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

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.

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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)
Inputs
InputsDefinitionUnit
cShare of claims between the attachment point and the cap that the insurer or reinsurer pays, entered as 1 for full reimbursementproportion
p_1Probability that the person's annual eligible claims are at the first level C_1probability
C_1Lowest of the three annual claim levels that can reach the attachment points comparedUS dollars
UAnnual claims level above which no further claims are reimbursed, entered as C_3 or more for a layer with no capUS dollars
AAnnual eligible claims of one covered person beyond which the insurer or reinsurer paysUS dollars
p_2Probability that the person's annual eligible claims are at the second level C_2probability
C_2Middle one of the three annual claim levelsUS dollars
p_3Probability that the person's annual eligible claims are at the third level C_3probability
C_3Highest of the three annual claim levelsUS dollars
Output
E_RExpected reimbursement from the specific stop-loss insurer or reinsurer for one covered person in the contract year, before any loadingUS dollars per member
M2_RExpected value of the squared reimbursement for one covered personUS dollars squared
V_RVariance of the reimbursement for one covered person in the contract yearUS dollars squared
SD_RStandard deviation of the reimbursement for one covered person in the contract yearUS dollars

Function

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

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.

Computational function

  • Computational function for specific stop-loss expected cost, spread and premium by attachment point from a claims distribution and group size

    Takes the figures an employer or analyst usually holds when choosing a specific attachment point, a claims distribution for one member, the number of members, the coinsurance rate, any cap and a premium loading, and returns for each attachment point the expected number of members above it, the expected and standard deviation of total reimbursement, an illustrative premium and the ratio of spread to mean, the table of the article's worked example. It applies HE-FM-ATTP-001 for one member, then HE-FM-ASL-003 for the total over n independent members, and prices the layer at expected reimbursement times one plus the loading theta, the article's illustrative convention; HE-FM-ACTA-002 sets a proportional loading as a share of the gross premium instead. Its inputs differ from the symbols of HE-FM-ATTP-001 because it needs the group size and loading, and the Excel, R and Python versions loop over a grid of attachment points, which the calculator line takes one at a time.

    Inputs and outputs: p_1 to p_3 are the probabilities of the claim levels C_1 to C_3 in US dollars; A is the specific attachment point and U the cap on claims, in US dollars; c is the coinsurance rate; n is the number of members; theta is the premium loading as a proportion of expected reimbursement. The outputs are E_R, M2_R, V_R and SD_R per member, the expected number of members above the attachment point N_A, expected total reimbursement E_T and its standard deviation SD_T in US dollars, the illustrative premium P_T in US dollars and the ratio CV_T of SD_T to E_T.

    Assumption: Members' claims are independent and follow the same distribution, the remaining probability sits at claims below the attachment point, and the loading is a fixed share of expected reimbursement. Shared shocks make members' claims move together, and the true standard deviation of total reimbursement is then larger than SD_T (HE-AS-ASL-005).

    Worked example (Article's USD 50,000 specific attachment point): Ten members are expected above the attachment point, expected reimbursement is USD 900,000 with a standard deviation of about USD 485,222, and the illustrative premium at a 30% loading is USD 1,170,000. c = 1; p_1 = 0.006; C_1 = 75000; p_2 = 0.003; C_2 = 150000; p_3 = 0.001; C_3 = 500000; n = 1000; theta = 0.3; A = 50000; U = 500000; E_R = 900; M2_R = 236250000; V_R = 235440000; SD_R = 15344.05; N_A = 10; E_T = 900000; SD_T = 485221.6; P_T = 1170000; CV_T = 0.5391

    Worked example (Article's USD 100,000 specific attachment point): Four members are expected above it, expected reimbursement is USD 550,000 with a standard deviation of about USD 408,898, and the premium is USD 715,000. c = 1; p_1 = 0.006; C_1 = 75000; p_2 = 0.003; C_2 = 150000; p_3 = 0.001; C_3 = 500000; n = 1000; theta = 0.3; A = 100000; U = 500000; E_R = 550; M2_R = 167500000; V_R = 167197500; SD_R = 12930.49; N_A = 4; E_T = 550000; SD_T = 408897.91; P_T = 715000; CV_T = 0.7435

