Calculate reserves for long-term life and health insurance and annuity contracts using profit testing.

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 $Pr_t < 0$. Negative emerging profits inflate the hurdle-rate discount and look like operating losses. The fix is to hold a year-end reserve ${}_tV$ 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:

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

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

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

Year 1 absorbs pre-contract expenses and starts from ${}_0V = 0$ (or whatever initial reserve the test prescribes).

Common mistakes

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

Bottom line

• Profit vector $Pr_t$ is profit per policy in force at start of year $t$; profit signature $\Pi_t = Pr_t \cdot {}_{t-1}p_x^{(\tau)}$ is profit per policy issued, where $p_x^{(\tau)}$ carries both mortality and lapse.
• Zeroization picks the smallest non-negative reserves making each $Pr_t \ge 0$ for $t \ge 2$, funding future losses from earlier surplus.
• Recursion per policy in force: $({}_{t-1}V + P_{t-1} - E_{t-1})(1+i) = q \cdot \text{claim} + p \cdot {}_tV + Pr_t$.
• Solve backwards from ${}_nV = 0$ (or the maturity benefit). Set $Pr_t = 0$, solve for ${}_{t-1}V$.

Exam shortcut

Start the backward solve at ${}_nV$. At each step compute $Pr_t$ with ${}_tV$ known and ${}_{t-1}V = 0$. If $Pr_t < 0$, solve ${}_{t-1}V = (q \cdot S + p \cdot {}_tV)/(1+i) - (P - E)$. If $Pr_t \ge 0$, keep ${}_{t-1}V = 0$. DECISION: All $Pr_t \ge 0$ unzeroized, no reserves needed. Any $Pr_t < 0$, 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 ${}_{t-1}p_x^{(\tau)}$ only at the signature step.

The full lesson (about 1,940 words, 13 min read) adds 2 worked examples, all 6 common mistakes, a self-check, free in the app.

Learning objectives

• 4e

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