Chapter 11 - Frequency-Severity Techniques

On this page, we will recreating Chapter 11 of Friendland. There is not a dedicated FrequencySeverity method in the package. Instead, we will chain together existing methods and use Triangle arithmetic to calculate the utimate.

We will begin by import packages and setting up some helper functions.

import numpy as np
import pandas as pd
import chainladder as cl
from IPython.display import display as nb_display

#create reported/paid summary exhibit
def summary_exh(reported_tri:cl.Triangle,paid_tri:cl.Triangle,ult:cl.Triangle) -> pd.DataFrame:
    output = reported_tri.latest_diagonal.to_frame()
    output.columns = ['Reported']
    output['Paid'] = paid_tri.latest_diagonal.to_frame()
    #using floating point offset to achieve standard rounding
    output['Ult Claims'] = (ult.latest_diagonal + 1e-9).round(0).to_frame()
    output['Case Outstanding'] = output["Reported"] - output["Paid"]
    output['IBNR'] = output["Ult Claims"] - output["Reported"]
    output['Unpaid'] = output["Ult Claims"] - output["Paid"]
    return output.round(0)

#create a dict of developed triangles for each of the selection assumptions on a given page
def average_dev(tri: cl.Triangle, avg_params: dict[str,int]) -> dict[cl.Triangle]:
    return {k:cl.Development(**v).fit_transform(tri) for k,v in avg_params.items()}

#combine a dict of triangles into a singla triangle
def combine_tri(devs: dict[cl.Triangle]) -> cl.Triangle:
    avgs = [v.rename('index',k) for k,v in devs.items()]
    return cl.concat(avgs,axis=0)

#combine the ldf_ of a dict of triangles into a singla triangle
def combine_ldf(devs: dict[cl.Triangle]) -> cl.Triangle:
    return combine_tri({k:v.ldf_ for k,v in devs.items()})

#create a dict of DisposalRate estimators for each of the selection assumptions on a given page
def average_dr(tri: cl.Triangle, w:cl.Triangle, avg_params: dict[str,int]) -> dict[cl.DisposalRate]:
    return {k:cl.DisposalRate(**v).fit(X=tri,sample_weight = w) for k,v in avg_params.items()}

#combine the disposal_rate_ of a dict of triangles into a singla triangle
def combine_disposal(devs: dict[cl.Triangle]) -> cl.Triangle:
    return combine_tri({k:v.disposal_rate_ for k,v in devs.items()})

#create a dict of weighted regression objects for each of the selection assumptions on a given page
def regs(tri: cl.Triangle, avg_params: dict[str,int]) -> dict[cl.WeightedRegression]:
    return {k:cl.WeightedRegression(
        axis=2,
        thru_orig=False,
        xp=tri.get_array_module()
    ).fit(
        np.ones(tri.shape) * tri.origin.year.values[None,None,:,None],
        np.log(tri.values),
        cl.TriangleWeight(**v).fit(tri).w_.values
    ) for k,v in avg_params.items()}

#combine all the trends and r-squared values into pandas DataFrame
def reg_outputs(regs: dict[cl.WeightedRegression],devs:pd.Series) -> pd.DataFrame:
    trend = {k:(np.exp(v.slope_)-1).flatten() for k,v in regs.items()}
    trend = pd.DataFrame.from_dict(trend,orient='index')
    trend.columns = devs
    rsq = {k:v.rsq_.flatten() for k,v in regs.items()}
    rsq = pd.DataFrame.from_dict(rsq,orient='index')
    rsq.columns = devs
    return trend,rsq

#create a dict of weighted regression objects for each of the selection assumptions on a given page
def average_sev(tri: cl.Triangle, avg_params: dict[str,int]) -> dict[cl.Triangle]:
    output = {}
    for k,v in avg_params.items():
        w = cl.TriangleWeight(**v).fit(tri)
        weighted_tri = tri * w.w_.values
        output[k] = weighted_tri.mean()
    return output

Exhibit I Analysis

Displaying all the triangles and dataframes take up quite a bit of code. We will distill the core analysis code at the top of each exhibit.

Exhibit I follows Approach 1. Count and Severity are developed separately and multiplied together.

#loading data and assumptions
e1_tri = cl.load_sample('friedland_auto_freq_sev')
e1_cnt_assumptions = {}
e1_cnt_assumptions['simple_5'] = {'n_periods':5, 'average':'simple'}
e1_cnt_assumptions['simple_3'] = {'n_periods':3, 'average':'simple'}
e1_cnt_assumptions['medial_5x1'] = {'n_periods':5, 'average':'simple','drop_high':1, 'drop_low':1}
e1_cnt_assumptions['volume_5'] = {'n_periods':5, 'average':'volume'}
e1_cnt_assumptions['volume_3'] = {'n_periods':3, 'average':'volume'}
e1_sev_assumptions = {}
e1_sev_assumptions['simple_5'] = {'n_periods':5, 'average':'simple'}
e1_sev_assumptions['simple_3'] = {'n_periods':3, 'average':'simple'}
e1_sev_assumptions['medial_5x1'] = {'n_periods':5, 'average':'simple','drop_high':1, 'drop_low':1}

#developing closed claim counts
e1_ccc_devs = average_dev(e1_tri['Closed Claim Counts'],e1_cnt_assumptions)
e1_ccc_selected = cl.TailConstant(tail = 1.0, projection_period = 0).fit_transform(e1_ccc_devs['simple_3'])
e1_ccc_selected.ldf_ = e1_ccc_selected.ldf_.round(3)

#developing reported claim counts
e1_rcc_devs = average_dev(e1_tri['Reported Claim Counts'],e1_cnt_assumptions)
e1_rcc_selected = cl.TailConstant(tail = 1.0, projection_period = 0).fit_transform(e1_rcc_devs['simple_3'])
e1_rcc_selected.ldf_ = e1_rcc_selected.ldf_.round(3)

#combining closed and reported claim counts
e1_ccc_cl = cl.Chainladder().fit(e1_ccc_selected)
e1_rcc_cl = cl.Chainladder().fit(e1_rcc_selected)
e1_cc_ult = (e1_ccc_cl.ultimate_ + e1_rcc_cl.ultimate_)/2
e1_cc_ult.iloc[:,:,-1,:] = e1_rcc_cl.ultimate_.iloc[:,:,-1,:]

#calculating and developing reported severity
e1_rsev = e1_tri['Reported Claims'] / e1_tri['Reported Claim Counts']
e1_rsev_devs = average_dev(e1_rsev,e1_sev_assumptions)
e1_rsev_selected = cl.TailConstant(tail = 1.0, projection_period = 0).fit_transform(e1_rsev_devs['medial_5x1'])
e1_rsev_selected.ldf_ = e1_rsev_selected.ldf_.round(3)
e1_rsev_cl = cl.Chainladder().fit(e1_rsev_selected)

#combining developed count and severity
e1_ult = e1_rsev_cl.ultimate_ * e1_cc_ult / 1000

/home/docs/checkouts/readthedocs.org/user_builds/chainladder-python/envs/experimental/lib/python3.11/site-packages/chainladder/utils/triangle_weight.py:367: UserWarning: Some exclusions have been ignored. At least 1 (use preserve = ...) link ratio(s) is required for development estimation.
  warnings.warn(warning)
/home/docs/checkouts/readthedocs.org/user_builds/chainladder-python/envs/experimental/lib/python3.11/site-packages/chainladder/development/base.py:272: UserWarning: Some exclusions have been ignored. At least 1 (use preserve = ...) link ratio(s) is required for development estimation.
  warnings.warn(warning)
/home/docs/checkouts/readthedocs.org/user_builds/chainladder-python/envs/experimental/lib/python3.11/site-packages/chainladder/utils/triangle_weight.py:367: UserWarning: Some exclusions have been ignored. At least 1 (use preserve = ...) link ratio(s) is required for development estimation.
  warnings.warn(warning)
/home/docs/checkouts/readthedocs.org/user_builds/chainladder-python/envs/experimental/lib/python3.11/site-packages/chainladder/development/base.py:272: UserWarning: Some exclusions have been ignored. At least 1 (use preserve = ...) link ratio(s) is required for development estimation.
  warnings.warn(warning)

/home/docs/checkouts/readthedocs.org/user_builds/chainladder-python/envs/experimental/lib/python3.11/site-packages/chainladder/utils/triangle_weight.py:367: UserWarning: Some exclusions have been ignored. At least 1 (use preserve = ...) link ratio(s) is required for development estimation.
  warnings.warn(warning)
/home/docs/checkouts/readthedocs.org/user_builds/chainladder-python/envs/experimental/lib/python3.11/site-packages/chainladder/development/base.py:272: UserWarning: Some exclusions have been ignored. At least 1 (use preserve = ...) link ratio(s) is required for development estimation.
  warnings.warn(warning)

Exhibit I Sheet 1

print('PART 1 - Data Triangle')
nb_display(e1_tri['Closed Claim Counts'])
print('PART 2 - Age-to-Age Factors')
nb_display(e1_tri['Closed Claim Counts'].age_to_age.round(3))
print('PART 3 - Average Age-to-Age Factor')
nb_display(combine_ldf(e1_ccc_devs).round(3).to_frame())
print('PART 4 - Selected Age-to-Age Factors')
print('Selected')
nb_display(e1_ccc_selected.ldf_)
print('CDF to Ultimate')
nb_display(e1_ccc_selected.cdf_.round(3))
print('Percent Closed')
nb_display(1/e1_ccc_selected.cdf_.round(3))

PART 1 - Data Triangle

	6	12	18	24	30	36	42	48	54	60
2003H2	2,547	3,262	3,287	3,291	3,292	3,292	3,292	3,292	3,292	3,292
2004H1	2,791	3,217	3,240	3,242	3,243	3,243	3,243	3,243	3,242
2004H2	2,099	2,677	2,695	2,697	2,697	2,698	2,698	2,698
2005H1	2,370	2,735	2,751	2,754	2,755	2,755	2,756
2005H2	1,966	2,609	2,630	2,634	2,634	2,634
2006H1	2,261	2,671	2,694	2,696	2,697
2006H2	1,949	2,637	2,659	2,662
2007H1	2,059	2,496	2,520
2007H2	2,083	2,732
2008H1	2,533

PART 2 - Age-to-Age Factors

	6-12	12-18	18-24	24-30	30-36	36-42	42-48	48-54	54-60
2003H2	1.2810	1.0080	1.0010	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000
2004H1	1.1530	1.0070	1.0010	1.0000	1.0000	1.0000	1.0000	1.0000
2004H2	1.2750	1.0070	1.0010	1.0000	1.0000	1.0000	1.0000
2005H1	1.1540	1.0060	1.0010	1.0000	1.0000	1.0000
2005H2	1.3270	1.0080	1.0020	1.0000	1.0000
2006H1	1.1810	1.0090	1.0010	1.0000
2006H2	1.3530	1.0080	1.0010
2007H1	1.2120	1.0100
2007H2	1.3120

PART 3 - Average Age-to-Age Factor

development	6-12	12-18	18-24	24-30	30-36	36-42	42-48	48-54	54-60
Total
simple_5	1.277	1.008	1.001	1.0	1.0	1.0	1.0	1.0	1.0
simple_3	1.292	1.009	1.001	1.0	1.0	1.0	1.0	1.0	1.0
medial_5x1	1.284	1.008	1.001	1.0	1.0	1.0	1.0	1.0	1.0
volume_5	1.274	1.008	1.001	1.0	1.0	1.0	1.0	1.0	1.0
volume_3	1.291	1.009	1.001	1.0	1.0	1.0	1.0	1.0	1.0

PART 4 - Selected Age-to-Age Factors
Selected

	6-12	12-18	18-24	24-30	30-36	36-42	42-48	48-54	54-60	60-66
(All)	1.2920	1.0090	1.0010	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000

CDF to Ultimate

	6-Ult	12-Ult	18-Ult	24-Ult	30-Ult	36-Ult	42-Ult	48-Ult	54-Ult	60-Ult
(All)	1.3050	1.0100	1.0010	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000

Percent Closed

	6-Ult	12-Ult	18-Ult	24-Ult	30-Ult	36-Ult	42-Ult	48-Ult	54-Ult	60-Ult
(All)	0.7663	0.9901	0.9990	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000

Exhibit I Sheet 2

print('PART 1 - Data Triangle')
nb_display(e1_tri['Reported Claim Counts'])
print('PART 2 - Age-to-Age Factors')
nb_display(e1_tri['Reported Claim Counts'].age_to_age.round(3))
print('PART 3 - Average Age-to-Age Factor')
nb_display(combine_ldf(e1_rcc_devs).round(3).to_frame())
print('PART 4 - Selected Age-to-Age Factors')
print('Selected')
nb_display(e1_rcc_selected.ldf_)
print('CDF to Ultimate')
nb_display(e1_rcc_selected.cdf_.round(3))
print('Percent Reported')
nb_display(1/e1_rcc_selected.cdf_.round(3))

PART 1 - Data Triangle

	6	12	18	24	30	36	42	48	54	60
2003H2	3,556	3,314	3,301	3,299	3,295	3,294	3,293	3,293	3,293	3,292
2004H1	3,492	3,262	3,250	3,247	3,247	3,245	3,245	3,244	3,243
2004H2	2,980	2,712	2,704	2,702	2,700	2,700	2,699	2,699
2005H1	2,896	2,768	2,761	2,758	2,758	2,758	2,757
2005H2	2,814	2,650	2,640	2,639	2,638	2,636
2006H1	2,808	2,712	2,704	2,701	2,700
2006H2	2,799	2,675	2,670	2,668
2007H1	2,578	2,533	2,529
2007H2	2,791	2,778
2008H1	3,139

