Functions & Formulae

Each applied formula has its own function page, with a signature, implementations, and tests.

Aggregate stop-loss payment and expected payment function

L = f(S, A, M); R = S - L; EL = g(A, E_S, SD_S)

Maps a self-funded employer's eligible claims for the contract year, net of specific stop-loss recoveries, to the payment due from the aggregate stop-loss carrier and the claims the employer retains, given the aggregate attachment point and any maximum aggregate payment. Applied to the distribution of the year's claims instead of one realised year, the same rule gives the carrier's expected payment, the net stop-loss premium before any loading. The records follow the notation of the Aggregate Stop-Loss article, where S is net claims for the year, A is the aggregate attachment point and L is the carrier's payment.

  • Aggregate stop-loss carrier payment and employer retention for one contract year

    L = min(max(S - A, 0), M); R = S - L

    Gives the aggregate stop-loss carrier's payment for a contract year as the excess of the employer's net claims over the aggregate attachment point, capped at the policy's maximum aggregate payment, and the employer's retained claims as the remainder. Net claims S are the sum over covered members of each member's eligible claims net of specific stop-loss recoveries, so the calculator takes the total directly. The functions min and max return the smaller and the larger of their arguments. With no maximum the retained claims equal the smaller of S and A, so they never exceed the attachment point.

  • Aggregate attachment point set as a multiple of expected claims

    A = a * E_S

    Sets the aggregate attachment point by applying an attachment factor to the group's expected annual claims, for example 1.25 for an attachment point at 125% of expected claims. Expected claims are defined on the same net basis as the claims that accumulate towards the attachment point. A lower factor transfers more risk to the carrier and costs more.

  • Mean, standard deviation and coefficient of variation of a group's total net claims

    E_S = n * mu; SD_S = sigma * sqrt(n); CV_S = sigma / (mu * sqrt(n))

    Gives the expected value, standard deviation and coefficient of variation of a group's total net claims when members' claims are independent with a common mean and standard deviation. Total claims grow in proportion to n but their spread only with the square root of n, so the relative spread falls as the group grows. The function sqrt is the square root.

  • Standardised distance from expected claims to the aggregate attachment point

    z = (A - E_S) / SD_S

    Expresses the gap between the aggregate attachment point and expected net claims in standard deviations of total claims. Under a normal approximation to total claims, the probability that claims exceed the attachment point is 1 minus Phi(z), where Phi is the standard normal cumulative distribution function; Excel returns it as 1-NORM.S.DIST(z,TRUE). The calculator on this page returns z, since it has no normal distribution function.

  • Expected aggregate stop-loss payment under a normal approximation

    z = (A - E_S) / SD_S; phi_z = exp(-z * z / 2) / sqrt(2 * 3.14159265); EL = SD_S * (phi_z - z * Q_z)

    Gives the expected excess of total net claims over the aggregate attachment point, the net stop-loss premium before loading, when total claims are approximately normal. phi_z is the standard normal density at z, written out with exp and sqrt, and Q_z is the standard normal upper-tail probability 1 minus Phi(z), entered from Excel's 1-NORM.S.DIST(z,TRUE) because the calculator has no normal distribution function. The formula has no maximum aggregate payment.