This post is a follow-up to ‘mlreserve: machine-learning loss reserving on ‘real-world’ triangles (based on a ‘ChainLadder’ fork)’, with better explanations of what’s being done; a better description of the experimental setting.
We simulate claims triangles whose true outstanding reserve is known, then ask how well two families of methods recover it:
- the Mack Chain-Ladder (
chainladder.MackChainladder), the actuarial benchmark; mlreserving.MLReserving(From Techtonique’s package mlreserving) wrapped around every scikit-learn regressor that can be built with default arguments (no tuning), with two feature sets.
The triangles are deliberately distorted in two ways that break the chain-ladder assumptions: settlement speed-up (recent accident years pay faster) and a calendar-year inflation shock (payments made after a given calendar period grow faster). Section 1 states the setting formally, Section 2 makes the two distortions visible, Section 3 runs the model sweep.
⚠️ Deliberate data leakage. Models are selected on the truth of the evaluation triangles. Section 3.4 re-scores the selected models on fresh triangles, and Section 4 discusses what that check does and does not prove.
1. Formal setting
1.1 Notation
A run-off triangle has accident (origin) years \(i = 1,\dots,n\) and development years \(j = 1,\dots,n\), with \(n = 10\).
The calendar period of cell \((i,j)\) is
\[t = i + j - 1 .\]Let \(X_{ij}\) be the incremental payment of accident year \(i\) in development year \(j\), and \(C_{ij} = \sum_{k \le j} X_{ik}\) the cumulative payment. Only cells with \(t \le n\) are observed:
\[\mathcal{D} = \{X_{ij} : i + j - 1 \le n\} \quad \text{(upper triangle)}, \qquad \mathcal{F} = \{(i,j) : i + j - 1 > n\} \quad \text{(future cells)}.\]The true reserve of accident year \(i\) and the total reserve are
\[R_i = \sum_{j:\,(i,j)\in\mathcal{F}} X_{ij}, \qquad R = \sum_{i=2}^{n} R_i .\]Because we simulate the full \(n \times n\) square, \(R\) is known exactly for every triangle (no tail beyond \(j = n\)).
1.2 Data-generating process (aggregate simulator)
The expected incremental payment factorises into an ultimate, a payment pattern and a calendar index:
\[\boxed{\;\mu_{ij} \;=\; U_i \cdot \pi_{ij} \cdot I_{t}\;}, \qquad t = i+j-1 .\]Ultimates (exposure). A 3% per year volume trend with lognormal noise:
\[U_i = 50\,000\,\bigl(1 + 0.03\,(i-1)\bigr)\,e^{\varepsilon_i}, \qquad \varepsilon_i \overset{iid}{\sim} \mathcal{N}(0,\,0.05^2).\]Payment pattern and settlement speed-up. The payment delay of a claim from accident year \(i\) is \(\mathrm{Gamma}(\kappa = 2,\ \theta_i = m_i/\kappa)\), with mean delay \(m_i\) (in years). Development year \(j\) collects the payments falling in \((j-1, j]\):
\[\pi_{ij} = G_i(j) - G_i(j-1), \qquad G_i = \text{c.d.f. of } \mathrm{Gamma}(2,\ m_i/2).\]The speed-up shortens the mean delay linearly for accident years after \(i_0 = 4\):
\[m_i \;=\; m_0\Bigl(1 - s\,\Bigl(\tfrac{i - i_0}{\,n - i_0\,}\Bigr)_{+}\Bigr), \qquad m_0 = 3,\ \ i_0 = 4,\ \ (x)_+ = \max(x,0).\]So accident years 1–4 share the same pattern, and the mean delay then falls linearly to \(m_0(1-s)\) for the latest year: with \(s = 0.3\), from 3 years to 2.1 years. (Truncating at \(j = n\) drops at most about 1% of the mass, for the slowest years.)
Calendar-year inflation. A 2% per period trend up to calendar period \(\tau = 7\), then growth at rate \(g\):
\[I_t = \exp\bigl(0.02\,\min(t, \tau) \;+\; g\,(t - \tau)_{+}\bigr), \qquad \tau = 7 .\]With \(g = 0.02\) this is a plain 2% trend. With \(g = 0.08\) there is a kink at \(t = 7\). Only three observed diagonals (\(t = 8, 9, 10\)) carry the higher rate, while every future diagonal (\(t = 11,\dots,19\)) does, up to \(I_{19}/I_{19}^{(g=0.02)} = e^{0.06 \times 12} \approx 2.05\).
Noise (over-dispersed Poisson). With dispersion \(\varphi = 5\),
\[X_{ij} = \varphi\, N_{ij}, \qquad N_{ij} \sim \mathrm{Poisson}(\mu_{ij}/\varphi) \quad\Longrightarrow\quad \mathbb{E}[X_{ij}] = \mu_{ij}, \quad \mathrm{Var}[X_{ij}] = \varphi\,\mu_{ij}.\]Scenarios.
| Scenario | speed-up \(s\) | post-\(\tau\) inflation \(g\) |
|---|---|---|
| Baseline (control, used for illustration only) | 0 | 0.02 |
| Inflation only (illustration only) | 0 | 0.08 |
| Settlement speed-up | 0.3 | 0.02 |
| Speed-up + inflation | 0.3 | 0.08 |
We also use two triangles aggregated from the Wang & Wüthrich individual-claims simulator (“Speed-up”, and “All combined” = speed-up + inflation shock + a few large, slowly paid claims). Their generating process is richer than the one above and is not restated here.