PART 2 - Age-to-Age Factors

	6-12	12-18	18-24	24-30	30-36	36-42	42-48	48-54	54-60
2003H2	0.9320	0.9960	0.9990	0.9990	1.0000	1.0000	1.0000	1.0000	1.0000
2004H1	0.9340	0.9960	0.9990	1.0000	0.9990	1.0000	1.0000	1.0000
2004H2	0.9100	0.9970	0.9990	0.9990	1.0000	1.0000	1.0000
2005H1	0.9560	0.9970	0.9990	1.0000	1.0000	1.0000
2005H2	0.9420	0.9960	1.0000	1.0000	0.9990
2006H1	0.9660	0.9970	0.9990	1.0000
2006H2	0.9560	0.9980	0.9990
2007H1	0.9830	0.9980
2007H2	0.9950

PART 3 - Average Age-to-Age Factor

development	6-12	12-18	18-24	24-30	30-36	36-42	42-48	48-54	54-60
Total
simple_5	0.968	0.997	0.999	1.0	1.0	1.0	1.0	1.0	1.0
simple_3	0.978	0.998	0.999	1.0	1.0	1.0	1.0	1.0	1.0
medial_5x1	0.968	0.998	0.999	1.0	1.0	1.0	1.0	1.0	1.0
volume_5	0.968	0.997	0.999	1.0	1.0	1.0	1.0	1.0	1.0
volume_3	0.978	0.998	0.999	1.0	1.0	1.0	1.0	1.0	1.0

PART 4 - Selected Age-to-Age Factors
Selected

	6-12	12-18	18-24	24-30	30-36	36-42	42-48	48-54	54-60	60-66
(All)	0.9780	0.9980	0.9990	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000

CDF to Ultimate

	6-Ult	12-Ult	18-Ult	24-Ult	30-Ult	36-Ult	42-Ult	48-Ult	54-Ult	60-Ult
(All)	0.9750	0.9970	0.9990	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000

Percent Reported

	6-Ult	12-Ult	18-Ult	24-Ult	30-Ult	36-Ult	42-Ult	48-Ult	54-Ult	60-Ult
(All)	1.0256	1.0030	1.0010	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000

Exhibit I Sheet 3

e1_ccc_df = cl.model_diagnostics(e1_ccc_cl).to_frame(keepdims=True,implicit_axis=True).set_index('origin')
e1_rcc_df = cl.model_diagnostics(e1_rcc_cl).to_frame(keepdims=True,implicit_axis=True).set_index('origin')
e1_s3 = e1_ccc_df[['development','Latest','CDF','Ultimate']].rename(columns={'Latest':'Closed Claim Counts','CDF':'Closed CDF','Ultimate':'Ult Count Using CCC'})
e1_s3[['Reported Claim Counts','Reported CDF','Ult Count Using RCC']] = e1_rcc_df[['Latest','CDF','Ultimate']]
e1_s3["Selected Ult CC"] = e1_cc_ult.latest_diagonal.to_frame()
e1_s3[['Closed CDF','Reported CDF']] = e1_s3[['Closed CDF','Reported CDF']].round(3)
#using floating point offset to achieve standard rounding
e1_s3[['Ult Count Using CCC','Ult Count Using RCC','Selected Ult CC']] = (e1_s3[['Ult Count Using CCC','Ult Count Using RCC','Selected Ult CC']] + 1e-9).round(0)
e1_s3[['development','Closed Claim Counts','Reported Claim Counts','Closed CDF','Reported CDF','Ult Count Using CCC','Ult Count Using RCC','Selected Ult CC']]

	development	Closed Claim Counts	Reported Claim Counts	Closed CDF	Reported CDF	Ult Count Using CCC	Ult Count Using RCC	Selected Ult CC
origin
2003-07-01	60	3292.0	3292.0	1.000	1.000	3292.0	3292.0	3292.0
2004-01-01	54	3242.0	3243.0	1.000	1.000	3242.0	3243.0	3243.0
2004-07-01	48	2698.0	2699.0	1.000	1.000	2698.0	2699.0	2699.0
2005-01-01	42	2756.0	2757.0	1.000	1.000	2756.0	2757.0	2757.0
2005-07-01	36	2634.0	2636.0	1.000	1.000	2634.0	2636.0	2635.0
2006-01-01	30	2697.0	2700.0	1.000	1.000	2697.0	2700.0	2699.0
2006-07-01	24	2662.0	2668.0	1.000	1.000	2662.0	2668.0	2665.0
2007-01-01	18	2520.0	2529.0	1.001	0.999	2523.0	2526.0	2524.0
2007-07-01	12	2732.0	2778.0	1.010	0.997	2759.0	2770.0	2765.0
2008-01-01	6	2533.0	3139.0	1.305	0.975	3305.0	3061.0	3061.0

Exhibit I Sheet 4

(e1_tri['Closed Claim Counts']/e1_tri['Reported Claim Counts']).round(3)

	6	12	18	24	30	36	42	48	54	60
2003H2	0.7160	0.9840	0.9960	0.9980	0.9990	0.9990	1.0000	1.0000	1.0000	1.0000
2004H1	0.7990	0.9860	0.9970	0.9980	0.9990	0.9990	0.9990	1.0000	1.0000
2004H2	0.7040	0.9870	0.9970	0.9980	0.9990	0.9990	1.0000	1.0000
2005H1	0.8180	0.9880	0.9960	0.9990	0.9990	0.9990	1.0000
2005H2	0.6990	0.9850	0.9960	0.9980	0.9980	0.9990
2006H1	0.8050	0.9850	0.9960	0.9980	0.9990
2006H2	0.6960	0.9860	0.9960	0.9980
2007H1	0.7990	0.9850	0.9960
2007H2	0.7460	0.9830
2008H1	0.8070

Exhibit I Sheet 5

nb_display(e1_tri['Reported Claims']/1000)
nb_display(e1_rsev)

	6	12	18	24	30	36	42	48	54	60
2003H2	14,235	14,959	14,921	14,911	14,926	14,863	14,861	14,855	14,851	14,847
2004H1	14,548	14,672	14,644	14,628	14,621	14,609	14,609	14,611	14,616
2004H2	12,129	12,576	12,541	12,532	12,523	12,523	12,510	12,502
2005H1	11,981	11,922	11,883	11,862	11,854	11,843	11,841
2005H2	11,284	11,843	11,806	11,788	11,771	11,770
2006H1	11,945	11,857	11,819	11,774	11,761
2006H2	12,503	12,762	12,707	12,697
2007H1	11,663	11,523	11,492
2007H2	12,646	12,854
2008H1	14,072

	6	12	18	24	30	36	42	48	54	60
2003H2	4,003	4,514	4,520	4,520	4,530	4,512	4,513	4,511	4,510	4,510
2004H1	4,166	4,498	4,506	4,505	4,503	4,502	4,502	4,504	4,507
2004H2	4,070	4,637	4,638	4,638	4,638	4,638	4,635	4,632
2005H1	4,137	4,307	4,304	4,301	4,298	4,294	4,295
2005H2	4,010	4,469	4,472	4,467	4,462	4,465
2006H1	4,254	4,372	4,371	4,359	4,356
2006H2	4,467	4,771	4,759	4,759
2007H1	4,524	4,549	4,544
2007H2	4,531	4,627
2008H1	4,483

Exhibit I Sheet 6

print('PART 1 - Data Triangle')
nb_display(e1_rsev)
print('PART 2 - Age-to-Age Factors')
nb_display(e1_rsev.age_to_age.round(3))
print('PART 3 - Average Age-to-Age Factors')
nb_display(combine_ldf(e1_rsev_devs).round(3).to_frame())
print('PART 4 - Selected Age-to-Age Factors')
nb_display(e1_rsev_selected.ldf_)
print('CDF to Ultimate')
nb_display(e1_rsev_selected.cdf_.round(3))

PART 1 - Data Triangle

	6	12	18	24	30	36	42	48	54	60
2003H2	4,003	4,514	4,520	4,520	4,530	4,512	4,513	4,511	4,510	4,510
2004H1	4,166	4,498	4,506	4,505	4,503	4,502	4,502	4,504	4,507
2004H2	4,070	4,637	4,638	4,638	4,638	4,638	4,635	4,632
2005H1	4,137	4,307	4,304	4,301	4,298	4,294	4,295
2005H2	4,010	4,469	4,472	4,467	4,462	4,465
2006H1	4,254	4,372	4,371	4,359	4,356
2006H2	4,467	4,771	4,759	4,759
2007H1	4,524	4,549	4,544
2007H2	4,531	4,627
2008H1	4,483

PART 2 - Age-to-Age Factors

	6-12	12-18	18-24	24-30	30-36	36-42	42-48	48-54	54-60
2003H2	1.1280	1.0010	1.0000	1.0020	0.9960	1.0000	1.0000	1.0000	1.0000
2004H1	1.0800	1.0020	1.0000	1.0000	1.0000	1.0000	1.0000	1.0010
2004H2	1.1390	1.0000	1.0000	1.0000	1.0000	0.9990	0.9990
2005H1	1.0410	0.9990	0.9990	0.9990	0.9990	1.0000
2005H2	1.1140	1.0010	0.9990	0.9990	1.0010
2006H1	1.0280	1.0000	0.9970	0.9990
2006H2	1.0680	0.9970	1.0000
2007H1	1.0060	0.9990
2007H2	1.0210

PART 3 - Average Age-to-Age Factors

development	6-12	12-18	18-24	24-30	30-36	36-42	42-48	48-54	54-60
Total
simple_5	1.047	0.999	0.999	0.999	0.999	1.0	1.0	1.0	1.0
simple_3	1.032	0.999	0.999	0.999	1.000	1.0	1.0	1.0	1.0
medial_5x1	1.039	0.999	0.999	0.999	1.000	1.0	1.0	1.0	1.0

PART 4 - Selected Age-to-Age Factors

	6-12	12-18	18-24	24-30	30-36	36-42	42-48	48-54	54-60	60-66
(All)	1.0390	0.9990	0.9990	0.9990	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000

CDF to Ultimate

	6-Ult	12-Ult	18-Ult	24-Ult	30-Ult	36-Ult	42-Ult	48-Ult	54-Ult	60-Ult
(All)	1.0360	0.9970	0.9980	0.9990	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000

Exhibit I Sheet 7

e1_rsev_ult_df = cl.model_diagnostics(e1_rsev_cl).to_frame(keepdims=True,implicit_axis=True).set_index('origin')
e1_s7 = e1_rsev_ult_df[['development','Latest','CDF','Ultimate']].rename(columns={'Latest':'Reported Severity','Ultimate':'Ult Severity'})
e1_s7['Selected Ult CC'] = e1_s3['Selected Ult CC']
e1_s7["Ult Claims"] = e1_ult.latest_diagonal.to_frame()
e1_s7['CDF'] = e1_s7['CDF'].round(3)
#using floating point offset to achieve standard rounding
e1_s7[['Reported Severity','Ult Severity','Ult Claims']] = (e1_s7[['Reported Severity','Ult Severity','Ult Claims']]+1e-9).round(0)
e1_s7

	development	Reported Severity	CDF	Ult Severity	Selected Ult CC	Ult Claims
origin
2003-07-01	60	4510.0	1.000	4510.0	3292.0	14847.0
2004-01-01	54	4507.0	1.000	4507.0	3243.0	14614.0
2004-07-01	48	4632.0	1.000	4632.0	2699.0	12499.0
2005-01-01	42	4295.0	1.000	4295.0	2757.0	11839.0
2005-07-01	36	4465.0	1.000	4465.0	2635.0	11765.0
2006-01-01	30	4356.0	1.000	4356.0	2699.0	11755.0
2006-07-01	24	4759.0	0.999	4754.0	2665.0	12670.0
2007-01-01	18	4544.0	0.998	4535.0	2524.0	11448.0
2007-07-01	12	4627.0	0.997	4613.0	2765.0	12753.0
2008-01-01	6	4483.0	1.036	4644.0	3061.0	14214.0

Exhibit I Sheet 8

summary_exh(e1_tri['Reported Claims']/1000,e1_tri['Paid Claims']/1000,e1_ult)

	Reported	Paid	Ult Claims	Case Outstanding	IBNR	Unpaid
2003-07-01	14847.0	14846.0	14847.0	1.0	0.0	1.0
2004-01-01	14616.0	14614.0	14614.0	2.0	-2.0	0.0
2004-07-01	12502.0	12502.0	12499.0	-0.0	-3.0	-3.0
2005-01-01	11841.0	11840.0	11839.0	1.0	-2.0	-1.0
2005-07-01	11770.0	11765.0	11765.0	5.0	-5.0	0.0
2006-01-01	11761.0	11755.0	11755.0	6.0	-6.0	0.0
2006-07-01	12697.0	12679.0	12670.0	18.0	-27.0	-9.0
2007-01-01	11492.0	11406.0	11448.0	86.0	-44.0	42.0
2007-07-01	12854.0	12648.0	12753.0	206.0	-101.0	105.0
2008-01-01	14072.0	11833.0	14214.0	2239.0	142.0	2381.0

At the end of each exhibit, we reconcile to hardcoded figures from Friedland using a series of assert statemenets. When any of these statements errors out, we know some bug has been introduced into the package.

#Exhibit I Sheet 1
assert np.all(e1_ccc_selected.cdf_.round(3).values == np.array([1.305, 1.010, 1.001, 1.000, 1.000, 1.000, 1.000, 1.000, 1.000, 1.000]))
#Exhibit I Sheet 2
assert np.all(e1_rcc_selected.cdf_.round(3).values == np.array([0.975, 0.997, 0.999, 1.000, 1.000, 1.000, 1.000, 1.000, 1.000, 1.000]))
#Exhibit I Sheet 3
assert np.allclose(
    e1_s3['Selected Ult CC'].values,
    np.array([3292, 3243, 2699, 2757, 2635, 2699, 2665, 2524, 2764, 3061]),
    atol=1
)
#Exhibit I Sheet 6
assert np.all(e1_rsev_selected.cdf_.round(3).values == np.array([1.036, 0.997, 0.998, 0.999, 1.000, 1.000, 1.000, 1.000, 1.000, 1.000]))
#Exhibit I Sheet 7
assert np.allclose(
    e1_s7['Ult Severity'].values,
    np.array([4510, 4507, 4632, 4295, 4465, 4356, 4754, 4535, 4613, 4644]),
    atol=1
)

Exhibit II Analysis

Exhibit II also follows Approach 1, and is largely similar to Exhibit I.

