Example 10: the pinned ratio, two ways

Some experience-rated contracts fix a loss ratio and solve the premium from it. That inverts the usual build-up — the ratio is the input, the rate is the output, and the renewal “selection” is formulaic: the one case where the indication is the action, because the contract says so. This page runs the two standard pins side by side on the same block. Under the gross pin, claims / premium = LR*, premium is a function of the claims projection alone and expenses are exhibit-only. Under the net pin, claims / (premium − expenses) = LR*, the dollar expenses are projected before the solve and feed the premium. Same nominal 0.85, materially different deals — including one that loses money.

The constructors here arrived in ratingmodels 0.7.3. Every number on this page is the output of this exact deterministic run, pinned by regression tests in the projectionmodels and ratingmodels suites.

The pins, as algebra

Both are parameterizations of the fundamental insurance equation. The gross pin leaves no degree of freedom for expenses:

\[ \frac{C}{P} = \text{LR}^* \;\Longrightarrow\; P = \frac{C}{\text{LR}^*}. \]

The net pin nets expenses \(E = F + VP\) out of the denominator — the dollar loads \(F\) ride the numerator, the percent-of-premium loads \(V\) cannot be “projected first” (they are circular in \(P\)) and divide instead:

\[ \frac{C}{P - F - VP} = \text{LR}^* \;\Longrightarrow\; P = \frac{C/\text{LR}^* + F}{1 - V}. \]

One consequence worth holding onto: the net pin’s margin is claims-proportional, \(P - E - C = C\,(1-\text{LR}^*)/\text{LR}^*\), while the gross pin makes expenses and margin fight over the same \(P(1-\text{LR}^*)\).

The claims, projected

Twelve months of mature experience, two claim types, type-keyed trend. The contract pins the annual ratio, so the contract-year grain — exposure weighted — is the one that matters.

import numpy as np
import pandas as pd
from actuarialpy import Experience
import projectionmodels as pm
import ratingmodels as rm

KEYS = ["group_id"]
RENEWAL = pd.Timestamp("2027-07-01")
horizon = pm.ProjectionHorizon("2026-07-01", periods=24)
exposure = pd.DataFrame({
    "group_id": "A",
    "projection_period": pd.period_range("2026-07", periods=24, freq="M").astype(str),
    "member_months": np.linspace(1_000.0, 1_046.0, 24).round(0),
})

months = pd.date_range("2025-07-01", periods=12, freq="MS")
history = pd.DataFrame(
    [{"group_id": "A", "claim_type": ct, "incurred_month": m,
      "reported_claims": base * (1 + tr) ** (i / 12) * 1_000.0,
      "member_months": 1_000.0}
     for ct, base, tr in (("inpatient", 260.0, 0.075),
                          ("outpatient", 190.0, 0.060))
     for i, m in enumerate(months)])

experience = Experience(
    history, expense="reported_claims", exposure="member_months",
    date="incurred_month", dimensions=[*KEYS, "claim_type"])
claim_trend = pm.TrendAssumption.from_values(
    "claim_trend",
    pd.DataFrame({"claim_type": ["inpatient", "outpatient"],
                  "annual_trend": [0.075, 0.060]}),
    lookup=["claim_type"], rate_col="annual_trend")

claims = pm.project(
    experience, exposure=exposure, exposure_col="member_months",
    horizon=horizon, trend=claim_trend,
).project().summarize(by=["group_id", "projection_period"],
                      measures=["member_months", "projected_claims"])
claims["cy"] = np.where(claims["projection_period"] < "2027-07", "CY1", "CY2")

annual = claims.groupby("cy")[["member_months", "projected_claims"]].sum()
c1, c2 = (annual["projected_claims"] / annual["member_months"])[["CY1", "CY2"]]
#  c1 = 497.3488, c2 = 531.5452 — blended claims trend +6.8757%

The expenses that don’t need premium

Three of the four expense bases are premium-independent, and all three are in this retention: a $25 flat fee held flat by a zero trend keyed to its expense type, a care-management fee trending at 3%, and a hospital surcharge at 8% of projected claims. The surcharge is percent_claims — the claims table joins per period, so each month’s surcharge is 8% of that month’s claims — and being claims-proportional makes it premium-independent too. Only the commission, a percent of premium, must wait.

def expense_trend(pairs):
    types, rates = map(list, zip(*pairs, strict=True))
    return pm.TrendAssumption.from_values(
        "expense_trend",
        pd.DataFrame({"expense_type": types, "expense_trend": rates}),
        lookup="expense_type")

dollar = pm.ExpenseProjection(
    pd.DataFrame({"group_id": ["A"] * 3,
                  "expense_type": ["admin_fee", "care_mgmt", "hcra_surcharge"],
                  "base_value": [25.0, 8.0, 0.08],
                  "basis": ["per_exposure", "per_exposure", "percent_claims"],
                  "base_date": pd.Timestamp("2026-07-01")}),
    projection_keys=KEYS, expense_type_col="expense_type",
    base_value_col="base_value", basis_col="basis", base_date_col="base_date",
    horizon=horizon,
    trend=expense_trend([("admin_fee", 0.0),        # contractually flat
                         ("care_mgmt", 0.03),
                         ("hcra_surcharge", 0.0)]),  # the 8% is a rate
    exposure=exposure, exposure_col="member_months",
    claims=claims[["group_id", "projection_period", "projected_claims"]],
).project().summarize(by=["group_id", "projection_period"],
                      measures=["projected_expense"])
dollar["cy"] = np.where(dollar["projection_period"] < "2027-07", "CY1", "CY2")

f1, f2 = (dollar.groupby("cy")["projected_expense"].sum()
          / annual["member_months"])[["CY1", "CY2"]]
#  f1 = 72.9073, f2 = 75.8866 per member-month

