Resolving Conflicting Indication Evidence

Why Indications Can Conflict

A rate review may produce different indicated changes from different methods, experience periods, data sources, or assumptions. A difference is not automatically evidence that one result is wrong. The methods may be measuring different populations, responding differently to unusual experience, or carrying different amounts of uncertainty. A defensible selection begins by identifying what each indication measures and why it differs from the others.

The goal is not to force every result into agreement. It is to select a rate change for a defined target portfolio and effective period, using evidence that is relevant to that target. The selected change should reflect the strength and limitations of the evidence, not simply the order in which results were calculated.

Establish Comparability Before Assigning Weight

An average of incomparable estimates has no clear interpretation. Before combining indications, verify that they refer to the same coverage, policyholder population, target loss and expense basis, and proposed effective period. Also check that they express changes from the same current rate level. If one indication is for a narrower segment or a different effective date, it may still be useful, but it cannot be treated as directly interchangeable without an adjustment or a clear explanation of its role.

Separate two questions that are easy to confuse. First, what does the indication estimate? Second, how much confidence should it receive for this target? For example, an indication based on a larger volume of experience may be statistically stable but less relevant if the target book has changed substantially. A more recent estimate may be relevant but uncertain if its data are immature. Volume, recency, method, and fit to the target all matter; none is a universal weighting rule by itself.

• Comparability: Are the indications on a common rate level, coverage, and effective-date basis?
• Reliability: Are the data complete, credible for the intended use, and processed consistently?
• Relevance: Does the experience represent the risks and conditions expected during the proposed period?
• Dependence: Do the indications use overlapping data or share assumptions, so that they are not independent evidence?

If a difference is caused by an identifiable scope or assumption difference, investigate and explain that difference before treating it as random estimation error. A large numerical gap alone does not tell the actuary which indication deserves more weight.

Precision Weighting as a Model-Based Rule

When several indications estimate the same unknown change, a precision-weighted average gives more weight to estimates with lower variance. This rule has a statistical justification, but only under explicit assumptions: each indication is unbiased for the same quantity, its variance is known or reasonably estimated, and the indication errors are independent. It is not a substitute for checking whether the indications are comparable.

Here \(\hat{\theta}_j\) is indication \(j\), \(\sigma_j^2\) is its error variance, and \(\hat{\theta}\) is the combined estimate. The normalized weight for indication \(j\) is \(w_j/\sum_j w_j\). If standard errors are provided, square each standard error to obtain the variance before taking its reciprocal.

Why the Precision-Weighted Average Works

Suppose each indication has the model \(\hat{\theta}_j=\theta+\varepsilon_j\), where the common target is \(\theta\), each error has mean zero and variance \(\sigma_j^2\), and errors are mutually independent. Consider a linear combination \(\sum_j a_j\hat{\theta}_j\), with \(\sum_j a_j=1\), so it remains unbiased for \(\theta\). Independence gives its variance as \(\sum_j a_j^2\sigma_j^2\).

To minimize that variance subject to \(\sum_j a_j=1\), form the Lagrangian \(L=\sum_j a_j^2\sigma_j^2-\lambda(\sum_j a_j-1)\). Differentiating with respect to \(a_j\) gives \(2a_j\sigma_j^2-\lambda=0\), so \(a_j=\lambda/(2\sigma_j^2)\). Applying the constraint yields \(a_j=(1/\sigma_j^2)/\sum_r(1/\sigma_r^2)\). Thus the minimum-variance unbiased linear combination assigns weight in proportion to inverse variance. If indications are biased, target different quantities, or have dependent errors, this justification does not apply as stated.

When Weights Require Judgment

In many practical selections, defensible variances are not available, or the model assumptions for precision weighting are not credible. The actuary may then use judgmental weights, but the weights should express an explained view of the evidence rather than conceal an arbitrary preference. State what factors drove the weights: for example, whether the data are mature, whether the experience reflects the target mix, or whether a method’s assumptions are better aligned with the intended use.

Judgmental weighting is not the same as declaring one source “more credible” without support. A useful explanation links each weight to a specific strength or limitation and shows how the result changes when plausible weights change. If the selection is highly sensitive to a small adjustment in weights, that sensitivity itself is important evidence to disclose.

Overlapping Evidence Is Not Independent Evidence

Two methods can produce different indications while relying on much of the same underlying experience. If their errors move together, treating them as independent overstates the amount of information available. In particular, giving each method a separate full weight can effectively count shared claims or assumptions more than once.

For two equally precise estimates with variance \(\sigma^2\) and correlation \(\rho\), the variance of their arithmetic mean is \(\sigma^2(1+\rho)/2\). If \(\rho=0\), this is \(\sigma^2/2\), the familiar benefit from averaging independent estimates. If \(\rho=1\), the variance remains \(\sigma^2\): averaging identical information does not reduce uncertainty. Correlation does not necessarily make an estimate unusable, but it changes how much independent evidence it contributes.

A Practical Evidence-Resolution Process

Common Exam Trap: Averaging Before Checking the Basis

Suppose one indication is a 10% increase for a broad portfolio, while another is a 6% increase for a smaller segment. An equal average of 8% is not automatically a valid overall indication. It implicitly gives each estimate equal influence, regardless of the segment’s share of the target portfolio, and assumes both estimates are already expressed on the same basis. The correct response is to establish how each indication relates to the target first. If the segment estimate is intended to represent only part of the portfolio, the selection needs a defensible aggregation that reflects the target, not an unexamined arithmetic mean.

Even when indications are comparable, equal weighting is a choice, not a neutral default. Precision weights require unbiased, independent estimates of one target. Judgmental weights require a reasoned explanation. Shared experience calls for attention to dependence. In every case, the weight should follow an evidence-based rationale.