#loading data and assumptions
e2_tri = cl.load_sample('friedland_xyz_auto_bi')
e2_cnt_assumptions = {}
e2_cnt_assumptions['simple_3'] = {'n_periods':3, 'average':'simple'}
e2_cnt_assumptions['simple_2'] = {'n_periods':2, 'average':'simple'}
e2_cnt_assumptions['medial_5x1'] = {'n_periods':5, 'average':'simple','drop_high':1, 'drop_low':1}
e2_cnt_assumptions['volume_3'] = {'n_periods':3, 'average':'volume'}
e2_cnt_assumptions['volume_2'] = {'n_periods':2, 'average':'volume'}
e2_sev_assumptions = {}
e2_sev_assumptions['simple_3'] = {'n_periods':3, 'average':'simple'}
e2_sev_assumptions['simple_2'] = {'n_periods':2, 'average':'simple'}
e2_sev_assumptions['medial_5x1'] = {'n_periods':5, 'average':'simple','drop_high':1, 'drop_low':1}

#developing closed claim counts
e2_ccc_devs = average_dev(e2_tri['Closed Claim Counts'],e2_cnt_assumptions)
e2_ccc_selected = cl.TailConstant(tail = 1.0, projection_period = 0).fit_transform(e2_ccc_devs['volume_2'])
e2_ccc_selected.ldf_ = e2_ccc_selected.ldf_.round(3)

#developing reported claim counts
e2_rcc_devs = average_dev(e2_tri['Reported Claim Counts'],e2_cnt_assumptions)
e2_rcc_selected = cl.TailConstant(tail = 1.0, projection_period = 0).fit_transform(e2_rcc_devs['volume_2'])
e2_rcc_selected.ldf_ = e2_rcc_selected.ldf_.round(3)

#combining closed and reported claim counts
e2_ccc_cl = cl.Chainladder().fit(e2_ccc_selected)
e2_rcc_cl = cl.Chainladder().fit(e2_rcc_selected)
e2_cc_ult = (e2_ccc_cl.ultimate_ + e2_rcc_cl.ultimate_)/2

#calculating and developing reported severity
e2_rsev = e2_tri['Reported Claims'] / e2_tri['Reported Claim Counts'] * 1000
e2_rsev_devs = average_dev(e2_rsev,e2_sev_assumptions)
e2_rsev_selected = cl.TailConstant(tail = 1.0, projection_period = 0).fit_transform(e2_rsev_devs['simple_2'])
e2_rsev_selected.ldf_ = e2_rsev_selected.ldf_.round(3)
e2_rsev_cl = cl.Chainladder().fit(e2_rsev_selected)

#combining developed count and severity
e2_ult = e2_rsev_cl.ultimate_ * e2_cc_ult / 1000

/home/docs/checkouts/readthedocs.org/user_builds/chainladder-python/envs/experimental/lib/python3.11/site-packages/chainladder/utils/triangle_weight.py:367: UserWarning: Some exclusions have been ignored. At least 1 (use preserve = ...) link ratio(s) is required for development estimation.
  warnings.warn(warning)
/home/docs/checkouts/readthedocs.org/user_builds/chainladder-python/envs/experimental/lib/python3.11/site-packages/chainladder/development/base.py:272: UserWarning: Some exclusions have been ignored. At least 1 (use preserve = ...) link ratio(s) is required for development estimation.
  warnings.warn(warning)
/home/docs/checkouts/readthedocs.org/user_builds/chainladder-python/envs/experimental/lib/python3.11/site-packages/chainladder/utils/triangle_weight.py:367: UserWarning: Some exclusions have been ignored. At least 1 (use preserve = ...) link ratio(s) is required for development estimation.
  warnings.warn(warning)
/home/docs/checkouts/readthedocs.org/user_builds/chainladder-python/envs/experimental/lib/python3.11/site-packages/chainladder/development/base.py:272: UserWarning: Some exclusions have been ignored. At least 1 (use preserve = ...) link ratio(s) is required for development estimation.
  warnings.warn(warning)
/home/docs/checkouts/readthedocs.org/user_builds/chainladder-python/envs/experimental/lib/python3.11/site-packages/chainladder/utils/triangle_weight.py:367: UserWarning: Some exclusions have been ignored. At least 1 (use preserve = ...) link ratio(s) is required for development estimation.
  warnings.warn(warning)
/home/docs/checkouts/readthedocs.org/user_builds/chainladder-python/envs/experimental/lib/python3.11/site-packages/chainladder/development/base.py:272: UserWarning: Some exclusions have been ignored. At least 1 (use preserve = ...) link ratio(s) is required for development estimation.
  warnings.warn(warning)

Exhibit II Sheet 1

print('PART 1 - Data Triangle')
nb_display(e2_tri['Closed Claim Counts'])
print('PART 2 - Age-to-Age Factors')
nb_display(e2_tri['Closed Claim Counts'].age_to_age.round(3))
print('PART 3 - Average Age-to-Age Factor')
nb_display(combine_ldf(e2_ccc_devs).round(3).to_frame())
print('PART 4 - Selected Age-to-Age Factors')
print('Selected')
nb_display(e2_ccc_selected.ldf_)
print('CDF to Ultimate')
nb_display(e2_ccc_selected.cdf_.round(3))
print('Percent Closed')
nb_display(1/e2_ccc_selected.cdf_.round(3))

PART 1 - Data Triangle

	12	24	36	48	60	72	84	96	108	120	132
1998				510.00	547.00	575.00	598.00	612.00	620.00	635.00	637.00
1999			686.00	819.00	910.00	980.00	1,007.00	1,036.00	1,039.00	1,044.00
2000		650.00	932.00	1,095.00	1,216.00	1,292.00	1,367.00	1,391.00	1,402.00
2001	304.00	681.00	936.00	1,092.00	1,225.00	1,357.00	1,432.00	1,446.00
2002	203.00	607.00	841.00	1,089.00	1,327.00	1,464.00	1,523.00
2003	181.00	614.00	941.00	1,263.00	1,507.00	1,568.00
2004	235.00	848.00	1,442.00	1,852.00	2,029.00
2005	295.00	1,119.00	1,664.00	1,946.00
2006	307.00	906.00	1,201.00
2007	329.00	791.00
2008	276.00

PART 2 - Age-to-Age Factors

	12-24	24-36	36-48	48-60	60-72	72-84	84-96	96-108	108-120	120-132
1998				1.0730	1.0510	1.0400	1.0230	1.0130	1.0240	1.0030
1999			1.1940	1.1110	1.0770	1.0280	1.0290	1.0030	1.0050
2000		1.4340	1.1750	1.1110	1.0620	1.0580	1.0180	1.0080
2001	2.2400	1.3740	1.1670	1.1220	1.1080	1.0550	1.0100
2002	2.9900	1.3860	1.2950	1.2190	1.1030	1.0400
2003	3.3920	1.5330	1.3420	1.1930	1.0400
2004	3.6090	1.7000	1.2840	1.0960
2005	3.7930	1.4870	1.1690
2006	2.9510	1.3260
2007	2.4040

PART 3 - Average Age-to-Age Factor

development	12-24	24-36	36-48	48-60	60-72	72-84	84-96	96-108	108-120	120-132
Total
simple_3	3.050	1.504	1.265	1.169	1.084	1.051	1.019	1.008	1.015	1.003
simple_2	2.678	1.406	1.227	1.144	1.072	1.048	1.014	1.005	1.015	1.003
medial_5x1	3.317	1.468	1.250	1.142	1.081	1.045	1.020	1.008	1.015	1.003
volume_3	3.025	1.499	1.251	1.157	1.081	1.051	1.018	1.007	1.012	1.003
volume_2	2.668	1.415	1.223	1.135	1.070	1.048	1.014	1.006	1.012	1.003

PART 4 - Selected Age-to-Age Factors
Selected

	12-24	24-36	36-48	48-60	60-72	72-84	84-96	96-108	108-120	120-132	132-144
(All)	2.6680	1.4150	1.2230	1.1350	1.0700	1.0480	1.0140	1.0060	1.0120	1.0030	1.0000

CDF to Ultimate

	12-Ult	24-Ult	36-Ult	48-Ult	60-Ult	72-Ult	84-Ult	96-Ult	108-Ult	120-Ult	132-Ult
(All)	6.0850	2.2810	1.6120	1.3180	1.1610	1.0850	1.0350	1.0210	1.0150	1.0030	1.0000

Percent Closed

	12-Ult	24-Ult	36-Ult	48-Ult	60-Ult	72-Ult	84-Ult	96-Ult	108-Ult	120-Ult	132-Ult
(All)	0.1643	0.4384	0.6203	0.7587	0.8613	0.9217	0.9662	0.9794	0.9852	0.9970	1.0000

Exhibit II Sheet 2

print('PART 1 - Data Triangle')
nb_display(e2_tri['Reported Claim Counts'])
print('PART 2 - Age-to-Age Factors')
nb_display(e2_tri['Reported Claim Counts'].age_to_age.round(3))
print('PART 3 - Average Age-to-Age Factor')
nb_display(combine_ldf(e2_rcc_devs).round(3).to_frame())
print('PART 4 - Selected Age-to-Age Factors')
print('Selected')
nb_display(e2_rcc_selected.ldf_)
print('CDF to Ultimate')
nb_display(e2_rcc_selected.cdf_.round(3))
print('Percent Reported')
nb_display(1/e2_rcc_selected.cdf_.round(3))

PART 1 - Data Triangle

	12	24	36	48	60	72	84	96	108	120	132
1998				634	635	635	637	637	637	637	637
1999			1,026	1,039	1,047	1,050	1,053	1,047	1,047	1,047
2000		1,354	1,397	1,411	1,410	1,408	1,408	1,408	1,408
2001	1,305	1,421	1,449	1,458	1,458	1,455	1,455	1,455
2002	1,342	1,514	1,548	1,557	1,549	1,552	1,554
2003	1,373	1,616	1,630	1,626	1,629	1,629
2004	1,932	2,168	2,234	2,249	2,258
2005	2,067	2,293	2,367	2,390
2006	1,473	1,645	1,657
2007	1,192	1,264
2008	1,036

PART 2 - Age-to-Age Factors

	12-24	24-36	36-48	48-60	60-72	72-84	84-96	96-108	108-120	120-132
1998				1.0020	1.0000	1.0030	1.0000	1.0000	1.0000	1.0000
1999			1.0130	1.0080	1.0030	1.0030	0.9940	1.0000	1.0000
2000		1.0320	1.0100	0.9990	0.9990	1.0000	1.0000	1.0000
2001	1.0890	1.0200	1.0060	1.0000	0.9980	1.0000	1.0000
2002	1.1280	1.0220	1.0060	0.9950	1.0020	1.0010
2003	1.1770	1.0090	0.9980	1.0020	1.0000
2004	1.1220	1.0300	1.0070	1.0040
2005	1.1090	1.0320	1.0100
2006	1.1170	1.0070
2007	1.0600

PART 3 - Average Age-to-Age Factor

development	12-24	24-36	36-48	48-60	60-72	72-84	84-96	96-108	108-120	120-132
Total
simple_3	1.096	1.023	1.005	1.000	1.000	1.000	0.998	1.0	1.0	1.0
simple_2	1.089	1.020	1.008	1.003	1.001	1.001	1.000	1.0	1.0	1.0
medial_5x1	1.116	1.021	1.006	1.000	1.000	1.001	1.000	1.0	1.0	1.0
volume_3	1.099	1.025	1.005	1.001	1.000	1.000	0.998	1.0	1.0	1.0
volume_2	1.092	1.022	1.008	1.003	1.001	1.001	1.000	1.0	1.0	1.0

PART 4 - Selected Age-to-Age Factors
Selected

	12-24	24-36	36-48	48-60	60-72	72-84	84-96	96-108	108-120	120-132	132-144
(All)	1.0920	1.0220	1.0080	1.0030	1.0010	1.0010	1.0000	1.0000	1.0000	1.0000	1.0000

CDF to Ultimate

	12-Ult	24-Ult	36-Ult	48-Ult	60-Ult	72-Ult	84-Ult	96-Ult	108-Ult	120-Ult	132-Ult
(All)	1.1310	1.0350	1.0130	1.0050	1.0020	1.0010	1.0000	1.0000	1.0000	1.0000	1.0000

Percent Reported

	12-Ult	24-Ult	36-Ult	48-Ult	60-Ult	72-Ult	84-Ult	96-Ult	108-Ult	120-Ult	132-Ult
(All)	0.8842	0.9662	0.9872	0.9950	0.9980	0.9990	1.0000	1.0000	1.0000	1.0000	1.0000

