Worked Profit Test for an Equity-Linked Policy

Purpose and Scope of an Equity-Linked Profit Test

A valuation of an equity-linked guarantee answers what the guarantee is worth on a specified basis. A profit test goes further: it projects the insurer’s period cash flows, tracks the guarantee reserve, and identifies when profit emerges. This matters because a contract can have a positive overall present value while still producing a loss in an early period, or a large loss under an adverse investment path.

The calculations below isolate the insurer’s guarantee and operating cash flows. The policyholder’s fund is projected separately and treated as a pass-through asset: the insurer pays the fund value as part of the benefit, but that amount is matched by the fund liability. The insurer’s guarantee claim is only the amount by which the guaranteed benefit exceeds the fund. All products and data are hypothetical. The methods are aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.

Definition: For an in-force policy, let $F_t$ be the policy fund immediately before a claim or maturity payment at time $t$, and let $K_t$ be the guaranteed minimum benefit. The insurer’s guarantee top-up is $G_t=\max(K_t-F_t,0)$. A guarantee reserve $V_t$ is the prospective value, per policy then in force, of future expected guarantee top-ups, discounted at the valuation interest rate. In this tutorial, the reserve excludes future charges and expenses; those cash flows are recognized in period profit when they occur.

Profit-Test Equation and Timing

Use the following timing convention. The effective annual valuation interest rate is $i$. Charges and expenses are paid at the end of each year, and claims are settled at that year-end. A policy pays the annual expense if it was in force during the year. Mortality is measured over each year: $q_t$ is the probability of death during year $t$, conditional on being in force at its start, and $p_t=1-q_t$. The reserve $V_t$ is conditional on being in force at time $t$, after that year’s claim settlement. Set the reserve after the final contract payment equal to zero.

For a cohort of one policy issued, let $S_{t-1}$ be the probability that it is in force at the start of year $t$. Thus $S_0=1$ and $S_t=S_{t-1}p_t$. Let $C_t$ and $E_t$ be the charge income and expense per policy in force during year $t$. Let $Q_t$ be the expected guarantee top-up claim during year $t$, per policy originally issued. The expected profit at the end of year $t$, per policy issued, is

\Pi_t=S_{t-1}(C_t-E_t)-Q_t+(1+i)S_{t-1}V_{t-1}-S_tV_t.

The two reserve terms are essential. The beginning reserve accumulates at the valuation rate; the ending reserve is the expected liability still held for policies that survive the year. Both are weighted by the relevant in-force probability. If death and maturity benefits differ, $Q_t$ must include the appropriate probability-weighted top-ups for each event rather than treating every event as having the same claim.

Derivation: Conditional on being in force at time $t-1$, the prospective reserve satisfies $(1+i)V_{t-1}=E[G_t\mid\text{in force at }t-1]+p_tV_t$, when $G_t$ includes the current-year top-up and $V_t$ is the survivor’s reserve after the claim. Multiplying by $S_{t-1}$ gives $(1+i)S_{t-1}V_{t-1}=Q_t+S_tV_t$. Therefore expected claims and reserve movement cancel in the profit equation when the reserve uses the same assumptions as the projection. Charges less expenses then emerge as expected profit. If projection and valuation assumptions differ, the cancellation need not hold.

The profit signature is the sequence $(\Pi_1,\Pi_2,\ldots,\Pi_n)$. Its net present value at issue is

\operatorname{NPV}=\sum_{t=1}^{n}\frac{\Pi_t}{(1+i)^t}.

This convention reports profits at year-end. If a problem places a cash flow at another time, its accumulation or discounting must reflect that timing.

Worked Profit Tests

Worked Example: Two-Year Guarantee With No Maturity Top-Up

A hypothetical two-year policy has an initial fund of $900 and a guaranteed death or maturity benefit of $1,000. The fund earns 10% in each year and a $30 fund deduction is taken at each year-end. Death probabilities are $q_1=0.10$ and $q_2=0.20$. An insurer charge of $50 and an insurer expense of $20 apply per policy in force each year. Assume a valuation rate $i=5\%$. Calculate the profit signature and NPV.

Method. Project the fund first, then calculate guarantee top-ups, prospective guarantee reserves, and the mortality-weighted profit for each year. The fund deduction reduces the policy fund; it is not an additional insurer expense in this test.

At the end of year 1, before a guarantee top-up, the fund is $900(1.10)-30=\$960$. A death claim then requires a top-up of $\max(1{,}000-960,0)=\$40$. The probability of death is 0.10, so the expected year-1 top-up per policy issued is $Q_1=0.10(40)=\$4$.

The year-2 fund is $960(1.10)-30=\$1{,}026$. The year-2 top-up is zero whether the policy ends in death or maturity. At time 1, therefore, $V_1=0$. At issue, the only expected top-up is the year-1 death claim, so

V_0=\frac{4}{1.05}=\$3.8095238.

All policies are in force during year 1, so the profit at time 1 is

\Pi_1=50-20-4+1.05(3.8095238)-0(0)=\$30.

The year-2 in-force probability is $S_1=0.90$. There is no year-2 top-up or reserve remaining after maturity, so

\Pi_2=0.90(50-20)=\$27.

The profit signature is $(\$30,\$27)$. Discounting both profits to issue gives

\operatorname{NPV}=30+\frac{27}{1.05} =30+25.7142857 =\$55.71.

The check is that the expected guarantee claim is fully funded by the initial reserve and its interest: $1.05V_0=\$4$, equal to $Q_1$. The remaining expected profit is the charge less expense in each period, adjusted for the probability of still being in force.

Worked Example: A Year-Two Investment Shock and Reserve Release

Keep the policy, mortality, charges, expenses, and 5% valuation rate from the preceding example, but suppose the year-2 fund return is $-10\%$ rather than 10%. Calculate both reserves, both profits, and the NPV. The year-1 fund is still $960.

