Profit Signatures for Limited-Term Contracts

Why the Contract Term Determines the Signature

A limited-term contract has a fixed number of covered durations. Its profit signature must therefore include the profit expected from each covered duration, and no profit from a duration after coverage has ended. Getting the duration count right is only part of the task: each expected profit must also be assigned to the time it emerges. In particular, profit from the final covered duration is not automatically a profit at the start of that duration.

The construction below uses annual durations and assumes annual profit emerges at the end of each duration. The method applies whether the profit for a duration comes from deterministic cash flows or is an expected profit that incorporates events during that duration. Other timing conventions require discounting to the actual emergence time.

Definition: Consider a contract issued at time 0 with a term of $n$ years and annual covered durations. Index the duration starting at time $k$ by $k=0,\ldots,n-1$; it ends at time $k+1$. Let $G_k$ be the expected profit for that duration, conditional on the policy being in force at its start. Let $I_k$ be the probability that a policy issued at time 0 is in force at time $k$. The unconditional profit-signature entry for the end of that duration is $\sigma_{k+1}=I_kG_k$. Thus the signature has $n$ entries, indexed $1,\ldots,n$, with its final entry emerging at time $n$.

If the only reason a policy leaves the in-force population is death, then $I_k={}_kp_x$, the probability that a life aged $x$ survives $k$ years. If lapses or other decrements can also remove a policy, $I_k$ must instead reflect the probability of remaining in force under the applicable decrement model. Do not use a mortality-only survival probability when the stated in-force condition includes other exit events.

For a cohort of $N$ identical policies, the expected aggregate profit at the end of duration $k$ is $NI_kG_k=N\sigma_{k+1}$. With an effective annual interest rate $i$, $v=1/(1+i)$, and end-of-duration profit emergence, the expected present value per issued policy is

\operatorname{EPV}=\sum_{k=0}^{n-1}v^{k+1}I_kG_k =\sum_{j=1}^{n}v^j\sigma_j.

The factor $v^{k+1}$ is used because the profit for duration $k$ emerges at its end, time $k+1$. The contract term determines the final index: $k=n-1$ is the last covered duration, and its profit is included at time $n$. There is no covered duration indexed $k=n$ in this $n$-year contract.

Why In-Force Weighting and Discounting Work

The signature records expected profits from the perspective of the original issued policy, not just from the perspective of policies still active at a later duration. A policy contributes a duration’s profit only if it is in force at the start of that duration. Conditional on being in force, its expected duration profit is $G_k$. Multiplying these quantities gives the expected contribution per originally issued policy.

Condition on being in force.
At the start of duration $k$, the probability of being in force is $I_k$, and the conditional expected profit for the duration is $G_k$.

Take the unconditional expectation.
The expected contribution from the duration, per issued policy, is $I_kG_k$. This is the signature entry $\sigma_{k+1}$.

Discount to time 0.
Since the duration’s profit emerges at time $k+1$, its present value is $v^{k+1}I_kG_k$. Summing over the $n$ covered durations gives the expected present value.

This argument is a conditioning argument: the duration profit contributes zero when the policy is not in force at the start, and has conditional expectation $G_k$ when it is. It does not require profits in different durations to be independent. It does require that $G_k$ is defined consistently as the conditional expected profit under the stated in-force condition and profit-timing convention.

Worked Construction and Valuation

Worked Example: A Three-Year Contract with Duration-Dependent Profits

A hypothetical three-year contract has annual profit emergence at each year-end. For a life aged $x$, assume independent annual mortality with probability of death $q=0.055$ in each year, so annual survival is $p=0.945$. There are no other decrements. The conditional expected profits per policy in force at the start of durations 0, 1, and 2 are $G_0=100$, $G_1=200$, and $G_2=300$ dollars. The effective annual interest rate is $i=0.05$. Construct the signature and calculate the expected present value per issued policy.

Method. There are exactly three covered durations, indexed 0, 1, and 2. Their profits emerge at times 1, 2, and 3. The in-force probabilities at the starts of these durations are $I_0=1$, $I_1=0.945$, and $I_2=0.945^2=0.893025$.

Weight each conditional expected profit by its in-force probability:

\sigma_1=1(100)=100,\qquad \sigma_2=0.945(200)=189,\qquad \sigma_3=0.893025(300)=267.9075.

Thus, the signature is $(100,189,267.9075)$ dollars, with entries emerging at times 1, 2, and 3. The discount factor is $v=1/1.05$. Discount each entry from its own emergence time:

\begin{aligned} \operatorname{EPV} &=\frac{100}{1.05}+\frac{189}{1.05^2}+\frac{267.9075}{1.05^3}\\ &=95.238095+171.428571+231.428571\\ &=498.095238. \end{aligned}

The expected present value is $498.10 per issued policy, rounded to the nearest cent. As a check, the combined one-year factor for survival and discounting is $0.945/1.05=0.9$. The three present-value terms are therefore $100/1.05$, $200(0.9)/1.05$, and $300(0.9^2)/1.05$, agreeing with the direct calculation.

Common mistake avoided: The last profit is not discounted for only two years because it belongs to duration 2. It emerges at the end of that duration, at time 3, so its discount factor is $v^3$.

Worked Example: Scaling a Four-Year Signature to a Cohort

A hypothetical four-year contract is issued to 500 identical lives. Annual mortality is independent with $q=0.05$ each year, with no other decrements. Conditional expected profits per policy in force at the start of the four durations are $30$, $40$, $50$, and $0$ dollars, respectively. The effective annual interest rate is $i=0.04$, and profits emerge at year-end. Calculate the cohort’s expected present value.

