=\mu /(\mu +\delta )$. Increasing $\delta$ makes the denominator larger, reducing ${\bar{A}}_x$.

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

• Interest up lowers both $A_x$ and ${\ddot{a}}_x$, since heavier discounting shrinks every present value
• Mortality up raises $A_x$ but lowers ${\ddot{a}}_x$; insurance and annuities move in opposite directions
• Mortality down lowers $A_x$ and raises ${\ddot{a}}_x$
• The recursion $A_x=v{q}_x+v{p}_x{A}_{x+1}$ 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 1,693 words, 11 min read) adds 3 worked examples, all 5 common mistakes, a self-check, free in the app.

Learning objectives

• 9d

Browse all free Exam FAM lessons or jump into free Exam FAM practice questions.