1.3 Why these distortions break the Chain-Ladder hypotheses
The Chain-Ladder estimates volume-weighted age-to-age factors from the observed triangle,
\[\hat f_j = \frac{\sum_{i=1}^{n-j} C_{i,j+1}}{\sum_{i=1}^{n-j} C_{ij}}, \qquad \hat C_{in} = C_{i,n+1-i}\prod_{j=n+1-i}^{n-1}\hat f_j, \qquad \hat R_i^{\,CL} = \hat C_{in} - C_{i,n+1-i},\]and is unbiased when the expected development ratio \(\mathbb{E}[C_{i,j+1}]/\mathbb{E}[C_{ij}]\) does not depend on \(i\). Under our DGP (ignoring noise),
\[\frac{\mathbb{E}[C_{i,j+1}]}{\mathbb{E}[C_{ij}]} = \frac{\sum_{k\le j+1}\pi_{ik}\,I_{i+k-1}}{\sum_{k\le j}\pi_{ik}\,I_{i+k-1}} .\]- Baseline (\(s=0\), constant 2% trend). \(\pi_{ik}\) does not depend on \(i\), and \(I_{i+k-1} = e^{0.02(i-1)}e^{0.02k}\) is separable, so the \(i\)-terms cancel and the Chain-Ladder is exactly right in expectation.
- Speed-up. \(\pi_{ik}\) depends on \(i\). Recent accident years are more developed at a given age than the older years the factors \(\hat f_j\) are estimated from. Applying those slow factors to fast years overstates the reserve.
- Inflation kink. \(I_{i+k-1}\) is no longer separable in \((i,k)\). Future diagonals grow at \(g\), while the factors mostly reflect the 2% regime. On its own this understates the reserve, so it partly offsets the speed-up.
1.4 The machine-learning reserving model
mlreserving regresses the arcsinh-transformed incremental payments on features of the cell:
with either
[log]features: \(\mathbf{x}_{ij} = (\log i,\ \log j)\), standardised; or[onehot]features: \(\mathbf{x}_{ij} = (\mathbf{e}_i,\ \mathbf{e}_j)\), one-hot origin and development dummies.
No calendar feature is used. Any regressor \(\hat f\) gives a reserve by back-transforming the predictions of the future cells:
\[\hat R_i = \sum_{j:\,(i,j)\in\mathcal{F}} \sinh\bigl(\hat f(\mathbf{x}_{ij})\bigr), \qquad \hat R = \sum_i \hat R_i .\]1.5 Evaluation and the selection step
For each triangle \(k\) we record the relative error of the total reserve, \(e_k = \hat R_k / R_k - 1\).
For a block \(b\) of triangles (a scenario, or a single Wang & Wüthrich triangle),
\[\mathrm{relRMSE}_b(m) = \Bigl(\tfrac{1}{|b|}\sum_{k\in b} e_k(m)^2\Bigr)^{1/2}, \qquad \mathrm{Score}(m) = \tfrac{1}{4}\sum_{b} \mathrm{relRMSE}_b(m),\]over the four blocks: two simulated scenarios with 20 triangles each, plus the two Wang & Wüthrich triangles.
We keep the top \(K = 5\) models using the true reserves of the very triangles we report on. For the selected models, \(\mathrm{Score}(\hat m;\mathcal{D}_{\text{sel}})\) is an optimistically biased estimate of the risk, because \(\mathbb{E}\bigl[\min_m \widehat{\mathrm{Score}}(m)\bigr] \le \min_m \mathbb{E}\bigl[\widehat{\mathrm{Score}}(m)\bigr]\). Section 3.4 re-scores \(\widehat{\mathcal{M}}\) on \(\mathcal{D}_{\text{fresh}}\), new seeds from the same generator.
0. Setup
%pip install -q mlreserving chainladder scikit-learn matplotlib joblib scipy pandas
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[?25h[31mERROR: pip's dependency resolver does not currently take into account all the packages that are installed. This behaviour is the source of the following dependency conflicts.