Method. The term gives four covered durations, with in-force probabilities $1$, $0.95$, $0.95^2=0.9025$, and $0.95^3=0.857375$. Weighting each conditional expected profit gives the per-policy signature:

(\sigma_1,\sigma_2,\sigma_3,\sigma_4) =(30,\;0.95(40),\;0.9025(50),\;0.857375(0)) =(30,\;38,\;45.125,\;0).

The fourth duration is covered, but its conditional expected profit is zero; its signature entry is therefore zero. With $v=1/1.04$, discount the entries to times 1 through 4:

\begin{aligned} \operatorname{EPV}_{\text{per policy}} &=\frac{30}{1.04}+\frac{38}{1.04^2} +\frac{45.125}{1.04^3}+\frac{0}{1.04^4}\\ &=28.846154+35.133136+40.115961+0\\ &=104.095251. \end{aligned}

For 500 policies, the expected present value is $500(104.095251)=52{,}047.6255$, or $52,047.63 to the nearest cent. Scaling the per-policy result is valid because the expected aggregate profit is the sum of the expected profits for the 500 policies; independence between policies is not needed for this expectation.

Common mistake avoided: A zero expected profit in the fourth duration does not mean the contract has only three covered durations. The duration count is set by the contract term; the profit amount determines the value of each entry.

The Final Covered Duration and Its Timing

For an $n$-year contract, duration $n-1$ starts at time $n-1$ and ends at time $n$. Its signature entry is $\sigma_n=I_{n-1}G_{n-1}$, discounted by $v^n$. This remains true even though it is the last entry. The final duration is not omitted, and its profit is not moved to time $n-1$ merely because that is when the duration begins.

The duration profit $G_{n-1}$ is defined under the profit-vector timing convention: it may already incorporate the expected effects of events during the final covered year. The signature then weights that conditional expected profit by the probability of being in force at the start of the year. If the problem instead specifies that profit emerges at a different point in the year, discount to that specified time rather than mechanically applying an end-of-year factor. A separate cash flow at contract expiry must likewise be placed at its stated time; it should not be silently treated as an extra annual covered duration.

Worked Example: Avoiding an Off-by-One Error at Expiry

A hypothetical two-year contract has end-of-year profit emergence, effective annual interest $i=0.10$, and independent annual mortality with $q=0.20$ and no other decrements. The conditional expected profits for durations 0 and 1 are $G_0=10$ and $G_1=50$ dollars per policy in force at the start of the duration. Find the signature and expected present value per issued policy.

Method. The two covered durations start at times 0 and 1. Their in-force probabilities are $I_0=1$ and $I_1=0.80$. Therefore:

\sigma_1=1(10)=10,\qquad \sigma_2=0.80(50)=40.

The first profit emerges at time 1 and the final profit at time 2. Thus,

\operatorname{EPV} =\frac{10}{1.10}+\frac{40}{1.10^2} =9.090909+33.057851 =42.148760.

The answer is $42.15 per issued policy, rounded to the nearest cent. There are two signature entries, not three: the second entry is the profit from the final covered year and emerges at expiry, time 2.

Common mistake and correction. Discounting the final entry for only one year gives the wrong calculation $10/1.10+40/1.10=45.454545$. The correction is to recognize that $G_1$ belongs to the duration from time 1 to time 2, so its present value uses $v^2$, not $v$.

Common Checks for Limited-Term Signatures

Check the number of entries. An $n$-year contract with annual durations has $n$ covered-duration entries, indexed by $j=1,\ldots,n$.
Check the in-force condition. The weight for the profit of duration $k$ is the probability of being in force at its start, not at its end.
Check the emergence time. Under year-end emergence, duration $k$'s profit is discounted $k+1$ years.
Keep term and profit amount distinct. A covered duration can have zero expected profit and still be one of the contract’s durations.
Check the model behind the weights. Use the in-force probabilities that match the stated decrements; survival from mortality alone is insufficient if another decrement also ends coverage.

These checks help keep the signature aligned with both the contract boundary and the timing convention. That alignment matters because a correct profit vector can still produce an incorrect expected present value if it is given the wrong in-force weight, omitted at expiry, or discounted to the wrong time.

Check Your Understanding

For each problem, construct or interpret the limited-term signature using the stated year-end profit emergence. Amounts are per policy unless a cohort size is given.

A three-year contract has $i=0.05$, independent annual mortality $q=0.10$, no other decrements, and conditional expected profits $G_0=20$, $G_1=30$, and $G_2=40$. Find the signature and expected present value per issued policy. (Answer: Signature $(20,27,32.4)$; EPV $=\$71.53$ per policy, to the nearest cent.)
A four-year contract has no decrements, $i=0$, and conditional expected profit of $\$25$ in each covered duration. State the signature and total expected present value per issued policy. (Answer: Signature $(25,25,25,25)$; EPV $=\$100$.)
A two-year contract has $i=0.10$, independent annual mortality $q=0.20$, no other decrements, and conditional expected profits $G_0=12$ and $G_1=30$. Find the signature and expected present value per issued policy. (Answer: Signature $(12,24)$; EPV $=\$30.74$, to the nearest cent.)
A three-year contract is issued to 200 policies. It has $i=0.05$, independent annual mortality $q=0.05$, no other decrements, and conditional expected profits $G_0=10$, $G_1=20$, and $G_2=30$. Find the per-policy signature and the cohort expected present value. (Answer: Signature $(10,19,27.075)$; cohort EPV $=\$10{,}029.15$, to the nearest cent.)
A five-year contract’s signature has four entries, each indexed as an end-of-year profit. Under the annual-duration convention used here, identify the inconsistency and give the correct final signature index and emergence time. (Answer: Five covered annual durations require five entries; the final entry is $\sigma_5$, emerging at time 5.)

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