    Worked example (Article's USD 200,000 specific attachment point): One member is expected above it, expected reimbursement is USD 300,000 with a standard deviation of about USD 299,850, and the premium is USD 390,000, USD 780,000 less than at USD 50,000; the ratio of spread to mean is about 1.00. c = 1; p_1 = 0.006; C_1 = 75000; p_2 = 0.003; C_2 = 150000; p_3 = 0.001; C_3 = 500000; n = 1000; theta = 0.3; A = 200000; U = 500000; E_R = 300; M2_R = 90000000; V_R = 89910000; SD_R = 9482.09; N_A = 1; E_T = 300000; SD_T = 299849.96; P_T = 390000; CV_T = 0.9995

    Excel: With ClaimLevels, ClaimProbs, ClaimCap and CoinsRate as in HE-IM-ATTP-001, the number of members in a cell named Members, the loading in PremLoading, the capped claim levels =MIN(B2,ClaimCap) filled down in E2 to E4 and named CappedLevels, and attachment points in F2 to F4, the row formulas =CoinsRate*SUMPRODUCT(ClaimProbs,(CappedLevels-F2)*(CappedLevels>F2)) in G2, =CoinsRate^2*SUMPRODUCT(ClaimProbs,((CappedLevels-F2)*(CappedLevels>F2))^2)-G2^2 in H2, =Members*G2 in I2, =SQRT(H2)*SQRT(Members) in J2, =(1+PremLoading)*I2 in K2, =J2/I2 in L2 and =Members*SUMPRODUCT(ClaimProbs,--(ClaimLevels>F2)) in M2, filled down to row 4, give the article's table.

    R: attp_grid <- function(p, C, A, U = Inf, cc = 1, n = 1000, theta = 0.3) { t(sapply(A, function(a) { x <- cc*pmax(pmin(C, U)-a, 0); E <- sum(p*x); V <- sum(p*x^2)-E^2; c(A = a, E_R = E, SD_R = sqrt(V), N_A = n*sum(p[C > a]), E_T = n*E, SD_T = sqrt(n*V), P_T = (1+theta)*n*E, CV_T = sqrt(n*V)/(n*E)) })) } Run as attp_grid(c(0.006, 0.003, 0.001), c(75000, 150000, 500000), c(50000, 100000, 200000)), which returns totals of 900,000, 550,000 and 300,000 with standard deviations of about 485,222, 408,898 and 299,850.

    Python: def attp_grid(p, C, A, U=float("inf"), c=1.0, n=1000, theta=0.3): import math; out = []; [out.append((a, E, math.sqrt(V), n*sum(pk for pk, ck in zip(p, C) if ck > a), n*E, math.sqrt(n*V), (1+theta)*n*E, math.sqrt(n*V)/(n*E))) for a in A for x in [[c*max(min(ck, U)-a, 0) for ck in C]] for E in [sum(pk*xk for pk, xk in zip(p, x))] for V in [sum(pk*xk**2 for pk, xk in zip(p, x))-E**2]]; return out Run as attp_grid([0.006, 0.003, 0.001], [75000, 150000, 500000], [50000, 100000, 200000]), which returns the same figures as the R version.

    Test (Article's group standard deviations reproduced): With the article's inputs, Members at 1000, CoinsRate at 1, ClaimCap at 500000 and attachment points of 50000, 100000 and 200000 in F2 to F4, J2 to J4 round to 485,222, 408,898 and 299,850 and I2 equals 900,000. Expected result TRUE. FALSE shows the squared mean left unsubtracted in column H, which gives about 486,056 in J2, or the per-member standard deviation multiplied by the number of members, about 15,344,054 in J2. Excel check =AND(ROUND(J2,0)=485222,ROUND(J3,0)=408898,ROUND(J4,0)=299850,I2=900000)