Exhibit II Sheet 3

e2_ccc_df = cl.model_diagnostics(e2_ccc_cl).to_frame(keepdims=True,implicit_axis=True).set_index('origin')
e2_rcc_df = cl.model_diagnostics(e2_rcc_cl).to_frame(keepdims=True,implicit_axis=True).set_index('origin')
e2_s3 = e2_ccc_df[['development','Latest','CDF','Ultimate']].rename(columns={'Latest':'Closed Claim Counts','CDF':'Closed CDF','Ultimate':'Ult Count Using CCC'})
e2_s3[['Reported Claim Counts','Reported CDF','Ult Count Using RCC']] = e2_rcc_df[['Latest','CDF','Ultimate']]
e2_s3["Selected Ult CC"] = e2_cc_ult.latest_diagonal.to_frame()
e2_s3[['Closed CDF','Reported CDF']] = e2_s3[['Closed CDF','Reported CDF']].round(3)
#using floating point offset to achieve standard rounding
e2_s3[['Ult Count Using CCC','Ult Count Using RCC','Selected Ult CC']] = (e2_s3[['Ult Count Using CCC','Ult Count Using RCC','Selected Ult CC']] + 1e-9).round(0)
e2_s3[['development','Closed Claim Counts','Reported Claim Counts','Closed CDF','Reported CDF','Ult Count Using CCC','Ult Count Using RCC','Selected Ult CC']]

	development	Closed Claim Counts	Reported Claim Counts	Closed CDF	Reported CDF	Ult Count Using CCC	Ult Count Using RCC	Selected Ult CC
origin
1998-01-01	132	637.0	637.0	1.000	1.000	637.0	637.0	637.0
1999-01-01	120	1044.0	1047.0	1.003	1.000	1047.0	1047.0	1047.0
2000-01-01	108	1402.0	1408.0	1.015	1.000	1423.0	1408.0	1416.0
2001-01-01	96	1446.0	1455.0	1.021	1.000	1477.0	1455.0	1466.0
2002-01-01	84	1523.0	1554.0	1.035	1.000	1577.0	1554.0	1565.0
2003-01-01	72	1568.0	1629.0	1.085	1.001	1701.0	1631.0	1666.0
2004-01-01	60	2029.0	2258.0	1.161	1.002	2356.0	2263.0	2309.0
2005-01-01	48	1946.0	2390.0	1.318	1.005	2564.0	2402.0	2483.0
2006-01-01	36	1201.0	1657.0	1.612	1.013	1936.0	1679.0	1807.0
2007-01-01	24	791.0	1264.0	2.281	1.035	1804.0	1309.0	1556.0
2008-01-01	12	276.0	1036.0	6.085	1.131	1679.0	1171.0	1425.0

Exhibit II Sheet 4

nb_display(e2_tri['Reported Claims'])
nb_display(e2_rsev)

	12	24	36	48	60	72	84	96	108	120	132
1998			11,171	12,380	13,216	14,067	14,688	16,366	16,163	15,835	15,822
1999		13,255	16,405	19,639	22,473	23,764	25,094	24,795	25,071	25,107
2000	15,676	18,749	21,900	27,144	29,488	34,458	36,949	37,505	37,246
2001	11,827	16,004	21,022	26,578	34,205	37,136	38,541	38,798
2002	12,811	20,370	26,656	37,667	44,414	48,701	48,169
2003	9,651	16,995	30,354	40,594	44,231	44,373
2004	16,995	40,180	58,866	71,707	70,288
2005	28,674	47,432	70,340	70,655
2006	27,066	46,783	48,804
2007	19,477	31,732
2008	18,632

	12	24	36	48	60	72	84	96	108	120	132
1998				19,527	20,813	22,153	23,058	25,692	25,374	24,859	24,838
1999			15,989	18,902	21,464	22,632	23,831	23,682	23,946	23,980
2000		13,847	15,676	19,237	20,913	24,473	26,242	26,637	26,453
2001	9,063	11,262	14,508	18,229	23,460	25,523	26,489	26,665
2002	9,546	13,454	17,220	24,192	28,673	31,380	30,997
2003	7,029	10,517	18,622	24,966	27,152	27,239
2004	8,797	18,533	26,350	31,884	31,128
2005	13,872	20,686	29,717	29,563
2006	18,375	28,440	29,453
2007	16,340	25,104
2008	17,985

Exhibit II Sheet 5

print('PART 1 - Data Triangle')
nb_display(e2_rsev)
print('PART 2 - Age-to-Age Factors')
nb_display(e2_rsev.age_to_age.round(3))
print('PART 3 - Average Age-to-Age Factors')
nb_display(combine_ldf(e2_rsev_devs).round(3).to_frame())
print('PART 4 - Selected Age-to-Age Factors')
nb_display(e2_rsev_selected.ldf_)
print('CDF to Ultimate')
nb_display(e2_rsev_selected.cdf_.round(3))

PART 1 - Data Triangle

	12	24	36	48	60	72	84	96	108	120	132
1998				19,527	20,813	22,153	23,058	25,692	25,374	24,859	24,838
1999			15,989	18,902	21,464	22,632	23,831	23,682	23,946	23,980
2000		13,847	15,676	19,237	20,913	24,473	26,242	26,637	26,453
2001	9,063	11,262	14,508	18,229	23,460	25,523	26,489	26,665
2002	9,546	13,454	17,220	24,192	28,673	31,380	30,997
2003	7,029	10,517	18,622	24,966	27,152	27,239
2004	8,797	18,533	26,350	31,884	31,128
2005	13,872	20,686	29,717	29,563
2006	18,375	28,440	29,453
2007	16,340	25,104
2008	17,985

PART 2 - Age-to-Age Factors

	12-24	24-36	36-48	48-60	60-72	72-84	84-96	96-108	108-120	120-132
1998				1.0660	1.0640	1.0410	1.1140	0.9880	0.9800	0.9990
1999			1.1820	1.1360	1.0540	1.0530	0.9940	1.0110	1.0010
2000		1.1320	1.2270	1.0870	1.1700	1.0720	1.0150	0.9930
2001	1.2430	1.2880	1.2560	1.2870	1.0880	1.0380	1.0070
2002	1.4090	1.2800	1.4050	1.1850	1.0940	0.9880
2003	1.4960	1.7710	1.3410	1.0880	1.0030
2004	2.1070	1.4220	1.2100	0.9760
2005	1.4910	1.4370	0.9950
2006	1.5480	1.0360
2007	1.5360

PART 3 - Average Age-to-Age Factors

development	12-24	24-36	36-48	48-60	60-72	72-84	84-96	96-108	108-120	120-132
Total
simple_3	1.525	1.298	1.182	1.083	1.062	1.033	1.005	0.997	0.991	0.999
simple_2	1.542	1.236	1.102	1.032	1.049	1.013	1.011	1.002	0.991	0.999
medial_5x1	1.527	1.379	1.269	1.120	1.079	1.044	1.011	0.993	0.991	0.999

PART 4 - Selected Age-to-Age Factors

	12-24	24-36	36-48	48-60	60-72	72-84	84-96	96-108	108-120	120-132	132-144
(All)	1.5420	1.2360	1.1020	1.0320	1.0490	1.0130	1.0110	1.0020	0.9910	0.9990	1.0000

CDF to Ultimate

	12-Ult	24-Ult	36-Ult	48-Ult	60-Ult	72-Ult	84-Ult	96-Ult	108-Ult	120-Ult	132-Ult
(All)	2.3100	1.4980	1.2120	1.1000	1.0660	1.0160	1.0030	0.9920	0.9900	0.9990	1.0000

Exhibit II Sheet 6

e2_rsev_df = cl.model_diagnostics(e2_rsev_cl).to_frame(keepdims=True,implicit_axis=True).set_index('origin')
e2_s6 = e2_rsev_df[['development','Latest','CDF','Ultimate']].rename(columns={'Latest':'Reported Severity','Ultimate':'Ult Severity'})
e2_s6['Selected Ult CC'] = e2_s3['Selected Ult CC']
e2_s6["Ult Claims"] = e2_ult.latest_diagonal.to_frame()
e2_s6['CDF'] = e2_s6['CDF'].round(3)
#using floating point offset to achieve standard rounding
e2_s6[['Reported Severity','Ult Severity','Ult Claims']] = (e2_s6[['Reported Severity','Ult Severity','Ult Claims']]+1e-9).round(0)
e2_s6

	development	Reported Severity	CDF	Ult Severity	Selected Ult CC	Ult Claims
origin
1998-01-01	132	24838.0	1.000	24838.0	637.0	15822.0
1999-01-01	120	23980.0	0.999	23956.0	1047.0	25083.0
2000-01-01	108	26453.0	0.990	26189.0	1416.0	37071.0
2001-01-01	96	26665.0	0.992	26452.0	1466.0	38772.0
2002-01-01	84	30997.0	1.003	31087.0	1565.0	48665.0
2003-01-01	72	27239.0	1.016	27674.0	1666.0	46106.0
2004-01-01	60	31128.0	1.066	33174.0	2309.0	76605.0
2005-01-01	48	29563.0	1.100	32514.0	2483.0	80739.0
2006-01-01	36	29453.0	1.212	35697.0	1807.0	64510.0
2007-01-01	24	25104.0	1.498	37607.0	1556.0	58528.0
2008-01-01	12	17985.0	2.310	41544.0	1425.0	59213.0

Exhibit II Sheet 7

summary_exh(e2_tri['Reported Claims'],e2_tri['Paid Claims'],e2_ult)

	Reported	Paid	Ult Claims	Case Outstanding	IBNR	Unpaid
1998-01-01	15822.0	15822.0	15822.0	0.0	0.0	0.0
1999-01-01	25107.0	24817.0	25083.0	290.0	-24.0	266.0
2000-01-01	37246.0	36782.0	37071.0	464.0	-175.0	289.0
2001-01-01	38798.0	38520.0	38772.0	278.0	-26.0	252.0
2002-01-01	48169.0	44438.0	48665.0	3731.0	496.0	4227.0
2003-01-01	44373.0	39320.0	46106.0	5053.0	1733.0	6786.0
2004-01-01	70288.0	52811.0	76605.0	17477.0	6317.0	23794.0
2005-01-01	70655.0	40026.0	80739.0	30629.0	10084.0	40713.0
2006-01-01	48804.0	22819.0	64510.0	25985.0	15706.0	41691.0
2007-01-01	31732.0	11865.0	58528.0	19867.0	26796.0	46663.0
2008-01-01	18632.0	3409.0	59213.0	15223.0	40581.0	55804.0

#Exhibit II Sheet 1
assert np.all(e2_ccc_selected.cdf_.round(3).values == np.array([6.085, 2.281, 1.612, 1.318, 1.161, 1.085, 1.035, 1.021, 1.015, 1.003, 1.000]))
#Exhibit II Sheet 2
assert np.all(e2_rcc_selected.cdf_.round(3).values == np.array([1.131, 1.035, 1.013, 1.005, 1.002, 1.001, 1.000, 1.000, 1.000, 1.000, 1.000]))
#Exhibit II Sheet 3
assert np.allclose(
    e2_s3['Selected Ult CC'].values,
    np.array([637, 1047, 1416, 1466, 1565, 1666, 2309, 2483, 1807, 1556, 1426]),
    atol=1
)
#Exhibit II Sheet 5
assert np.all(e2_rsev_selected.cdf_.round(3).values == np.array([2.310, 1.498, 1.212, 1.100, 1.066, 1.016, 1.003, 0.992, 0.990, 0.999, 1.000]))
#Exhibit II Sheet 6
assert np.allclose(
    e2_s6['Ult Severity'].values,
    np.array([24839, 23956, 26189, 26452, 31090, 27675, 33183, 32519, 35697, 37606, 41544]),
    rtol=0.001
)

Exhibit III Analysis

Exhibit III follows Approach 2, which adds explicit selections for the most recent frequency and severity. To calculate various averages for frequency and severity, we use the TriangleWeight utility class, and supply n_periods and drop_high as we would with Development. We also leverage the Trend adjustment method.

#loading data and assumptions
e3_tri = cl.load_sample('friedland_wc_self_insurer')
e3_cnt_assumptions = {}
e3_cnt_assumptions['simple_5'] = {'n_periods':5, 'average':'simple'}
e3_cnt_assumptions['simple_3'] = {'n_periods':3, 'average':'simple'}
e3_cnt_assumptions['medial_5x1'] = {'n_periods':5, 'average':'simple','drop_high':1, 'drop_low':1}
e3_cnt_assumptions['volume_5'] = {'n_periods':5, 'average':'volume'}
e3_cnt_assumptions['volume_3'] = {'n_periods':3, 'average':'volume'}
e3_sev_assumptions = {}
e3_sev_assumptions['simple_5'] = {'n_periods':5, 'average':'simple'}
e3_sev_assumptions['simple_3'] = {'n_periods':3, 'average':'simple'}
e3_sev_assumptions['medial_5x1'] = {'n_periods':5, 'average':'simple','drop_high':1, 'drop_low':1}
e3_sevavg_assumptions = {}
e3_sevavg_assumptions['all_years'] = {}
e3_sevavg_assumptions['all_years_excl_hilo'] = {'drop_high':1, 'drop_low':1}
e3_sevavg_assumptions['latest_3'] = {'n_periods':3}

#developing closed claim counts
#There is a typo in the text. We will use a tail of 1.002 to match the 84-ult factor'
e3_ccc_devs = average_dev(e3_tri['Closed Claim Counts'],e3_cnt_assumptions)
e3_ccc_selected = cl.TailConstant(tail = 1.002, projection_period = 0).fit_transform(e3_ccc_devs['volume_5'])
e3_ccc_selected.ldf_ = e3_ccc_selected.ldf_.round(3)

#developing reported claim counts
e3_rcc_devs = average_dev(e3_tri['Reported Claim Counts'],e3_cnt_assumptions)
e3_rcc_selected = cl.TailConstant(tail = 1.000, projection_period = 0).fit_transform(e3_rcc_devs['volume_5'])
e3_rcc_selected.ldf_ = e3_rcc_selected.ldf_.round(3)

#combining closed and reported claim counts
e3_ccc_cl = cl.Chainladder().fit(e3_ccc_selected)
e3_rcc_cl = cl.Chainladder().fit(e3_rcc_selected)
e3_cc_ult = (e3_ccc_cl.ultimate_ + e3_rcc_cl.ultimate_)/2

#calculating trended frequency
e3_freq_trend = cl.Trend(-.01).fit(e3_tri['Reported Claim Counts']).trend_
e3_payroll_trend = cl.Trend(.025).fit(e3_tri['Payroll']).trend_
e3_freq_2008 = .0036
e3_freq_2007 = e3_freq_2008 / e3_freq_trend.values[0,0,-2,0] * e3_payroll_trend.latest_diagonal.values[0,0,-2,0]