google-colab 1.0.0 requires pandas==2.2.3, but you have pandas 3.0.6 which is incompatible.[0m[31m
[0m
import os
import time
import warnings
import chainladder as cl
import matplotlib.pyplot as plt
import numpy as np
import pandas as pd
from joblib import Parallel, delayed
from matplotlib.colors import TwoSlopeNorm
from mlreserving import MLReserving
from scipy.stats import gamma
from sklearn.base import clone
from sklearn.utils import all_estimators
warnings.filterwarnings("ignore")
os.environ["PYTHONWARNINGS"] = "ignore" # also silences joblib workers
pd.set_option("display.width", 160)
plt.rcParams.update({"figure.dpi": 110, "axes.spines.top": False,
"axes.spines.right": False, "font.size": 10})
n = 10 # triangle size
BASE_YEAR = 2000 # chainladder needs calendar-like origins
TAU = 7 # calendar period where the inflation regime changes
M0, I0, KAPPA = 3.0, 4, 2.0 # base mean delay, first sped-up AY, gamma shape
PHI = 5 # ODP dispersion
N_REP = 20 # triangles per scenario
SELECT_SEEDS = range(1, N_REP + 1)
FRESH_SEEDS = range(101, 101 + N_REP)
TOP_K = 5
N_JOBS = -1
SCENARIOS = { # used in the benchmark
"Settlement speed-up": dict(speedup=0.3, inflation=0.02),
"Speed-up + inflation": dict(speedup=0.3, inflation=0.08),
}
ILLUSTRATE = { # used only in Section 2
"Baseline": dict(speedup=0.0, inflation=0.02),
"Inflation only": dict(speedup=0.0, inflation=0.08),
**SCENARIOS,
}
INK, GREY, ACCENT, MACK_C = "#0b0b0b", "#b8b8b8", "#1baf7a", "#eb6834"
SCEN_COLOURS = {"Baseline": "#8a8a8a", "Inflation only": "#7a4fd6",
"Settlement speed-up": "#2a78d6", "Speed-up + inflation": "#eb6834"}
Data-generating process
dgp_components returns the deterministic pieces of Section 1.2: \(m_i\), \(\pi_{ij}\), \(I_t\) and \(\mu_{ij}\), with \(\varepsilon_i = 0\).
simulate_case adds the lognormal ultimate noise and the over-dispersed Poisson noise.
AY = np.arange(1, n + 1)
DEV = np.arange(1, n + 1)
CAL = np.add.outer(AY, DEV) - 1 # t = i + j - 1
FUTURE = CAL > n
def mean_delay(speedup):
return M0 * (1 - speedup * np.maximum(0, (AY - I0) / (n - I0)))
def pattern(speedup):
m = mean_delay(speedup)
return np.array([np.diff(gamma.cdf(np.arange(n + 1), a=KAPPA, scale=mi / KAPPA))
for mi in m]) # pi_ij
def inflation_index(g, t=CAL):
return np.exp(0.02 * np.minimum(t, TAU) + g * np.maximum(t - TAU, 0))
def dgp_components(speedup, inflation, eps=None):
eps = np.zeros(n) if eps is None else eps
U = 5e4 * (1 + 0.03 * (AY - 1)) * np.exp(eps)
pi = pattern(speedup)
I = inflation_index(inflation)
return dict(m=mean_delay(speedup), U=U, pi=pi, I=I, mu=U[:, None] * pi * I)
def make_case(full):
full = np.asarray(full, dtype=float)
cum = np.where(FUTURE, np.nan, np.cumsum(np.where(FUTURE, 0, full), axis=1))
i, j = np.indices((n, n))
long = pd.DataFrame({"origin": i.ravel() + 1, "dev": j.ravel() + 1,
"values": cum.ravel()}).dropna()
long["development"] = long["origin"] + long["dev"] - 1
cl_df = long.assign(origin=long["origin"] + BASE_YEAR,
development=long["development"] + BASE_YEAR)
tri = cl.Triangle(cl_df, origin="origin", development="development",
columns="values", cumulative=True)
return {"full": full, "upper_long": long[["origin", "development", "values"]],
"upper_cl": tri, "true_total": float((full * FUTURE).sum())}
def simulate_case(seed, speedup=0.3, inflation=0.08, phi=PHI):
rng = np.random.default_rng(seed)
eps = rng.normal(0, 0.05, n)
mu = dgp_components(speedup, inflation, eps)["mu"]
return make_case(phi * rng.poisson(mu / phi))
2. Making the distortions visible
2.1 Settlement speed-up: payment delays and patterns
Left: the delay density \(\mathrm{Gamma}(2, m_i/2)\) for the slowest and fastest accident years. Middle: the mean delay \(m_i\) by accident year. Right: the cumulative share paid, \(G_i(j)\), by development year. Each curve is an accident year, and darker means more recent. Recent years under speed-up are further along at every age. The Chain-Ladder averages the old, slow curves.
