From Annual Profit Vectors to Profit Signatures

From Annual Profits to a Signature

An annual profit vector records expected profit by policy duration, but the basis of each entry matters. A profit for a policy that is in force at the start of a year is not automatically the same as expected profit per policy originally issued: some policies may have terminated before that year began. A profit signature puts the duration-by-duration amounts on a consistent original-policy basis.

The conversion uses two ingredients: conditional annual profit at each duration and the probability that an original policy is still in force at the beginning of that duration. This tutorial uses one in-force state and annual policy years. It does not discount the entries across durations; the focus is how they are defined and weighted.

Definition: Let $g_k$ be the expected profit in policy year $k$, conditional on the policy being in force at the start of that year. The vector $(g_1,\ldots,g_n)$ is a conditional annual profit vector. Let $r_{k-1}$ be the probability that a policy is in force at the start of year $k$, given issue. The profit signature per policy issued is the sequence $(\sigma_1,\ldots,\sigma_n)$, where $\sigma_k=r_{k-1}g_k$.

The conditional profit $g_k$ is an expected value, not necessarily the profit that every policy actually realizes. It can include the expected effects of cash flows during that year, conditional on being in force at its start. For example, if death claims or lapse payments can occur during the year, their expected effects may already be included in $g_k$. The signature then accounts for the chance that the policy reached the start of the year.

For a single in-force state, suppose $t_k$ is the probability of termination during year $k$, conditional on being in force at the start of that year. If every policy is in force at issue, then $r_0=1$, and

r_k=r_{k-1}(1-t_k), \qquad r_{k-1}=\prod_{j=1}^{k-1}(1-t_j).

Here $t_k$ can represent an all-cause termination probability. If the model has several mutually exclusive causes of termination, it is the total probability of leaving the in-force state during that year. The formula requires consistent conditional probabilities: each year’s termination probability applies to policies in force at that year’s start.

Why the Signature Uses the Start-of-Year Probability

The weighting follows directly from expectation. Define $I_k$ to equal 1 if a policy is in force at the start of year $k$, and 0 otherwise. Let $Z_k$ be the policy’s profit attributable to year $k$, with $Z_k=0$ when it is not in force at the start of that year. Conditional on being in force, its expected annual profit is $g_k$. Therefore,

E[Z_k] = P(I_k=1)E[Z_k\mid I_k=1] +P(I_k=0)E[Z_k\mid I_k=0] =r_{k-1}g_k+ (1-r_{k-1})\cdot 0 =r_{k-1}g_k.

Thus $\sigma_k$ is the expected profit in year $k$ per policy originally issued. This argument does not require the year’s profit to be independent of termination. Any relationship between events during the year and profit belongs in the conditional expectation $g_k$; the factor $r_{k-1}$ accounts for whether the policy reached the start of that year.

Establish the basis.
Check whether each annual profit is conditional on being in force at the start of its year, or is already stated per policy issued.

Find the in-force probabilities.
Start with $r_0=1$ at issue and apply each year’s conditional termination probability to obtain the next start-of-year probability.

Form the signature.
For conditional entries, multiply $g_k$ by $r_{k-1}$, not by the probability of remaining in force through year $k$.

Check the interpretation.
Each signature entry is an expected profit per original policy for that duration. Multiply by the number issued only if an expected cohort total is required.

Worked Applications

Worked Example: A Three-Year Mortality-Only Contract

A hypothetical insurer issues a three-year contract. The effective annual interest rate is $i=4\%$. There is no lapse, and the conditional probabilities of death during years 1, 2, and 3 are $0.10$, $0.20$, and $0.25$, respectively. Assume a Bernoulli death-or-survival outcome in each year conditional on being in force at its start. After allowing for the contract’s cash flows and reserves at the stated interest rate, the conditional annual profit vector is $(120,-50,40)$ dollars. Find the profit signature per policy issued.

Method. These entries are conditional on being in force at the start of the applicable year. First calculate the start-of-year in-force probabilities. With no lapse, the termination probabilities are the stated mortality probabilities.

