Quantifying Uncertainty In Retention Forecasts

Why a Renewal Forecast Needs an Uncertainty Range

An expected renewal count is a useful summary, but it is not a promise that exactly that many policies will renew. If a portfolio forecast is 700 renewals, decision-makers also need to understand whether counts near 680 or 720 are plausible, and what assumptions drive that range. This tutorial develops uncertainty calculations for aggregate renewal forecasts, aligned with the CAS Exam 8 (Advanced Ratemaking) syllabus.

There are several distinct sources of variation. Individual policyholders may renew or lapse even when their renewal probabilities are known. A cohort may also share an uncertain renewal environment, so that a market or model scenario changes many probabilities together. Finally, the future portfolio may contain a different mix of segments than expected. These sources should be identified before combining them; otherwise a forecast can omit important variation or count the same uncertainty twice.

Policy-Level Variation When Probabilities Are Fixed

Suppose the portfolio contains \(N\) policies and policy \(i\) has a known renewal probability \(p_i=P(R_i=1)\). If renewal outcomes are conditionally independent and the probabilities are treated as fixed, then \(R_i\) is Bernoulli with mean \(p_i\) and variance \(p_i(1-p_i)\). Linearity of expectation gives the expected renewal count; independence allows the individual variances to be added.

The distribution of \(M\) in this case is a Poisson-binomial distribution, because the independent Bernoulli outcomes can have different probabilities. Replacing all policy probabilities with their average does not generally preserve the variance. If every policy has the same probability \(p\), then \(M\) is binomial and its variance is \(Np(1-p)\). But with varying probabilities, the individual variances should be summed.

Shared Uncertainty and the Law of Total Variance

A different situation arises when a probability is uncertain for the whole cohort. For example, suppose renewal outcomes are independent conditional on a market scenario, but the scenario determines a common probability for all policies. The outcomes are then not independent unconditionally: observing a favorable scenario makes many renewals more likely at once.

The identity follows by writing \(M-E[M]\) as \([M-E(M\mid\Theta)]+[E(M\mid\Theta)-E(M)]\). When the square is expanded and its expectation taken, the cross-product has expectation zero: conditional on \(\Theta\), the first bracket has mean zero, while the second is fixed. The two remaining squared terms give the conditional variance averaged over scenarios and the variance of the conditional mean. Thus scenario variation cannot be discarded just because the overall average probability is known.

More generally, if all \(N\) policies share a random probability \(\Theta\), and outcomes are independent conditional on \(\Theta\), then \(E[M\mid\Theta]=N\Theta\) and \(\operatorname{Var}(M\mid\Theta)=N\Theta(1-\Theta)\). Substitution into the total-variance identity gives \(\operatorname{Var}(M)=N E[\Theta(1-\Theta)]+N^2\operatorname{Var}(\Theta)\). Equivalently, this is \(N\mu(1-\mu)+N(N-1)\operatorname{Var}(\Theta)\), where \(\mu=E[\Theta]\). The additional term reflects positive dependence induced by the shared probability. This formula does not apply if policies have independent, policy-specific uncertainty rather than a common random probability.

Uncertainty in Future Portfolio Composition

A renewal forecast can also be uncertain because the future portfolio's segment counts are not fixed. Suppose there are segments \(j=1,\ldots,K\), with renewal probability \(p_j\) for a policy in segment \(j\). Let \(N_j\) be the future number of policies in segment \(j\), so the aggregate count is \(M=\sum_j M_j\). Conditional on the segment counts, and assuming independent renewal outcomes with fixed segment probabilities, the conditional mean and variance are

If segment counts themselves are uncertain, apply the law of total variance again: \(\operatorname{Var}(M)=E[\operatorname{Var}(M\mid N_1,\ldots,N_K)]+\operatorname{Var}(E[M\mid N_1,\ldots,N_K])\). The second term is the variability of the expected renewal count as the portfolio mix changes. It is zero when the segment counts are fixed. If segment counts move together, their covariances also affect this term; treating them as independent without justification can misstate the range.

Interpreting and Communicating a Forecast Range

A predictive range is conditional on its assumptions. It is not automatically a confidence interval for an average renewal probability, nor does it guarantee that the observed count will fall inside it. A confidence interval for a model parameter addresses uncertainty in that parameter; a predictive range for renewals additionally includes future Bernoulli outcomes. If model-parameter uncertainty is included, it should be propagated into the predictive distribution rather than added informally.

For an applied forecast, state which uncertainties are included. A calculation based on fixed policy probabilities includes policy-level outcome variation. A scenario model can add shared uncertainty in the probabilities. A simulation with future segment assignments can add composition variation. If the same uncertainty is represented both in scenario-specific probabilities and in a separate variance adjustment, it may be counted twice.

• Fixed mix, fixed probabilities: sum policy-level variances when outcomes are independent.
• Shared scenario: include both average conditional variance and variance between scenario-specific expected counts.
• Random composition: account for variation in the expected count across possible segment mixes.
• Approximate ranges: check that a normal approximation is plausible; otherwise use an appropriate discrete distribution or simulation.

The practical value of this decomposition is diagnostic as well as numerical. A wide range driven mainly by outcome variation may respond to a larger portfolio, while a range driven by a shared market scenario or uncertain mix requires better scenario or composition assumptions. Reporting only a point forecast conceals that distinction.