Analyze and interpret gains by source.

A life insurer projected zero surplus on its 2026 term block. Actual surplus came in materially above plan. Where did it come from? Mortality, interest, expense, or lapse? Gains-by-source answers that.

Per policy in force at time t, the expected reserve recursion balances on the pricing basis:

$$(V_t + P - e)(1+i) = q_x S + w_x CV_{t+1} + (1 - q_x - w_x) V_{t+1}$$

On the expected basis the equation balances. Replace each expected value with its actual to obtain the total gain G per starting policy:

$$G = (V_t + P - e')(1+i') - q'_x S - w'_x CV_{t+1} - (1 - q'_x - w'_x) V_{t+1}$$

Because the expected-basis equation balances to zero, this expression is really the actual right-hand side minus that zero. G is the residual surplus left over once the priced balance has been subtracted, not the full year-end fund.

A single G tells management nothing about which assumption is mis-set. Decomposition attributes G to specific lines so pricing, reserving, and ALM can be repaired separately.

Common mistakes

• Not stating the order. Different orders shift gain across lines. Always declare the order; the exam usually declares it for you.
• Using $i$ instead of $i'$ in expense accumulation. Expense gain uses $(1+i')$ because interest is already actual at this step. Under E-I-M it flips to $(1+i)$ since interest hasn't switched.
• Using $V_t$ in DSAR. DSAR is $S − V_{t+1}$, the END-of-year reserve, not the start-of-year reserve.

Bottom line

• Total gain splits into interest, expense, mortality, and withdrawal pieces that sum to the same total no matter the order chosen.
• Order matters per line. Standard order is interest, expense, mortality, withdrawal, with each step using actual for prior items and expected for later ones.
• Interest gain equals $G_i = (i' − i)(V_t + P − e)$; expense stays expected because expense comes later in the order.
• Expense gain equals $G_e = (e − e')(1 + i')$, accumulated at the actual rate because interest has already switched.

Exam shortcut

Write each event line as (expected − actual) × (gain-when-fewer-events). For mortality the multiplier is $S − V_{t+1}$; for withdrawal it is $CV_{t+1} − V_{t+1}$. Interest is the exception: (actual − expected) × base. DECISION: Asked for one line only? Three formulas cover most items: $G_i = (i'-i)(V_t + P - e)$, $G_e = (e - e')(1+i')$, $G_m = (q - q')(S - V_{t+1})$. The $(1+i')$ factor on expense and the year-end reserve in DSAR are the two slip points. "IEMW" for the standard order: Interest, Expense, Mortality, Withdrawal.

The full lesson (about 6,000 words, 40 min read) adds 5 worked examples, all 11 common mistakes, a self-check, free in the app.

Learning objectives

• 4b

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