Calculate and interpret expected profit and actual profit.

A profit-test cash flow that looks healthy on paper can still bleed money once real mortality, interest, and expense experience emerge. Profit analysis is the bridge between the priced expectation and the realized outcome.

Pick any policy year $t$. At time $t-1$ the insurer holds reserve ${}_{t-1}V$ per in-force policy. The insurer collects premium $P_t$, pays expense $e_t$, earns interest at rate $i$, pays death benefit $S_t$ plus claim expense $E_t$ on deaths, and sets up reserve ${}_{t}V$ on survivors. The year-end accounting gives the profit vector:

$$Pr_t = ({}_{t-1}V + P_t - e_t)(1+i) - q_{x+t-1}(S_t + E_t) - p_{x+t-1} \cdot {}_{t}V$$

$Pr_0$ captures any pre-issue cost not absorbed in $e_1$ (acquisition, commission). Many tables push all front-end load into $e_1$ and set $Pr_0 = 0$; follow the convention given in the problem.

KEY: The reserve at $t-1$ is treated as if it were income, and the reserve at $t$ as if it were an outgo. Profit is what is left after the insurer rebuilds the liability for survivors.

Common mistakes

• Forgetting interest on premium net of expense. Premium and expense flow at time $t-1$; both earn interest for the full year. Computing $Pr_t$ as $(V + P)(1+i) - e$ misses the wash on $e$ and overstates profit by $e \cdot i$.
• Recalculating reserves on actual basis. Reserves stay on the valuation basis. Substituting actual rates into the reserve formula double-counts the experience deviation.
• Mixing profit vector and profit signature when computing NPV. NPV uses $\Pi_t$, not $Pr_t$. Discounting the vector ignores in-force decay and overstates expected profit at issue, especially for long durations.

Bottom line

• Profit vector $Pr_t$: the per-policy year-end cash flow, conditional on being in force at the start of year $t$.
• Profit signature $\Pi_t = {}_{t-1}p_x \cdot Pr_t$ weights by the probability of being in force at the start of year $t$, measured from issue.
• Expected profit at issue is the NPV of $\Pi_t$ at the hurdle rate $r$ (not the earned rate $i$); the IRR is the $r$ making NPV zero.
• Actual profit uses realized expenses, interest, deaths, and lapses; reserves stay on the valuation basis. Reserve at $t-1$ is income, reserve at $t$ on survivors is outgo.

Exam shortcut

Build a single table with columns $t$, ${}_{t-1}V$, $P_t$, $e_t$, $(V+P-e)(1+i)$, $q \cdot S$, $p \cdot {}_{t}V$, $Pr_t$, ${}_{t-1}p_x$, $\Pi_t$, discount factor, $\Pi_t v^t$. NPV, payback, and margin all read off this table. DECISION: Question asks for "profit signature" → multiply by ${}_{t-1}p_x$. Question asks for "profit vector" → leave conditional. Mislabeling loses easy points. For gain by source, write the profit vector with three named slots $(e, i, q)$.

The full lesson (about 5,751 words, 38 min read) adds 4 worked examples, all 11 common mistakes, a self-check, free in the app.

Learning objectives

• 4a

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