Calculate premiums for long-term life and health insurance and annuity contracts based on a specified profit objective.

A profit-tested premium is the gross premium that makes the insurer's projected future profits hit a stated target. You build a cash-flow projection, discount it at the shareholder's hurdle rate, and solve for the premium that makes the profit measure equal the goal.

Project the policy year by year. For each year $t = 1, 2, \ldots, n$, the profit vector entry is the expected end-of-year cash flow assuming the policy is in force at the start of the year:

$$\text{Pr}_t = ({}_{t-1}V + P_t - e_t)(1+i) - q_{x+t-1} \cdot S_t - p_{x+t-1} \cdot {}_tV$$

Here $i$ is the earned investment rate, $P_t$ the premium, $e_t$ the per-policy expense, $S_t$ the death benefit. ${}_0V$ is typically zero. Pre-contract acquisition expense $I$ gives $\text{Pr}_0 = -I$.

The vector conditions on survival to time $t-1$. The signature unconditions back to issue:

$$\Pi_0 = \text{Pr}_0, \qquad \Pi_t = {}_{t-1}p_x^{\overline{00}} \cdot \text{Pr}_t \quad (t \geq 1)$$

For multi-state contracts (disability income, joint-life, long-term care), replace ${}_{t-1}p_x$ by the probability of being in the premium-paying state at $t-1$. Lapses enter through ${}_{t-1}p_x^{\overline{00}}$, the probability of remaining active.

Common mistakes

• Discounting the profit vector instead of the signature. Skipping the ${}_{t-1}p_x^{\overline{00}}$ factor inflates NPV. The vector is per survivor; only the signature aggregates to per policy issued.
• Using the RDR inside $\text{Pr}_t$. Reserves and premiums accumulate at the earned rate $i$. The RDR $r$ appears only in the outer discount. A common wrong answer mixes the rates and produces a premium 5 to 10 percent low.
• Forgetting the time-zero acquisition expense. Issue expense $I$ makes $\Pi_0 = -I$. Omitting it overstates NPV by exactly $I$ and gives a premium that is too low.

Bottom line

• Profit vector $\text{Pr}_t$: end-of-year cash flow per policy in force at start. Profit signature $\Pi_t$: $\Pi_0 = \text{Pr}_0$, and $\Pi_t = {}_{t-1}p_x^{\overline{00}} \cdot \text{Pr}_t$ for $t \geq 1$; discount the signature, not the vector.
• Profit vector formula: $\text{Pr}_t = ({}_{t-1}V + P_t - e_t)(1+i) - q \cdot S - p \cdot {}_tV$, with reserves and premiums accumulated at the earned rate $i$.
• NPV at risk discount rate $r$: $\text{NPV} = \sum_{t \geq 0} v_r^t \Pi_t$, where $r > i$. Set NPV equal to the target and solve for $P$.
• Profit margin: NPV divided by EPV of premiums at the RDR; common target 5 to 15 percent.

Exam shortcut

Write each year's profit vector as "income (premium plus reserve) accumulated at $i$, minus expected outgo (benefits plus end reserve)." If the table gives ${}_tV$, do not reinvent it from Thiele. DECISION: Target is dollar NPV: solve $A + BP = \text{target}$. Target is margin $m$: solve $A + BP = m \cdot P \cdot \ddot a^r$. Target is IRR: solve $\text{NPV}(r_{\text{hurdle}}) = 0$. Pre-compute the in-force vector ${}_{t-1}p_x^{\overline{00}}$ and the discount vector $v_r^t$ once. Then signature, NPV, and EPV reduce to dot products.

The full lesson (about 1,902 words, 13 min read) adds 2 worked examples, all 6 common mistakes, a self-check, free in the app.

Learning objectives

• 4d

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