The solve, per contract

The only stage where the two contracts differ. Full credibility, because the contract has already made the selection; the RateIndication call is identical either way — retention= absorbs the pin.

gross = rm.RetentionLoad.from_gross_loss_ratio(
    0.85, variable_items={"commission": 0.03})
net_cy1 = rm.RetentionLoad.from_net_loss_ratio(
    0.85, fixed_expense=f1, variable_items={"commission": 0.03})
net_cy2 = rm.RetentionLoad.from_net_loss_ratio(
    0.85, fixed_expense=f2, variable_items={"commission": 0.03})

def solve(contract_cy1, contract_cy2):
    rate1 = contract_cy1.gross_rate(c1)
    action = rm.RateIndication(
        experience_loss_cost=c2, manual_loss_cost=c2, credibility=1.0,
        current_rate=rate1, retention=contract_cy2,
    ).indicated_rate_change()
    return rate1, action

gross_rate1, gross_action = solve(gross, gross)
#  585.1162, +6.8757% — the action IS the claims trend, identically
net_rate1, net_action = solve(net_cy1, net_cy2)
#  678.3747, +6.5667% — the slower expense trend dilutes the action

The gross action reproducing the claims trend is not a coincidence, it is the contract: \(P_2/P_1 = C_2/C_1\) when both are \(C/\text{LR}^*\). The net action sits 31 basis points lower because \(F\) trends at roughly 1%–3% while claims trend at 6.9% — expense discipline moderates the renewal, mechanically.

Premium under the action, and the booking view

Stages three and four never change between pins. Premium projects under the selected action; then the full expense schedule — commission now included — runs against the projected premium for the view the year will book.

def project_premium(rate1, action):
    return pm.PremiumProjection(
        premium_data=pd.DataFrame({"group_id": ["A"],
                                   "current_premium_rate": [rate1],
                                   "renewal_date": [RENEWAL]}),
        projection_keys=KEYS, exposure=exposure, exposure_col="member_months",
        horizon=horizon,
        rate_actions=pm.RenewalRateActions(
            frame=pd.DataFrame({"group_id": ["A"], "effective_date": [RENEWAL],
                                "rate_action": [action]}),
            projection_keys=KEYS),
    ).project().summarize(by=["group_id", "projection_period"],
                          measures=["premium"])

def project_booking(premium):
    return pm.ExpenseProjection(
        pd.DataFrame({"group_id": ["A"] * 4,
                      "expense_type": ["admin_fee", "care_mgmt",
                                       "hcra_surcharge", "commission"],
                      "base_value": [25.0, 8.0, 0.08, 0.03],
                      "basis": ["per_exposure", "per_exposure",
                                "percent_claims", "percent_premium"],
                      "base_date": pd.Timestamp("2026-07-01")}),
        projection_keys=KEYS, expense_type_col="expense_type",
        base_value_col="base_value", basis_col="basis", base_date_col="base_date",
        horizon=horizon,
        trend=expense_trend([("admin_fee", 0.0), ("care_mgmt", 0.03),
                             ("hcra_surcharge", 0.0), ("commission", 0.0)]),
        exposure=exposure, exposure_col="member_months",
        claims=claims[["group_id", "projection_period", "projected_claims"]],
        premium=premium[["group_id", "projection_period", "premium"]],
    ).project().summarize(by=["group_id", "projection_period"],
                          measures=["projected_expense"])

The exhibit

Aggregate to the contract year, check the ratio the contract actually pins, and let the margin say what kind of deal was signed.

def exhibit(pin, rate1, action):
    premium = project_premium(rate1, action)
    booking = project_booking(premium)
    for frame in (premium, booking):
        frame["cy"] = np.where(frame["projection_period"] < "2027-07", "CY1", "CY2")
    C = annual["projected_claims"]
    P = premium.groupby("cy")["premium"].sum()
    E = booking.groupby("cy")["projected_expense"].sum()
    ratio = C / P if pin == "gross" else C / (P - E)
    margin = P * 0.15 - E if pin == "gross" else P - E - C
    return ratio.round(4), margin.round(0)

exhibit("gross", gross_rate1, gross_action)
#  ratio 0.8500 / 0.8500     margin  −32,676 / −10,495
exhibit("net", net_rate1, net_action)
#  ratio 0.8500 / 0.8500     margin  1,064,794 / 1,165,022

	gross pin	net pin
in-force rate	585.12	678.37
renewal action	+6.88%	+6.57%
contract ratio, CY1 / CY2	0.8500 / 0.8500	0.8500 / 0.8500
margin, CY1	−32,676	1,064,794
margin, CY2	−10,495	1,165,022

Both contracts hold their ratio to the fourth decimal — annually. Monthly, the ratio wobbles as trend accrues against a rate that steps only at renewal; asserting the pin at any finer grain than the contract’s own is a false alarm generator. The margins are the story: the same nominal 0.85 is a 9% premium difference and, with a hospital surcharge riding the expense stack, the gross-pinned deal is underwater — fifteen retention points cannot cover roughly ninety per-member-month of expenses. The gross pin never consults the expense projection when setting premium, which is exactly why the exhibit has to; the net pin prices the expenses in and banks a claims-proportional margin. Which ratio a contract pins is not a bookkeeping detail. It is the deal.