Profit Signatures and NPV for Equity-Linked Products

From Profit Vectors to Profit Signatures

An equity-linked policy’s profit can emerge unevenly over its term. A favorable investment return may increase the value of the policy fund, the associated benefit, and the reserve needed to support future obligations. A poor return can produce a different pattern. Thus, an overall profit figure alone can conceal when profit or loss arises and how much of it is expected to emerge after policyholders have left the contract.

A profit vector records profit by duration for a specified projection, such as one investment path. A profit signature brings those duration-specific results back to the policy’s issue date by allowing for the probability that the policy is still in force. Discounting the signature then gives the expected present value of profit, or NPV, on the stated valuation basis.

Definition: Let $\Pi_{t+1}$ be profit at the end of policy year $t+1$, conditional on the policy being in force at the start of year $t+1$. Its conditional expected profit is $g_t=E[\Pi_{t+1}\mid\text{in force at time }t]$. If ${}_tp_x$ is the probability of remaining in force to time $t$, the duration-$(t+1)$ profit signature is $\psi_{t+1}={}_tp_xg_t$. At effective annual valuation interest rate $i$, the NPV is $\sum_{t=0}^{n-1}v^{t+1}\psi_{t+1}$, where $v=1/(1+i)$. Here the signature is stated before discounting; some reporting conventions display the discounted contributions $v^{t+1}\psi_{t+1}$ as the signature instead.

This definition treats a policy that has left the contract as contributing no later profit. The in-force probability must reflect the exits included in the model. In the examples below, exits are by death only; if surrender or another decrement is included, its probability and associated cash flow must also be represented. Investment-scenario weights are valuation weights specified for the calculation. They are not automatically risk-neutral probabilities.

Profit Calculation and Timing

For a simple annual profit test, suppose an in-force policy starts the year with reserve $V_t$. A premium $P_t$ is received and an expense $E_t$ is paid at the start of the year. These amounts earn the year’s investment return $j_t$. At year-end, a death benefit $D_t$ is paid if death occurs, while a closing reserve $V_{t+1}$ is held for a surviving policy. The closing reserve is zero after a terminal benefit has been paid. Assume the investment return is deterministic within the path being projected, and that the year-end death probability is $q_t$, with $p_t=1-q_t$. Then expected profit per policy in force at the start of the year is

g_t=(V_t+P_t-E_t)(1+j_t)-q_tD_t-p_tV_{t+1}.

This formula uses a single-year timing convention: premium and expense occur at the start; investment accumulation, death claims, and the closing reserve are accounted for at the end. A contract with different payment timing requires a correspondingly adjusted cash-flow calculation. If a maturity benefit $M$ is paid at the end of the year to every survivor, the survivor outflow becomes $p_t(M+V_{t+1})$. Do not also include a closing reserve that covers a benefit already paid.

If there are investment scenarios $s$, with weights $w_s$ satisfying $\sum_s w_s=1$, calculate the profit vector separately in each scenario. When scenario and mortality are independent, the expected conditional profit is $g_t=\sum_s w_sg_{t,s}$. If they are dependent, use the appropriate joint probabilities instead; multiplying separate marginal expectations would then be incorrect.

Why Survival Weighting and Discounting Give NPV

The signature follows directly from conditioning on whether a policy reaches each duration. At time $t$, the policy contributes an expected profit only if it remains in force. The probability-weighted expected profit at the end of the following year is therefore ${}_tp_xg_t$. Its value at issue is this amount multiplied by the discount factor for $t+1$ years.

Condition on being in force.
Calculate $g_t$, the expected profit for the next year per policy in force at time $t$, incorporating the year’s claims and closing reserve.

Weight by reaching the duration.
Multiply by ${}_tp_x$. This converts conditional profit per in-force policy into expected profit per policy issued.

Discount to issue.
Multiply the duration-$(t+1)$ signature by $v^{t+1}$, because the profit is measured at the end of that year.

Add across durations.
NPV is the sum of the discounted expected profit contributions. Linearity of expectation permits this addition without assuming profits at different durations are independent.

In symbols, the justification is

\operatorname{NPV} =\sum_{t=0}^{n-1}v^{t+1}E\!\left[{\boldsymbol{1}}_{\{\text{in force at }t\}}\Pi_{t+1}\right] =\sum_{t=0}^{n-1}v^{t+1}{}_tp_xg_t,

where ${\boldsymbol{1}}_{\{\text{in force at }t\}}$ equals 1 if the policy is in force at time $t$, and 0 otherwise. The second equality uses the definition of conditional expectation. With investment scenarios, take the scenario-weighted expectation inside the same expression. If mortality probabilities are also scenario-dependent, use the joint expectation rather than applying a single survival factor to an already averaged profit vector.