At issue, $r_0=1$. The probability of surviving year 1 is $1-0.10=0.90$, so $r_1=1(0.90)=0.90$. The probability of surviving year 2, conditional on reaching its start, is $1-0.20=0.80$, so $r_2=0.90(0.80)=0.72$.

Now weight each conditional profit by the probability of being in force at that year’s start:

\sigma_1=1(120)=120,\qquad \sigma_2=0.90(-50)=-45,\qquad \sigma_3=0.72(40)=28.80.

The signature is $(120,-45,28.80)$ dollars per policy issued. As a check, the year 3 amount uses the probability of reaching the start of year 3, $0.72$, which is the product of survival probabilities for years 1 and 2. No year 3 survival factor is applied to its conditional profit.

Worked Example: Competing Termination Causes and a Year-End Profit

A hypothetical two-year product has an effective annual interest rate of $i=3\%$. In year 1, a policy in force at the start can terminate by death with probability $0.02$, lapse with probability $0.08$, or remain in force; in year 2, the corresponding probabilities are $0.05$, $0.05$, and the remaining probability. The causes are mutually exclusive within each year. The conditional profit vector, including the expected effects of any death or lapse cash flows during each year, is $(30,80)$ dollars. Find the signature and identify the error in using end-of-year persistence as the year 2 weight.

Method. Add the mutually exclusive termination probabilities to find each year’s all-cause termination probability. In year 1, $t_1=0.02+0.08=0.10$, so $r_1=1(1-0.10)=0.90$. In year 2, $t_2=0.05+0.05=0.10$, so $r_2=0.90(1-0.10)=0.81$.

The year 1 conditional profit is weighted by $r_0=1$, while the year 2 conditional profit is weighted by $r_1=0.90$:

\sigma_1=1(30)=30,\qquad \sigma_2=0.90(80)=72.

The signature is $(30,72)$ dollars per policy issued. A common mistake is to use $r_2=0.81$ for the year 2 profit, giving $0.81(80)=64.80$. That incorrectly requires the policy to remain in force through year 2 before crediting profit arising during year 2. The correct weight is $r_1=0.90$, because the conditional profit is defined for policies in force at the start of year 2 and already includes the expected effects of termination during that year.

The year 2 termination probabilities still matter for whether a policy reaches the start of a later year. This contract has no year 3, so $r_2$ is not used to weight either of its two annual profits.

Worked Example: From a Per-Policy Signature to a Cohort Total

A hypothetical insurer issues 250 identical four-year policies. The effective annual interest rate is $i=5\%$. There is no lapse, and the conditional annual mortality probabilities for years 1, 2, and 3 are $0.10$, $0.20$, and $0.25$. The conditional annual profit vector is $(40,20,-10,4)$ dollars per policy in force at the start of the year. Find the per-policy signature and the expected aggregate profit by duration for the original cohort.

Method. The annual death probabilities are also the termination probabilities. The successive start-of-year in-force probabilities are

r_0=1,\quad r_1=1(0.90)=0.90,\quad r_2=0.90(0.80)=0.72,\quad r_3=0.72(0.75)=0.54.

Weight the four annual profits by $r_0,r_1,r_2,r_3$, in that order:

(\sigma_1,\sigma_2,\sigma_3,\sigma_4) =(1(40),\,0.90(20),\,0.72(-10),\,0.54(4)) =(40,18,-7.20,2.16).

These are expected profits per policy issued. For 250 policies, multiply each entry by 250:

(250(40),\,250(18),\,250(-7.20),\,250(2.16)) =(10{,}000,\,4{,}500,\,-1{,}800,\,540).

The expected aggregate duration profits are therefore $(\$10{,}000,\ \$4{,}500,\ -\$1{,}800,\ \$540)$. The negative third entry remains negative after weighting: in-force adjustment changes its expected magnitude, not its sign. This cohort calculation is linear because the expected total is the sum of the expected profits for the individual policies; independence between policies is not needed for that expectation.

