Building Equity-Linked Profit Vectors

What an Annual Profit Vector Measures

An equity-linked contract can generate several different amounts in the same year: investment income on backing assets, charges received by the insurer, expenses, benefits paid, and changes in the reserve held for future benefits. A profit vector puts these items on one consistent annual basis. The central task is to state the timing and the unit of account before combining them.

The annual profits below are measured per policy in force at the start of each year. Death is assumed to occur at the year-end, after the fund value has been projected for that year. For a multi-year contract, a policy that survives the year remains in force at the next year’s start. These conventions make the expected claims and the reserve movement unambiguous.

Definition: Let $V_t$ be the reserve per policy in force at the start of year $t$, $i_t$ the deterministic return earned on the opening backing assets, $C_t$ external charges received during the year, $E_t$ expenses, $B_t$ the benefit payable on death during the year, and $M_t$ any maturity benefit payable at the year-end. Let $q_t$ be the probability of death during the year and $p_t=1-q_t$ the probability of survival. If $V_{t+1}$ is the reserve for a policy surviving to the next year, expected profit per starting policy is

\pi_t=(1+i_t)V_t+C_t-E_t-q_tB_t-p_tM_t-p_tV_{t+1}.

The maturity term is zero in a year with no maturity payment. At the contract’s final year, $V_{t+1}=0$ after all benefits have been paid.

This formula uses a gross benefit reserve: future charges are not deducted from $V_t$. A fund deduction that reduces the policy’s account value is reflected in the projected account and thus in the reserve; it is not also entered as an external charge $C_t$. By contrast, a flat policy fee collected separately from the account is included in $C_t$.

Constructing the Vector Consistently

The calculation is easiest to audit if each year is built in the same order. First project the account value under the deterministic scenario, then determine the death and maturity benefits, then calculate the reserves, and finally combine investment income, charges, expenses, claims, and reserve movements. The scenario and valuation basis must be stated explicitly; a result from one assumed return or mortality basis is not automatically applicable to another.

Project the account.
Apply the stated investment return and any deductions from the fund using their specified timing.

Identify the year’s outcomes.
Apply the death probability to death benefits and the survival probability to any maturity benefit or continuing reserve.

Value the reserve.
Use a stated gross-benefit reserve basis, including the relevant future benefits and excluding charges when the reserve is defined as gross.

Assemble and check profit.
Add investment income and external charges; subtract expenses, expected claims, and the reserve carried forward for survivors.

The reserve movement appears as $V_t-p_tV_{t+1}$ when the formula is written as opening reserve plus investment income less closing survivor reserves. This is not an additional cash payment: it is the accounting adjustment for the liability that remains after the year’s claims. The derivation is a direct balance of resources and obligations.

Justification: At the start of the year, the insurer has assets $V_t$. They grow to $(1+i_t)V_t$. During the year the insurer receives $C_t$, incurs $E_t$, and pays expected benefits $q_tB_t+p_tM_t$. At year-end, only surviving policies carry the next reserve, with expected amount $p_tV_{t+1}$. The remaining amount is profit:

\pi_t=(1+i_t)V_t+C_t-E_t-q_tB_t-p_tM_t-p_tV_{t+1}.

This balance requires the asset return, cash-flow timing, benefit definition, and reserve basis to be mutually consistent.

Worked Projection with a Death Guarantee

The following hypothetical two-year contract illustrates a complete annual vector. Its account value is invested in a fund earning a deterministic 5% annually. At each year-end, a fund charge equal to 2% of the opening account value is deducted. The insurer also collects a separate flat fee of $30 at each year-end and incurs an expense of $12 per policy-year. At the end of either year, a policyholder who dies receives the account value plus a $100 top-up. A policyholder who survives to the end of year 2 receives the account value as the maturity benefit.

The mortality assumptions are $q_1=0.02$ and $q_2=0.03$, conditional on being in force at the start of the relevant year. Thus $p_1=0.98$ and $p_2=0.97$. The reserve is the account value plus the prospective value of future $100 death top-ups, valued at 5% using these same mortality assumptions. There are no further benefits after year 2.

Worked Example: Two-Year Profit Vector and Guarantee Reserve

Method. Project the account first, calculate the guarantee reserves recursively, and then apply the annual profit balance per policy in force at the start of each year. This method avoids counting the fund charge both as a cash receipt and through its effect on the account reserve.