    Test (Layer between two attachment points by the difference route): With the cap at or above the attachment point in F4, G2 less G4 equals the expected reimbursement of the layer between the attachment points in F2 and F4 computed directly with the higher one as a cap, the excess layer as the difference of two first-dollar expected values in equation 11 of Miccolis 1977. Expected result TRUE. FALSE shows the whole claim paid once it passes the attachment point, which gives a difference of 900 against 600, or G2 halved, which gives 150. Excel check =ABS((G2-G4)-CoinsRate*SUMPRODUCT(ClaimProbs,(((ClaimLevels<F4)*ClaimLevels+(ClaimLevels>=F4)*F4)-F2)*(ClaimLevels>F2)))<1E-6

    Common error (Whole premium counted as a resource cost): The article notes that from a health system or societal perspective the reimbursement is a transfer between payers and only the administration and capital costs within the loading, with any change in care, are real resource effects. At USD 100,000 the illustrative premium of USD 715,000 holds USD 550,000 of expected reimbursement, so at most USD 165,000 of it can be a resource cost; counting the whole premium overstates that cost more than fourfold.

    Source: Miccolis RS. On the theory of increased limits and excess of loss pricing. Proceedings of the Casualty Actuarial Society. 1977;64:27-59. Equations 9 and 10 (page 34) for the excess of loss cost function and its expected value, equation 11 (page 34) for the excess layer as the difference of two limited expected values, and section Variance as a Measure of Risk with equation 31 (pages 42 to 44) for the variance and second moment. National Association of Insurance Commissioners. Stop Loss Insurance, Self-Funding and the ACA. White paper. 2015. Section V on premiums developed from an actuary's determination of expected losses.

    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); N_A = n * (p_1 * (C_1 > A) + p_2 * (C_2 > A) + p_3 * (C_3 > A)); E_T = n * E_R; SD_T = SD_R * sqrt(n); P_T = (1 + theta) * E_T; CV_T = SD_T / E_T

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Implementations

  • Excel

    Specific stop-loss reimbursement mean and variance from a claims table in Excel

    The three claim levels are in B2 to B4, named ClaimLevels, and their probabilities in C2 to C4, named ClaimProbs, with named cells AttachPoint, ClaimCap and CoinsRate. The first formula, entered in D2 and filled down to D4, gives each level's reimbursement in the range named LevelPayments. The next three formulas return the expected reimbursement in a cell named ExpReimb, the variance in VarReimb and the standard deviation in SDReimb. SUMPRODUCT needs no array entry, and further claim levels can be added as rows.

    =CoinsRate*MAX(MIN(B2,ClaimCap)-AttachPoint,0); =SUMPRODUCT(ClaimProbs,LevelPayments); =SUMPRODUCT(ClaimProbs,LevelPayments^2)-ExpReimb^2; =SQRT(VarReimb)

Assumptions

  • Remaining claims probability below every specific attachment point compared

    The three levels carry all the probability of claims that can reach the attachment point. The remaining probability, 1 minus p_1 minus p_2 minus p_3, sits at claims at or below A and adds nothing to E_R or M2_R, although it still counts in the variance through the squared mean. In the article's example 0.990 of members claim less than USD 50,000, so the formula holds for attachment points of USD 50,000 or more; a lower attachment point needs the claims below USD 50,000 written out as further levels.

  • Cap at or above the specific attachment point and full reimbursement entered as 1

    U is at least A, so U minus A is the layer limit of HE-FM-ASL-001, and a contract with no cap is entered with U at least C_3. The coinsurance rate c is the share paid inside the layer, as 45 CFR 153.20 defines the coinsurance rate and 45 CFR 153.230(c) applies it, and is entered at least 0 and at most 1.

  • One covered person and one contract year on the specific contract's own basis

    C is one person's eligible claims for one contract year, counted on the contract's basis. The NAIC white paper notes that, unless a provision or endorsement extends cover, a stop-loss policy reimburses only claims incurred and paid in the same policy year, and that an insurer may set a higher laser specific attachment point for individuals with known high-cost conditions, so a lasered person is computed with that person's own A. Claims outside the contract year or the covered services are not in C.