#calculating and developing paid severity
e3_psev = e3_tri["Paid Claims"] / e3_tri['Closed Claim Counts']
e3_psev_devs = average_dev(e3_psev,e3_sev_assumptions)
e3_psev_selected = cl.TailConstant(tail = 1.150, projection_period = 0).fit_transform(e3_psev_devs['medial_5x1'])
e3_psev_selected.ldf_ = e3_psev_selected.ldf_.round(3)

#calculating and developing reported severity
e3_rsev = e3_tri["Reported Claims"] / e3_tri['Reported Claim Counts']
e3_rsev_devs = average_dev(e3_rsev,e3_sev_assumptions)
e3_rsev_selected = cl.TailConstant(tail = 1.025, projection_period = 0).fit_transform(e3_rsev_devs['medial_5x1'])
e3_rsev_selected.ldf_ = e3_rsev_selected.ldf_.round(3)

#combining developed count and severity
e3_psev_cl = cl.Chainladder().fit(e3_psev_selected)
e3_resv_cl = cl.Chainladder().fit(e3_rsev_selected)
e3_sev_ult = (e3_psev_cl.ultimate_ + e3_resv_cl.ultimate_)/2

#calculated trended severity
e3_sev_trend = cl.Trend(.075).fit(e3_tri['Reported Claims']).trend_
e3_sev_ult_trended = e3_sev_ult * e3_sev_trend.latest_diagonal.values
#prep ult severity for average calculations
e3_sev_ult_trended_2006 = e3_sev_ult_trended[e3_sev_ult_trended.origin<='2006']
e3_sevs = average_sev(e3_sev_ult_trended_2006,e3_sevavg_assumptions)
#calculate selected severities
e3_sev_2008 = 7100.
e3_sev_2007 = e3_sev_2008 / e3_sev_trend.values[0,0,-2,0]

/home/docs/checkouts/readthedocs.org/user_builds/chainladder-python/envs/experimental/lib/python3.11/site-packages/chainladder/utils/triangle_weight.py:367: UserWarning: Some exclusions have been ignored. At least 1 (use preserve = ...) link ratio(s) is required for development estimation.
  warnings.warn(warning)
/home/docs/checkouts/readthedocs.org/user_builds/chainladder-python/envs/experimental/lib/python3.11/site-packages/chainladder/development/base.py:272: UserWarning: Some exclusions have been ignored. At least 1 (use preserve = ...) link ratio(s) is required for development estimation.
  warnings.warn(warning)
/home/docs/checkouts/readthedocs.org/user_builds/chainladder-python/envs/experimental/lib/python3.11/site-packages/chainladder/utils/triangle_weight.py:367: UserWarning: Some exclusions have been ignored. At least 1 (use preserve = ...) link ratio(s) is required for development estimation.
  warnings.warn(warning)
/home/docs/checkouts/readthedocs.org/user_builds/chainladder-python/envs/experimental/lib/python3.11/site-packages/chainladder/development/base.py:272: UserWarning: Some exclusions have been ignored. At least 1 (use preserve = ...) link ratio(s) is required for development estimation.
  warnings.warn(warning)
/home/docs/checkouts/readthedocs.org/user_builds/chainladder-python/envs/experimental/lib/python3.11/site-packages/chainladder/utils/triangle_weight.py:367: UserWarning: Some exclusions have been ignored. At least 1 (use preserve = ...) link ratio(s) is required for development estimation.
  warnings.warn(warning)
/home/docs/checkouts/readthedocs.org/user_builds/chainladder-python/envs/experimental/lib/python3.11/site-packages/chainladder/development/base.py:272: UserWarning: Some exclusions have been ignored. At least 1 (use preserve = ...) link ratio(s) is required for development estimation.
  warnings.warn(warning)
/home/docs/checkouts/readthedocs.org/user_builds/chainladder-python/envs/experimental/lib/python3.11/site-packages/chainladder/utils/triangle_weight.py:367: UserWarning: Some exclusions have been ignored. At least 1 (use preserve = ...) link ratio(s) is required for development estimation.
  warnings.warn(warning)
/home/docs/checkouts/readthedocs.org/user_builds/chainladder-python/envs/experimental/lib/python3.11/site-packages/chainladder/development/base.py:272: UserWarning: Some exclusions have been ignored. At least 1 (use preserve = ...) link ratio(s) is required for development estimation.
  warnings.warn(warning)

Exhibit III Sheet 1

print('PART 1 - Data Triangle')
nb_display(e3_tri["Closed Claim Counts"])
print('PART 2 - Age-to-Age Factors')
nb_display(e3_tri['Closed Claim Counts'].age_to_age.round(3))
print('PART 3 - Average Age-to-Age Factor')
nb_display(combine_ldf(e3_ccc_devs).round(3).to_frame())
print('PART 4 - Selected Age-to-Age Factors')
print('Selected')
nb_display(e3_ccc_selected.ldf_)
print('CDF to Ultimate')
nb_display(e3_ccc_selected.cdf_.round(3))
print('Percent Closed')
nb_display((1/e3_ccc_selected.cdf_).round(3))

PART 1 - Data Triangle

	12	24	36	48	60	72	84	96
2001	789	1,196	1,255	1,310	1,324	1,327	1,332	1,343
2002	978	1,506	1,609	1,629	1,669	1,676	1,683
2003	1,070	1,557	1,665	1,721	1,738	1,748
2004	1,029	1,525	1,618	1,688	1,717
2005	974	1,459	1,532	1,597
2006	1,746	2,632	2,761
2007	1,683	2,572
2008	1,560

PART 2 - Age-to-Age Factors

	12-24	24-36	36-48	48-60	60-72	72-84	84-96
2001	1.5160	1.0490	1.0440	1.0110	1.0020	1.0040	1.0080
2002	1.5400	1.0680	1.0120	1.0250	1.0040	1.0040
2003	1.4550	1.0690	1.0340	1.0100	1.0060
2004	1.4820	1.0610	1.0430	1.0170
2005	1.4980	1.0500	1.0420
2006	1.5070	1.0490
2007	1.5280

PART 3 - Average Age-to-Age Factor

development	12-24	24-36	36-48	48-60	60-72	72-84	84-96
Total
simple_5	1.494	1.060	1.035	1.016	1.004	1.004	1.008
simple_3	1.511	1.053	1.040	1.017	1.004	1.004	1.008
medial_5x1	1.496	1.060	1.040	1.014	1.004	1.004	1.008
volume_5	1.499	1.058	1.035	1.016	1.004	1.004	1.008
volume_3	1.513	1.053	1.040	1.017	1.004	1.004	1.008

PART 4 - Selected Age-to-Age Factors
Selected

	12-24	24-36	36-48	48-60	60-72	72-84	84-96	96-108
(All)	1.4990	1.0580	1.0350	1.0160	1.0040	1.0040	1.0080	1.0020

CDF to Ultimate

	12-Ult	24-Ult	36-Ult	48-Ult	60-Ult	72-Ult	84-Ult	96-Ult
(All)	1.6980	1.1330	1.0710	1.0340	1.0180	1.0140	1.0100	1.0020

Percent Closed

	12-Ult	24-Ult	36-Ult	48-Ult	60-Ult	72-Ult	84-Ult	96-Ult
(All)	0.5890	0.8830	0.9340	0.9670	0.9820	0.9860	0.9900	0.9980

Exhibit III Sheet 2

print('PART 1 - Data Triangle')
nb_display(e3_tri["Reported Claim Counts"])
print('PART 2 - Age-to-Age Factors')
nb_display(e3_tri['Reported Claim Counts'].age_to_age.round(3))
print('PART 3 - Average Age-to-Age Factor')
nb_display(combine_ldf(e3_rcc_devs).round(3).to_frame())
print('PART 4 - Selected Age-to-Age Factors')
nb_display(e3_rcc_selected.ldf_)
print('CDF to Ultimate')
nb_display(e3_rcc_selected.cdf_.round(3))
print('Percent Reported')
nb_display((1/e3_rcc_selected.cdf_).round(3))

PART 1 - Data Triangle

	12	24	36	48	60	72	84	96
2001	1,235	1,321	1,342	1,349	1,350	1,350	1,350	1,350
2002	1,555	1,660	1,685	1,695	1,700	1,700	1,700
2003	1,628	1,740	1,762	1,771	1,775	1,775
2004	1,600	1,714	1,740	1,747	1,750
2005	1,510	1,612	1,639	1,647
2006	2,750	2,941	2,985
2007	2,650	2,842
2008	2,438

PART 2 - Age-to-Age Factors

	12-24	24-36	36-48	48-60	60-72	72-84	84-96
2001	1.0700	1.0160	1.0050	1.0010	1.0000	1.0000	1.0000
2002	1.0680	1.0150	1.0060	1.0030	1.0000	1.0000
2003	1.0690	1.0130	1.0050	1.0020	1.0000
2004	1.0710	1.0150	1.0040	1.0020
2005	1.0680	1.0170	1.0050
2006	1.0690	1.0150
2007	1.0720

PART 3 - Average Age-to-Age Factor

development	12-24	24-36	36-48	48-60	60-72	72-84	84-96
Total
simple_5	1.07	1.015	1.005	1.002	1.0	1.0	1.0
simple_3	1.07	1.016	1.005	1.002	1.0	1.0	1.0
medial_5x1	1.07	1.015	1.005	1.002	1.0	1.0	1.0
volume_5	1.07	1.015	1.005	1.002	1.0	1.0	1.0
volume_3	1.07	1.015	1.005	1.002	1.0	1.0	1.0

PART 4 - Selected Age-to-Age Factors

	12-24	24-36	36-48	48-60	60-72	72-84	84-96	96-108
(All)	1.0700	1.0150	1.0050	1.0020	1.0000	1.0000	1.0000	1.0000

CDF to Ultimate

	12-Ult	24-Ult	36-Ult	48-Ult	60-Ult	72-Ult	84-Ult	96-Ult
(All)	1.0940	1.0220	1.0070	1.0020	1.0000	1.0000	1.0000	1.0000

Percent Reported

	12-Ult	24-Ult	36-Ult	48-Ult	60-Ult	72-Ult	84-Ult	96-Ult
(All)	0.9140	0.9780	0.9930	0.9980	1.0000	1.0000	1.0000	1.0000

Exhibit III Sheet 3

e3_ccc_df = cl.model_diagnostics(e3_ccc_cl).to_frame(keepdims=True,implicit_axis=True).set_index('origin')
e3_rcc_df = cl.model_diagnostics(e3_rcc_cl).to_frame(keepdims=True,implicit_axis=True).set_index('origin')
e3_s3 = e3_ccc_df[['development','Latest','CDF','Ultimate']].rename(columns={'Latest':'Closed Claim Counts','CDF':'Closed CDF','Ultimate':'Ult Count Using CCC'})
e3_s3[['Reported Claim Counts','Reported CDF','Ult Count Using RCC']] = e3_rcc_df[['Latest','CDF','Ultimate']]
e3_s3["Selected Ult CC"] = e3_cc_ult.latest_diagonal.to_frame()
e3_s3[['Closed CDF','Reported CDF']] = e3_s3[['Closed CDF','Reported CDF']].round(3)
#using floating point offset to achieve standard rounding
e3_s3[['Ult Count Using CCC','Ult Count Using RCC','Selected Ult CC']] = (e3_s3[['Ult Count Using CCC','Ult Count Using RCC','Selected Ult CC']] + 1e-9).round(0)
e3_s3

	development	Closed Claim Counts	Closed CDF	Ult Count Using CCC	Reported Claim Counts	Reported CDF	Ult Count Using RCC	Selected Ult CC
origin
2001-01-01	96	1343.0	1.002	1346.0	1350.0	1.000	1350.0	1348.0
2002-01-01	84	1683.0	1.010	1700.0	1700.0	1.000	1700.0	1700.0
2003-01-01	72	1748.0	1.014	1773.0	1775.0	1.000	1775.0	1774.0
2004-01-01	60	1717.0	1.018	1748.0	1750.0	1.000	1750.0	1749.0
2005-01-01	48	1597.0	1.034	1652.0	1647.0	1.002	1650.0	1651.0
2006-01-01	36	2761.0	1.071	2956.0	2985.0	1.007	3006.0	2981.0
2007-01-01	24	2572.0	1.133	2913.0	2842.0	1.022	2905.0	2909.0
2008-01-01	12	1560.0	1.698	2649.0	2438.0	1.094	2666.0	2658.0

Exhibit III Sheet 4

e3_s4 = e3_s3[['Selected Ult CC']].copy()
e3_s4['CC Trend'] = e3_freq_trend.latest_diagonal.to_frame()
e3_s4['Trended Ult CC'] = e3_s4['Selected Ult CC'] * e3_s4['CC Trend']
e3_s4['Payroll'] = e3_tri['Payroll'].latest_diagonal.to_frame()
e3_s4['Payroll Trend'] = e3_payroll_trend.latest_diagonal.to_frame()
e3_s4['Trended Payroll'] = e3_s4['Payroll'] * e3_s4['Payroll Trend']
e3_s4['Trended Ultimate Freq'] = e3_s4['Trended Ult CC'] / e3_s4['Trended Payroll']
e3_s4[['CC Trend','Payroll Trend']] = e3_s4[['CC Trend','Payroll Trend']].round(3)
#using floating point offset to achieve standard rounding
e3_s4[['Trended Ult CC','Trended Payroll']] = (e3_s4[['Trended Ult CC','Trended Payroll']] + 1e-9).round(0)
e3_s4['Trended Ultimate Freq'] = e3_s4["Trended Ultimate Freq"].map("{:.2%}".format)
nb_display(e3_s4)
print(f"Selected Frequency at 2008 level {e3_freq_2008:.2%}")
print(f"Selected Frequency at 2007 level {float(e3_freq_2007):.2%}")

