Calculate and interpret profit signature, profit vector, net present value, internal rate of return, profit margin, and discounted payback period for long-term life and health insurance, and annuity contracts.

Profit testing compresses a year-by-year projection into a handful of decision numbers. The board wants one figure to accept or reject a product, and you need to know which figure answers which question.

Run a year-by-year projection for one policy assumed in force at the start of each year. For year $t$, collect the premium, pay expenses, earn interest on the fund, pay death claims, and release the reserve for survivors. The end-of-year balance is the profit vector entry.

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

Here ${}_{t-1}V$ is the held reserve per in-force policy, $P_t$ the premium at start, $e_t$ start-of-year expense, $i$ the earned interest assumption, $S_t$ the death benefit, $E_t$ claim settlement cost, and ${}_tV$ the end-of-year reserve. The entry $Pr_0$ captures issue-time outflows (commission, underwriting) that occur before year 1 begins.

KEY: $Pr_t$ is conditional on the policy being in force at $t-1$. To get the cash an insurer expects per policy ISSUED, you must weight by survival.

Common mistakes

• Forgetting survival weighting. Reporting $Pr_t$ directly as the signature ignores that not every issued policy is in force at year-start. Always multiply by ${}_{t-1}p_x$.
• Discounting at the earned rate. NPV uses the risk discount rate $r$, not the asset earned rate $i$. The hurdle rewards shareholders, not the bond portfolio. Mixing them up typically inflates NPV by 30 to 100 percent.
• Confusing IRR with hurdle. IRR is computed from the signature; the hurdle is set by the insurer. The accept rule is IRR > hurdle, never the other way.

Bottom line

• Profit vector $Pr_t$ is end-of-year cash per policy in force at start of year $t$; profit signature $\Pi_t = {}_{t-1}p_x \cdot Pr_t$ is per policy issued and is the series you discount.
• NPV discounts $\Pi_t$ at the risk discount rate $r$, never the earned rate $i$; accept when NPV > 0. The roll-up inside $Pr_t$ uses $1+i$.
• The hurdle $r$ exceeds the earned rate $i$ because shareholders demand a return above the asset yield.
• IRR solves $\mathrm{NPV}(r^{*}) = 0$; accept when IRR exceeds the hurdle $r$. Multiple sign flips in $\Pi_t$ can yield several IRRs, so prefer NPV.

Exam shortcut

Build one cumulative-discounted column from the signature. Read NPV as the last entry; read DPP as the first non-negative row. Two metrics, one computation. DECISION: Single hurdle rate given, ranking absolute profit → use NPV and DPP. Comparing contracts of very different sizes → use profit margin or IRR. For quick IRR, find two trial rates that bracket NPV = 0 and linearly interpolate.

The full lesson (about 11,482 words, 77 min read) adds 7 worked examples, all 15 common mistakes, a self-check, free in the app.

Learning objectives

• 4c

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