Account values. The first-year fund charge is $0.02(1000)=20$. The account earns $0.05(1000)=50$, so its end-year value is $1000+50-20=1030$. In year 2, the fund charge is $0.02(1030)=20.60$, investment growth is $0.05(1030)=51.50$, and the account becomes $1030+51.50-20.60=1060.90$.

Guarantee reserves. Write $G_t$ for the reserve for future top-ups alone. At the end of year 2 there is no future top-up, so $G_2=0$. At time 1, only a year-2 death top-up can be paid:

G_1=\frac{0.03(100)}{1.05}=2.8571429.

At time 0, the expected year-1 top-up is $0.02(100)=2$, and the year-2 reserve is required only if the policy survives year 1. Therefore:

G_0=\frac{0.02(100)+0.98G_1}{1.05} =\frac{2+0.98(2.8571429)}{1.05} =\frac{4.8}{1.05} =4.5714286.

The total reserves are $V_0=1000+4.5714286=1004.5714286$ and $V_1=1030+2.8571429=1032.8571429$. At the end of year 2, $V_2=0$.

Year 1 profit. Opening-asset investment income is $0.05V_0=50.2285714$. The expected death claim is $0.02(1030+100)=22.60$. There is no maturity claim in year 1. The expected reserve carried forward is $0.98V_1=0.98(1032.8571429)=1012.20$. Substitution gives:

\begin{aligned} \pi_1 &=50.2285714+30-12-22.60-1012.20+1004.5714286\\ &=38.00. \end{aligned}

Year 2 profit. Investment income is $0.05V_1=51.6428571$. A death claim is $1060.90+100=1160.90$, and the maturity benefit for a survivor is $1060.90$. Expected total benefits are $0.03(1160.90)+0.97(1060.90)=34.827+1029.073=1063.90$. There is no reserve after the final benefits, so:

\begin{aligned} \pi_2 &=51.6428571+30-12-1063.90+1032.8571429\\ &=38.60. \end{aligned}

Thus the annual profit vector per policy in force at each year’s start is $(38.00,\ 38.60)$ dollars. As a check, the fund deductions are $20 and $20.60. With the guarantee reserve valued consistently at the same investment and mortality basis, each deduction produces the corresponding account-charge margin; the external fee less expense adds $30-12=18$ each year. The resulting profits are $20+18=38$ and $20.60+18=38.60$.

Fund Deductions, External Charges, and Reserve Changes

The key distinction in a fund-linked profit test is whether a charge is deducted from the account or paid separately to the insurer. A deduction from the account lowers the benefit-linked account value and therefore affects both claims and reserves. An external fee is a direct insurer receipt. Treating both as separate cash receipts when the account deduction is already reflected in the reserve overstates profit.

Similarly, the investment-linked part of a benefit should be included in claims using the account value at the benefit date, not the value at the beginning of the year. The opening asset earns the scenario return during the year, while the policy account may grow at a different net rate because of fund deductions. The reserve change reconciles these amounts; it does not justify replacing the stated benefit with a different amount.

Worked Example: One-Year Contract with a Maturity Payment

A hypothetical one-year policy starts with an account value of $2,000. The deterministic annual return is 4%, and a fund charge of 1% of the opening account is deducted at year-end. A separate $25 fee is collected, and the insurer incurs $10 of expense. Death probability is 0.01. Death benefit is the end-year account value plus $80; a surviving policy receives the end-year account value as maturity benefit. The gross reserve is the account value plus the reserve for the death top-up, and there is no reserve after maturity.

Account and reserve. Investment growth is $0.04(2000)=80$, the fund charge is $0.01(2000)=20$, and the end-year account value is $2000+80-20=2060$. The top-up reserve is $G_0=0.01(80)/1.04=0.7692308$, so $V_0=2000.7692308$ and $V_1=0$.

Expected benefits. The death benefit is $2060+80=2140$, while the maturity benefit is $2,060. Expected benefits are $0.01(2140)+0.99(2060)=21.40+2039.40=2060.80$. Investment income on the opening reserve assets is $0.04(2000.7692308)=80.0307692$.

