PTF Residuals

import chainladder as cl
import pandas as pd
import numpy as np

This example replicates the diagnostic residuals from Barnett and Zehnwirth’s “Best Estimates for Reserves” paper in which they describe the Probabilistic Trend Family (PTF) model. With the “ABC” triangle, they show that the basic chainladder, which ignores trend along the valuation axis, fails to have iid weighted standardized residuals along the valuation of the Triangle.

We fit a “diagnostic” model that deliberately ignores modeling the valuation vector. This is done by specifying the patsy formula C(origin)+C(development), which fits origin and development as categorical features. The data is adjusted using a vector of exposures prior to performing the regression.

abc = cl.load_sample('abc')
exposure = np.array([[2.2], [2.4], [2.2], [2.0], [1.9], [1.6], [1.6], [1.8], [2.2], [2.5], [2.6]])
model = cl.BarnettZehnwirth(formula='C(origin) + C(development)').fit(abc/exposure)

plot1a = model.std_residuals_.T
plot1b = plot1a.T.mean()

plot2a = model.std_residuals_
plot2b = plot2a.T.mean()

plot3a = model.std_residuals_.dev_to_val().T
plot3b = model.std_residuals_.dev_to_val().mean('origin').T

plot4 = pd.concat((
    model.triangle_ml_[model.triangle_ml_.valuation<=abc.valuation_date].log().unstack().rename('Fitted Values'),
    model.std_residuals_.unstack().rename('Residual')), axis=1).dropna()

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import matplotlib.pyplot as plt
plt.style.use('ggplot')
%config InlineBackend.figure_format = 'retina'

fig, ((ax00, ax01), (ax10, ax11)) = plt.subplots(ncols=2, nrows=2, figsize=(14,10))
fig.suptitle("Barnett Zehnwirth\nStandardized residuals of the statistical chainladder model\n(Figure 23)");

plot1a.plot(
    style='.', color='gray', legend=False, ax=ax00,
    xlabel='Development Month', ylabel='Weighted Standardized Residuals')
plot1b.plot(color='red', legend=False, ax=ax00)

plot2a.plot(style='.', color='gray', legend=False, ax=ax01, xlabel='Origin Period')
plot2b.plot(color='red', legend=False, ax=ax01,xlim=(6.5,17.5))
plot3a.plot(
    style='.', color='gray', legend=False, ax=ax10,
    xlabel='Valuation Date', ylabel='Weighted Standardized Residuals')
plot3b.plot(color='red', legend=False, ax=ax10)

plot4.plot(kind='scatter', marker='o', color='gray', 
           x='Fitted Values', y='Residual', ax=ax11, sharey=True);