Use deterministic profit testing to assess emerging surplus for equity-linked life insurance cash flows, 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.

An equity-linked policy throws off two streams the actuary must reconcile: the policyholder's separate account, which rides the market, and the insurer's general-account cash flow from M&E charges minus expenses minus guarantee claims. Deterministic profit testing locks down a scenario, projects both streams, and asks whether the deal earns its capital.

Equity-linked life insurance (unit-linked, variable life, variable annuity) splits the policy into:

• Separate account (SA). Policyholder's fund. Receives net premium, grows at gross equity return $i^{SA}$ less fund management charge, debits M&E and policy fees.
• Insurer general account. Receives M&E charges and policy fees. Pays acquisition and maintenance expenses plus any guarantee top-up at death or surrender.

The separate account evolves recursively:

$$A{V}_t=\left[(A{V}_{t-1}+P_t-c_t)(1+i^{SA})\right](1-m)$$

where $c_t$ is the front-end allocation charge and $m$ is the M&E rate applied to year-end fund.

$P{r}_t$ is the cash surplus emerging to the insurer at the END of year $t$, PER policy still in force at the START of year $t$:

Common mistakes

• Confusing $P{r}_t$ and ${\Pi }_t$. Forgetting to multiply by ${}_{t-1}{p}_x^{(\tau )}$ overstates each year's contribution. Issue-year ${\Pi }_0=P{r}_0$ since no survival adjustment applies at duration zero.
• Discounting $P{r}_t$ directly. NPV discounts ${\Pi }_t$, not $P{r}_t$. Doing the former double-counts in-force lives.
• Using earned rate as RDR. The earned rate $i$ is the assumed investment return; the RDR $r$ is the shareholder hurdle. They are different numbers, usually $r>i$.

Bottom line

• Profit vector $P{r}_t$ is per-in-force end-of-year emerging surplus. Profit signature ${\Pi }_t={}_{t-1}{p}_x^{(\tau )}\cdot P{r}_t$ is per-issue, with ${\Pi }_0=P{r}_0$.
• NPV discounts ${\Pi }_t$ at the risk discount rate (not the earned rate). IRR is the RDR that zeroes NPV. Profit margin equals NPV divided by EPV of premiums.
• Issue strain $P{r}_0$ goes negative when acquisition cost exceeds initial charges and lengthens the DPP, the first $t$ at which cumulative discounted $\Pi$ turns non-negative.
• Best estimate uses central assumptions. Stress tests flex equity return, lapse, mortality, or expense and re-run; combined stresses can turn a profitable deal into a loss.

Exam shortcut

Build the spreadsheet column by column: AV roll-forward, then charges and expenses, then expected DB and SB cost, then $P{r}_t$, then ${}_{t-1}{p}_x^{(\tau )}$, then ${\Pi }_t$, then discount factor at the RDR. Sum the last column for NPV. DECISION: Stress equity for GMDB exposure. Stress mortality for term and whole life. Stress lapse for products with surrender charges or front-loaded acquisition. For DPP, run a cumulative discounted $\Pi$ column.

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Learning objectives

• 7e

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