Worked examples

  • Specific attachment point of USD 50,000 in the article's illustrative claims distribution

    The article's illustrative distribution has annual claims of USD 75,000, USD 150,000 and USD 500,000 with probabilities 0.006, 0.003 and 0.001, full reimbursement and no cap (U entered as USD 500,000). At an attachment point of USD 50,000 the expected reimbursement is USD 900 per member and the variance 235,440,000 dollars squared, as in the article, with a standard deviation of about USD 15,344 per member. For 1,000 independent members HE-FM-ASL-003 gives a standard deviation of total reimbursement of about USD 485,222, the article's figure.

    c = 1; p_1 = 0.006; C_1 = 75000; U = 500000; A = 50000; p_2 = 0.003; C_2 = 150000; p_3 = 0.001; C_3 = 500000; E_R = 900; M2_R = 236250000; V_R = 235440000; SD_R = 15344.05

  • Specific attachment point of USD 100,000 in the article's illustrative claims distribution

    At USD 100,000 the USD 75,000 level no longer reaches the layer. The expected reimbursement is USD 550 per member and the standard deviation about USD 12,930, which for 1,000 independent members gives about USD 408,898, the article's figure.

    c = 1; p_1 = 0.006; C_1 = 75000; U = 500000; A = 100000; p_2 = 0.003; C_2 = 150000; p_3 = 0.001; C_3 = 500000; E_R = 550; M2_R = 167500000; V_R = 167197500; SD_R = 12930.49

  • Specific attachment point of USD 200,000 in the article's illustrative claims distribution

    At USD 200,000 only the USD 500,000 level reaches the layer. The expected reimbursement is USD 300 per member and the standard deviation about USD 9,482, about USD 299,850 for 1,000 members. The mean is a third of its value at USD 50,000, while the standard deviation is still about 62% of it, the slower fall the article describes.

    c = 1; p_1 = 0.006; C_1 = 75000; U = 500000; A = 200000; p_2 = 0.003; C_2 = 150000; p_3 = 0.001; C_3 = 500000; E_R = 300; M2_R = 90000000; V_R = 89910000; SD_R = 9482.09

  • Layer between USD 50,000 and USD 200,000 given up by raising the specific attachment point

    The layer the employer takes back by raising the attachment point is the same formula with A at USD 50,000 and the cap U at USD 200,000. Its expected reimbursement is USD 600 per member, the difference of USD 900 and USD 300, and its standard deviation about USD 7,476, about USD 236,411 for 1,000 members, the article's figure. The ratio of standard deviation to mean for the group is about 0.39 against about 1.00 for the layer above USD 200,000.

    c = 1; p_1 = 0.006; C_1 = 75000; U = 200000; A = 50000; p_2 = 0.003; C_2 = 150000; p_3 = 0.001; C_3 = 500000; E_R = 600; M2_R = 56250000; V_R = 55890000; SD_R = 7475.96

  • Coinsurance and cap of the 2015 ACA reinsurance parameters on the illustrative claims distribution

    The 2015 benefit year parameters of the ACA transitional reinsurance programme after HHS lowered the attachment point, USD 45,000 with a USD 250,000 cap and a 50% coinsurance rate as set, are applied here to the article's illustrative claims distribution for illustration only. Because USD 45,000 is below the USD 50,000 that HE-AS-ATTP-001 requires, the example also takes the 0.990 of members who claim less than USD 50,000 to claim no more than USD 45,000; any of them with claims above USD 45,000 would add to the figures. Each level pays half of its claims between USD 45,000 and USD 250,000, so the expected reimbursement is USD 350 per member and the standard deviation about USD 4,472.

    c = 0.5; p_1 = 0.006; C_1 = 75000; U = 250000; A = 45000; p_2 = 0.003; C_2 = 150000; p_3 = 0.001; C_3 = 500000; E_R = 350; M2_R = 20125000; V_R = 20002500; SD_R = 4472.42

  • Specific attachment point at the highest claim level as a limiting case

    With the attachment point at USD 500,000, the highest claim level, no claim exceeds it, so the expected reimbursement, the second moment and the variance are all zero. This limiting case checks the implementation.

    c = 1; p_1 = 0.006; C_1 = 75000; U = 500000; A = 500000; p_2 = 0.003; C_2 = 150000; p_3 = 0.001; C_3 = 500000; E_R = 0; M2_R = 0; V_R = 0; SD_R = 0

Common errors

  • Mean specific reimbursement per claimant used as the mean per member

    At a USD 100,000 attachment point the members who reach the layer, 0.4% of the group, receive on average USD 137,500 each, but the expected reimbursement per member is USD 550. Pricing the layer at the per-claimant mean overstates its expected cost 250-fold; the probability of claims above A has to stay in the expectation, as it does in E_R.