s = SCENARIOS["Settlement speed-up"]["speedup"]
fig, axes = plt.subplots(1, 3, figsize=(15, 4.3))
x = np.linspace(0, 10, 400)
for mi, lab, col in [(M0, f"AY 1–{I0}: mean {M0:.1f} y", GREY),
(M0 * (1 - s), f"AY {n}: mean {M0*(1-s):.1f} y", SCEN_COLOURS["Settlement speed-up"])]:
axes[0].plot(x, gamma.pdf(x, a=KAPPA, scale=mi / KAPPA), lw=2, color=col, label=lab)
axes[0].axvline(mi, color=col, lw=1, ls=":")
axes[0].set(title="Payment-delay density", xlabel="Delay (years)", ylabel="Density")
axes[0].legend(frameon=False)
axes[1].plot(AY, mean_delay(0), "o-", color=GREY, lw=2, label="No speed-up")
axes[1].plot(AY, mean_delay(s), "o-", color=SCEN_COLOURS["Settlement speed-up"], lw=2,
label=f"Speed-up s = {s}")
axes[1].axvline(I0, color=INK, lw=0.8, ls="--")
axes[1].annotate(f"i₀ = {I0}", (I0 - 0.15, 2.0), ha="right", fontsize=9)
axes[1].set(title="Mean delay mᵢ by accident year", xlabel="Accident year i",
ylabel="Mean delay (years)", xticks=AY, ylim=(1.9, 3.4))
axes[1].legend(frameon=False, loc="upper center", ncol=2)
cum = np.cumsum(pattern(s), axis=1)
cmap = plt.cm.Blues(np.linspace(0.3, 1, n))
for i in range(n):
axes[2].plot(DEV, cum[i], color=cmap[i], lw=1.8, label=f"AY {i+1}" if i in (0, n - 1) else None)
axes[2].plot(DEV, np.cumsum(pattern(0), axis=1)[0], color=INK, lw=1.2, ls="--",
label="AY 1–4 (unchanged)")
axes[2].set(title="Cumulative share paid Gᵢ(j), speed-up", xlabel="Development year j",
ylabel="Share of ultimate paid", xticks=DEV)
axes[2].legend(frameon=False, loc="lower right")
fig.tight_layout(); plt.show()
print("Share paid by end of development year 1 and 2, AY 1 vs AY 10:")
print(pd.DataFrame(cum[[0, -1], :2], index=["AY 1", "AY 10"], columns=["j=1", "j=2"]).round(3))
Share paid by end of development year 1 and 2, AY 1 vs AY 10:
j=1 j=2
AY 1 0.144 0.385
AY 10 0.247 0.568
2.2 Calendar-year inflation: the index \(I_t\)
The shaded area is the future (calendar periods \(t > n\)), where the reserve sits. With \(g = 0.08\) the kink at \(\tau = 7\) leaves only three observed diagonals in the new regime. The right panel is the ratio of the two indices, which is the extra loading that inflation puts on each calendar period.
t = np.arange(1, 2 * n)
fig, axes = plt.subplots(1, 2, figsize=(13, 4.2))
for g, col, lab in [(0.02, GREY, "g = 0.02 (2% throughout)"),
(0.08, SCEN_COLOURS["Speed-up + inflation"], "g = 0.08 after τ = 7")]:
axes[0].plot(t, inflation_index(g, t), "o-", ms=4, lw=2, color=col, label=lab)
ratio = inflation_index(0.08, t) / inflation_index(0.02, t)
axes[1].bar(t, ratio, color=np.where(t > n, SCEN_COLOURS["Speed-up + inflation"], GREY), width=0.7)
for ax in axes:
ax.axvspan(n + 0.5, 2 * n - 0.5, color=INK, alpha=0.06)
ax.axvline(TAU, color=INK, lw=0.8, ls="--")
ax.set_xticks(t)
ax.set_xlabel("Calendar period t = i + j − 1")
axes[0].text(n + 0.7, axes[0].get_ylim()[1] * 0.97, "future diagonals", va="top", fontsize=9)
axes[0].set(title="Calendar index Iₜ", ylabel="Iₜ")
axes[0].legend(frameon=False, loc="upper left")
axes[1].set(title="Extra loading from inflation: Iₜ(g=0.08) / Iₜ(g=0.02)", ylabel="Ratio")
for tt, r in zip(t, ratio):
if tt in (TAU, n, 2 * n - 1):
axes[1].annotate(f"{r:.2f}", (tt, r), ha="center", va="bottom", fontsize=9)
fig.tight_layout(); plt.show()
2.3 Where each distortion hits the triangle
Each heatmap shows \(\log\bigl(\mu_{ij}^{\text{scenario}} / \mu_{ij}^{\text{baseline}}\bigr)\) cell by cell. Blue means less expected payment than the baseline and red means more. The staircase marks the boundary between the observed upper triangle and the future cells.
- Speed-up moves payments of recent accident years to earlier development years: red on the left, blue on the right. The blue cells are almost all in the future, which is the part Chain-Ladder overstates.
- Inflation loads the bottom-right diagonals and leaves the observed upper triangle almost untouched.
