In-Force Probabilities as Signature Weights

Why In-Force Probabilities Weight Profits

A profit amount calculated for a policy in force at a particular duration is not automatically the profit expected from every policy originally issued. Some policies may have terminated before that duration. To construct a cohort profit signature, the conditional profit for a policy that is in force must be weighted by the probability that an original policy is still in force.

The essential distinction is between a probability conditional on being in force at the start of a year and a probability measured from issue. Annual persistency rates are conditional; duration-specific in-force probabilities accumulate those rates over all earlier years.

Definition: Let $r_t$ be the annual persistency rate for duration $t$: the probability that a policy in force at the start of duration $t$ remains in force at the start of duration $t+1$. Let $w_t$ be the probability that a policy issued at time zero is in force at the start of duration $t$. With all policies in force at the start of duration 1, $w_1=1$, and $$ w_t=\prod_{j=1}^{t-1} r_j,\qquad t\geq 2. $$ For a cohort of $N_0$ policies, the expected number in force at the start of duration $t$ is $N_0w_t$.

A persistency rate is not necessarily a mortality rate or a lapse rate by itself. It is the probability of remaining in force over the year under the model being used. It may already combine several ways of leaving the in-force state, such as death, lapse, or another termination. If the supplied rates are already annual persistency rates, do not apply an additional decrement adjustment to them.

From a Conditional Profit to a Cohort Duration Amount

Suppose $g_t$ is the expected profit for one policy during duration $t$, conditional on that policy being in force at the start of the duration. It may incorporate the possible events and cash flows within that year. The expected number of policies exposed to that conditional profit is $N_0w_t$. Thus the cohort amount for duration $t$, before any discounting across durations, is

$$ C_t=N_0w_tg_t. $$

This calculation weights the annual conditional profit by the probability of reaching the duration in force. It does not discount a duration amount to time zero. Discounting and the present value of a profit signature are separate questions.

Why the Product of Persistency Rates Is Correct

Let $I_t$ be an indicator equal to 1 if a policy is in force at the start of duration $t$, and 0 otherwise. For the policy to be in force at the start of duration $t+1$, it must be in force at the start of duration $t$ and persist through duration $t$. By the definition of conditional probability,

P(I_{t+1}=1)=P(I_t=1)P(I_{t+1}=1\mid I_t=1)=w_t r_t.

Therefore $w_{t+1}=w_tr_t$. Starting with $w_1=1$ and applying this recursion gives $w_t=r_1r_2\cdots r_{t-1}$. This reasoning does not require the annual persistency rates to be equal; it only requires that each $r_t$ be conditional on being in force at the start of its stated duration.

The cohort weighting follows from the same indicator idea. If the conditional expected profit for an in-force policy is $g_t$, then the expected profit contribution from one original policy at duration $t$ is $P(I_t=1)g_t=w_tg_t$. Summing over $N_0$ policies gives $N_0w_tg_t$. Independence between policies is not needed for this expectation, although independence is commonly assumed when a model also requires a probability distribution for the number of policies in force.

Worked Example: Applying Three Annual Persistency Rates

A hypothetical insurer issues 1,000 identical policies. The annual persistency rates for durations 1, 2, and 3 are 0.90, 0.92, and 0.95, respectively. These rates already reflect all exits from the in-force state. Conditional expected profits per policy in force at the start of the respective durations are $-\$80$, $\$120$, and $\$50$. Assume policies have independent persistency indicators with the stated probabilities. No discounting is applied; the effective annual interest rate for this calculation is $i=0\%$.

Method. First calculate each probability from issue by multiplying all earlier annual persistency rates. Then multiply the expected in-force count at each duration by that duration’s conditional profit.

At the start of duration 1, all policies are in force, so $w_1=1$ and the expected count is $1{,}000(1)=1{,}000$.

At the start of duration 2,

w_2=0.90,\qquad 1{,}000w_2=1{,}000(0.90)=900.

At the start of duration 3,

w_3=0.90(0.92)=0.828,\qquad 1{,}000w_3=828.

The cohort duration amounts are therefore:

C_1=1{,}000(-\$80)=-\$80{,}000,

C_2=900(\$120)=\$108{,}000,

C_3=828(\$50)=\$41{,}400.

