Reading Positive and Negative Duration Profits

Introduction

A profit signature records the time-zero value of expected profit arising at each duration. Its entries answer a specific question: how much does the profit expected at this duration contribute to the contract’s present value? A positive entry represents an expected contribution that raises NPV; a negative entry represents one that lowers it.

The sign of an individual entry does not, by itself, determine whether the contract has positive NPV. The overall NPV depends on the sum of all entries. Further, the size of an entry is not simply the profit expected for a policy that reaches that duration: the probability of remaining in force and the discount factor also matter. This interpretation is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.

What a Signature Entry Represents

Use the following convention throughout this tutorial. The contract has annual durations $k=1,2,\ldots,n$, and duration $k$'s profit is recognized at the end of that duration. Let $w_{k-1}$ be the probability that the policy is in the relevant in-force state at the start of duration $k$. Let $g_k$ be the expected profit for duration $k$, conditional on being in that state at its start. The conditional expectation includes the duration’s applicable cash flows, including any termination cash flows recognized during the duration. Let $i$ be the effective annual discount rate and $v=1/(1+i)$.

Definition: Under the stated end-of-duration convention, the discounted profit signature entry for duration $k$ is $\sigma_k=v^k w_{k-1}g_k$. The contract’s NPV is the sum of the entries, $\operatorname{NPV}=\sum_{k=1}^{n}\sigma_k$. A signature entry is an expected present-value contribution, not a profit realized on every policy.

For a simple life-contingent contract with no other decrement, $w_{k-1}$ may be the probability of surviving in force to the start of duration $k$. For a contract with lapses or other decrements, it is the relevant in-force probability under those assumptions. When profit depends on which state the policy occupies, state-specific contributions must be represented appropriately; a single in-force probability is not enough to describe distinct states.

The definition separates three quantities that are easy to conflate. The conditional expected profit is $g_k$. The probability weight $w_{k-1}$ accounts for the chance that the policy can generate that duration’s profit. Discounting by $v^k$ converts an end-of-duration amount to time zero. Since $v^k$ and $w_{k-1}$ are nonnegative, the signature entry has the same sign as $g_k$ whenever $w_{k-1}>0$. Its magnitude, however, need not resemble the magnitude of $g_k$.

Why the Formula Holds

The probability and discount factors in a signature entry follow directly from expected present value. Let $I_k$ equal 1 if the policy is in force at the start of duration $k$, and 0 otherwise. Let $G_k$ be the duration’s profit, conditional on the policy being in force at that point. The time-zero contribution is $v^k I_kG_k$. Taking its expectation gives

E[v^k I_kG_k] =v^k E[I_kG_k] =v^k P(I_k=1)E[G_k\mid I_k=1] =v^k w_{k-1}g_k =\sigma_k.

The middle equality uses the fact that $I_kG_k$ is zero when the policy is not in force and equals $G_k$ when it is in force. Adding these expected contributions across durations gives the NPV. This justification assumes the stated timing convention and that all relevant duration profit is included in $G_k$. If profits are instead recorded at a different time, the discount factor must reflect that timing.

Identify conditional profit.
Determine the expected net profit for the duration conditional on being in force at its start.

Apply the in-force weight.
Multiply by the probability of being in the relevant state at the start of the duration.

Discount to time zero.
For an end-of-duration profit, multiply by $v^k$.

Interpret the entry and total.
Read the entry’s sign as the direction of that duration’s expected contribution, then sum all entries for NPV.

Worked Interpretations

Worked Example: A Negative Entry Followed by Positive Entries

A hypothetical three-year contract has annual effective interest $i=5\%$. In a discrete annual mortality model, the conditional probability of death during duration 1 is $q_1=0.10$; conditional on survival to duration 2, it is $q_2=0.20$. There are no other decrements. The conditional expected profits at the ends of durations 1, 2, and 3 are respectively $-\$40$, $\$30$, and $\$50$. Calculate and interpret the signature. Treat the given conditional profits as expected amounts, with any relevant duration cash flows included.

Method. Calculate the probability of reaching each duration, then weight and discount each conditional profit. The annual survival probabilities are $p_1=1-q_1=0.90$ and $p_2=1-q_2=0.80$. Thus the start-of-duration weights are $w_0=1$, $w_1=0.90$, and $w_2=0.90(0.80)=0.72$. The discount factor is $v=1/1.05$.