base_mu = dgp_components(**ILLUSTRATE["Baseline"])["mu"]
names = ["Settlement speed-up", "Inflation only", "Speed-up + inflation"]
logr = {k: np.log(dgp_components(**ILLUSTRATE[k])["mu"] / base_mu) for k in names}
lim = max(np.abs(v).max() for v in logr.values())
norm = TwoSlopeNorm(vmin=-lim, vcenter=0, vmax=lim)
fig, axes = plt.subplots(1, 3, figsize=(16, 5))
for ax, k in zip(axes, names):
im = ax.imshow(logr[k], cmap="RdBu_r", norm=norm, origin="upper",
extent=(0.5, n + 0.5, n + 0.5, 0.5))
# staircase: right edge of the last observed cell j = n + 1 - i in each row
xs, ys = [], []
for i in range(1, n + 1):
xs += [n + 1.5 - i, n + 1.5 - i]
ys += [i - 0.5, i + 0.5]
ax.plot(xs, ys, color=INK, lw=1.6)
ax.set(title=k, xlabel="Development year j", ylabel="Accident year i",
xticks=DEV, yticks=AY)
cbar = fig.colorbar(im, ax=axes, shrink=0.85, pad=0.015)
cbar.set_label("log( μᵢⱼ scenario / μᵢⱼ baseline )")
fig.suptitle("Where the distortions act (staircase = last observed diagonal)", x=0.01, ha="left")
plt.show()
2.4 What this does to the Chain-Ladder, without noise
We feed the expected triangle \(\mu_{ij}\) (no noise, \(\varepsilon_i = 0\)) to the chain-ladder formulas of Section 1.3 and compare the resulting reserve with the true expected reserve \(\sum_{(i,j)\in\mathcal F}\mu_{ij}\). This isolates the structural bias of Chain-Ladder in each scenario. In the baseline the ratio is exactly 1, as Section 1.3 predicts.
def expected_cl_vs_truth(speedup, inflation):
mu = dgp_components(speedup, inflation)["mu"]
C = np.cumsum(mu, axis=1)
f = np.array([C[: n - j - 1, j + 1].sum() / C[: n - j - 1, j].sum() for j in range(n - 1)])
latest = np.array([C[i, n - 1 - i] for i in range(n)])
cl_ult = np.array([latest[i] * np.prod(f[n - 1 - i:]) for i in range(n)])
return cl_ult - latest, (mu * FUTURE).sum(axis=1)
fig, axes = plt.subplots(1, 2, figsize=(14, 4.5), gridspec_kw={"width_ratios": [2.2, 1]})
totals = {}
for k, par in ILLUSTRATE.items():
cl_res, true_res = expected_cl_vs_truth(**par)
axes[0].plot(AY[1:], cl_res[1:] / true_res[1:], "o-", lw=2, color=SCEN_COLOURS[k], label=k)
totals[k] = cl_res.sum() / true_res.sum()
axes[0].axhline(1, color=INK, lw=0.8)
axes[0].set(title="Chain-ladder reserve / true reserve, by accident year (expected triangle)",
xlabel="Accident year i", ylabel="Ratio", xticks=AY[1:])
axes[0].legend(frameon=False)
k = list(totals)
axes[1].barh(k[::-1], [totals[x] for x in k[::-1]], color=[SCEN_COLOURS[x] for x in k[::-1]], height=0.6)
axes[1].axvline(1, color=INK, lw=0.8)
for y, x in enumerate(k[::-1]):
axes[1].annotate(f" {totals[x]:.2f}", (totals[x], y), va="center", fontsize=9)
axes[1].set(title="Total reserve ratio", xlabel="CL / truth")
fig.tight_layout(); plt.show()
2.5 From expectations to simulated triangles: the spread of the truth and of Mack
With noise added, each scenario gives a distribution of true total reserves \(R\) over seeds. Below, for the 20 selection seeds of each benchmark scenario, are the true total reserves (black) and the Mack estimates (orange). The gap between the two clouds is the chain-ladder bias, and it is much larger than the seed-to-seed noise.
fig, axes = plt.subplots(1, 2, figsize=(14, 4.2), sharey=True)
rng = np.random.default_rng(0)
for ax, (sc, par) in zip(axes, SCENARIOS.items()):
cases = [simulate_case(seed, **par) for seed in SELECT_SEEDS]
truth = np.array([c["true_total"] for c in cases]) / 1000
mack = np.array([float(cl.MackChainladder().fit(c["upper_cl"]).ibnr_.sum()) for c in cases]) / 1000
for y, vals, col, lab in [(1, truth, INK, "True reserve R"), (0, mack, MACK_C, "Mack estimate")]:
ax.scatter(vals, y + rng.uniform(-0.15, 0.15, len(vals)), color=col, s=28, alpha=0.8,
edgecolor="white", linewidth=0.5)
ax.vlines(vals.mean(), y - 0.3, y + 0.3, color=col, lw=2)
ax.annotate(f"mean {vals.mean():,.0f}", (vals.mean(), y + 0.33), ha="center", fontsize=9)
ax.set_yticks([0, 1], ["Mack estimate", "True reserve R"])
ax.set(title=f"{sc} ({N_REP} seeds)", xlabel="Total reserve (millions)", ylim=(-0.6, 1.6))
fig.tight_layout(); plt.show()
3. The benchmark: untuned scikit-learn regressors in mlreserving