The duration-specific cohort amounts are $-\$80{,}000$, $\$108{,}000$, and $\$41{,}400$. As a check, the duration 3 weight is below both 0.90 and 0.92 because it is the product of two probabilities less than 1. The third-year persistency rate of 0.95 would be used to find the probability in force at the start of duration 4; it does not affect the number in force at the start of duration 3.

Common mistake avoided: Using $1{,}000(0.92)=920$ as the duration 3 count treats the duration 2 persistency rate as if it were measured from issue. The correct count is $1{,}000(0.90)(0.92)=828$.

When Mortality and Lapse Are Supplied Separately

If death and lapse are the only exits during a year and they are mutually exclusive outcomes for a policy in force at the start of that year, the annual persistency rate is one minus the sum of their probabilities. This formula applies only when the probabilities refer to competing exits over the same year and cannot both occur as the policy’s first exit. If the given rates use a different convention, the persistency rate must be derived using that convention instead.

r_t=1-q_t^{D}-q_t^{L},

Here $q_t^{D}$ is the probability of death during duration $t$, conditional on being in force at its start, and $q_t^{L}$ is the corresponding probability of lapse. Once $r_t$ is found, the duration weights are calculated in the same way as when persistency is given directly.

Worked Example: Combining Mutually Exclusive Death and Lapse

A hypothetical cohort contains 500 policies at issue. For policies in force at the start of duration 1, the probabilities of death and lapse during that year are 0.02 and 0.04. For policies in force at the start of duration 2, they are 0.03 and 0.03. For policies in force at the start of duration 3, they are 0.03 and 0.02. Death and lapse are mutually exclusive first exits in each year, and there are no other exits. Assume independent policy transitions. Conditional expected profits per policy in force at the start of durations 1, 2, and 3 are $\$30$, $-\$60$, and $\$100$. No discounting is applied, so $i=0\%$.

Method. Find each annual persistency rate by subtracting the two mutually exclusive exit probabilities from 1. Then cumulate the rates to obtain the expected number in force at each duration.

The annual persistency rates are

r_1=1-0.02-0.04=0.94,\quad r_2=1-0.03-0.03=0.94,\quad r_3=1-0.03-0.02=0.95.

The probabilities of being in force at the start of durations 1, 2, and 3 are

w_1=1,\qquad w_2=0.94,\qquad w_3=0.94(0.94)=0.8836.

The corresponding expected in-force counts are $500$, $500(0.94)=470$, and $500(0.8836)=441.8$. Fractional expected counts are appropriate: they are expectations, not a claim that a realized cohort must contain a fractional policy.

Now weight each conditional profit:

C_1=500(\$30)=\$15{,}000,\qquad C_2=470(-\$60)=-\$28{,}200,

C_3=441.8(\$100)=\$44{,}180.

The duration 3 count also checks by sequential application: $470(0.94)=441.8$. The third-year persistency rate affects the next duration’s in-force probability, not the duration 3 starting count.

Common mistake avoided: Adding death and lapse probabilities is justified here because the exits are stipulated to be mutually exclusive. If they were overlapping events rather than competing first exits, $1-q_t^D-q_t^L$ could be incorrect.

Different Classes and Changing Survivor Mix

For a cohort with different classes, each class generally has its own duration-specific in-force probability. Calculate the expected in-force count class by class, then combine the counts and profit amounts. In general, the initial class proportions cannot be reused as the proportions among policies still in force later: different persistency rates change the cohort mix over time.

Worked Example: Weighting Profits for Two Classes

A hypothetical cohort has 600 policies: 360 in class A and 240 in class B. Class A has annual persistency rates 0.90 in duration 1 and 0.70 in duration 2. Class B has rates 0.75 and 0.90. Conditional expected profits per in-force policy are, for class A, $\$100$, $-\$20$, and $\$150$ in durations 1, 2, and 3; for class B they are $\$50$, $\$200$, and $-\$25$. Assume independent policy transitions within each class. No discounting is applied, with $i=0\%$.

Method. Calculate expected in-force counts separately by class. Multiply each count by the matching class-and-duration conditional profit, then add the class amounts.

At the start of duration 1, the counts are 360 in class A and 240 in class B. At the start of duration 2, they are

N_{A,2}=360(0.90)=324,\qquad N_{B,2}=240(0.75)=180.