  • Whole claim reimbursed once it passes the specific attachment point

    Paying the full claim of a person whose claims exceed A, instead of the excess over A, turns the excess layer into a franchise. At USD 100,000 it gives 0.003 times 150,000 plus 0.001 times 500,000, USD 950 per member, about 73% more than the USD 550 of the excess rule of 45 CFR 153.230(c) and HE-FM-ASL-001.

  • Assuming the spread of transferred claims falls in step with expected reimbursement

    Raising the attachment point from USD 50,000 to USD 200,000 cuts the expected reimbursement of 1,000 members by two thirds, from USD 900,000 to USD 300,000. Scaling the standard deviation down in the same proportion would put it at about USD 161,741, but the variance formula gives about USD 299,850, because the rare USD 500,000 claim carries most of the variance. Treating a higher attachment point as giving up mainly predictable spending, as the article does, rests on this difference.

Sources

  • Miccolis on the expected value and second moment of an excess of loss layer

    Miccolis RS. On the theory of increased limits and excess of loss pricing. Proceedings of the Casualty Actuarial Society. 1977;64:27-59. Section Excess of Loss Coverage (page 34), equation 9, which defines the excess of loss cost function h(x; r, j) as zero for losses up to the retention r, the loss minus r up to r plus the maximum liability j, and j above that, and equation 10, which gives its expected value; and section Variance as a Measure of Risk (pages 42 to 44), which writes the variance of a cost function as its second moment less the square of its first moment and gives the second moment of the excess of loss cost function in equation 31. The article's attachment point A is the retention r and its cap U is r plus j; the coinsurance factor c and the three-level discrete form are applied here.

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  • 45 CFR 153.230 payment rule with coinsurance between attachment point and cap

    US Code of Federal Regulations. Title 45, part 153 (Standards related to reinsurance, risk corridors, and HHS risk adjustment under the Affordable Care Act). eCFR text as at 1 October 2026, read 18 October 2026. Section 153.20 defines the attachment point, the coinsurance rate as the rate of reimbursement for claims costs after the attachment point and before the reinsurance cap, and the reinsurance cap; section 153.230(c) calculates each payment as the national coinsurance rate multiplied by the issuer's claims costs for an individual enrollee in the benefit year between the national attachment point and the national reinsurance cap.

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  • HHS 2016 Payment Notice on the 2015 reinsurance attachment point, cap and coinsurance rate

    US Department of Health and Human Services. Patient Protection and Affordable Care Act; HHS Notice of Benefit and Payment Parameters for 2016. Federal Register. 2015;80:10750 (27 February 2015). Section I, Executive Summary, which finalises the 2016 uniform reinsurance payment parameters of a USD 90,000 attachment point, a USD 250,000 reinsurance cap and a 50 percent coinsurance rate, and decreases the attachment point for the 2015 benefit year from USD 70,000 to USD 45,000 while retaining the USD 250,000 cap and the 50 percent coinsurance rate; and section III.C.3.j, Uniform Reinsurance Payment Parameters for 2015, which finalises that reduction as proposed.

    View source →

  • NAIC white paper on specific attachment points, lasers and the policy year basis

    National Association of Insurance Commissioners. Stop Loss Insurance, Self-Funding and the ACA. White paper. 2015. Section II, which describes a specific attachment point, or retention level, that protects against claim severity for a single covered individual; section III, which describes the laser specific attachment point, a higher attachment point for individuals with preexisting high-cost conditions or other identified risk factors, and states that unless a provision or endorsement provides extended cover the policy reimburses only claims incurred and paid during the same policy year; and section V, which states that stop-loss premiums are developed from an actuary's determination of the expected losses of the self-funded group.

    View source →

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