Profit Testing: Profit Signature, NPV, IRR, Embedded Value

For a policy in force at the start of year t, the year's expected cashflows include premium income, expected claim outgo (sum insured times mortality rate), expected expenses, expected surrender outgo, the year-end reserve, and interest on assets backing premiums and reserves during the year. Profit in year t per policy in force at start is the sum of cashflows plus the released reserve minus the closing reserve, all rolled forward at the assumed earned rate.

The standard profit-test recursion sets PR_t as the year-end accumulation of (P_t − e_t + interest) minus expected death benefit and surrender outgo, minus the change in reserve. ALTAM and ILA 201 problems usually provide P_t, e_t, q_{x+t-1}, interest rate, and reserve schedule directly. The mechanical part of the test is then arithmetic.

The profit vector is (PR_1, PR_2, ..., PR_n), profit per policy in force at start of each year. The profit signature converts this to profit per policy issued by weighting each PR_t by the probability the policy reaches year t. For an alive-only policy the weight is tp_x; with lapses and other decrements, use the survival of the policy across all relevant decrements.

Why two vectors: PR_t lets you reason about year-by-year economics conditional on still being on the books. The profit signature aggregates across the whole portfolio expected from one initial sale and is what gets discounted into NPV.

Discount the profit signature back to time 0 at a risk discount rate r that reflects the cost of capital the company demands for new business. For traditional life insurance r is normally well above the assumed earned rate, often 8-12% in current practice. The resulting NPV is the value of new business per policy issued (VNB per policy).

Equivalence-principle pricing sets the gross premium so that NPV at r equals zero. Profitability-driven pricing sets the premium so NPV at r meets an internal hurdle, typically a positive VNB target.

The internal rate of return is the discount rate at which NPV equals zero. For a policy with a large negative profit in year 1 (acquisition strain) followed by positive profits, IRR is well defined and is widely used as a profitability metric. For policies whose profit signature changes sign multiple times, IRR can be multi-valued and is unreliable; rely on NPV at the risk discount rate instead.

The break-even period is the smallest t at which the cumulative undiscounted profit signature turns non-negative. It is a simple liquidity-style measure of how long the policy takes to recover its first-year cost.

Embedded value is the shareholder-value perspective on the whole company: free surplus plus the in-force book. Free surplus, called adjusted net worth (ANW), is the market value of assets backing capital that is not required to support the in-force book. Value of in-force (VIF) is the present value of future shareholder cashflows from the existing book, discounted at the risk discount rate, with the cost of required capital subtracted. The two add up.

Value of new business (VNB) is the NPV of the profit signature of one year's new sales. Reporting EV with VNB lets management distinguish what was on the books at year start from what was added during the year. ILA 201 develops this decomposition; CFE 101 uses it lightly in the context of insurer financial reporting.

MCEV replaces the company-specific risk discount rate with the risk-free yield curve and adds explicit allowances for non-hedgeable risk (a frictional cost of required capital and a cost of residual non-hedgeable risks). The motivation is to make EV comparable across companies that would otherwise pick different risk discount rates.

MCEV is the standard in European insurance reporting. Under U.S. GAAP, similar ideas show up in fair-value liability frameworks and in long-duration insurance contract disclosures.

Three-year term insurance, sum insured 1,000, level annual premium P = 20 paid at the start of each year. Mortality q_{x} = 0.005, q_{x+1} = 0.008, q_{x+2} = 0.010. Acquisition expense e_0 = 30 in year 1, maintenance expense 2 in years 2 and 3. Assumed earned interest i = 0.05. No reserve held (so the profit-test recursion collapses to cash flow).

Profit per policy in force: PR_1 = (20 − 30)(1.05) − 0.005(1000) = -10.5 − 5 = -15.5. PR_2 = (20 − 2)(1.05) − 0.008(1000) = 18.9 − 8 = 10.9. PR_3 = (20 − 2)(1.05) − 0.010(1000) = 18.9 − 10 = 8.9.

Profit signature: σ_1 = -15.5. σ_2 = (1 − 0.005)(10.9) = 10.846. σ_3 = (1 − 0.005)(1 − 0.008)(8.9) = 0.98704 × 8.9 = 8.785. Cumulative undiscounted profit signature: -15.5, -4.654, +4.131. Break-even occurs during year 3, approximately at 4.654 / 8.785 = 0.530 of the way through, so a little over 2.5 years after issue.

Same profit signature as above. NPV at the risk discount rate r = 0.08: σ_1 / 1.08 + σ_2 / 1.08^2 + σ_3 / 1.08^3 = -14.352 + 9.298 + 6.974 = 1.920. So the value of new business per policy is about 1.92.

IRR: search for r with NPV(r) = 0. At r = 0.12, NPV ≈ 1.06; at r = 0.20, NPV ≈ -0.30; at r = 0.18, NPV ≈ 0.00. The IRR is about 18%. With a risk discount rate of 8%, the policy clears the hurdle by 10 percentage points of IRR; the same observation reads as NPV(0.08) = 1.92 per policy issued.

Same policy with mortality scaled up uniformly by 10%: q_x = 0.0055, q_{x+1} = 0.0088, q_{x+2} = 0.011. Recomputing: PR_1 = -10.5 − 5.5 = -16.00. PR_2 = 18.9 − 8.8 = 10.10. PR_3 = 18.9 − 11.0 = 7.90.

Profit signature: σ_1 = -16.00. σ_2 = (0.9945)(10.10) = 10.044. σ_3 = (0.9945)(0.9912)(7.90) = 7.788. NPV at r = 0.08: -16.00 / 1.08 + 10.044 / 1.1664 + 7.788 / 1.2597 = -14.815 + 8.612 + 6.183 ≈ -0.02. A 10% mortality shock essentially wipes out the entire VNB for this term-insurance product. The implied mortality elasticity of VNB is on the order of -10 to -20 for typical term insurance; longer-duration term has even higher sensitivity.

ALTAM topic 4 (profit analysis) tests the mechanical profit-vector and profit-signature recursion, NPV at a stated risk discount rate, and the directional sensitivities to mortality, lapse, expense, and interest assumptions. Expect either a hand-computed three-to-five-year cashflow table or an Excel task that builds the same projection over a longer horizon.

ILA 201 (LP topic) tests the same machinery inside a fuller pricing-and-valuation context, including embedded value, value of new business, and assumption-change attribution. CFE 101 covers the financial-reporting framing of EV but does not require the deep cashflow recursion.

First trap: dropping the reserve term from PR_t when the problem specifies a non-zero reserve. The full recursion includes both the prior reserve rolled forward and the new closing reserve. Missing this is the single biggest source of wrong PR_t values on past exams.

Second trap: discounting at the assumed earned rate instead of the risk discount rate. The profit signature already accumulates at the earned rate inside each PR_t. Discounting the signature back to time zero must use the risk discount rate, not the earned rate.

Third trap: reporting IRR for a profit signature that changes sign more than once. Multiple sign changes produce multiple IRRs and the metric loses its usual interpretation. Switch to NPV at the risk discount rate and document the sign pattern.

Standard textbook: Dickson, Hardy, and Waters, Actuarial Mathematics for Life Contingent Risks, 3rd edition, Chapter 12 (profit testing) and Chapter 13 (participating products and embedded options). The recursion above and the worked-example structure follow that text.

Exam topics: ALTAM topic 4 (profit analysis), ALTAM topic 6 (universal life insurance, where profit testing on universal life is more delicate), ILA 201 (LP topic on pricing), CFE 101 (light treatment within insurer financial reporting).