At the start of duration 3, they are

N_{A,3}=324(0.70)=226.8,\qquad N_{B,3}=180(0.90)=162.

The class-specific contributions to the cohort profit amounts are:

C_1=360(\$100)+240(\$50)=\$36{,}000+\$12{,}000=\$48{,}000,

C_2=324(-\$20)+180(\$200)=-\$6{,}480+\$36{,}000=\$29{,}520,

C_3=226.8(\$150)+162(-\$25)=\$34{,}020-\$4{,}050=\$29{,}970.

The combined in-force counts at the starts of durations 1, 2, and 3 are 600, 504, and 388.8. Relative to the original 600 policies, the duration 2 and duration 3 weights are $504/600=0.84$ and $388.8/600=0.648$. The duration 3 check is $226.8+162=388.8$, and the total duration 3 profit is the sum of the two class contributions.

Common mistake avoided: The initial class mix is 60% A and 40% B, but it is not the duration 2 mix. At duration 2, the expected counts are 324 and 180, so class A’s share is $324/504$, not 60%. For duration 3, applying the initial mix to a single average second-year rate would also be wrong: the duration 2 survivor mix has already changed.

Checks That Prevent Weighting Errors

Before using in-force weights, check the meaning and timing of each input. The annual persistency rate must be conditional on being in force at the start of its duration. The duration $t$ weight includes persistency through durations 1 to $t-1$, but not through duration $t$. If a cohort has multiple classes, accumulate each class’s rates separately before combining the results.

Check the starting point. If all issued policies are in force at duration 1, use $w_1=1$, not the first annual persistency rate.
Check the duration index. The probability in force at the start of duration $t$ uses $t-1$ annual rates.
Check for double adjustment. If mortality and lapse are already included in a supplied persistency rate, do not subtract them again.
Check cohort composition. For distinct classes, use class-specific in-force counts and profits rather than applying initial proportions at every duration.
Keep weighting distinct from discounting. In-force weights account for persistence; they do not move profits between time periods.

Key takeaway: Start with $w_1=1$, multiply by each conditional annual persistency rate to reach later durations, and weight each conditional profit using the in-force count at the start of the duration to which that profit belongs.

Check Your Understanding

For each problem, treat the stated annual profits as conditional expected amounts per policy in force at the start of the stated duration. No discounting is required.

A hypothetical cohort has 2,500 policies and annual persistency rates 0.96 in duration 1 and 0.90 in duration 2. Find the probabilities in force at the starts of durations 1, 2, and 3, and the corresponding expected counts. (Answer: weights $1,\ 0.96,\ 0.864$; counts $2{,}500,\ 2{,}400,\ 2{,}160$.)
A cohort of 800 policies has mutually exclusive death and lapse probabilities of 0.01 and 0.04 in duration 1, and 0.02 and 0.03 in duration 2. Find the duration 3 expected in-force count. (Answer: persistency rates $0.95,\ 0.95$; duration 3 count $800(0.95)(0.95)=722$.)
A cohort begins with 300 policies in class A and 200 in class B. Their duration 1 persistency rates are 0.80 and 0.90. Find the combined duration 2 in-force probability relative to the original 500 and the expected duration 2 count. (Answer: count $300(0.80)+200(0.90)=420$; weight $420/500=0.84$.)
Using the cohort in question 3, suppose the duration 2 persistency rates are 0.75 for class A and 0.80 for class B. Conditional duration 3 profits are $\$40$ for class A and $-\$10$ for class B. Find each expected duration 3 class count and the combined duration 3 cohort profit amount. (Answer: counts $300(0.80)(0.75)=180$ and $200(0.90)(0.80)=144$; profit $180(\$40)+144(-\$10)=\$5{,}760$.)
A cohort has 400 policies in class A and 600 in class B. Their duration 1 persistency rates are 0.90 and 0.70, and their duration 2 rates are 0.80 and 0.90. Find the correct duration 3 expected in-force count. Explain why using the initial class proportions to average duration 2 persistency is not correct. (Answer: $400(0.90)(0.80)+600(0.70)(0.90)=288+378=666$; the survivor counts entering duration 2 are 360 and 420, not the initial 40% and 60% mix.)

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