For duration 1, $\sigma_1=v(-\$40)=-\$40/1.05=-\$38.095238\ldots$. For duration 2, $\sigma_2=v^2(0.90)(\$30)=\$27/1.1025=\$24.489795\ldots$. For duration 3, $\sigma_3=v^3(0.72)(\$50)=\$36/1.157625=\$31.098153\ldots$.

The first entry is negative: duration 1 reduces expected present value. The next two entries are positive: their expected profits increase it. Adding without rounding intermediate amounts gives $-38.095238\ldots+24.489795\ldots+31.098153\ldots=\$17.492711\ldots$. Rounded to the nearest cent, the NPV is $\$17.49$, which is positive despite the first-duration loss.

Check. The entries sum to the reported NPV. A common mistake would be to call the contract unprofitable because its first entry is negative; the correction is to assess the total by summing all duration contributions.

Worked Example: Conditional Profit Size Versus Signature Size

A hypothetical three-year policy has annual effective interest $i=10\%$. In each year, a policy in force at the start has a $0.50$ probability of lapsing at year-end and a $0.50$ probability of remaining in force; these annual probabilities apply conditionally at each duration, with no other decrement. The conditional expected profits at the ends of the three durations are $\$60$, $\$100$, and $\$100$. Calculate the entries and explain why the largest conditional profit does not produce the largest signature entry.

Method. The start-of-duration in-force weights are $w_0=1$, $w_1=0.50$, and $w_2=0.50^2=0.25$. With $v=1/1.10$, the entries are:

$\sigma_1=v(1)(\$60)=\$60/1.10=\$54.545454\ldots$. Next, $\sigma_2=v^2(0.50)(\$100)=\$50/1.21=\$41.322314\ldots$. Finally, $\sigma_3=v^3(0.25)(\$100)=\$25/1.331=\$18.782870\ldots$.

All three entries are positive, so each duration contributes positively to NPV. The conditional profit is larger in durations 2 and 3 than in duration 1, but the signature entries are smaller: fewer policies are expected to reach those durations, and the later profits are discounted for longer. The total is $\$54.545454\ldots+\$41.322314\ldots+\$18.782870\ldots=\$114.650638\ldots$, or $\$114.65$ to the nearest cent.

Check. The total is positive and equals the sum of the entries. A common mistake is to rank signature entries using only the conditional profits $\$60,\$100,\$100$. The correction is to compare the weighted, discounted entries, not the conditional amounts alone.

Worked Example: Positive Entries Do Not Guarantee Positive NPV

A hypothetical three-year contract has annual effective interest $i=25\%$. At the end of each duration, a policy in force at the start exits with probability $0.20$ and remains in force with probability $0.80$; there are no other decrements. Conditional expected profits are $-\$100$, $\$60$, and $\$70$. Calculate the signature and decide whether the contract’s NPV is positive.

Method. The start-of-duration weights are $1$, $0.80$, and $0.80^2=0.64$. The discount factor is $v=1/1.25=0.80$, so $v^2=0.64$ and $v^3=0.512$. Therefore,

$\sigma_1=(1)(-\$100)(0.80)=-\$80$. For duration 2, $\sigma_2=(0.80)(\$60)(0.64)=\$30.72$. For duration 3, $\sigma_3=(0.64)(\$70)(0.512)=\$22.9376$.

The last two entries are positive, but their sum is $\$30.72+\$22.9376=\$53.6576$, less than the initial $\$80$ loss in present-value terms. Thus $\operatorname{NPV}=-\$80+\$30.72+\$22.9376=-\$26.3424$, or $-\$26.34$ to the nearest cent. The contract has a negative NPV under these assumptions.

Check. The entries sum to the negative amount reported. The common mistake is to infer positive NPV merely because there are more positive entries than negative ones. The correction is to compare their amounts and sum them; the count of positive and negative durations is not decisive.

Interpreting Signs and Sizes Carefully

A negative signature entry identifies a duration whose expected discounted contribution is a loss to NPV under the stated assumptions. A positive entry identifies a duration whose expected contribution is a gain. These are statements about expected values, not guarantees about the profit of an individual policy or the outcome in every possible experience. Actual outcomes may vary around the assumptions used to calculate the expected profits and in-force probabilities.

