Functions & Formulae

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

Selection incentive function for payments that do not match expected cost

F(R_i, EC_i, R, m_k, EC_k, N, R_g, C_g) = (pi_i, pi_bar, Pi, OC_g, PR_g)

Maps the payment an insurer or provider receives for a member and the member's expected cost of care to the expected profit or loss on that member, the expected profit per member from the mix of risks a plan enrols, and the group-level over- or undercompensation and predictive ratio that evaluators of payment systems use to find groups likely to be targets of selection. Cream skimming is profitable where these measures differ predictably between groups that a plan can identify and influence. Community rating creates such differences through the premium, as the cross-subsidies on the Community Rating page show (HE-FM-CRAT-002), and the risk equalisation transfers that offset them are on that page (HE-FM-CRAT-003).

  • Expected profit per member under a fixed or risk-adjusted payment

    pi_i = R_i - EC_i

    The payment received for a member, whether a community-rated premium, a capitation payment or a risk-adjusted payment, minus the member's expected cost of care, written pi_i = R_i minus E[C_i] in the article. A plan has an incentive to attract members with a positive expected profit and deter those with a negative one whenever it can tell them apart and influence who joins. Under community rating with no risk equalisation the payment is the premium, and the expected profit equals the cross-subsidy of HE-FM-CRAT-002.

  • Expected profit per member from an enrolled risk mix

    pi_bar = R - sum_(k=1)^K [m_k * EC_k]; Pi = N * pi_bar

    The payment per member minus the average expected cost of the members a plan actually enrols, where the payment is the same for every member of the payment cell and the plan's mix of risk groups may differ from the cell's population. The profit comes entirely from the mix, since the expected cost of each type of member is unchanged. Multiplying by the number of members gives the plan's total expected profit, which is an upper bound on the gain from selection because attracting the mix has costs of its own.

  • Over- or under-compensation and predictive ratio for a risk group

    OC_g = R_g - C_g; PR_g = R_g / C_g

    Two group-level measures of the selection incentive a payment system leaves. Over- or undercompensation is the average payment minus the average cost for the group, in money, and is the usual European measure; the predictive ratio is payment divided by cost, without units, and is the usual US measure. A positive difference or a ratio above 1 indicates overcompensation, and a negative difference or a ratio below 1 undercompensation, so a group with a ratio well below 1 is a likely target for selection. Both are computed over the same members, so the ratio of total payment to total cost equals the ratio of the averages.