	Selected Ult CC	CC Trend	Trended Ult CC	Payroll	Payroll Trend	Trended Payroll	Trended Ultimate Freq
origin
2001-01-01	1348.0	0.932	1256.0	195000.0	1.189	231794.0	0.54%
2002-01-01	1700.0	0.941	1601.0	260000.0	1.160	301520.0	0.53%
2003-01-01	1774.0	0.951	1687.0	280000.0	1.131	316794.0	0.53%
2004-01-01	1749.0	0.961	1680.0	280000.0	1.104	309068.0	0.54%
2005-01-01	1651.0	0.970	1602.0	350000.0	1.077	376912.0	0.43%
2006-01-01	2981.0	0.980	2922.0	790000.0	1.051	829994.0	0.35%
2007-01-01	2909.0	0.990	2880.0	780000.0	1.025	799500.0	0.36%
2008-01-01	2658.0	1.000	2658.0	740000.0	1.000	740000.0	0.36%

Selected Frequency at 2008 level 0.36%
Selected Frequency at 2007 level 0.37%

Exhibit III Sheet 5

print('Paid Claims')
nb_display(e3_tri["Paid Claims"])
print('Paid Severities')
nb_display(e3_psev)
print('Reported Claims')
nb_display(e3_tri["Reported Claims"])
print('Reported Severities')
nb_display(e3_rsev)

Paid Claims

	12	24	36	48	60	72	84	96
2001	1,318,000	2,842,000	3,750,000	4,300,000	4,650,000	4,850,000	5,050,000	5,200,000
2002	1,780,000	3,817,000	5,016,000	5,750,000	6,100,000	6,300,000	6,555,000
2003	1,890,000	4,184,000	5,500,000	6,300,000	6,800,000	7,100,000
2004	1,900,000	4,100,000	5,560,000	6,430,000	6,950,000
2005	1,960,000	4,290,000	5,688,000	6,570,000
2006	4,030,000	8,650,000	11,400,000
2007	4,200,000	9,043,000
2008	4,170,000

Paid Severities

	12	24	36	48	60	72	84	96
2001	1,670	2,376	2,988	3,282	3,512	3,655	3,791	3,872
2002	1,820	2,535	3,117	3,530	3,655	3,759	3,895
2003	1,766	2,687	3,303	3,661	3,913	4,062
2004	1,846	2,689	3,436	3,809	4,048
2005	2,012	2,940	3,713	4,114
2006	2,308	3,286	4,129
2007	2,496	3,516
2008	2,673

Reported Claims

	12	24	36	48	60	72	84	96
2001	3,200,000	4,300,000	4,900,000	5,200,000	5,300,000	5,400,000	5,550,000	5,650,000
2002	4,300,000	5,900,000	6,600,000	6,950,000	7,200,000	7,400,000	7,500,000
2003	4,800,000	6,600,000	7,400,000	7,800,000	8,100,000	8,300,000
2004	4,900,000	6,700,000	7,700,000	8,150,000	8,600,000
2005	5,200,000	7,100,000	7,900,000	8,350,000
2006	10,100,000	13,800,000	15,500,000
2007	10,500,000	14,400,000
2008	10,300,000

Reported Severities

	12	24	36	48	60	72	84	96
2001	2,591	3,255	3,651	3,855	3,926	4,000	4,111	4,185
2002	2,765	3,554	3,917	4,100	4,235	4,353	4,412
2003	2,948	3,793	4,200	4,404	4,563	4,676
2004	3,062	3,909	4,425	4,665	4,914
2005	3,444	4,404	4,820	5,070
2006	3,673	4,692	5,193
2007	3,962	5,067
2008	4,225

Exhibit III Sheet 6

print('PART 1 - Data Triangle')
nb_display(e3_psev)
print('PART 2 - Age-to-Age Factors')
nb_display(e3_psev.age_to_age.round(3))
print('PART 3 - Average Age-to-Age Factor')
nb_display(combine_ldf(e3_psev_devs).round(3).to_frame())
print('PART 4 - Selected Age-to-Age Factors')
print('Selected')
nb_display(e3_psev_selected.ldf_)
print('CDF to Ultimate')
nb_display(e3_psev_selected.cdf_.round(3))

PART 1 - Data Triangle

	12	24	36	48	60	72	84	96
2001	1,670	2,376	2,988	3,282	3,512	3,655	3,791	3,872
2002	1,820	2,535	3,117	3,530	3,655	3,759	3,895
2003	1,766	2,687	3,303	3,661	3,913	4,062
2004	1,846	2,689	3,436	3,809	4,048
2005	2,012	2,940	3,713	4,114
2006	2,308	3,286	4,129
2007	2,496	3,516
2008	2,673

PART 2 - Age-to-Age Factors

	12-24	24-36	36-48	48-60	60-72	72-84	84-96
2001	1.4230	1.2570	1.0990	1.0700	1.0410	1.0370	1.0210
2002	1.3930	1.2300	1.1320	1.0350	1.0280	1.0360
2003	1.5210	1.2290	1.1080	1.0690	1.0380
2004	1.4560	1.2780	1.1090	1.0630
2005	1.4610	1.2630	1.1080
2006	1.4240	1.2560
2007	1.4090

PART 3 - Average Age-to-Age Factor

development	12-24	24-36	36-48	48-60	60-72	72-84	84-96
Total
simple_5	1.454	1.251	1.111	1.059	1.036	1.037	1.021
simple_3	1.431	1.266	1.108	1.056	1.036	1.037	1.021
medial_5x1	1.447	1.250	1.108	1.066	1.038	1.037	1.021

PART 4 - Selected Age-to-Age Factors
Selected

	12-24	24-36	36-48	48-60	60-72	72-84	84-96	96-108
(All)	1.4470	1.2500	1.1080	1.0660	1.0380	1.0370	1.0210	1.1500

CDF to Ultimate

	12-Ult	24-Ult	36-Ult	48-Ult	60-Ult	72-Ult	84-Ult	96-Ult
(All)	2.7000	1.8660	1.4930	1.3470	1.2640	1.2180	1.1740	1.1500

Exhibit III Sheet 7

print('PART 1 - Data Triangle')
nb_display(e3_rsev)
print('PART 2 - Age-to-Age Factors')
nb_display(e3_rsev.age_to_age.round(3))
print('PART 3 - Average Age-to-Age Factor')
nb_display(combine_ldf(e3_rsev_devs).round(3).to_frame())
print('PART 4 - Selected Age-to-Age Factors')
print('Selected')
nb_display(e3_rsev_selected.ldf_)
print('CDF to Ultimate')
nb_display(e3_rsev_selected.cdf_.round(3))

PART 1 - Data Triangle

	12	24	36	48	60	72	84	96
2001	2,591	3,255	3,651	3,855	3,926	4,000	4,111	4,185
2002	2,765	3,554	3,917	4,100	4,235	4,353	4,412
2003	2,948	3,793	4,200	4,404	4,563	4,676
2004	3,062	3,909	4,425	4,665	4,914
2005	3,444	4,404	4,820	5,070
2006	3,673	4,692	5,193
2007	3,962	5,067
2008	4,225

PART 2 - Age-to-Age Factors

	12-24	24-36	36-48	48-60	60-72	72-84	84-96
2001	1.2560	1.1220	1.0560	1.0180	1.0190	1.0280	1.0180
2002	1.2850	1.1020	1.0470	1.0330	1.0280	1.0140
2003	1.2860	1.1070	1.0490	1.0360	1.0250
2004	1.2760	1.1320	1.0540	1.0530
2005	1.2790	1.0940	1.0520
2006	1.2780	1.1070
2007	1.2790

PART 3 - Average Age-to-Age Factor

development	12-24	24-36	36-48	48-60	60-72	72-84	84-96
Total
simple_5	1.280	1.108	1.051	1.035	1.024	1.021	1.018
simple_3	1.278	1.111	1.052	1.041	1.024	1.021	1.018
medial_5x1	1.278	1.105	1.052	1.035	1.025	1.021	1.018

PART 4 - Selected Age-to-Age Factors
Selected

	12-24	24-36	36-48	48-60	60-72	72-84	84-96	96-108
(All)	1.2780	1.1050	1.0520	1.0350	1.0250	1.0210	1.0180	1.0250

CDF to Ultimate

	12-Ult	24-Ult	36-Ult	48-Ult	60-Ult	72-Ult	84-Ult	96-Ult
(All)	1.6790	1.3140	1.1890	1.1300	1.0920	1.0650	1.0430	1.0250

Exhibit III Sheet 8

e3_psev_df = cl.model_diagnostics(e3_psev_cl).to_frame(keepdims=True,implicit_axis=True).set_index('origin')
e3_resv_df = cl.model_diagnostics(e3_resv_cl).to_frame(keepdims=True,implicit_axis=True).set_index('origin')
e3_s8 = e3_psev_df[['development','Latest','CDF','Ultimate']].rename(columns={'Latest':'Paid Sev','CDF':'Paid CDF','Ultimate':'Ult Paid Sev'})
e3_s8[['Reported Sev','Reported CDF','Ult Reported Sev']] = e3_resv_df[['Latest','CDF','Ultimate']]
e3_s8["Selected Ult Sev"] = e3_sev_ult.latest_diagonal.to_frame()
e3_s8[['Paid CDF','Reported CDF']] = e3_s8[['Paid CDF','Reported CDF']].round(3)
#using floating point offset to achieve standard rounding
e3_s8[['Paid Sev','Reported Sev','Ult Paid Sev','Ult Reported Sev','Selected Ult Sev']] = (e3_s8[['Paid Sev','Reported Sev','Ult Paid Sev','Ult Reported Sev','Selected Ult Sev']] + 1e-9).round(0)
e3_s8.iloc[:-2,:][['development','Paid Sev','Reported Sev','Paid CDF','Reported CDF','Ult Paid Sev','Ult Reported Sev','Selected Ult Sev']]

	development	Paid Sev	Reported Sev	Paid CDF	Reported CDF	Ult Paid Sev	Ult Reported Sev	Selected Ult Sev
origin
2001-01-01	96	3872.0	4185.0	1.150	1.025	4453.0	4290.0	4371.0
2002-01-01	84	3895.0	4412.0	1.174	1.043	4573.0	4603.0	4588.0
2003-01-01	72	4062.0	4676.0	1.218	1.065	4946.0	4982.0	4964.0
2004-01-01	60	4048.0	4914.0	1.264	1.092	5116.0	5366.0	5241.0
2005-01-01	48	4114.0	5070.0	1.347	1.130	5543.0	5730.0	5636.0
2006-01-01	36	4129.0	5193.0	1.493	1.189	6164.0	6174.0	6169.0

Exhibit III Sheet 9

e3_s9 = e3_s8[['Selected Ult Sev']].copy()
e3_s9['Sev Trend'] = e3_sev_trend.latest_diagonal.to_frame()
e3_s9['Trended Ult Sev'] = e3_sev_ult_trended.latest_diagonal.to_frame()
e3_s9[['Sev Trend']] = e3_s9[['Sev Trend']].round(3)
#using floating point offset to achieve standard rounding
e3_s9[['Trended Ult Sev']] = (e3_s9[['Trended Ult Sev']] + 1e-9).round(0)
nb_display(e3_s9.iloc[:-2,:])
print('Average Trended Severity at 2008 Cost Level')
for k,v in e3_sevs.items():
    print(f"\t{k:<20} \t{v.round(0)}")
print(f"Selected 2008 Severity \t\t{e3_sev_2008:.0f}")
print(f"Estimated 2007 Severity \t{e3_sev_2007:.0f}")

	Selected Ult Sev	Sev Trend	Trended Ult Sev
origin
2001-01-01	4371.0	1.659	7252.0
2002-01-01	4588.0	1.543	7081.0
2003-01-01	4964.0	1.436	7126.0
2004-01-01	5241.0	1.335	6999.0
2005-01-01	5636.0	1.242	7002.0
2006-01-01	6169.0	1.156	7129.0

Average Trended Severity at 2008 Cost Level
	all_years            	7098.0
	all_years_excl_hilo  	7084.0
	latest_3             	7043.0
Selected 2008 Severity 		7100
Estimated 2007 Severity 	6605

Exhibit III Sheet 10

e3_s10 = e3_s4[['Payroll']].copy()[-2:]
e3_s10['Selected Frequency'] = [e3_freq_2007,e3_freq_2008]
e3_s10['Ult CC'] = e3_s10['Payroll'] * e3_s10['Selected Frequency']
e3_s10['Selected Severity'] = [e3_sev_2007,e3_sev_2008]
e3_s10['Ult Claims'] = e3_s10['Ult CC'] * e3_s10['Selected Severity']
e3_s10['Reported Claims'] = e3_tri['Reported Claims'].latest_diagonal.to_frame()
e3_s10['Paid Claims'] = e3_tri['Paid Claims'].latest_diagonal.to_frame()
e3_s10['Case Outstanding'] = e3_s10['Reported Claims'] - e3_s10['Paid Claims']
e3_s10['IBNR'] = e3_s10['Ult Claims'] - e3_s10['Reported Claims']
e3_s10['Unpaid'] = e3_s10['IBNR'] + e3_s10['Case Outstanding']
e3_s10.T

origin	2007-01-01	2008-01-01
Payroll	7.800000e+05	7.400000e+05
Selected Frequency	3.727273e-03	3.600000e-03
Ult CC	2.907273e+03	2.664000e+03
Selected Severity	6.604651e+03	7.100000e+03
Ult Claims	1.920152e+07	1.891440e+07
Reported Claims	1.440000e+07	1.030000e+07
Paid Claims	9.043000e+06	4.170000e+06
Case Outstanding	5.357000e+06	6.130000e+06
IBNR	4.801522e+06	8.614400e+06
Unpaid	1.015852e+07	1.474440e+07

#Exhibit III Sheet 1
assert np.all(e3_ccc_selected.cdf_.round(3).values[...,:7] == np.array([1.698, 1.133, 1.071, 1.034, 1.018, 1.014, 1.010]))
#Exhibit III Sheet 2
assert np.all(e3_rcc_selected.cdf_.round(3).values == np.array([1.094, 1.022, 1.007, 1.002, 1.000, 1.000, 1.000, 1.000]))
#Exhibit III Sheet 3
assert np.allclose(
    e3_s3['Selected Ult CC'].values[...,1:],
    np.array([1700, 1774, 1749, 1651, 2982, 2909, 2658]),
    atol=1
)
#Exhibit III Sheet 4
assert np.all(e3_s4['Trended Ultimate Freq'].values[...,1:] == np.array(['0.53%', '0.53%', '0.54%', '0.43%', '0.35%', '0.36%', '0.36%']))
#Exhibit III Sheet 6
assert np.allclose(
    e3_psev_selected.ldf_.round(3).values,
    np.array([1.447, 1.249, 1.108, 1.066, 1.038, 1.037, 1.021, 1.150]),
    atol=.001
)
#Exhibit III Sheet 7
assert np.all(e3_rsev_selected.cdf_.round(3).values == np.array([1.679, 1.314, 1.189, 1.130, 1.092, 1.065, 1.043, 1.025]))
#Exhibit III Sheet 8
assert np.allclose(
    e3_s8['Selected Ult Sev'].values[:-2],
    np.array([4371, 4587, 4963, 5242, 5635, 6169]),
    atol=1
)
#Exhibit III Sheet 9
assert np.allclose(
    e3_s9['Trended Ult Sev'].values[:-2],
    np.array([7251, 7079, 7125, 7000, 7001, 7129]),
    rtol=0.001
)

Exhibit IV Analysis

Exhibit IV continues to follow Approach 2. We demonstrate two separate ways in the package to achievement the on-leveling and law-change adjustments. Also, there are no full-size triangles in this exhibit. This gives us an opportunity to demontrate utilizing columns in Triangle, as if we were in Pandas.