3.1 Evaluation triangles
squares = {
"Speed-up": [4172.1, 16153.4, 14800.2, 7561.1, 3275, 1856.3, 726.7, 410.7, 35.8, 140, 4983.2, 18064.1, 13574.5, 8350.7, 3954.9, 1499, 510.3, 9.5, 0, 11.3, 4698.1, 19428.8, 14864, 7343, 3710.5, 1378.6, 384.7, 92.2, 6.8, 27.1, 5467.1, 18481, 14847.6, 8089.7, 2794.1, 1379.2, 549.5, 152.2, 6.8, 0, 5985.2, 21516.6, 15980.6, 8353, 2954.2, 1655.5, 636.4, 191.3, 211.3, 28.9, 8274.1, 22014.4, 15981.4, 7425.8, 2362.7, 731, 242.1, 41.6, 52.8, 0, 8148.4, 24712.9, 16329.1, 6587.5, 2804.9, 753.6, 49.5, 0, 0, 0, 9974.2, 27573.4, 14523, 5141.6, 1123.8, 350, 73.3, 0, 0, 0, 10698.8, 29877.8, 14303.3, 3200.3, 978.3, 239.1, 0, 0, 0, 0, 13669.1, 30541.7, 10984, 1837.5, 157.3, 0, 0, 0, 0, 0],
"All combined": [4236.8, 16766.9, 15671.5, 8229, 3582.8, 2143.9, 838.8, 548.6, 57.6, 261.6, 5195.7, 19008.8, 15926.9, 9857.6, 4643.8, 1746.5, 712.4, 26.4, 0, 22.5, 4956.8, 20861.6, 16283.4, 8704.2, 4321.7, 1876.2, 666.5, 161.8, 14.7, 61.4, 5908.2, 20336.4, 16580.1, 9331.1, 3673.8, 2276.2, 1205.8, 332.2, 15.4, 0, 6586.1, 24045.6, 18499, 11218.3, 4576, 3174.8, 1307, 460.6, 665, 96.3, 9292, 26084.3, 21055.7, 11396.7, 4203.8, 1531.8, 585.4, 115, 163.7, 0, 9702.9, 33010, 25202.7, 12133, 6404.4, 1956.8, 385.9, 0, 0, 0, 13752.5, 42686.8, 25857.7, 10698.4, 2706.5, 3213, 4232.5, 0, 0, 0, 17094.3, 53962.4, 30132.7, 8527.6, 3087.9, 746.2, 0, 0, 0, 0, 25375.8, 63473.7, 26195.3, 5737.2, 507, 674.2, 0, 0, 0, 0],
}
def build_cases(seeds):
return [(sc, seed, simulate_case(seed, **par))
for sc, par in SCENARIOS.items() for seed in seeds]
select_cases = build_cases(SELECT_SEEDS) + [
(f"W&W: {k}", 0, make_case(np.reshape(v, (n, n)))) for k, v in squares.items()]
fresh_cases = build_cases(FRESH_SEEDS)
print(f"selection set: {len(select_cases)} triangles, fresh set: {len(fresh_cases)} triangles")
selection set: 42 triangles, fresh set: 40 triangles
3.2 Candidates and scoring
Every scikit-learn regressor that builds with default arguments, under both feature sets \(\times\) [log], [onehot].
Wrappers and multi-output-only estimators are excluded. A pair that errors or returns a non-finite reserve on any
triangle is dropped. Residuals are taken in-sample (residual_source="in_sample"); only point estimates are needed here.
EXCLUDE = {"CCA", "PLSCanonical", "IsotonicRegression", "MultiOutputRegressor",
"RegressorChain", "StackingRegressor", "VotingRegressor",
"MultiTaskElasticNet", "MultiTaskElasticNetCV", "MultiTaskLasso",
"MultiTaskLassoCV", "TransformedTargetRegressor"}
def candidates():
out = {}
for name, Est in all_estimators(type_filter="regressor"):
if name in EXCLUDE:
continue
try:
est = Est()
except Exception:
continue
if "random_state" in est.get_params():
est.set_params(random_state=1)
out[name] = est
return out
def total_ibnr_ml(est, case, use_factors):
m = MLReserving(model=clone(est), use_factors=use_factors,
residual_source="in_sample", random_state=1)
m.fit(case["upper_long"], origin_col="origin", development_col="development",
value_col="values", cumulated=True)
m.predict()
return float(m.get_ibnr().mean.sum())
def total_ibnr_mack(case):
return float(cl.MackChainladder().fit(case["upper_cl"]).ibnr_.sum())
def score_one(label, est, use_factors, cases):
import warnings; warnings.filterwarnings("ignore")
rows = []
for sc, seed, case in cases:
try:
pred = total_ibnr_mack(case) if est is None else total_ibnr_ml(est, case, use_factors)
except Exception:
return []
if not np.isfinite(pred):
return []
rows.append({"model": label, "scenario": sc, "seed": seed,
"error": pred - case["true_total"],
"rel_error": pred / case["true_total"] - 1})
return rows
def summarise(df):
g = df.groupby(["model", "scenario"])
s = g["error"].agg(bias="mean", RMSE=lambda e: np.sqrt(np.mean(e ** 2)))
s["relRMSE"] = g["rel_error"].agg(lambda e: np.sqrt(np.mean(e ** 2)))
return s.reset_index()
def pivot_rel(s):
p = s.pivot(index="model", columns="scenario", values="relRMSE")
p["mean relRMSE"] = p.mean(axis=1)
return p.sort_values("mean relRMSE")
cands = candidates()
jobs = [("Mack Chain-Ladder", None, False)] + [
(f"{name} [{'onehot' if f else 'log'}]", est, f)