Method. Recalculate the fund and top-up under the specified deterministic path, then value the year-2 claim conditional on surviving year 1. A common error is to subtract the claim but omit the reserve release that finances it.

The year-2 fund is $960(0.90)-30=\$834$. A top-up of $\max(1{,}000-834,0)=\$166$ is due at death or maturity. Conditional on being in force at time 1, the policy ends in year 2 either by death or, if it survives, by maturity. In either case the top-up is $166. Thus $V_1=166/1.05=\$158.0952381$.

At issue, the year-1 death claim has probability 0.10 and top-up $40. The year-2 claim has probability $0.90$ of being reached from issue and then has top-up $166. Hence

V_0=\frac{0.10(40)}{1.05} +\frac{0.90(166)}{(1.05)^2} =3.8095238+135.5102041 =\$139.3197279.

Year-1 expected profit includes the expected $4 claim and the reserve movement:

\begin{aligned} \Pi_1 &=50-20-4+1.05(139.3197279)-0.90(158.0952381)\\ &=26+146.2857143-142.2857143\\ &=\$30. \end{aligned}

For year 2, the expected charge less expense is $0.90(50-20)=\$27$, and the expected claim is $0.90(166)=\$149.40$. The accumulated reserve for the surviving cohort is $1.05(0.90)(158.0952381)=\$149.40$. Therefore

\Pi_2=27-149.40+149.40=\$27.

The profit signature is $(\$30,\$27)$, and the NPV is $30+27/1.05=\$55.71$. As a check, omitting the year-2 reserve release would give the incorrect profit $-\$122.40$. The reserve release is not optional: it is the liability held for the claim now being paid.

Worked Example: Scenario-Weighted Guarantee Profit

A hypothetical one-year policy has a fund of $900 and a maturity guarantee of $1,000. There is no fund deduction. At the end of the year the fund return is either $+20\%$, with probability 0.75, or $-20\%$, with probability 0.25. A death benefit at the same year-end has the same $1,000 floor as maturity, so the guarantee top-up is the same for either terminal event. The insurer receives a $50 charge and pays a $20 expense per policy. Use $i=5\%$. Find the scenario profits, expected profit, and NPV of expected profit.

Method. Calculate the top-up in each deterministic investment scenario, then set the issue reserve using the probability-weighted expected top-up. The reserve is not the top-up in just one selected scenario.

In the high-return scenario, the fund is $900(1.20)=\$1{,}080$, so the top-up is zero. In the low-return scenario, the fund is $900(0.80)=\$720$, so the top-up is $1{,}000-720=\$280$. The expected year-end top-up is

E[G_1]=0.75(0)+0.25(280)=\$70, \qquad V_0=\frac{70}{1.05}=\$66.6666667.

There is no reserve after the terminal payment. The high-scenario profit is $50-20+1.05(66.6666667)=\$100$. The low-scenario profit is $50-20-280+1.05(66.6666667)=-\$180$. Weighting the profits gives

E[\Pi_1]=0.75(100)+0.25(-180)=75-45=\$30.

The NPV of expected profit is $30/1.05=\$28.57$. Equivalently, discount the scenario profits first: $0.75(100/1.05)+0.25(-180/1.05)=\$28.57$. The agreement checks both the scenario weighting and the timing convention.

Interpreting the Results and Avoiding Traps

A profit vector should be read together with its assumptions. A deterministic investment path gives a deterministic sequence of fund values and top-ups, but mortality can still make the expected profit vector probability-weighted. If several investment scenarios are used, calculate the guarantee top-up separately in each scenario before weighting; in general, the top-up of an average fund is not the average top-up because the maximum function is nonlinear.

The reserve basis must also match the stated purpose of the test. In the examples, the reserve covers guarantee top-ups only, while charges and expenses enter current profit. If a problem defines a reserve net of future charges or expenses, those items must be included in the prospective reserve and not counted a second time as though the reserve covered only claims. Likewise, the policy fund is not itself a guarantee claim: paying the fund value and then adding the top-up as though both were insurer costs double counts the fund.

A reliable calculation sequence is to project fund values and event-specific top-ups, apply mortality to obtain expected claims, calculate conditional reserves under the stated valuation rate, and then apply the profit equation period by period. Finally, discount the profit signature using the timing specified. This makes the reserve movement visible and helps identify whether a negative period profit reflects a genuine emergence pattern or a missing cash flow.

Check Your Understanding

For each problem, use the stated timing convention: end-of-year profits, and reserves accumulated at the stated effective annual valuation rate.

A one-year guarantee has charge income $42, expenses $17, an expected top-up claim of $10, and an issue reserve of $20. The valuation rate is 4%, and the terminal reserve is zero. Find the year-end profit. (Answer: $35.80)
For year 1, charge income is $30, expense is $12, and expected claims per policy issued are $5. The issue reserve is $50, $i=5\%$, and 90% of policies survive to time 1, each with conditional reserve $V_1=\$40$. Find $\Pi_1$. (Answer: $29.50)
A one-year policy has charge income $60 and expense $15. At year-end, the guarantee top-up is zero with probability 0.60 and $100 with probability 0.40. The valuation rate is 5%. Find the scenario profits, expected profit, and NPV of expected profit. (Answer: scenario profits $85 and $-\$15$; expected profit $45; NPV $42.86)
A two-year guarantee has $q_1=0.20$, $p_1=0.80$, a year-1 death top-up of $30, and a year-2 top-up of $50 for every policy reaching year 2. At $i=5\%$, calculate $V_0$, taking the year-2 payment to occur at time 2. (Answer: $V_0=\$42.00$, to the nearest cent)

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