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.

For a unit whole life benefit at positive interest with a fixed nonnegative premium, higher mortality at each future age raises $A_{x+t}$ and lowers $\ddot{a}_{x+t}$. Both effects push the reserve $A_{x+t} - P_x\ddot{a}_{x+t}$ higher.

KEY: At positive interest with the contract premium fixed and nonnegative, uniformly higher future mortality increases a whole life reserve. Both the benefit and premium-income effects reinforce.

Both $A_{x+t}$ and $\ddot{a}_{x+t}$ increase as discounting becomes lighter. The reserve increases only if the benefit-value increase exceeds the increase in premium-income value. The direction depends on the cash flows and fixed premium.

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

• At positive interest with a fixed nonnegative premium, uniformly higher future mortality raises the whole life reserve through both benefit and premium-income values.
• Lower interest can raise or lower the reserve. Compute the benefit-value change minus the fixed premium times the annuity-value change.
• Re-valuation: the premium is fixed at original assumptions; only future $A_{x+t}$ and $\ddot{a}_{x+t}$ change
• Simultaneous mortality and interest changes require revaluation on the combined basis; their reserve effects need not reinforce.

Exam shortcut

For each assumption change, keep the contract premium fixed and mark the effect on the benefit value and premium annuity. Compute their difference to verify the reserve direction and magnitude. Keep the premium fixed. For interest changes, compare the change in benefit value with the change in premium-income value before deciding the reserve direction.

The full lesson (about 1,605 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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