Use deterministic profit testing to calculate premiums or reserves, and assess emerging surplus for Universal Life insurance, including profit signature, profit vector, net present value, internal rate of return, profit margin, and discounted payback periods, under best estimate or stress test assumptions.

Universal Life profit testing pushes the policy's account value through a yearly cashflow recursion, asks how much surplus the insurer keeps after paying claims and expenses, and discounts that emerging surplus to judge whether to issue the contract.

Profit testing a UL contract starts from the account value recursion. With premium $P_t$ paid at the start of year $t$, an expense charge $E{C}_t$, a cost-of-insurance deduction $Co{I}_t$, and credited interest $i_c$ on the post-deduction balance:

$$A{V}_t=(A{V}_{t-1}+P_t-E{C}_t-Co{I}_t)(1+i_c)$$

The cost-of-insurance charge funds the net amount at risk (NAR), the slice of the death benefit above AV. A standard year-start charge is $Co{I}_t=q_{x+t-1}^{CoI}\cdot v_{CoI}\cdot NA{R}_t$, where $NA{R}_t=D{B}_t-(A{V}_{t-1}+P_t-E{C}_t)$. Follow the question's exact timing of deductions.

Reserve Choice. For a Type A (level) or Type B (level plus AV) UL, the policyholder reserve held by the insurer is usually the AV itself.

Common mistakes

• Discounting the profit vector instead of the profit signature. NPV uses ${\Pi }_t={}_{t-1}{p}_x^{(\tau )}\cdot P{r}_t$. Summing $P{r}_t\cdot v_r^t$ ignores decrements and overstates NPV.
• Using credited rate where earned rate belongs. The shareholder accumulates net cashflows at the earned rate $i_e$. The credited rate $i_c$ only grows AV. Confusing them deletes the spread profit.
• Computing CSV from the pre-credit balance. Cash surrender value is end-of-year $A{V}_t$ minus surrender charge, not the working balance before crediting.

Bottom line

• Profit vector $P{r}_t$ is profit per in-force policy at start of year $t$; profit signature ${\Pi }_t={}_{t-1}{p}_x^{(\tau )}\cdot P{r}_t$ weights by survival from issue.
• NPV discounts the signature (not the vector) at the risk discount rate (RDR); positive NPV means accept.
• IRR solves NPV equal to zero; profit margin is NPV divided by EPV of premiums; DPP is the first year cumulative discounted signature turns non-negative.
• Account value evolves as premium minus charges accumulated at the credited rate $i_c$; UL reserve typically equals AV, and cash surrender value is AV minus surrender charge.

Exam shortcut

Lay out profit testing in a single table with columns $A{V}_{t-1}$, $P$, $EC$, $CoI$, $A{V}_t$, $q^d$, $q^w$, $p^{(\tau )}$, $DB$, $CSV$, $P{r}_t$, ${}_{t-1}p$, ${\Pi }_t$, $v_r^t$, ${\Pi }_t{v}_r^t$. The running cumulative sum of the last column gives partial NPV at every duration, so DPP and total NPV drop out of the same sheet. DECISION: Premium solve → linear interpolation between two trial NPVs. Reserve solve → invert the $P{r}_t$ equation for $V_t$.

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Learning objectives

• 6c

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