Exam Workshop: Diagnose Inconsistent Loss and Treaty Bases

Introduction

A loss record can be correct on its own terms and still be unusable beside another record. A ground-up claim count includes losses that produce no policy payment; a count of paid claims does not. Likewise, a treaty may apply to the policy’s covered payment rather than to the original loss. Combining these quantities without first aligning their bases can distort both estimated frequency and expected cost.

This workshop follows hypothetical claims from ground-up loss through a policy deductible and limit to a per-claim reinsurance layer. The central discipline is to label every count and amount by the event or payment it represents. The calculations use an exponential severity model to make the conversions explicit; the same basis checks matter with other severity models. This approach is aligned with the SOA Exam ASTAM (Advanced Short-Term Actuarial Mathematics) syllabus.

State the Basis Before Calculating

For a simple model, suppose ground-up losses are independent and identically distributed, and each has the same policy terms. Write \(q=P(Y^L>0)\) for the probability that one ground-up loss produces a positive policy payment. If there are \(m\) ground-up claims, the expected number of positive policy payments is \(mq\). This is an expectation, not a claim that the realized count must equal \(mq\).

If \(N\) is a random ground-up claim count with \(E[N]=\lambda\), and the loss distribution and policy terms are the same for each claim, the expected number of positive policy payments is \(\lambda q\). Independence between claims’ payment indicators is not needed for this expected-count relation: conditional on \(N\), linearity of expectation gives \(E[\sum_{j=1}^{N} I_j\mid N]=Nq\) when each indicator \(I_j\) has probability \(q\). If claim probabilities differ, sum the individual probabilities instead. Independence assumptions may matter for the distribution or variance of the paid count, but not for this mean.

The amounts also need a basis label. \(E[Y^L]\) is the average policy payment per ground-up claim, including zero payments. \(E[Y^P]\) is the average amount among claims that do produce a payment. They satisfy \(E[Y^L]=qE[Y^P]\). Multiplying a paid-claim count by \(E[Y^L]\), or a ground-up count by \(E[Y^P]\), mixes bases.

Policy Payments Under an Exponential Model

Assume \(X\) has an exponential distribution with mean \(\theta>0\), so \(P(X>x)=e^{-x/\theta}\) for \(x\geq 0\). For a positive deductible \(d\) and payment limit \(u>0\), the policy pays only when \(X>d\), giving \(q=e^{-d/\theta}\). The formula below is for a per-claim payment cap \(u\), not a cap on the original loss.

The integral follows from the tail-integral identity for a nonnegative random variable: its expectation is the integral of its survival probability. For \(0\leq t<u\), a policy payment exceeds \(t\) exactly when \(X>d+t\); beyond \(u\), it cannot exceed \(t\). Integrating \(e^{-(d+t)/\theta}\) from \(0\) to \(u\) gives the stated mean. Dividing by \(q\) conditions on a positive payment. Equivalently, given \(X>d\), the exponential memoryless property says that \(X-d\) is again exponential with mean \(\theta\), and the policy payment is \(\min(X-d,u)\).

Convert Paid-Claim Frequency to Ground-Up Frequency

When an exposure record reports only claims that generated a positive payment, it cannot be used directly as a ground-up frequency. If the model gives a positive-payment probability \(q\), then an expected positive-payment count \(m_P\) corresponds to a ground-up expected count \(m_P/q\), provided the same severity and coverage assumptions apply to the claims in question. This conversion is about expected counts. For realized data, the observed counts need not match their expectations exactly.

Apply the Treaty to the Covered Payment

A per-claim excess-of-loss treaty may attach above the policy’s payment, not above \(X\). Let \(a\) be the treaty attachment and \(w\) its maximum recovery per claim. If it covers the policy payment \(Y^L\), its recovery per ground-up claim is \(R=\min((Y^L-a)_+,w)\). A recovery count, like a paid-claim count, refers only to claims with positive recovery unless explicitly stated otherwise.

For the exponential model and policy limit \(u>a\), define the effective treaty width \(b=\min(w,u-a)\). The treaty can pay only after the policy payment exceeds \(a\), and the maximum payable amount is \(b\). Applying the same tail-integral argument to the recovery gives the result below. If \(u\leq a\), there is no recovery. The formula does not apply unchanged if the treaty attaches to a different base, such as ground-up loss.

The threshold for any recovery is \(X>d+a\), because the policy first subtracts \(d\) and the treaty then requires the policy payment to exceed \(a\). For \(0\leq t<b\), \(R>t\) exactly when \(X>d+a+t\). Integrating \(P(R>t)=e^{-(d+a+t)/\theta}\) from zero to \(b\) yields the expected recovery. This derivation also shows why applying the attachment directly to \(X\) changes the threshold.

Common Basis Errors and When the Formulas Need Adjustment

A frequent error is to pair a positive-payment count with an average that includes zero payments. In the frequency example, multiplying 800 positive payments by \(E[Y^L]=\$5{,}721.3379\) gives only \(800(5721.3379)=\$4{,}577{,}070.31\). That is not the expected annual payment total for 800 positive claims. The correction is to use the conditional mean \(E[Y^P]=\$6{,}988.0579\), giving \(\$5{,}590{,}446.32\). Alternatively, convert 800 positive claims to 977.1222 expected ground-up claims and use \(E[Y^L]\).

Another common error is to apply the treaty attachment to ground-up loss when its contract says it attaches to covered policy payment. In the treaty example, a \(\$3{,}000\) attachment above the policy payment requires \(X>\$2{,}000+\$3{,}000=\$5{,}000\). Treating the attachment as \(\$3{,}000\) above \(X\) would instead use the wrong threshold and overstate expected recovery. Always translate contract wording into the quantity to which the attachment applies before calculating a layer.

The exponential formulas rely on the stated exponential severity model and identical policy terms. Different deductibles, limits, or severity distributions require claim-specific probabilities and expectations or an appropriately modified model. If claim counts are random, the expected aggregate amount can be calculated as expected count times the mean amount per claim when the claim severity and count are independent, as in the usual compound frequency-severity model. For non-identical claims, use the corresponding sum of expectations rather than a single common mean. Do not infer that a paid count must be Poisson merely from its expected value; Poisson thinning requires additional distributional and independence assumptions.