Reading the Basis Correctly

A signature is meaningful only when its basis is stated. The conditional vector $g_k$ answers: given that the policy is in force at the start of year $k$, what is its expected profit for that year? The signature entry $\sigma_k$ answers: per policy originally issued, what is the expected profit attributable to year $k$? The two are equal in the first year when all policies are in force at issue, but generally differ in later years.

Before applying an in-force probability, inspect how the annual profit was constructed. If an entry is already an expected profit per policy issued, multiplying by an in-force probability again double-counts termination. Likewise, if the entry is conditional on surviving to the end of the year rather than being in force at its start, the simple start-of-year weighting formula does not apply without first reconciling the entry’s definition and timing.

Annual entry basis	Meaning	Action when forming a signature
Conditional on being in force at the start of year $k$	Expected profit during year $k$ for a policy that reached its start	Multiply by $r_{k-1}$
Already per policy originally issued	Expected duration profit with termination exposure already allowed for	Do not apply the in-force probability again
Conditional on surviving to the end of year $k$	Expected profit on a different conditioning basis	Do not treat it as $g_k$ without adjusting the basis

The most important exam check is the conditioning event. If profit during a policy year includes an expected claim or surrender payment on termination, it can still be conditional on being in force at the start of that year. In that case, the signature weight is the start-of-year in-force probability. Applying an extra probability of surviving the year would remove policies whose year-$k$ profit is already represented in the conditional expectation.

Key takeaway: For a conditional annual profit vector, the profit signature per policy issued has entry $\sigma_k=r_{k-1}g_k$. Use the probability of being in force at the start of the year, and verify that the annual profit has not already been placed on an original-policy basis.

Check Your Understanding

For each problem, identify the conditioning basis before calculating.

A hypothetical two-year policy has $i=4\%$, no lapse, mortality probabilities $0.10$ in year 1 and $0.20$ in year 2, and conditional profits of $\$50$ and $\$30$. Find the per-policy signature. (Answer: $(\$50,\ \$27)$)
A hypothetical three-year contract has $i=3\%$, no mortality, and lapse probabilities $0.05$, $0.10$, and $0.20$ in years 1, 2, and 3. Conditional profits are $(20,-10,15)$. Find its signature per policy issued. (Answer: $(20,-9.50,12.825)$)
For a two-year hypothetical product, $i=5\%$. In year 1, mutually exclusive death and lapse probabilities are $0.03$ and $0.07$; in year 2 they are $0.04$ and $0.06$. Conditional annual profits are $(-40,100)$. Find the signature and state the year 2 weight. (Answer: $(-40,90)$; year 2 weight $0.90$)
A hypothetical insurer issues 400 policies. The conditional annual profit vector is $(\$12,\ \$8,\ -\$5)$, and the probabilities of being in force at the starts of years 1, 2, and 3 are $1,\ 0.95,\ 0.76$. Find the expected aggregate profit for each duration. (Answer: $(\$4{,}800,\ \$3{,}040,\ -\$1{,}520)$)
A hypothetical two-year product has conditional profits of $\$25$ in year 1 and $\$40$ in year 2. Its year 1 termination probability is $0.20$, and the year 2 entry is already stated per policy originally issued, after allowing for termination. What is the signature, and should the year 2 entry be multiplied by $0.80$ again? (Answer: $(\$25,\ \$40)$; no, the year 2 entry is already on the original-policy basis.)

Annual entry basis	Meaning	Action when forming a signature
Conditional on being in force at the start of year \(k\)	Expected profit during year \(k\) for a policy that reached its start	Multiply by \(r_{k-1}\)
Already per policy originally issued	Expected duration profit with termination exposure already allowed for	Do not apply the in-force probability again
Conditional on surviving to the end of year \(k\)	Expected profit on a different conditioning basis	Do not treat it as \(g_k\) without adjusting the basis

What You'll Learn