Signature
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
| Inputs | Definition | Unit |
|---|---|---|
p_1 | Probability that a member's annual eligible claims are at the first level C_1 | probability |
C_1 | Lowest of the three annual claim levels at current prices | US dollars |
A | Specific attachment point, unchanged in money while claims rise | US dollars |
p_2 | Probability that a member's annual eligible claims are at the second level C_2 | probability |
C_2 | Middle one of the three annual claim levels at current prices | US dollars |
p_3 | Probability that a member's annual eligible claims are at the third level C_3 | probability |
C_3 | Highest of the three annual claim levels at current prices | US dollars |
g | Proportional rise in every claim level over the period, entered as 0.1 for 10%, above zero | proportion |
E_0 | Expected reimbursement per member of the uncapped layer above the attachment point at current claim levels | US dollars per member |
|---|---|---|
E_g | Expected reimbursement per member of the same layer after every claim level is multiplied by 1 plus g | US dollars per member |
rise_E | E_g divided by E_0, less 1 | proportion |
mult_E | Proportional rise in expected layer reimbursement divided by the claims inflation rate, 1 when the layer grows at the same rate as claims | ratio |
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.
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Implementations
Excel
Expected excess layer reimbursement before and after uniform claims inflation in Excel
With ClaimLevels and ClaimProbs as in HE-IM-ATTP-001, the attachment point in AttachPoint and the inflation rate in a cell named ClaimsInflation, the four formulas return the expected reimbursement before inflation in a cell named ExpReimbBase, after inflation in ExpReimbInflated, the proportional rise in LayerRise and its ratio to the inflation rate in InflationMultiple. The comparisons inside SUMPRODUCT return 1 or 0 and keep only the levels above the attachment point.
=SUMPRODUCT(ClaimProbs,(ClaimLevels-AttachPoint)*(ClaimLevels>AttachPoint)); =SUMPRODUCT(ClaimProbs,((1+ClaimsInflation)*ClaimLevels-AttachPoint)*((1+ClaimsInflation)*ClaimLevels>AttachPoint)); =ExpReimbInflated/ExpReimbBase-1; =LayerRise/ClaimsInflation
Assumptions
Uniform claims inflation with the specific attachment point fixed in money
Every claim level rises by the same factor 1 plus g, the simple case that Lee 1988 and Miccolis 1977 analyse, while the attachment point keeps its money value, as a contract or regulatory amount does until it is changed. A contract with an index clause that raises the attachment point with claims is a different case.
Claims below the three levels stay below the specific attachment point after inflation
The remaining probability sits at claims that stay at or below A after inflation. In the article's example claims below USD 50,000 stay below USD 55,000 after 10% inflation and the USD 75,000 level becomes USD 82,500, so at an attachment point of USD 100,000 only the two upper levels pay; at USD 50,000 the formula would need the claims below USD 50,000 written out.
Uncapped layer, positive inflation and a layer that pays before inflation
The layer has no cap, g is above zero and E_0 is above zero, so both ratios are defined. With a cap, part of the inflated claims falls above it; for an illustrative layer between USD 100,000 and USD 150,000 on the article's distribution, 10% inflation leaves the expected reimbursement unchanged at USD 200 per member (derived), because both upper claim levels already exhaust the layer.
Worked examples
Ten per cent claims inflation at a USD 100,000 specific attachment point
The article's step 4. With every claim level 10% higher the USD 150,000 level becomes USD 165,000 and the USD 500,000 level USD 550,000, so the expected reimbursement rises from USD 550 to USD 645 per member, about 17.3%, about 1.73 times the rate of claims inflation.
p_1 = 0.006; C_1 = 75000; A = 100000; p_2 = 0.003; C_2 = 150000; p_3 = 0.001; C_3 = 500000; g = 0.1; E_0 = 550; E_g = 645; rise_E = 0.1727; mult_E = 1.7273
Ten per cent claims inflation at a USD 200,000 specific attachment point
At USD 200,000 only the top level pays, and 10% inflation raises its excess from USD 300,000 to USD 350,000, so the expected reimbursement rises from USD 300 to USD 350 per member, about 16.7%. This figure is computed here from the article's distribution and is not in the article.
