Analyze and interpret gains by source. | Free Exam ALTAM Lesson

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:

$(V_{t} + P - e) (1 + i) = q_{x} S + w_{x} C V_{t + 1} + (1 - q_{x} - w_{x}) V_{t + 1}$

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

$G = (V_{t} + P - e^{'}) (1 + i^{'}) - q_{x}^{'} S - w_{x}^{'} C V_{t + 1} - (1 - q_{x}^{'} - w_{x}^{'}) V_{t + 1}$

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.

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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 $C V_{t + 1} - V_{t + 1}$ . 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.

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Learning objectives

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