#loading data and assumptions
xyz_tort_data = pd.DataFrame({
    "EffDate": ["2006-01-01", "2007-01-01"],
    "RateChange": [(0.67/0.75) - 1, 0.75-1]
})
xyz_tort_adjustment = cl.ParallelogramOLF(
    rate_history=xyz_tort_data,
    change_col="RateChange",
    date_col="EffDate",
    vertical_line=True
)
#Both Chapter 6 and Chapter 8 contain the code to derive these on-level factors from underlying rate changes. We will simply use the factors as given
olf = cl.Triangle(
    data = {
        'origin':[1998,2002,2003,2004,2005,2006,2007,2008],
        'valuation':[2008,2008,2008,2008,2008,2008,2008,2008],
        'On-Level Adjustment':[0,.914,.87,.81,.704,.64,.8,1],
    },
    origin='origin',
    development='valuation',
    columns='On-Level Adjustment'
)
e4_freq_assumptions = {}
e4_freq_assumptions['all_years'] = {}
e4_freq_assumptions['all_years_excl_hilo'] = {'drop_high':1, 'drop_low':1}
e4_freq_assumptions['latest_2'] = {'n_periods':2}
e4_sev_assumptions = {}
e4_sev_assumptions['latest_5_years'] = {'n_periods':5}
e4_sev_assumptions['latest_5_years_excl_hilo'] = {'n_periods':5,'drop_high':1, 'drop_low':1}
e4_sev_assumptions['latest_3'] = {'n_periods':3}

# In Exhibit II, the selected ultimate was actually the average of reported and closed ultimates.
e4_s1_tri = e2_rcc_cl.ultimate_.copy()
e4_s1_tri = e4_s1_tri.rename(axis='columns',value='Ult CC')
e4_s1_tri['CC Trend'] = cl.Trend(-.015,dates=('2008-12-31','2002-01-01')).fit(e4_s1_tri).trend_
e4_s1_tri['Trended Ult CC'] = e4_s1_tri['CC Trend'] * e4_s1_tri['Ult CC']
e4_s1_tri['Earned Premium'] = e2_tri['Earned Premium'].latest_diagonal
e4_s1_tri['On-Level Adjustment'] = olf['On-Level Adjustment']
e4_s1_tri['On-Level Premium'] = e4_s1_tri['On-Level Adjustment'] * e4_s1_tri['Earned Premium']
e4_s1_tri['Trended Ult Freq'] = e4_s1_tri['Trended Ult CC'] / e4_s1_tri['On-Level Premium']
e4_s1_tri_2006 = e4_s1_tri.iloc[:,:,4:9,:]
#calculate average frequencies
e4_freq = average_sev(e4_s1_tri_2006['Trended Ult Freq'],e4_freq_assumptions)
#calculate selected frequencies
e4_freq_2008 = e4_freq['latest_2'].round(4)
e4_freq_2007 = e4_freq_2008 / e4_s1_tri['CC Trend'].values[0,0,-2,0] * e4_s1_tri['On-Level Adjustment'].values[0,0,-2,0]
e4_freq_2007 = e4_freq_2007.round(4)

e4_s2_tri = e2_rsev_cl.ultimate_.copy()
e4_s2_tri = e4_s2_tri.rename(axis='columns',value='Ult Sev')
e4_s2_tri['Sev Trend'] = cl.Trend(.05,dates=('2008-12-31','1998-01-01')).fit(e4_s2_tri).trend_.round(3)
#from Chapter 8 Exhibit III sheet 1
e4_s2_tri['Tort Reform Factors'] = xyz_tort_adjustment.fit(e4_s2_tri['Ult Sev']).olf_
e4_s2_tri['Trended Ult Sev'] = e4_s2_tri['Ult Sev'] * e4_s2_tri['Sev Trend'] * e4_s2_tri['Tort Reform Factors']
e4_s2_tri_2006 = e4_s2_tri[e4_s2_tri.origin <= '2006']
#calculate average severities
e4_sev = average_sev(e4_s2_tri_2006['Trended Ult Sev'],e4_sev_assumptions)
#calculate selected severities
e4_sev_2008 = e4_sev['latest_5_years_excl_hilo'].round(0)
e4_sev_2007 = e4_sev_2008 / e4_s2_tri['Sev Trend'].values[0,0,-2,0] / e4_s2_tri['Tort Reform Factors'].values[0,0,-2,0]
e4_sev_2007 = e4_sev_2007.round(0)

/home/docs/checkouts/readthedocs.org/user_builds/chainladder-python/envs/experimental/lib/python3.11/site-packages/chainladder/core/triangle.py:480: UserWarning: 
                The cumulative property of your triangle is not set. This may result in
                undesirable behavior. In a future release this will result in an error.
                
  warnings.warn(

Exhibit IV Sheet 1

e4_s1 = e4_s1_tri.to_frame().T
e4_s1[['CC Trend','On-Level Adjustment']] = e4_s1[['CC Trend','On-Level Adjustment']].round(3)
#using floating point offset to achieve standard rounding
e4_s1[['Trended Ult CC','On-Level Premium']] = (e4_s1[['Trended Ult CC','On-Level Premium']] + 1e-9).round(0)
e4_s1['Trended Ult Freq'] = e4_s1["Trended Ult Freq"].map("{:.2%}".format)
nb_display(e4_s1.iloc[4:9])
print('Average Trended Frequency at 2008 Cost Level')
for k,v in e4_freq.items():
    print(f"\t{k:<20} \t{v:.2%}")
print(f"Selected 2008 Frequency \t{e4_freq_2008:.2%}")
print(f"Estimated 2007 Frequency \t{e4_freq_2007:.2%}")

	Ult CC	CC Trend	Trended Ult CC	Earned Premium	On-Level Adjustment	On-Level Premium	Trended Ult Freq
origin
2002	1554.000000	0.913	1419.0	61183.0	0.914	55921.0	2.54%
2003	1630.629000	0.927	1512.0	69175.0	0.870	60182.0	2.51%
2004	2262.518258	0.941	2130.0	99322.0	0.810	80451.0	2.65%
2005	2401.966737	0.956	2295.0	138151.0	0.704	97258.0	2.36%
2006	1678.618977	0.970	1629.0	107578.0	0.640	68850.0	2.37%

Average Trended Frequency at 2008 Cost Level
	all_years            	2.48%
	all_years_excl_hilo  	2.47%
	latest_2             	2.36%
Selected 2008 Frequency 	2.36%
Estimated 2007 Frequency 	1.92%

Exhibit IV Sheet 2

e4_s2 = e4_s2_tri.to_frame().T
e4_s2[['Sev Trend','Tort Reform Factors']] = e4_s2[['Sev Trend','Tort Reform Factors']].round(3)
#using floating point offset to achieve standard rounding
e4_s2[['Trended Ult Sev','Ult Sev']] = (e4_s2[['Trended Ult Sev','Ult Sev']] + 1e-9).round(0)
nb_display(e4_s2.iloc[:9])
print('Average Trended Severity at 2008 Cost Level')
for k,v in e4_sev.items():
    print(f"\t{k:<25} \t{v.round(0)}")
print(f"Selected 2008 Severity \t\t\t{e4_sev_2008}")
print(f"Estimated 2007 Severity \t\t{e4_sev_2007}")

	Ult Sev	Sev Trend	Tort Reform Factors	Trended Ult Sev
origin
1998	24838.0	1.629	0.67	27109.0
1999	23956.0	1.551	0.67	24894.0
2000	26189.0	1.477	0.67	25916.0
2001	26452.0	1.407	0.67	24936.0
2002	31087.0	1.340	0.67	27910.0
2003	27674.0	1.276	0.67	23659.0
2004	33174.0	1.216	0.67	27028.0
2005	32514.0	1.158	0.67	25226.0
2006	35697.0	1.102	0.75	29504.0

Average Trended Severity at 2008 Cost Level
	latest_5_years            	26665.0
	latest_5_years_excl_hilo  	26721.0
	latest_3                  	27253.0
Selected 2008 Severity 			26721.0
Estimated 2007 Severity 		25449.0

Exhibit IV Sheet 3

e4_s3 = e4_s1_tri['Earned Premium'].to_frame(keepdims=True).set_index('origin').iloc[-2:][['Earned Premium']]
e4_s3['Selected Frequency'] = [e4_freq_2007,e4_freq_2008]
e4_s3['Ult CC'] = e4_s3['Earned Premium'] * e4_s3['Selected Frequency']
e4_s3['Selected Severity'] = [e4_sev_2007,e4_sev_2008]
e4_s3['Ult Claims'] = e4_s3['Ult CC'] * e4_s3['Selected Severity']
e4_s3['Reported Claims'] = (e2_tri['Reported Claims']*1000).latest_diagonal.to_frame()
e4_s3['Paid Claims'] = (e2_tri['Paid Claims']*1000).latest_diagonal.to_frame()
e4_s3['Case Outstanding'] = e4_s3['Reported Claims'] - e4_s3['Paid Claims']
e4_s3['IBNR'] = e4_s3['Ult Claims'] - e4_s3['Reported Claims']
e4_s3['Unpaid'] = e4_s3['IBNR'] + e4_s3['Case Outstanding']
e4_s3.T

origin	2007-01-01	2008-01-01
Earned Premium	6.243800e+04	4.779700e+04
Selected Frequency	1.920000e-02	2.360000e-02
Ult CC	1.198810e+03	1.128009e+03
Selected Severity	2.544900e+04	2.672100e+04
Ult Claims	3.050851e+07	3.014153e+07
Reported Claims	3.173200e+07	1.863200e+07
Paid Claims	1.186472e+07	3.408876e+06
Case Outstanding	1.986728e+07	1.522312e+07
IBNR	-1.223494e+06	1.150953e+07
Unpaid	1.864379e+07	2.673266e+07

#Exhibit IV Sheet 1
assert np.all(e4_s1.iloc[4:9]['Trended Ult Freq'].values == np.array(['2.54%', '2.51%', '2.65%', '2.36%', '2.37%']))
#Exhibit IV Sheet 2
assert np.allclose(
    list(e4_sev.values()),
    np.array([26669., 26720., 27254.]),
    rtol=0.001
)
#Exhibit IV Sheet 3
assert np.allclose(
    e4_s3.iloc[:,4].values,
    np.array([30512152., 30140260.]),
    rtol=0.001
)

Exhibit V Analysis

Exhibit V follows Approach 3, which uses full incremental closed count and paid triangles to estimate the ultimate. Full incremental closed count triangle is estimated after first estimating ultimate count using reported count. We use the DisposalRate adjustment method to perform this calculation. Full incremental paid severity triangle is estimated through trending.

