Expected claims cost per covered person from utilisation and unit cost

Builds the expected claims cost per covered person per year service by service, as the sum over benefit categories of the utilisation rate times the unit cost, and multiplies it by the number of covered persons to give total expected claims for the year. The per-person cost is the net or risk premium for the group.

Signature

C_bar = sum_(i=1)^k [f_i * u_i]; T = N * C_bar
Inputs
InputsDefinitionUnit
f_iExpected number of units of benefit category i (inpatient days, visits, courses, prescriptions) per covered person per year, one value per categoryunits per person per year
u_iCost to the scheme of one unit of category i after any member cost sharing, one value per categorycurrency per unit
NNumber of persons covered by the scheme or rating group over the yearpersons
Output
C_barExpected cost of claims to the scheme for one covered person over one yearcurrency per person per year, for example pounds
TTotal expected claims cost of the covered group for the yearcurrency per year
  • k Number of benefit categories, equal to the length of the f_i and u_i lists (count)

Function

Actuarial claims costing, premium loading and claims reserving function

Maps utilisation rates, unit costs and the size of a covered group to the expected claims cost of a health insurance scheme, turns that expected cost into a gross premium or contribution with loadings, measures how predictable the group's average cost is, and estimates the claims incurred but not yet paid at a reporting date. These are the calculations of the Actuarial Analysis article. The plan generosity measure is covered separately on the Actuarial Value page.

Computational function

  • Computational function: new benefit costing from base use to gross premium in actuarial analysis

    Takes the figures an analyst usually holds when costing a new benefit, a base utilisation rate per 1,000 members, an allowance for induced use, the allowed cost per unit, the member coinsurance rate, the loadings and the membership, and returns the expected claims cost, the gross premium, the scheme totals and the pooled standard deviation of the average cost. It chains the expected claims cost formula HE-FM-ACTA-001, the gross premium formula HE-FM-ACTA-002 and the pooled standard deviation formula HE-FM-ACTA-003, with a two-point (use or no use) model for one member's cost. Its inputs differ from the formulae's variables: the utilisation rate per person and the unit cost to the scheme are built inside the function.

    Inputs and outputs: r_base: Base utilisation in units per 1,000 members a year, observed where members paid in full; required, zero or above. Unit: units per 1,000 members per year.; m: Proportional rise in use under the new cover, for example 0.25 for a quarter; required, zero or above. Unit: proportion.; a: Allowed cost per unit of the benefit; required, zero or above. Unit: currency per unit.; c: Member coinsurance rate as a decimal; required, at least 0 and at most 1. Unit: proportion.; beta: Fixed loading per member; required, zero or above. Unit: currency per member per year.; gamma: Proportional loading as a share of the gross premium; required, at least 0 and below 1. Unit: proportion.; N: Number of members; required, above zero. Unit: persons.; f: Utilisation rate per member under the new cover, an intermediate output. Unit: units per member per year.; u: Cost to the scheme per unit, an intermediate output. Unit: currency per unit.; P: Net premium, the expected claims cost per member. Unit: currency per member per year.; G: Gross premium per member. Unit: currency per member per year.; T: Total expected claims of the scheme. Unit: currency per year.; R_inc: Total gross premium income of the scheme. Unit: currency per year.; sigma: Standard deviation of one member's annual cost under the two-point model. Unit: currency per member per year.; SD_N: Standard deviation of the average cost per member across N members. Unit: currency per member per year.

    Assumption: Each member uses either no unit or one unit of the benefit in the year, with probability f, so f is at most 1 and sigma follows the two-point formula. Members' costs are independent, and the loadings are defined as in HE-FM-ACTA-002. For benefits used more than once a year, sigma is estimated from claims data and passed to HE-FM-ACTA-003 directly.