for name, est in cands.items() for f in (False, True)]
print(f"{len(jobs) - 1} (estimator, feature-set) pairs + Mack")
86 (estimator, feature-set) pairs + Mack
3.3 Sweep on the selection set (≈ 2–3 minutes on 2 cores)
t0 = time.time()
res = Parallel(n_jobs=N_JOBS)(delayed(score_one)(lbl, est, f, select_cases) for lbl, est, f in jobs)
sel = pd.DataFrame([r for rr in res for r in rr])
print(f"done in {time.time() - t0:.0f}s; {sel['model'].nunique() - 1} pairs ran without error")
sel_rank = pivot_rel(summarise(sel))
mack_rank = list(sel_rank.index).index("Mack Chain-Ladder") + 1
print(f"Mack Chain-Ladder rank: {mack_rank} / {len(sel_rank)}\n")
print("Relative RMSE of the total reserve (W&W columns are single triangles, i.e. |relative error|):")
sel_rank.round(3).head(20)
done in 516s; 82 pairs ran without error
Mack Chain-Ladder rank: 60 / 83
Relative RMSE of the total reserve (W&W columns are single triangles, i.e. |relative error|):
| scenario | Settlement speed-up | Speed-up + inflation | W&W: All combined | W&W: Speed-up | mean relRMSE |
|---|---|---|---|---|---|
| model | |||||
| MLPRegressor [onehot] | 0.039 | 0.127 | 0.229 | 0.022 | 0.104 |
| AdaBoostRegressor [onehot] | 0.211 | 0.100 | 0.241 | 0.182 | 0.183 |
| ExtraTreeRegressor [onehot] | 0.139 | 0.069 | 0.159 | 0.395 | 0.191 |
| ExtraTreesRegressor [onehot] | 0.139 | 0.070 | 0.171 | 0.404 | 0.196 |
| DecisionTreeRegressor [onehot] | 0.140 | 0.069 | 0.176 | 0.430 | 0.204 |
| ExtraTreeRegressor [log] | 0.345 | 0.109 | 0.052 | 0.340 | 0.211 |
| BaggingRegressor [onehot] | 0.177 | 0.046 | 0.215 | 0.418 | 0.214 |
| DecisionTreeRegressor [log] | 0.137 | 0.337 | 0.052 | 0.340 | 0.217 |
| RandomForestRegressor [onehot] | 0.206 | 0.046 | 0.171 | 0.458 | 0.220 |
| OrthogonalMatchingPursuitCV [onehot] | 0.090 | 0.158 | 0.202 | 0.437 | 0.222 |
| RandomForestRegressor [log] | 0.190 | 0.142 | 0.183 | 0.397 | 0.228 |
| BaggingRegressor [log] | 0.194 | 0.128 | 0.185 | 0.408 | 0.229 |
| AdaBoostRegressor [log] | 0.178 | 0.138 | 0.069 | 0.561 | 0.237 |
| OrthogonalMatchingPursuit [log] | 0.067 | 0.168 | 0.590 | 0.317 | 0.286 |
| SGDRegressor [log] | 0.581 | 0.349 | 0.144 | 0.164 | 0.310 |
| GradientBoostingRegressor [log] | 0.212 | 0.155 | 0.376 | 0.520 | 0.316 |
| ARDRegression [log] | 0.655 | 0.425 | 0.192 | 0.003 | 0.319 |
| ExtraTreesRegressor [log] | 0.305 | 0.232 | 0.265 | 0.492 | 0.324 |
| HistGradientBoostingRegressor [log] | 0.532 | 0.281 | 0.005 | 0.525 | 0.336 |
| LassoCV [log] | 0.677 | 0.432 | 0.174 | 0.104 | 0.347 |
picks = [m for m in sel_rank.index if m != "Mack Chain-Ladder"][:TOP_K]
print("Picked:", picks)
# Top 40 pairs, plus Mack appended below a gap if it ranks outside the top 40
TOP_N = 40
score = sel_rank["mean relRMSE"]
top = score.head(TOP_N)
labels = list(top.index)
values = list(top.values)
ypos = list(range(len(labels)))
if "Mack Chain-Ladder" not in top.index:
labels.append(f"Mack Chain-Ladder (rank {mack_rank} / {len(score)})")
values.append(score["Mack Chain-Ladder"])
ypos.append(len(top) + 1) # leave one empty row as a gap
colors = [MACK_C if m.startswith("Mack") else ACCENT if m in picks else GREY for m in labels]
fig, ax = plt.subplots(figsize=(10, 10.5))
ax.barh(ypos, values, color=colors, height=0.7)
ax.set_yticks(ypos, labels)
ax.invert_yaxis()
if ypos[-1] > len(top):
ax.axhline(len(top), color=INK, lw=0.8, ls=":")
ax.annotate(f"… {mack_rank - TOP_N - 1} pairs not shown …", (0.5, len(top)),
xycoords=("axes fraction", "data"), ha="center", va="center",
fontsize=9, color=INK, backgroundcolor="white")
ax.set_xscale("log")
ax.set(title=f"{TOP_N} best (estimator, features) pairs on the selection set, and Mack\n"
"Score = mean relative RMSE of total reserve, log scale (green: picked, orange: Mack)",
xlabel="Score")
ax.grid(axis="x", alpha=0.25)
fig.tight_layout(); plt.show()
Picked: ['MLPRegressor [onehot]', 'AdaBoostRegressor [onehot]', 'ExtraTreeRegressor [onehot]', 'ExtraTreesRegressor [onehot]', 'DecisionTreeRegressor [onehot]']
3.4 Re-scoring the picks on fresh triangles
These are new seeds (101–120) from the same two scenarios. No model was selected on them.