Profit and check. The annual profit is $80.0307692+25-10-2060.80+2000.7692308=35.00$. The same result follows by separating the components: the fund charge contributes $20, and the external fee less expense contributes $25-10=15$. The guarantee reserve exactly funds the expected top-up cost on the stated basis, so total profit is $20+15=\$35.00$.

From Per-Policy Values to a Cohort Vector

A per-policy profit vector and the expected aggregate profit for an original group are different quantities. To obtain the latter, multiply each year’s profit per starting in-force policy by the expected number of policies in force at the start of that year. With no lapses and independent mortality, the expected in-force proportion is the product of the preceding annual survival probabilities. This weighting changes the cohort totals, not the per-policy profit calculation.

Worked Example: Scaling Annual Profits to an Original Cohort

Suppose 1,000 hypothetical policies have the two-year contract and assumptions in the earlier worked projection. The annual per-policy profits are $38.00 and $38.60. There are no lapses, and the death probabilities are 0.02 in year 1 and 0.03 in year 2.

Year 1. All 1,000 policies are in force at the start of year 1. Expected aggregate profit is $1000(38.00)=\$38,000.00$.

Year 2 in-force count. The expected number surviving year 1 is $1000(0.98)=980$. Therefore expected aggregate year-2 profit is $980(38.60)=\$37,828.00$.

The cohort’s undiscounted annual profit amounts are therefore $(\$38,000.00,\ \$37,828.00)$. The second component uses 980 expected policies, not the original 1,000. This scaling is only a conversion from per-policy profits to expected cohort amounts; it does not apply interest discounting.

Common Exam Trap

A frequent error is to add an account deduction as an external charge while also using an account-based reserve that already reflects the deduction. In the first worked projection, adding the $20 year-1 fund deduction a second time would produce a reported profit of $58.00 instead of $38.00. The extra $20 is not a separate receipt: the account reserve has already fallen relative to the value it would have had without that deduction.

The correction is to choose and follow one consistent treatment. Either reflect the fund deduction in the account and its resulting benefit reserve, as in these examples, or explicitly construct a cash-flow model that records the deduction as a receipt and adjusts the reserve and asset flows so it is not counted twice. Before accepting a vector, check the benefit date, the population to which each profit applies, and whether each charge is already embedded in the projected fund value.

Key takeaway: An annual equity-linked profit vector is a sequence of annual balances, each measured on a stated per-policy or cohort basis. State the investment scenario, mortality, payment timing, charge treatment, and reserve definition; then reconcile investment income and receipts against expenses, expected benefits, and surviving reserves. This approach is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.

Check Your Understanding

For each problem, use the stated year-end timing and gross-benefit reserve convention. Amounts are per policy unless the question specifies a cohort.

A one-year policy has an opening account value of $500, a deterministic return of 6%, and a fund charge of 2% of the opening account. Its death probability is 0.02 and the death top-up is $50; survivors receive the account value at maturity. The external fee is $12, expenses are $7, and the gross reserve is the account value plus the prospective top-up reserve at 6%. Find the annual profit. (Answer: $15.00)
A two-year policy starts with an account of $800. The return is 5% each year and the fund deduction is 1.5% of opening account each year. Death probabilities are 0.01 and 0.04; the death top-up is $60 and survivors receive the account at year-2 maturity. External fees are $18 and expenses are $9 per year. Using the same gross top-up reserve convention, find $G_1$, $G_0$, and the two annual per-policy profits. (Answer: $G_1=\$2.2857$, $G_0=\$2.7265$, profit vector $(\$21.00,\ \$21.42)$)
A cohort of 2,400 policies has first-year death probability 0.015 and second-year death probability 0.025. Per-policy profits are $16.20 in year 1 and $19.50 in year 2, measured per policy in force at the year’s start. There are no lapses. Find the expected aggregate profit in each year. (Answer: $(\$38,880.00,\ \$46,098.00)$)
A one-year policy has an opening account of $1,200, a 5% return, and a 1% fund deduction. Death probability is 0.01 and the death top-up is $100; survivors receive the account at maturity. The insurer collects an external fee of $20 and incurs $8 expense. If the gross reserve includes the account and the top-up liability, find profit and explain why adding the $12 fund deduction again as an external receipt is incorrect. (Answer: $24.00; the $12 deduction is already reflected in the end-year account and reserve.)

What You'll Learn