Why the Valuation Date Changes the Denominator
A retrospective calculation starts with cash flows associated with a cohort of policies issued at an earlier date. But a reserve at a later valuation date is usually stated per policy still in force, not per policy originally issued. Those are different quantities: some insureds have died, and some policies may have ended for other reasons. Dividing by the original cohort size when the question asks for an amount per surviving policy leaves the reserve on the wrong basis.
The key is to identify the population represented by the numerator and the denominator. A cohort-wide expected fund refers to the original group; a reserve per in-force policy refers to the group remaining at the valuation date. Survival conditioning provides the bridge between them. This distinction is aligned with the SOA Exam FAM (Fundamentals of Actuarial Mathematics) syllabus.
From a Cohort Fund to an Amount per In-Force Policy
Consider a homogeneous cohort of \(N_0\) policies issued to independent lives aged \(x\). Suppose there are no terminations other than death. For each policy, the indicator of survival to time \(t\) has probability \({}_tp_x\). Thus the number \(K_t\) of policies in force has a binomial distribution:
This binomial result requires independent in-force indicators and a common in-force probability. In this particular no-lapse model, that probability is the survival probability. If policies can also lapse, the binomial model may still apply to a homogeneous independent cohort, but its parameter must be the probability a policy is actually in force, not merely the probability the insured is alive.
A cohort retrospective calculation may first produce an expected fund for the original cohort at time \(t\). Under the cohort convention, the corresponding expected amount per policy in force is the expected cohort fund divided by the expected number in force. If \(C_t\) denotes the expected cohort fund, this conversion is:
This is a conversion of expected cohort amounts. It is not, in general, the expected value of the random ratio of the actual fund to the actual number of survivors. Those two quantities can differ when both the fund and the survivor count vary. Use the ratio specified by the cohort reserve convention and the problem’s assumptions.
Record the original number of policies and the valuation time. Keep the original cohort size distinct from the number still in force.
Use survival probability only when death is the sole way coverage can end. Include any other termination condition if the model permits it.
Multiply the cohort size by the appropriate in-force probability.
Divide the expected cohort fund by the expected number in force, and label the result as an amount per in-force policy.
Why Survival Conditioning Works
The conversion follows from comparing two expected quantities. If each original policy has the same probability \(p\) of being in force at time \(t\), the expected number in force is \(N_0p\). If the cohort’s expected fund is \(C_t\), then \(C_t/(N_0p)\) is the expected fund per in-force policy under the cohort allocation convention. The original cohort size alone cannot give that per-survivor amount, because it includes policies that are no longer part of the in-force group.
A related conditional-probability identity explains the role of survival directly. For any random quantity \(W\) associated with one original policy, and an event \(A\) with positive probability, the conditional expectation satisfies:
where \(\mathbf{1}_A\) is 1 when \(A\) occurs and 0 otherwise. To see why, \(W\mathbf{1}_A\) contributes only on outcomes in \(A\); its expectation is the probability of \(A\) multiplied by the average value of \(W\) on those outcomes. Dividing by \(P(A)\) therefore leaves that conditional average.
Take \(A\) to be survival to time \(t\), so \(P(A)={}_tp_x\). This identity applies when the quantity being averaged is explicitly restricted to outcomes in which the policy survives. A pooled cohort fund may also include claims paid on policies that ended before \(t\). In that case, do not describe the quotient of the pooled fund by expected survivors as the conditional expectation of one survivor’s personal cash-flow history. It is a cohort fund allocated across the in-force policies under the stated convention.
Worked Examples
Worked Example: A One-Year Cohort Fund
A hypothetical insurer issues 1,000 identical one-year policies to independent lives aged \(x\). The effective annual interest rate is \(i=5\%\). Each policy collects a premium of $100 at issue. At the end of the year, the insurer pays a $300 death benefit for each death during the year. Assume \({}_1p_x=0.80\), no lapses, no expenses, and no other cash flows. Find the expected fund at year-end and the expected fund per policy still in force. Use the cohort allocation convention.
Method. First calculate the expected premiums accumulated to year-end and the expected death claims. Then divide the expected cohort fund by the expected number of survivors. This method avoids treating every policy originally issued as still in force.
The expected number of survivors is:
The death probability is \(1-0.80=0.20\), so the expected number of deaths is:
Premium income accumulated at \(5\%\) is \(1{,}000(100)(1.05)=\$105{,}000\). Expected death claims are \(200(300)=\$60{,}000\). Therefore the expected year-end cohort fund is:
The expected fund per policy in force is:
Check. Multiplying the per-survivor amount by the expected survivor count gives \(800(\$56.25)=\$45{,}000\), the expected cohort fund. The common mistake is to divide by 1,000 and report $45 per policy issued; that is an amount per original policy, not per policy in force.
