per-loss vs per-payment variable
Suppose your policy has a 500 deductible. There are two natural ways to talk about the insurer's payments. One: 'over all losses that occurred, including the small ones we paid nothing on, what is the average we paid?' Two: 'looking only at the losses big enough to trigger a cheque, what is the average cheque?' These are the per-loss and the per-payment views, and confusing them is one of the most common errors in loss modelling.
The per-loss variable looks at every loss event, assigning a payment of zero to losses below the deductible. With deductible d it is max(X - d, 0), counting the zeros. The per-payment variable conditions on a payment actually being made — it considers only losses above d, so it is (X - d) given that X is greater than d. The per-payment average is higher than the per-loss average, because the per-loss version is diluted by all those zero payments. Numerically, if 30% of losses fall below the deductible and pay nothing, the per-loss mean is only 70% of the way toward the per-payment mean. The two are linked through the probability of a payment.
The distinction is essential for pricing and especially for fitting. Insurance data are usually per-payment: you only have records of claims that exceeded the deductible (losses below it were never reported), so the data are left-truncated. Fitting a severity model as if the data were per-loss, ignoring the missing small losses, biases the estimates. Likewise, when you compute expected costs you must be clear which average you mean: expected aggregate cost uses the per-loss expectation, while 'average claim paid' is per-payment. A clear answer to 'per loss or per payment?' must precede almost every severity calculation involving a deductible.
With a 500 deductible, losses of 200, 400, 700, 1,500 occur. Per-loss payments are 0, 0, 200, 1,000, averaging 300 over all four events. Per-payment, we drop the two zeros and average only 200 and 1,000, giving 600. The per-payment figure is double the per-loss figure here, purely because half the losses paid nothing.
Per-loss counts the zeros; per-payment conditions on a cheque being written.
Real claim data are usually per-payment and left-truncated — small losses below the deductible were never reported. Fitting them as if they were complete per-loss data omits the missing small losses and biases the severity estimate.