The sign interpretation relies on using a positive discount factor and a nonnegative in-force probability. When the policy has positive probability of being in force, the sign of $\sigma_k$ matches the sign of its conditional expected duration profit $g_k$. If the in-force probability is zero, the entry is zero even if a hypothetical conditional profit is specified for that duration, because no policies are expected to generate it. A zero entry may also arise from zero conditional expected profit.

The size of a signature entry should be read as a present-value contribution. It combines conditional profit, exposure to the duration, and discounting. Consequently, two durations with equal conditional profit need not have equal entries, and a duration with a larger conditional profit need not have a larger entry. For example, later durations can have lower in-force weights and stronger discounting. Conversely, when comparing entries, do not remove those weights: they are part of what the signature measures.

What is being interpreted?	What it tells you	What it does not tell you by itself
Sign of $\sigma_k$	Whether duration $k$ raises or lowers expected present value	Whether the contract’s total NPV is positive
Size of $\sigma_k$	The discounted, in-force-weighted contribution of duration $k$	The conditional profit amount alone
Sum of all $\sigma_k$	The contract’s NPV under the modeled assumptions	That every policy will realize the expected result

Common Mistakes

Equating one negative entry with a negative contract NPV. A negative duration contribution can be offset by positive entries elsewhere. Add every entry before deciding the sign of NPV.
Equating positive entries with positive NPV. Positive entries may be smaller in total than the negative entries. Use the signed sum, not the number of positive durations.
Reading a signature entry as conditional profit. The conditional amount is $g_k$; the entry also includes the in-force weight and discount factor.
Using the wrong discount timing. With profits recognized at the end of duration $k$, discount by $v^k$. A different recognition time requires its corresponding discount factor.
Treating an expected entry as a guaranteed individual outcome. Signature entries summarize expected contributions under the model; individual policies can experience different outcomes.

Key takeaway: Read each signature entry as a duration-specific expected present-value contribution: its sign identifies whether it raises or lowers NPV, and its size reflects profit, in-force probability, and discounting. Only the sum of all entries determines the contract’s NPV.

Check Your Understanding

For each problem, use end-of-duration profits and calculate the discounted signature entries before interpreting their signs. Round dollar answers to the nearest cent where needed.

A two-year hypothetical contract has $i=0\%$, a $0.25$ probability of exit during duration 1, and no other decrements. Conditional expected profits are $-\$20$ and $\$40$. Find the entries and NPV. (Answer: entries $-\$20.00$ and $+\$30.00$; NPV $+\$10.00$.)
A three-year contract has $i=5\%$, conditional annual exit probability $0.20$, and conditional expected profits $\$50$, $\$0$, and $\$50$. Find the NPV. (Answer: entries $+\$47.62$, $\$0.00$, and $+\$27.64$; NPV $+\$75.26$.)
A three-year contract has $i=0\%$, with conditional exit probabilities $0.10$ in durations 1 and 2 and no other decrements. Conditional expected profits are $-\$30$, $+\$50$, and $+\$20$. Identify the negative duration contribution and calculate NPV. (Answer: duration 1 is negative; entries $-\$30.00$, $+\$45.00$, and $+\$16.20$; NPV $+\$31.20$.)
A three-year contract has $i=25\%$, with conditional exit probability $0.40$ at each year-end and no other decrements. Conditional expected profits are $+\$40$, $+\$80$, and $-\$50$. Find all entries and NPV. (Answer: entries $+\$32.00$, $+\$30.72$, and $-\$9.22$; NPV $+\$53.50$.)
A four-duration hypothetical contract has discounted signature entries of $-\$12{,}000$, $+\$8{,}000$, $+\$10{,}000$, and $-\$3{,}000$. Identify the loss-making and profit-making durations and determine overall NPV. (Answer: loss-making durations 1 and 4; profit-making durations 2 and 3; NPV $+\$3{,}000$.)

What is being interpreted?	What it tells you	What it does not tell you by itself
Sign of \(\sigma_k\)	Whether duration \(k\) raises or lowers expected present value	Whether the contract’s total NPV is positive
Size of \(\sigma_k\)	The discounted, in-force-weighted contribution of duration \(k\)	The conditional profit amount alone
Sum of all \(\sigma_k\)	The contract’s NPV under the modeled assumptions	That every policy will realize the expected result

What You'll Learn