    Worked example (Physiotherapy benefit for 200,000 members): The article's example: 40 courses per 1,000 members a year rising by a quarter, an allowed cost of £300 with 20% coinsurance and a 10% proportional loading. The net premium is £12, the gross premium about £13.33, expected claims £2.4 million, premium income about £2.67 million and the pooled standard deviation about £0.117. r_base = 40; m = 0.25; a = 300; c = 0.20; beta = 0; gamma = 0.10; N = 200000; f = 0.05; u = 240; P = 12; G = 13.333333; T = 2400000; R_inc = 2666666.67; sigma = 52.306787; SD_N = 0.116962

    Worked example (Physiotherapy benefit without induced use): With no induced use the net premium falls to £9.60 and the gross premium to about £10.67. r_base = 40; m = 0; a = 300; c = 0.20; beta = 0; gamma = 0.10; N = 200000; f = 0.04; u = 240; P = 9.6; G = 10.666667; T = 1920000; R_inc = 2133333.33; sigma = 47.030203; SD_N = 0.105163

    Worked example (Same benefit in a scheme of 2,000 members): The premium is unchanged, but the pooled standard deviation rises to about £1.17, nearly 10% of the net premium. r_base = 40; m = 0.25; a = 300; c = 0.20; beta = 0; gamma = 0.10; N = 2000; f = 0.05; u = 240; P = 12; G = 13.333333; T = 24000; R_inc = 26666.67; sigma = 52.306787; SD_N = 1.169615

    Excel: =(BaseRate/1000*(1+Uplift)*Allowed*(1-Coins)+FixedLoad)/(1-PropLoad) Returns the gross premium in one cell from named cells BaseRate, Uplift, Allowed, Coins, FixedLoad and PropLoad. The net premium is =BaseRate/1000*(1+Uplift)*Allowed*(1-Coins), and the pooled standard deviation is =Allowed*(1-Coins)*SQRT(UseRate*(1-UseRate))/SQRT(Members) with the per-member rate in UseRate.

    R: cost_benefit <- function(r_base, m, a, c, beta, gamma, n) { f <- r_base / 1000 * (1 + m); u <- a * (1-c); p <- f * u; g <- (p + beta) / (1-gamma); sigma <- u * sqrt(f * (1-f)); list(f = f, u = u, P = p, G = g, T = n * p, R_inc = n * g, sigma = sigma, SD_N = sigma / sqrt(n)) } Returns a named list; the arithmetic is vectorised, so r_base or m may hold several scenarios at once.

    Python: def cost_benefit(r_base, m, a, c, beta, gamma, n): f = r_base / 1000 * (1 + m); u = a * (1-c); p = f * u; g = (p + beta) / (1-gamma); sigma = u * math.sqrt(f * (1-f)); return dict(f=f, u=u, P=p, G=g, T=n * p, R_inc=n * g, sigma=sigma, SD_N=sigma / math.sqrt(n)) Uses the math module for single values.

    Test (Gross premium income covers expected claims plus loadings): Premium income less expected claims equals the fixed loading plus the proportional loading on the gross premium, for all members. Expected result: TRUE. Excel check: =ABS(PremiumIncome-ExpectedClaims-Members*(FixedLoad+PropLoad*GrossPremium))<1E-6

    Test (Scheme and members share the allowed cost): The scheme's expected cost per member plus the members' expected coinsurance equals the utilisation rate times the allowed cost. Expected result: TRUE. Excel check: =ABS(NetPremium+UseRate*Allowed*Coins-UseRate*Allowed)<1E-9

    Common error (Pricing at the use observed under full payment): Leaving out induced use prices the benefit at £9.60 per member instead of £12, so premium income falls short of expected claims by about £480,000 a year across 200,000 members before any loading.

    Source: Cichon M, Newbrander W, Yamabana H, Weber A, Normand C, Dror D, Preker A. Modelling in Health Care Finance: A Compendium of Quantitative Techniques for Health Care Financing. Geneva: International Labour Office; 1999. Chapter 5, printed p. 123 (expenditure from utilisation rate, unit cost and covered persons) and Issue Brief 3, section 3.4, printed p. 272 (net premium, fixed and variable cost supplements).

    f = r_base / 1000 * (1 + m); u = a * (1 - c); P = f * u; G = (P + beta) / (1 - gamma); T = N * P; R_inc = N * G; sigma = u * sqrt(f * (1 - f)); SD_N = sigma / sqrt(N)

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Implementations

  • Excel

    Expected claims cost per person and in total with SUMPRODUCT

    With one row per benefit category, the utilisation rates per person in a range named UseRate and the unit costs to the scheme in a range named UnitCost, the first formula returns the expected claims cost per person and the second the total for the number of covered persons in a cell named Members.