Worked Example: A Two-Year Profit Signature

A hypothetical two-year policy has annual premium $P=100$ and expense $E=10$, both at the start of each year. The investment return is 0% in both years. The reserves are $V_0=0$, $V_1=90$, and $V_2=0$. The death probability in each year, conditional on being in force at its start, is 0.10. The death benefit in year 1 is 100. At the end of year 2, either a death claim or a maturity benefit of 100 is paid, and no reserve remains. The valuation rate is $i=5\%$. Calculate the profit vector, signature, and NPV. Mortality is independent of investment because the investment path is deterministic.

Method. First calculate conditional profit for each year using the reserve-based cash-flow formula, then apply survival weighting and discounting.

Year 1 conditional profit. Assets after the start-of-year premium and expense are $(0+100-10)(1.00)=90$. Expected death claims are $0.10(100)=10$. The expected closing reserve for survivors is $0.90(90)=81$. Therefore $g_0=90-10-81=-1$.

Year 2 conditional profit. Assets after premium and expense are $(90+100-10)(1.00)=180$. Since a benefit of 100 is paid whether the policyholder dies during the year or survives to maturity, expected total benefit outgo is $0.10(100)+0.90(100)=100$. The terminal reserve is zero, so $g_1=180-100=80$.

Signature and NPV. Every policy is in force at time 0, so $\psi_1=1(-1)=-1$. Only 0.90 are expected to reach year 2, so $\psi_2=0.90(80)=72$. Discounting gives

\operatorname{NPV} =\frac{-1}{1.05}+\frac{72}{1.05^2} =-0.952381+65.306122 =64.353741.

The profit vector is $(-1,80)$; the pre-discount signature is $(-1,72)$. The NPV is $\$64.35$ per policy, rounded to the nearest cent. As a check, the second-year signature is below the conditional profit of 80 because only 90% of policies are expected to remain in force. The common mistake is to use 80 directly as the second-year profit per policy issued, omitting survival weighting.

Worked Example: Reversing the Order of Investment Returns

A hypothetical two-year equity-linked policy has opening reserves $V_0=V_1=100$, no premiums or expenses, and no reserve after the terminal payment at time 2. In each year the conditional death probability is 0.10. A benefit of 100 is paid at the end of year 1 on death; a reserve of 100 is held for survivors. At the end of year 2, the death or maturity benefit is 100. Consider two investment paths with equal weights: path A has returns $+20\%$ in year 1 and $-10\%$ in year 2; path B has returns $-10\%$ in year 1 and $+20\%$ in year 2. The valuation rate is 4%. Calculate the scenario profits, expected profit signature, and NPV.

Method. Apply the annual profit formula within each path before averaging. This preserves the duration at which each return affects profit.

Year 1, path A. Accumulated assets are $100(1.20)=120$. Expected year-end outgo is $0.10(100)+0.90(100)=100$, including the death claim and the reserve for survivors. Thus $g_{0,A}=120-100=20$. For path B, accumulated assets are $100(0.90)=90$, so $g_{0,B}=90-100=-10$.

Year 2, path A. Assets are $100(0.90)=90$. All policies still in force pay a terminal benefit of 100, whether death occurs during the year or maturity is reached. Hence $g_{1,A}=90-100=-10$. For path B, assets are $100(1.20)=120$, so $g_{1,B}=120-100=20$.

Average, survival-weighted profits. The scenario-weighted conditional profit is $(20-10)/2=5$ in each year. The signature is therefore $\psi_1=5$ and $\psi_2=0.90(5)=4.5$. Discounting:

\operatorname{NPV} =\frac{5}{1.04}+\frac{4.5}{1.04^2} =4.807692+4.160503 =8.968195.

The NPV is $\$8.97$ per policy, rounded to the nearest cent. The two paths have the same average return in each duration, but their profits arise in opposite years. Averaging scenario returns alone would not generally reproduce expected profit when benefit amounts or reserves depend nonlinearly on the fund value. The common mistake is to average the path returns first and then use that single return path without checking whether the profit calculation is linear in returns.

