Calculate reserves for long-term life and health insurance and annuity contracts using profit testing. | Free Exam ALTAM Lesson

Profit testing tells you when the cash arrives. Reserves tell you when to recognize it. Set reserves correctly and the profit signature smooths out; set them wrong and you book a loss in year five that should have lived in year one.

Without reserves, the office holds nothing at year-end and next year's expected outflow (claims and expenses minus premium) may exceed available cash, producing $P r_{t} < 0$ . Negative emerging profits inflate the hurdle-rate discount and look like operating losses. The fix is to hold a year-end reserve $_{t} V$ large enough to fund next year's deficit.

For an insurance contract paying sum insured $S$ on death within the year, on a per-policy-in-force basis:

$(_{t - 1} V + P_{t - 1} - E_{t - 1}) (1 + i) = q_{x + t - 1} \cdot S + p_{x + t - 1} \cdot_{t} V + P r_{t}$

Rearranging gives the per-policy-in-force profit:

$P r_{t} = (_{t - 1} V + P_{t - 1} - E_{t - 1}) (1 + i) - q_{x + t - 1} \cdot S - p_{x + t - 1} \cdot_{t} V$

Year 1 absorbs pre-contract expenses and starts from $_{0} V = 0$ (or whatever initial reserve the test prescribes).

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Common mistakes

Posting negative reserves. A backward step that returns $_{t} V < 0$ means zeroization is unnecessary that year. Set $_{t} V = $0$ and let the surplus emerge as positive $P r_{t}$ , not as a negative reserve.
Confusing profit vector and profit signature. $P r_{t}$ is per policy in force at start of year $t$ ; $Π_{t}$ multiplies by $_{t - 1} p_{x}^{(τ)}$ for per-policy-issued. NPV uses the signature.
Discounting at the earned rate. Reserve recursion uses $i$ (earned). NPV of $Π_{t}$ uses $r$ (risk discount rate). Mixing them double-counts the spread.

Bottom line

Profit vector $P r_{t}$ is profit per policy in force at start of year $t$ ; profit signature $Π_{t} = P r_{t} \cdot_{t - 1} p_{x}^{(τ)}$ is profit per policy issued, where $p_{x}^{(τ)}$ carries both mortality and lapse.
Zeroization picks the smallest non-negative reserves making each $P r_{t} \geq 0$ for $t \geq 2$ , funding future losses from earlier surplus.
Recursion per policy in force: $(_{t - 1} V + P_{t - 1} - E_{t - 1}) (1 + i) = q \cdot claim + p \cdot_{t} V + P r_{t}$ .
Solve backwards from $_{n} V = 0$ (or the maturity benefit). Set $P r_{t} = 0$ , solve for $_{t - 1} V$ .

Exam shortcut

Start the backward solve at $_{n} V$ . At each step compute $P r_{t}$ with $_{t} V$ known and $_{t - 1} V = 0$ . If $P r_{t} < 0$ , solve $_{t - 1} V = (q \cdot S + p \cdot_{t} V) / (1 + i) - (P - E)$ . If $P r_{t} \geq 0$ , keep $_{t - 1} V = 0$ . DECISION: All $P r_{t} \geq 0$ unzeroized, no reserves needed. Any $P r_{t} < 0$ , zeroize that year then check the previous one. Stop the moment reserves would go negative. Memorize the per-policy-in-force recursion first, then multiply by $_{t - 1} p_{x}^{(τ)}$ only at the signature step.

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Learning objectives

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