p_1 = 0.006; C_1 = 75000; A = 200000; p_2 = 0.003; C_2 = 150000; p_3 = 0.001; C_3 = 500000; g = 0.1; E_0 = 300; E_g = 350; rise_E = 0.1667; mult_E = 1.6667
Claims inflation with a zero attachment point as a limiting case
With an illustrative distribution in which every member claims USD 10,000, USD 20,000 or USD 50,000 with probabilities 0.5, 0.3 and 0.2 and the attachment point set to zero, the layer is the whole claim and grows at exactly the inflation rate, so the ratio is 1. This limiting case checks the implementation.
p_1 = 0.5; C_1 = 10000; A = 0; p_2 = 0.3; C_2 = 20000; p_3 = 0.2; C_3 = 50000; g = 0.1; E_0 = 21000; E_g = 23100; rise_E = 0.1; mult_E = 1
Common errors
Projecting a fixed specific attachment point layer at the general claims inflation rate
Projecting the USD 550 expected reimbursement at a USD 100,000 attachment point forward at the 10% claims inflation rate gives USD 605 per member, USD 40 below the USD 645 that the inflated distribution gives, about USD 40,000 too little for 1,000 members in one year. The gap widens with each further year at a fixed attachment point, to about USD 84 per member after two years of 10% inflation (derived, USD 749.50 against USD 665.50).
Holding a money attachment point or regulatory floor constant across years of claims inflation
A multi-year model in constant prices that keeps a money threshold, such as the USD 20,000 minimum annual attachment point per individual of the NAIC Stop Loss Insurance Model Act, at its face value treats it as if it rose in step with claims. With illustrative claims inflation of 5% a year and the threshold unchanged in money, after five years USD 20,000 corresponds to about USD 15,671 of current claims (derived), and the uncapped layer above it grows faster than claims each year (derived). The model act lets the commissioner amend its dollar amounts upon consideration of the medical components of the Consumer Price Index, and its drafting note says a state may wish to adjust them to reflect medical costs in that state.
Sources
Lee on the effect of inflation on an excess of loss layer with a fixed retention
Lee YS. The mathematics of excess of loss coverages and retrospective rating, a graphical approach. Proceedings of the Casualty Actuarial Society. 1988;75:49-77. Section 3, Trend (pages 61 to 63), which treats inflation as a uniform multiplicative factor on every loss, states that an excess of loss coverage is more seriously affected by inflation and that if the retention remains fixed the expected excess loss payment is increased proportionally much more than indicated by the general rate of inflation (Figure 11), and gives in equation 3.5 the inflated expected basic limits loss as the factor times the earlier expected loss limited at the limit divided by the factor.
Miccolis on trend with limits and retentions fixed while loss severity shifts
Miccolis RS. On the theory of increased limits and excess of loss pricing. Proceedings of the Casualty Actuarial Society. 1977;64:27-59. Section Trend (pages 35 to 37), which notes that limits and retentions remain fixed while loss severity is shifting, models trend as a transformation of loss size with each loss increased by the same multiplicative factor above one, and derives in equation 17 that the expected loss limited at k after trend equals the factor times the earlier expected loss limited at k divided by the factor, with the excess of loss form in equation 19. The uncapped excess form used here follows from equation 17 and the scaling of the mean (derived).
NAIC Stop Loss Insurance Model Act minimum specific attachment point and CPI amendment
National Association of Insurance Commissioners. Stop Loss Insurance Model Act (Model 92). Adopted 1995, amended 1999. Section 3A(1)(a), which bars an insurer from issuing a stop-loss policy with an annual attachment point for claims incurred per individual lower than USD 20,000, section 3A(3), which lets the commissioner amend the dollar amounts upon consideration of the medical components of the Consumer Price Index and publish any change at least six months before it takes effect, and the drafting note that a state may wish to adjust the dollar amounts to reflect medical costs in the particular state.
Canonical Identity
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