Functions & Formulae

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Specific attachment point expected reimbursement, variance and claims inflation response function

s(p_k, C_k, A, U, c, g) = (E_R, M2_R, V_R, SD_R, E_g, rise_E)

Maps the distribution of one covered person's eligible claims for a contract year, a specific attachment point A, a cap on claims U and a coinsurance rate c to the mean, second moment, variance and standard deviation of the stop-loss or reinsurance reimbursement for that person, and maps a uniform claims inflation rate g to the proportional rise in the expected reimbursement of an uncapped excess layer whose attachment point stays fixed. The one-year payment rule of the article, R equal to c times the positive part of min(C, U) minus A, is the payment rule of HE-FM-ASL-001 applied to one person's claims C in place of the group's net claims S, with a layer limit M equal to U minus A, multiplied by c, because for any cap at or above the attachment point min(max(C minus A, 0), U minus A) equals max(min(C, U) minus A, 0) (derived, checked numerically), so the rule is cited and not restated. The aggregate side of a stop-loss policy is live as HE-FM-ASL-001 to HE-FM-ASL-005 and HE-CF-ASL-001, including the normal approximation to an expected excess (HE-FM-ASL-005), and the standard deviation of a total over independent members is the square-root rule of HE-FM-ASL-003. A probability-weighted mean in general is HE-FN-EV-001 (HE-FM-CHN-001 at a chance node), a member's cost sharing under a deductible, coinsurance and out-of-pocket maximum is HE-FM-AV-002, a gross premium with loadings is HE-FM-ACTA-002 and moving a cost from one price year to another with an index is HE-FM-COST-002. This package adds the variance of the specific reimbursement over a discrete claims distribution, with its mean, and the response of an excess layer to claims inflation. Notation follows the Attachment Point article.

  • Mean and variance of specific stop-loss reimbursement over a three-level claims distribution

    E_R = c * (p_1 * max(min(C_1, U) - A, 0) + p_2 * max(min(C_2, U) - A, 0) + p_3 * max(min(C_3, U) - A, 0)); M2_R = c^2 * (p_1 * max(min(C_1, U) - A, 0)^2 + p_2 * max(min(C_2, U) - A, 0)^2 + p_3 * max(min(C_3, U) - A, 0)^2); V_R = M2_R - E_R^2; SD_R = sqrt(V_R)

    Gives the expected reimbursement per covered person, its second moment, variance and standard deviation when the person's annual eligible claims take one of three levels C_1 to C_3 with probabilities p_1 to p_3, and the remaining probability sits at claims that never reach the attachment point. Each level's payment is the specific layer rule of HE-FM-ASL-001 with a layer limit of U minus A, times the coinsurance rate c, and E_R is the probability-weighted mean of HE-FN-EV-001 over those payments, the net premium per person for the layer before any loading, carried here because the variance needs it. V_R is the second moment less the squared mean, following the expected value and second moment of the excess of loss cost function in Miccolis 1977. The record works from the discrete distribution of the article's worked example; the normal closed form for an aggregate total is the separate record HE-FM-ASL-005. For a group of n independent members the expected total is n times E_R and its standard deviation SD_R times the square root of n, the rule of HE-FM-ASL-003, which the computational function below applies. Raising A by an amount that passes no claim level and stays below the cap lowers E_R by c times that amount times the probability that claims exceed the old attachment point (derived), so expected reimbursement falls steeply while, as the worked examples show, the standard deviation falls more slowly.

  • Proportional rise in expected reimbursement of an uncapped specific excess layer under uniform claims inflation

    E_0 = p_1 * max(C_1 - A, 0) + p_2 * max(C_2 - A, 0) + p_3 * max(C_3 - A, 0); E_g = p_1 * max((1 + g) * C_1 - A, 0) + p_2 * max((1 + g) * C_2 - A, 0) + p_3 * max((1 + g) * C_3 - A, 0); rise_E = E_g / E_0 - 1; mult_E = rise_E / g

    Gives the expected reimbursement per member of an uncapped excess layer with full reimbursement before and after every claim level rises by the same inflation rate g while the attachment point stays fixed in money, the proportional rise rise_E in that expected reimbursement, and its ratio mult_E to g. Lee 1988 states that with the retention fixed the expected excess loss payment rises proportionally much more than the general rate of inflation. Miccolis 1977 shows that under a uniform inflation factor the expected loss limited at k equals the factor times the earlier expected loss limited at k divided by the factor (equation 17), from which E_g equals (1 + g) times the expected excess of current claims over A / (1 + g) (derived), so inflating claims has the same effect as lowering the attachment point in real terms, and without a cap mult_E is at least 1 (derived, checked over a grid of inputs). A coinsurance rate multiplies E_0 and E_g alike and cancels from rise_E; with a cap the top of the layer can already be used up and the rise can fall below g. The general price-year adjustment of a single cost is HE-FM-COST-002.

Attachment Point — Functions & Formulae | HealthEconomics.wiki