Functions & Formulae

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

Medicare Shared Savings Program annual settlement of an accountable care organisation

v(B, E, N, s, q, MSR, c_P, r_L, MLR, rho_R, Rev, rho_B, Q) = (P, SL, SL_E)

Maps an accountable care organisation's per capita benchmark, its assigned beneficiaries' per capita Medicare Parts A and B spending for the performance year and the parameters of its track to the shared savings payment CMS makes to it or the shared losses it repays, as set in 42 CFR 425.605 for the BASIC track and 425.610 for the ENHANCED track. HE-FM-ACO-001 gives the shared savings payment, gated by the minimum savings rate and capped by the performance payment limit, HE-FM-ACO-002 the shared losses of a two-sided BASIC level with its minimum loss rate and loss recoupment limit, and HE-FM-ACO-003 the shared losses of the ENHANCED track, whose loss rate falls as the quality score rises. Each settlement is gated, so the amount jumps from zero once the gap between spending and benchmark reaches the minimum rate and is then shared from the first dollar. This differs from a deductible layer such as the aggregate stop-loss payment HE-FM-ASL-001 (claims above an attachment point, capped at a maximum), the mean and variance of specific stop-loss reimbursement over a claims distribution HE-FM-ATTP-001 or the medical loss ratio rebate HE-FM-ACR-004 (the shortfall of the ratio below the standard times adjusted premium, floored at zero), all continuous in their claims or ratio argument. Only the final min step, a cap on the shared amount, takes the form of HE-FM-ASL-001 with no attachment point. The fee-for-service payments that continue underneath the settlement are not modelled, the per case blend of a fixed amount and a cost share is HE-FM-CTH-001, the margin reading of shared savings as a blend belongs to the blended-payment concept, and the expected profit per member that drives selection, payment minus expected cost, is HE-FM-CSKM-001. Notation follows the Accountable Care Organization article.

  • Shared savings payment of an accountable care organisation with the minimum savings rate gate and payment limit

    g_S = (B - E) / B; S_raw = s * q * N * (B - E); PPL = c_P * N * B; P = (g_S >= MSR) * min(S_raw, PPL)

    Computes the payment CMS makes to an accountable care organisation for a performance year in which its assigned beneficiaries' average per capita spending E falls below the updated benchmark B. The savings ratio g_S must reach the minimum savings rate MSR, and once it does the final sharing rate, s times the quality factor q, applies to all the savings on a first dollar basis, as 42 CFR 425.605(d) and 425.610(e) state. The payment may not exceed the performance payment limit PPL, a share c_P of total benchmark expenditure, 10% in the BASIC track and 20% in ENHANCED; that last step is the cap form of HE-FM-ASL-001 with no attachment point. Because the gate is a test and not a deductible, the payment jumps from zero to s q MSR N B at the MSR when the cap does not bind (derived), USD 2.4 million for the article's Level E ACO. A low revenue BASIC ACO that beats its benchmark but not its MSR is paid under 425.605(h) at half the sharing rate, entered as s halved with MSR set to 0.

  • Shared losses of a two-sided BASIC accountable care organisation with the minimum loss rate and loss recoupment limit

    g_L = (E - B) / B; L_raw = r_L * N * (E - B); LRL = min(rho_R * Rev, rho_B * N * B); SL = (g_L >= MLR) * min(L_raw, LRL)

    Computes what an ACO in BASIC Level C, D or E repays CMS when its per capita spending E exceeds the updated benchmark B. The loss ratio g_L must reach the minimum loss rate MLR, which mirrors the MSR, and the fixed 30% shared loss rate r_L then applies to the whole excess, the loss counterpart of HE-FM-ACO-001. The amount may not exceed the loss recoupment limit LRL, a share rho_R of the ACO participants' Medicare fee-for-service revenue, never more than a share rho_B of total benchmark expenditure, as 42 CFR 425.605(d)(1)(iii) to (v) and (i)(2) set out. The final min step is the cap form of HE-FM-ASL-001 with no attachment point. With the article's case 3 figures a 3% overspend gives USD 2.16 million, the article's result, whenever the limit is at least that amount.

  • Shared losses of an ENHANCED track accountable care organisation with the quality-based loss rate

    r_E = min(max(1 - 0.75 * Q, 0.4), 0.75); g_L = (E - B) / B; SL_E = (g_L >= MLR) * min(r_E * N * (E - B), 0.15 * N * B)

    Computes what an ACO in the ENHANCED track repays CMS when per capita spending E exceeds the benchmark B by at least the MLR. The shared loss rate r_E is 1 minus 75% of the quality score Q, held to at least 40% and at most 75%, and is 75% when the ACO meets neither quality standard, as 42 CFR 425.610(f)(4) sets for performance years from 2024. Entering Q = 0 for an ACO that meets neither standard returns the same 75%, since the ceiling binds for every score up to one third, and the floor of 40% binds for every score of 0.8 or more (derived, checked on a grid of 1,001 scores). Losses apply to the whole excess and may not exceed 15% of total benchmark expenditure under 425.610(g), the cap form of HE-FM-ASL-001 with no attachment point; there is no revenue-based limit in this track.