Effect of Changes in Underlying Assumptions on Reserves and Premiums

Problem Preview: A whole life policy issued at 5% interest ($A_{40}=0.20$, $A_{50}=0.30$). Re-value at 4%: $A_{50}^{4\%}=0.36$. The reserve jumps from 0.125 to 0.162, a 29.6% increase. For 10,000 policies at $100,000 face, that is $37 million in additional required reserves from one percentage point of interest rate change.

HIGH-FREQUENCY: The direction and magnitude of reserve changes when interest or mortality changes.

$A_{x+t}$ increases (benefits paid sooner). ${\ddot{a}}_{x+t}$ decreases (premium stream shorter). Both push the reserve $A_{x+t}-P_x{\ddot{a}}_{x+t}$ higher.

KEY: Higher mortality always increases the whole life reserve. Both components reinforce: $A_{x+t}$ goes up (more benefit liability) and ${\ddot{a}}_{x+t}$ goes down (less premium income). No exceptions for whole life.

Both $A_{x+t}$ and ${\ddot{a}}_{x+t}$ increase (all PVs grow with lighter discounting). But the insurance effect dominates, the death benefit is a single lump sum far in the future, more duration-sensitive than the shorter premium stream. Reserve increases.

Common mistakes

• Changing both the premium and the reserve. Re-valuation keeps the original premium. Recomputing $P$ under new assumptions gives zero (the new-issue reserve), not the re-valued reserve.
• Getting the direction wrong for interest on annuities. Lower interest rates increase present values of annuities, not decrease them. Do not confuse PVs with accumulated values.
• Ignoring the reinforcing effect of mortality on reserves. Higher mortality increases $A_{x+t}$ AND decreases ${\ddot{a}}_{x+t}$. Both push the reserve up. Analyzing only the insurance effect misses half the impact.

Bottom line

• Higher mortality always increases the whole life reserve: $A_{x+t}\uparrow$ and ${\ddot{a}}_{x+t}\downarrow$, both push the reserve up
• Lower interest generally increases the reserve: the benefit EPV is more interest-sensitive than the premium annuity
• Re-valuation: the premium is fixed at original assumptions; only future $A_{x+t}$ and ${\ddot{a}}_{x+t}$ change
• Most dangerous scenario: higher mortality plus lower interest, since both push reserves up

Exam shortcut

For directional questions, the annuity ratio $1-{\ddot{a}}_{x+t}/{\ddot{a}}_x$ is the fastest path. If the ratio decreases, the reserve increases. For quantitative problems, write the reserve formula and mark arrows on each component before computing. "HM-LI = Reserve Up: Higher Mortality, Lower Interest." Direction table: "Mortality up: A up, annuity down, reserve up. Interest up: A down, annuity down, reserve down." "Fix the premium", the golden rule of re-valuation.

The full lesson (about 1,697 words, 11 min read) adds 3 worked examples, all 5 common mistakes, a self-check, free in the app.

Learning objectives

• 10d

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