Worked Example: A Later Loss Can Reverse a Positive Early Profit

A hypothetical two-year equity-linked policy has $V_0=50$, $V_1=50$, and $V_2=0$, with no premiums or expenses. Its investment returns are $+10\%$ in year 1 and 0% in year 2. The conditional death probabilities are 0.20 in year 1 and 0.10 in year 2. A death benefit of 60 is paid in either year; survivors at the end of year 1 carry reserve 50, while survivors at the end of year 2 receive maturity benefit 60 and have no remaining reserve. The valuation rate is 5%. Find the profit signature and NPV.

Year 1 conditional profit. Accumulated assets are $50(1.10)=55$. Expected year-end outgo is $0.20(60)+0.80(50)=12+40=52$. Therefore $g_0=55-52=3$, and $\psi_1=3$.

Year 2 conditional profit. Assets are $50(1.00)=50$. A death benefit of 60 or a maturity benefit of 60 is paid to every policy in force at the start of the year, so expected benefit outgo is $0.10(60)+0.90(60)=60$. Thus $g_1=50-60=-10$. Only $0.80$ of the original policies are expected to reach year 2, giving $\psi_2=0.80(-10)=-8$.

NPV and check.

\operatorname{NPV} =\frac{3}{1.05}+\frac{-8}{1.05^2} =2.857143-7.256236 =-4.399093.

The NPV is $-\$4.40$ per policy, rounded to the nearest cent. The positive first-year profit does not imply a positive NPV: the later loss is larger after allowing for the probability of reaching year 2 and discounting. The calculation also checks the terminal treatment: no closing reserve is deducted after the maturity or death benefit has been paid.

Interpreting Results and Avoiding Traps

The signature is useful because it separates three effects that a single NPV combines: the conditional profit pattern, the probability of reaching each duration, and the time value of money. In equity-linked projections, the conditional pattern can change with the investment path because fund values, benefit amounts, and reserves can all respond to the path. A positive NPV on a specified basis is a valuation result for that model and those assumptions; it is not a guarantee of realized profit.

Do not confuse conditional and issue-level profit. $g_t$ is per policy in force at time $t$; $\psi_{t+1}$ is per policy issued and includes the probability of reaching that duration.
Use the correct discount time. Profit at the end of year $t+1$ is discounted by $v^{t+1}$, not $v^t$.
Do not count terminal outgo twice. If the terminal benefit is paid, do not also hold a reserve for that same benefit after payment.
Average profits, not just returns, when scenario profit is nonlinear. Floors, caps, and guarantees can make the profit calculated at an average return differ from the average profit across paths.
Keep scenario weights and mortality assumptions explicit. Their interpretation depends on the valuation basis, and separate weighting is valid only when the modeled quantities are independent.

Key takeaway: Calculate each path’s duration-specific conditional profit first. Then weight by the probability of reaching the duration, apply the end-of-period discount factor, and sum. This order makes the profit emergence and its assumptions visible.

This treatment is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus: candidates should be able to connect a profit vector to duration-specific expected profits and interpret the resulting discounted NPV without losing track of timing or scenario assumptions.

Check Your Understanding

For each problem, state whether profits are conditional or already weighted as indicated, and use the specified year-end timing. Answers are per policy issued unless the problem asks for conditional profit.

A one-year policy has expected conditional profit of 12 at year-end, no decrements during the year, and valuation interest $i=4\%$. Calculate its NPV. (Answer: $\$11.54$)
A two-year policy has conditional profits $g_0=8$ and $g_1=-3$, annual death probability 0.10, and valuation interest $i=5\%$. Calculate the NPV, assuming death is the only exit and the same conditional death probability applies in each year. (Answer: $\$5.17$)
Two investment scenarios have weights 0.25 and 0.75. Their conditional profit vectors are $(20,-8)$ and $(-4,12)$, respectively. The annual death probability is 0.20, and valuation interest is 10%. Calculate the scenario-weighted NPV, assuming mortality is independent of the scenario. (Answer: $\$6.45$)
With no decrements and $i=5\%$, path A produces profits $(0,20)$ at the ends of years 1 and 2, while path B produces $(20,0)$. Which path has the higher NPV, and by how much? (Answer: Path B, by $\$0.91$)
At the start of a year, $V_t=80$. A premium of 5 and expense of 2 occur immediately; the investment return is 5%; the death probability is 0.10; the death benefit is 90; and the closing reserve for survivors is 80. Calculate the conditional expected profit for this year. (Answer: $\$6.15$)

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