pick_jobs = [j for j in jobs if j[0] in picks or j[0] == "Mack Chain-Ladder"]
res = Parallel(n_jobs=N_JOBS)(delayed(score_one)(lbl, est, f, fresh_cases) for lbl, est, f in pick_jobs)
fresh = pd.DataFrame([r for rr in res for r in rr])
fresh_rank = pivot_rel(summarise(fresh))
sim_cols = list(SCENARIOS)
compare = pd.DataFrame({
"selection set": sel_rank.loc[picks + ["Mack Chain-Ladder"], sim_cols].mean(axis=1),
"fresh triangles": fresh_rank.loc[picks + ["Mack Chain-Ladder"], "mean relRMSE"],
}).sort_values("selection set")
fig, ax = plt.subplots(figsize=(10, 4.8))
y = np.arange(len(compare))
ax.barh(y + 0.2, compare["selection set"], height=0.38, color=GREY, label="Selection set (models chosen here)")
ax.barh(y - 0.2, compare["fresh triangles"], height=0.38, color=INK, label="Fresh triangles (seeds never used)")
ax.set_yticks(y, compare.index); ax.invert_yaxis()
ax.set(title="Does the advantage survive on triangles not used for selection?",
xlabel="Mean relative RMSE over the two simulated scenarios (lower is better)")
ax.legend(frameon=False, loc="upper center", bbox_to_anchor=(0.5, -0.16), ncol=2)
ax.grid(axis="x", alpha=0.25)
fig.tight_layout(); plt.show()
compare.round(3)
| selection set | fresh triangles | |
|---|---|---|
| model | ||
| MLPRegressor [onehot] | 0.083 | 0.079 |
| ExtraTreeRegressor [onehot] | 0.104 | 0.105 |
| ExtraTreesRegressor [onehot] | 0.105 | 0.106 |
| DecisionTreeRegressor [onehot] | 0.105 | 0.105 |
| AdaBoostRegressor [onehot] | 0.155 | 0.177 |
| Mack Chain-Ladder | 0.551 | 0.549 |
# Error distribution of the picks vs Mack on the fresh triangles
fig, axes = plt.subplots(1, 2, figsize=(14, 4.5), sharey=True)
order = list(compare.index)
for ax, sc in zip(axes, SCENARIOS):
d = fresh[fresh["scenario"] == sc]
data = [d.loc[d["model"] == m, "rel_error"].values * 100 for m in order]
bp = ax.boxplot(data, vert=False, widths=0.55, patch_artist=True,
medianprops=dict(color=INK, lw=1.5))
for patch, m in zip(bp["boxes"], order):
patch.set_facecolor(MACK_C if m == "Mack Chain-Ladder" else ACCENT)
patch.set_alpha(0.75)
ax.axvline(0, color=INK, lw=0.8)
ax.set_yticks(range(1, len(order) + 1), order)
ax.set(title=f"{sc}: fresh triangles", xlabel="Relative error of total reserve (%)")
axes[0].invert_yaxis()
fig.tight_layout(); plt.show()
4. Reading the results
- Chain-Ladder. As Section 2.4 predicts, the speed-up makes Mack over-reserve heavily. The inflation kink pulls the other way but only partly offsets it, so Mack sits near the bottom of the ranking in both scenarios.
- What “works”. The top of the ranking is dominated by
[onehot]features. Origin dummies let a learner absorb part of the speed-up as an accident-year effect, and non-linear learners (MLP, trees) are not tied to the multiplicative origin × development structure that the Chain-Ladder shares with linear models. - What the fresh-seed check shows. The scores of the picks barely move on new seeds, so the selection is not overfitting individual seeds. It is not evidence that the selection generalises. The fresh triangles come from the same generator with the same \(s\), \(g\), \(\tau\) and \(i_0\). Selecting on the truth tuned the model choice to these distortions. A fair test would change the data-generating process: no distortion (the Baseline), a slow-down (\(s < 0\)), a different \(\tau\), or the individual-claims simulator with other settings.
- Where the picks still miss. In the boxplot of Section 3.4, the learners over-reserve under speed-up alone and under-reserve once inflation is added. None of them sees a calendar feature, so the future growth of \(I_t\) (Section 2.2) is extrapolated from the development pattern rather than modelled. The selected models look good partly because the two errors roughly cancel in the score.
- Fragile pairs. Some linear models on
[onehot]features (e.g.Lars) blow up after the \(\sinh\) back-transform, with near-singular fits giving astronomically large reserves. Clip or exclude them before averaging ranks.
