Example 5: triangle to indicated change¶

The indication workflow end to end, with the uncertainty carried through: develop a claims triangle to ultimate with Mack standard errors (actuarialpy), restate premium at current rate level with exact parallelogram on-level factors, assemble the experience worksheet, and read off the indicated change — then ask how much the development uncertainty moves the answer. Every number on this page is the output of this exact run, pinned by a regression test in the ratingmodels suite.

The triangle, to ultimate with error bars¶

Three origin years of cumulative incurred, developed volume-weighted:

import numpy as np
import pandas as pd
from actuarialpy.reserving import ChainLadder
import ratingmodels as rm

triangle = pd.DataFrame(
    {12: [4_120_000.0, 4_390_000.0, 4_610_000.0],
     24: [5_230_000.0, 5_650_000.0, np.nan],
     36: [5_490_000.0, np.nan, np.nan]},
    index=pd.Index([2022, 2023, 2024], name="origin"),
)
cl = ChainLadder.fit(triangle)
cl.age_to_age            # 12->24: 1.2785, 24->36: 1.0497
mack = cl.mack_standard_errors(triangle)

origin	latest	ultimate	ibnr	se
2022	5,490,000	5,490,000	0	0
2023	5,650,000	5,930,880	280,880	87,959
2024	4,610,000	6,186,869	1,576,869	115,787
Total	15,750,000	17,607,748	1,857,748	171,830

A total reserve of 1.86m with a standard error of 172k — a CV of about 9%. That last number is new information the point estimate never carried, and it comes back at the end of the page.

Premium, restated¶

Two rate changes during the experience window; annual policies, so the transitions are parallelograms, computed in closed form:

olf = rm.on_level_factors(
    periods=[("2023-01-01", "2023-12-31"), ("2024-01-01", "2024-12-31")],
    rate_changes=[("2023-07-01", 0.08), ("2024-04-01", 0.05)],
    policy_term=1.0,
)
olf["on_level_factor"]   # CY2023: 1.1227, CY2024: 1.0448

The worksheet¶

ExperienceExhibit composes what the two packages produced — every adjustment a visible column:

proj = cl.project(triangle).loc[[2023, 2024]]
ex = rm.ExperienceExhibit(
    earned_premium=[7_450_000.0, 7_980_000.0],
    losses=proj["latest"].to_numpy(),
    on_level_factors=olf["on_level_factor"].to_numpy(),
    development_factors=proj["development_factor"].to_numpy(),
    trend_factors=[1.045**2, 1.045],     # trend to the rating period
    period_labels=["CY2023", "CY2024"],
)
ex.exhibit()

period	on_level_premium	adjusted_losses	loss_ratio
CY2023	8,364,027	6,476,669	0.7743
CY2024	8,337,712	6,465,278	0.7754

Weighted experience loss ratio: 0.7749.

The indication¶

The exhibit’s totals wire straight into RateIndication; the gross-up is the same RetentionLoad algebra as everywhere else in the package:

retention = rm.RetentionLoad(variable_expense_ratio=0.11,
                             profit_margin=0.03, lae_ratio=0.05)
ind = ex.to_indication(manual_loss_cost=395.0, credibility=0.7,
                       current_rate=455.0, exposure=33_600.0,
                       retention=retention)
ind.indicated_rate()          # 473.87
ind.indicated_rate_change()   # +4.15%

What the development uncertainty is worth¶

Re-run the exhibit with each origin’s ultimate shifted by one Mack standard error (the shifted ultimates are the developed losses, so the development factors drop out):

for shift in (-1.0, +1.0):
    bumped = proj["ultimate"].to_numpy() + shift * mack.loc[[2023, 2024], "se"].to_numpy()
    ex_s = rm.ExperienceExhibit(
        earned_premium=[7_450_000.0, 7_980_000.0], losses=bumped,
        on_level_factors=olf["on_level_factor"].to_numpy(),
        trend_factors=[1.045**2, 1.045],
        period_labels=["CY2023", "CY2024"])
    ex_s.to_indication(395.0, 0.7, 455.0, 33_600.0,
                       retention=retention).indicated_rate_change()

The +4.15% indication becomes +2.93% at −1σ and +5.36% at +1σ. One standard error of development uncertainty is worth about ±1.2 points of rate — which is the difference between a routine filing and a fight, and now it is a number on the exhibit instead of a feeling.