Worked Example: Two Classes with Different Survival Rates
A hypothetical cohort contains 600 policies in class A and 400 in class B. At the valuation date, the effective annual interest rate is \(i=0\%\). Assume no lapses and independent lives. Class A has survival probability \(0.90\) and an expected fund allocation of $80 for each class A policy in force. Class B has survival probability \(0.60\) and an expected fund allocation of $140 for each class B policy in force. Find the total expected cohort fund and the expected fund per policy in force across both classes.
Method. Calculate expected survivors and expected fund by class, then aggregate both before taking the per-in-force ratio. This avoids applying a simple unweighted average to the two classes.
Expected class A survivors are \(600(0.90)=540\), and expected class B survivors are \(400(0.60)=240\). The total expected number in force is \(540+240=780\).
The expected class A fund is \(540(\$80)=\$43{,}200\). The expected class B fund is \(240(\$140)=\$33{,}600\). Thus the total expected cohort fund is:
The expected amount per policy in force across the whole cohort is:
Check. The result is between the two class amounts, $80 and $140, as expected. It is weighted by the numbers expected to remain in force: class A contributes 540 policies and class B contributes 240. The effective annual interest rate is \(0\%\), so no further interest adjustment is needed at the valuation date.
Worked Example: Lapses Change the In-Force Probability
A hypothetical cohort has 2,000 policies at issue. At the valuation date, each insured has survival probability \(0.90\). Conditional on survival, a policy has probability \(0.80\) of remaining active; assume lapse and mortality are independent. The effective annual interest rate is \(i=3\%\), and the expected cohort fund, already valued at the valuation date under that rate, is $144,000. Find the expected number of policies in force and the expected fund per policy in force.
Method. Calculate the probability of both survival and continued policy activity, then use it to find the expected in-force count. This avoids using survival probability as a substitute for policy persistence.
A policy is in force only if the insured survives and the policy remains active. By independence, its in-force probability is:
With independent, identical in-force indicators, the number in force has a binomial distribution with parameters 2,000 and 0.72. Its expected value is:
Therefore the expected fund per policy in force is:
Check. Using survival alone would give an expected count of \(2{,}000(0.90)=1{,}800\), incorrectly including policies that lapsed. Dividing by that count would give $80, which does not match the specified in-force population. Here the rate is \(3\%\), but the cohort fund is already valued at the valuation date, so the rate is not applied a second time.
Common Traps and Practical Checks
- Using the wrong denominator. \(N_0\) is the number originally issued. Under a no-lapse model, the expected number in force is \(N_0{}_tp_x\), not \(N_0\).
- Equating survival with policy activity. A living insured may have a lapsed or otherwise terminated policy. Use the probability of being in force whenever coverage can end for reasons besides death.
- Ignoring a mixed cohort. If classes have different survival probabilities, calculate expected survivors and expected fund by class before aggregating.
- Confusing pooled balances with personal histories. A cohort fund can include premiums and claims from members who are no longer in force. Its allocated amount per survivor is not automatically the conditional expected value of one survivor’s own cash flows.
- Silently replacing a ratio of expectations with an expectation of a ratio. In general, \(E[C_t]/E[K_t]\) and \(E[C_t/K_t]\) are not the same. Follow the cohort reserve definition being used.
A reliable solution labels each quantity before calculating: expected cohort fund, expected policies in force, and expected fund per policy in force. This simple discipline keeps the cohort-wide perspective separate from the survivor-level perspective.
Check Your Understanding
For each problem, use the stated cohort convention and distinguish survival from policy activity where needed.
- A no-lapse cohort has 500 policies, each with the same survival probability \(0.80\) at the valuation date. Its expected cohort fund is $24,000. Find the expected number in force and the expected fund per policy in force. (Answer: 400 policies; $60.00 per policy)
- A no-lapse cohort has 800 policies and survival probability \(0.90\). The expected cohort fund is $54,000. What is the expected fund per policy in force? (Answer: $75.00)
- A cohort has 1,000 policies. The survival probability is \(0.85\), and conditional on survival the probability of remaining active is \(0.90\). Assume independence and an expected cohort fund of $61,200. Find the expected in-force count and fund per in-force policy. (Answer: 765 policies; $80.00 per policy)
- A cohort has 300 class A policies with survival probability \(0.80\) and an expected fund allocation of $50 per survivor, and 700 class B policies with survival probability \(0.50\) and an expected fund allocation of $110 per survivor. With no lapses, find the total expected fund and its expected amount per policy in force. (Answer: $50,500 total; $85.59 per policy in force)
- A homogeneous no-lapse cohort has 900 policies, survival probability \(0.70\), and an expected cohort fund of $44,100. State the expected number in force and the cohort-convention amount per policy in force. Explain briefly why the original cohort size is not the denominator for that amount. (Answer: 630 policies; $70.00 per policy; the amount is per policy in force, not per policy originally issued)