    =SUMPRODUCT(UseRate,UnitCost); =Members*SUMPRODUCT(UseRate,UnitCost)

Assumptions

  • Rates and unit costs for the same group, benefits and year

    Each utilisation rate and unit cost refers to the same covered population, benefit definition and year. Rates taken from past claims are adjusted for changes in prices, cover and membership before they describe a new benefit or population.

  • Utilisation under the cover being priced

    The utilisation rate is the one expected under the cover being priced. Use observed under less generous cover understates use under more generous cover, so the rate allows for induced use.

  • Unit cost net of member cost sharing

    The unit cost is the part of the allowed cost that the scheme pays. Where members pay coinsurance, the allowed cost per unit is multiplied by the scheme's share before it enters the sum.

  • Rating group large enough for a reliable average

    The covered group, or each rating group by age and sex, is large enough for its average cost to be a reliable estimate of the expected value. The spread of that average is given by the pooled standard deviation formula on this page.

Worked examples

  • Physiotherapy benefit for 200,000 members with induced use

    The article's example: 50 courses of physiotherapy per 1,000 members a year, or 0.05 per member, at a cost to the scheme of £240 a course (80% of an allowed £300). Expected claims are £12 per member per year and £2.4 million for the scheme.

    f_i = [0.05]; u_i = [240]; k = 1; N = 200000; C_bar = 12; T = 2400000

  • Physiotherapy benefit priced without induced use

    With no allowance for induced use the rate stays at 40 courses per 1,000 members, 0.04 per member, and expected claims fall to £9.60 per member and £1.92 million for the scheme, the lower end of the article's range.

    f_i = [0.04]; u_i = [240]; k = 1; N = 200000; C_bar = 9.6; T = 1920000

  • Physiotherapy benefit priced with induced use of a half

    If more generous cover raises use by a half, to 60 courses per 1,000 members, expected claims rise to £14.40 per member and £2.88 million for the scheme, the upper end of the article's range.

    f_i = [0.06]; u_i = [240]; k = 1; N = 200000; C_bar = 14.4; T = 2880000

Common errors

  • Entering a rate per 1,000 members as a rate per person

    Utilisation is often reported per 1,000 members. Entering 50 courses per 1,000 as f_i = 50 gives an expected cost of £12,000 per member instead of £12, a thousandfold overstatement that is easy to miss in a multi-category sum.

  • Using the allowed cost where members pay coinsurance

    Pricing the physiotherapy course at the allowed £300 rather than the £240 the scheme pays gives £15 per member instead of £12. The extra £3 is the members' coinsurance, which the scheme does not pay.

Sources

  • ILO projection of expenditure from utilisation rate, unit cost and covered persons

    Cichon M, Newbrander W, Yamabana H, Weber A, Normand C, Dror D, Preker A. Modelling in Health Care Finance: A Compendium of Quantitative Techniques for Health Care Financing. Geneva: International Labour Office; 1999. Chapter 5, printed p. 123: the expenditure in each category of care is projected by multiplying a utilisation rate by the unit cost and the number of covered persons.

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  • ILO risk premium as the average per capita claim of a rating group

    Cichon M, Newbrander W, Yamabana H, Weber A, Normand C, Dror D, Preker A. Modelling in Health Care Finance: A Compendium of Quantitative Techniques for Health Care Financing. Geneva: International Labour Office; 1999. Issue Brief 3, section 3.2, printed pp. 267-268: the risk premium as the expected per capita claim, estimated by the average for a large enough group, with claim amounts set by age and sex.

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