#loading data and assumptions
e5_tri = cl.load_sample('friedland_gl_insurer')
e5_cnt_assumptions = {}
e5_cnt_assumptions['simple_5'] = {'n_periods':5, 'average':'simple'}
e5_cnt_assumptions['simple_3'] = {'n_periods':3, 'average':'simple'}
e5_cnt_assumptions['medial_5x1'] = {'n_periods':5, 'average':'simple','drop_high':1, 'drop_low':1}
e5_cnt_assumptions['volume_5'] = {'n_periods':5, 'average':'volume'}
e5_cnt_assumptions['volume_3'] = {'n_periods':3, 'average':'volume'}
e5_disp_assumptions = {}
e5_disp_assumptions['simple_5'] = {'n_periods':5, 'average':'simple'}
e5_disp_assumptions['simple_3'] = {'n_periods':3, 'average':'simple'}
e5_disp_assumptions['medial_5x1'] = {'n_periods':5, 'average':'simple','drop_high':1, 'drop_low':1}
e5_trend_assumptions = {}
e5_trend_assumptions['all_years'] = {}
e5_trend_assumptions['latest_6'] = {'n_periods':6}
e5_trend_assumptions['latest_4'] = {'n_periods':4}
e5_sev_assumptions = {}
e5_sev_assumptions['simple_5'] = {'n_periods':5}
e5_sev_assumptions['simple_3'] = {'n_periods':3}
e5_sev_assumptions['medial_5x1'] = {'n_periods':5,'drop_high':1, 'drop_low':1}

#developing closed claim counts
e5_ccc_devs = average_dev(e5_tri['Closed Claim Counts'],e5_cnt_assumptions)
e5_ccc_selected = cl.TailConstant(tail = 1.100, projection_period = 0).fit_transform(e5_ccc_devs['volume_3'])
e5_ccc_selected.ldf_ = e5_ccc_selected.ldf_.round(3)

#developing reported claim counts
e5_rcc_devs = average_dev(e5_tri['Reported Claim Counts'],e5_cnt_assumptions)
e5_rcc_selected = cl.TailConstant(tail = 1.0, projection_period = 0).fit_transform(e5_rcc_devs['volume_3'])
e5_rcc_selected.ldf_ = e5_rcc_selected.ldf_.round(3)

#combining closed and reported claim counts
e5_ccc_cl = cl.Chainladder().fit(e5_ccc_selected)
e5_rcc_cl = cl.Chainladder().fit(e5_rcc_selected)
e5_cc_ult = ((e5_ccc_cl.ultimate_ + e5_rcc_cl.ultimate_)/2)

#calculating disposal rate and complete the closed count triangle
e5_drs = average_dr(e5_tri['Closed Claim Counts'],e5_cc_ult,e5_disp_assumptions)
e5_dr_selected = e5_drs['medial_5x1']
e5_dr_selected.disposal_rate_ = e5_dr_selected.disposal_rate_.round(3)
e5_dr_tri = e5_dr_selected.transform(e5_tri['Closed Claim Counts'],sample_weight = e5_cc_ult)

#calculate and select incremental paid severity
e5_ipsev = e5_tri["Paid Claims"].cum_to_incr() / e5_tri["Closed Claim Counts"].cum_to_incr()
e5_regs = regs(e5_ipsev,e5_trend_assumptions)
e5_trend,e5_rsq = reg_outputs(e5_regs,e5_ipsev.development)
e5_regs_ex2001 = regs(e5_ipsev[e5_ipsev.origin>'2001'],{'all_years_ex_2001':{}})
e5_trend_ex2001,e5_rsq_ex2001 = reg_outputs(e5_regs_ex2001,e5_ipsev.development)
e5_sevs = average_sev(e5_ipsev.trend(0.05),e5_sev_assumptions)
e5_sevs_sel = e5_sevs['simple_3'].copy()
e5_sevs_sel.origin = ['2008']

#calculate and select tail paid severity
e5_incr_ccc = e5_tri["Closed Claim Counts"].cum_to_incr()
e5_incr_paid = e5_tri["Paid Claims"].cum_to_incr()
e5_incr_paid_trended = e5_incr_paid.trend(0.05)

#complete the incremental paid severity triangle
e5_ipsev_full = e5_ipsev.copy()
#extending the severity triangle to 108 months
e5_ipsev_full = cl.concat((e5_ipsev_full,e5_ipsev_full.latest_diagonal.rename("development",[9999])),axis=3)
#setting the before 60 months to selected incremental
e5_ipsev_full.iloc[:,:,:,:5] = e5_sevs_sel.iloc[:,:,:,:5]
#setting the after 72 months to selected tail
e5_ipsev_full.iloc[:,:,:,5:] = (
    e5_incr_paid_trended[e5_incr_paid_trended.development >= 72].sum().sum()
    /
    e5_incr_ccc[e5_incr_ccc.development >= 72].sum().sum() 
)
#setting the valuation_date to the future
e5_ipsev_full.valuation_date = pd.to_datetime(cl.options.ULT_VAL)
#trending back from latest accident year
e5_ipsev_full_detrended = e5_ipsev_full.trend(1/1.05-1,start='2008-12-31',end='2001-01-01')
#compositing the actual incremental severities with dtrended selected to complete the incremental paid severity triangle
e5_ipsev_full_complete = e5_ipsev[e5_ipsev.valuation <= e5_ipsev.valuation_date] + e5_ipsev_full_detrended[e5_ipsev_full_detrended.valuation > e5_ipsev.valuation_date]

#complete the incremental paid triangle
e5_s11_ccc = e5_dr_tri.full_triangle_.cum_to_incr()
e5_s11_ip = e5_s11_ccc * e5_ipsev_full_complete

/home/docs/checkouts/readthedocs.org/user_builds/chainladder-python/envs/experimental/lib/python3.11/site-packages/chainladder/utils/triangle_weight.py:367: UserWarning: Some exclusions have been ignored. At least 1 (use preserve = ...) link ratio(s) is required for development estimation.
  warnings.warn(warning)
/home/docs/checkouts/readthedocs.org/user_builds/chainladder-python/envs/experimental/lib/python3.11/site-packages/chainladder/development/base.py:272: UserWarning: Some exclusions have been ignored. At least 1 (use preserve = ...) link ratio(s) is required for development estimation.
  warnings.warn(warning)

/home/docs/checkouts/readthedocs.org/user_builds/chainladder-python/envs/experimental/lib/python3.11/site-packages/chainladder/utils/triangle_weight.py:367: UserWarning: Some exclusions have been ignored. At least 1 (use preserve = ...) link ratio(s) is required for development estimation.
  warnings.warn(warning)
/home/docs/checkouts/readthedocs.org/user_builds/chainladder-python/envs/experimental/lib/python3.11/site-packages/chainladder/development/base.py:272: UserWarning: Some exclusions have been ignored. At least 1 (use preserve = ...) link ratio(s) is required for development estimation.
  warnings.warn(warning)
/home/docs/checkouts/readthedocs.org/user_builds/chainladder-python/envs/experimental/lib/python3.11/site-packages/chainladder/utils/triangle_weight.py:367: UserWarning: Some exclusions have been ignored. At least 1 (use preserve = ...) link ratio(s) is required for development estimation.
  warnings.warn(warning)

/home/docs/checkouts/readthedocs.org/user_builds/chainladder-python/envs/experimental/lib/python3.11/site-packages/chainladder/utils/triangle_weight.py:367: UserWarning: Some exclusions have been ignored. At least 1 (use preserve = ...) link ratio(s) is required for development estimation.
  warnings.warn(warning)

Exhibit V Sheet 1

print('PART 1 - Data Triangle')
nb_display(e5_tri['Closed Claim Counts'])
print('PART 2 - Age-to-Age Factors')
nb_display(e5_tri['Closed Claim Counts'].age_to_age.round(3))
print('PART 3 - Average Age-to-Age Factor')
nb_display(combine_ldf(e5_ccc_devs).round(3).to_frame())
print('PART 4 - Selected Age-to-Age Factors')
print('Selected')
nb_display(e5_ccc_selected.ldf_)
print('CDF to Ultimate')
nb_display(e5_ccc_selected.cdf_.round(3))
print('Percent Closed')
nb_display((1/e5_ccc_selected.cdf_).round(3))

PART 1 - Data Triangle

	12	24	36	48	60	72	84	96
2001	195.00	375.00	510.00	625.00	702.00	752.00	780.00	796.00
2002	199.00	349.00	445.00	508.00	563.00	594.00	626.00
2003	106.00	294.00	383.00	453.00	499.00	542.00
2004	126.00	281.00	377.00	445.00	494.00
2005	114.00	249.00	315.00	403.00
2006	114.00	229.00	300.00
2007	79.00	188.00
2008	127.00

PART 2 - Age-to-Age Factors

	12-24	24-36	36-48	48-60	60-72	72-84	84-96
2001	1.9230	1.3600	1.2250	1.1230	1.0710	1.0370	1.0210
2002	1.7540	1.2750	1.1420	1.1080	1.0550	1.0540
2003	2.7740	1.3030	1.1830	1.1020	1.0860
2004	2.2300	1.3420	1.1800	1.1100
2005	2.1840	1.2650	1.2790
2006	2.0090	1.3100
2007	2.3800

PART 3 - Average Age-to-Age Factor

development	12-24	24-36	36-48	48-60	60-72	72-84	84-96
Total
simple_5	2.315	1.299	1.202	1.111	1.071	1.046	1.021
simple_3	2.191	1.306	1.214	1.107	1.071	1.046	1.021
medial_5x1	2.265	1.296	1.196	1.109	1.071	1.046	1.021
volume_5	2.302	1.298	1.199	1.112	1.070	1.045	1.021
volume_3	2.169	1.307	1.210	1.107	1.070	1.045	1.021

PART 4 - Selected Age-to-Age Factors
Selected

	12-24	24-36	36-48	48-60	60-72	72-84	84-96	96-108
(All)	2.1690	1.3070	1.2100	1.1070	1.0700	1.0450	1.0210	1.1000

CDF to Ultimate

	12-Ult	24-Ult	36-Ult	48-Ult	60-Ult	72-Ult	84-Ult	96-Ult
(All)	4.7690	2.1990	1.6820	1.3900	1.2560	1.1740	1.1230	1.1000

Percent Closed

	12-Ult	24-Ult	36-Ult	48-Ult	60-Ult	72-Ult	84-Ult	96-Ult
(All)	0.2100	0.4550	0.5940	0.7190	0.7960	0.8520	0.8900	0.9090

Exhibit V Sheet 2

print('PART 1 - Data Triangle')
nb_display(e5_tri['Reported Claim Counts'])
print('PART 2 - Age-to-Age Factors')
nb_display(e5_tri['Reported Claim Counts'].age_to_age.round(3))
print('PART 3 - Average Age-to-Age Factor')
nb_display(combine_ldf(e5_rcc_devs).round(3).to_frame())
print('PART 4 - Selected Age-to-Age Factors')
nb_display(e5_rcc_selected.ldf_)
print('CDF to Ultimate')
nb_display(e5_rcc_selected.cdf_.round(3))
print('Percent Reported')
nb_display((1/e5_rcc_selected.cdf_).round(3))

PART 1 - Data Triangle

	12	24	36	48	60	72	84	96
2001	1,299.00	1,077.00	1,057.00	965.00	930.00	917.00	864.00	870.00
2002	847.00	945.00	864.00	787.00	784.00	743.00	731.00
2003	800.00	831.00	762.00	704.00	669.00	636.00
2004	823.00	862.00	797.00	728.00	684.00
2005	828.00	850.00	765.00	687.00
2006	824.00	809.00	734.00
2007	604.00	620.00
2008	812.00

PART 2 - Age-to-Age Factors

	12-24	24-36	36-48	48-60	60-72	72-84	84-96
2001	0.8290	0.9810	0.9130	0.9640	0.9860	0.9420	1.0070
2002	1.1160	0.9140	0.9110	0.9960	0.9480	0.9840
2003	1.0390	0.9170	0.9240	0.9500	0.9510
2004	1.0470	0.9250	0.9130	0.9400
2005	1.0270	0.9000	0.8980
2006	0.9820	0.9070
2007	1.0260

PART 3 - Average Age-to-Age Factor

development	12-24	24-36	36-48	48-60	60-72	72-84	84-96
Total
simple_5	1.024	0.913	0.912	0.962	0.961	0.963	1.007
simple_3	1.012	0.911	0.912	0.962	0.961	0.963	1.007
medial_5x1	1.031	0.913	0.912	0.957	0.951	0.963	1.007
volume_5	1.024	0.913	0.912	0.963	0.963	0.961	1.007
volume_3	1.010	0.911	0.912	0.963	0.963	0.961	1.007

PART 4 - Selected Age-to-Age Factors

	12-24	24-36	36-48	48-60	60-72	72-84	84-96	96-108
(All)	1.0100	0.9110	0.9120	0.9630	0.9630	0.9610	1.0070	1.0000

CDF to Ultimate

	12-Ult	24-Ult	36-Ult	48-Ult	60-Ult	72-Ult	84-Ult	96-Ult
(All)	0.7530	0.7460	0.8180	0.8970	0.9320	0.9680	1.0070	1.0000

Percent Reported

	12-Ult	24-Ult	36-Ult	48-Ult	60-Ult	72-Ult	84-Ult	96-Ult
(All)	1.3280	1.3410	1.2220	1.1140	1.0730	1.0330	0.9930	1.0000

Exhibit V Sheet 3

e5_ccc_df = cl.model_diagnostics(e5_ccc_cl).to_frame(keepdims=True,implicit_axis=True).set_index('origin')
e5_rcc_df = cl.model_diagnostics(e5_rcc_cl).to_frame(keepdims=True,implicit_axis=True).set_index('origin')
e5_s3 = e5_ccc_df[['development','Latest','Ultimate']].rename(columns={'Latest':'Closed Claim Counts','Ultimate':'Ult Count Using CCC'})
e5_s3[['Reported Claim Counts','Ult Count Using RCC']] = e5_rcc_df[['Latest','Ultimate']]
e5_s3["Selected Ult CC"] = e5_cc_ult.round(0).latest_diagonal.to_frame()
e5_s3

	development	Closed Claim Counts	Ult Count Using CCC	Reported Claim Counts	Ult Count Using RCC	Selected Ult CC
origin
2001-01-01	96	796.0	875.600000	870.0	870.000000	873.0
2002-01-01	84	626.0	703.060600	731.0	736.117000	720.0
2003-01-01	72	542.0	636.112609	636.0	615.474372	626.0
2004-01-01	60	494.0	620.362367	684.0	637.434033	629.0
2005-01-01	48	403.0	560.236193	687.0	616.541294	588.0
2006-01-01	36	300.0	504.629623	734.0	600.753529	553.0
2007-01-01	24	188.0	413.318575	620.0	462.285570	438.0
2008-01-01	12	127.0	605.606248	812.0	611.499422	609.0

Exhibit V Sheet 4

print('Part 1 - Disposal Rate Triangle')
nb_display(cl.DisposalRate().fit_transform(e5_tri['Closed Claim Counts'],sample_weight = e5_cc_ult).disposal_rate_tri.round(3))
print('PART 2 - Average Disposal Rate Factors')
nb_display(combine_disposal(e5_drs).round(3).to_frame())
print('PART 3 - Selected Disposal Rate Factors')
nb_display(e5_dr_selected.disposal_rate_)

Part 1 - Disposal Rate Triangle

	12-Ult	24-Ult	36-Ult	48-Ult	60-Ult	72-Ult	84-Ult	96-Ult
2001	0.2230	0.4300	0.5840	0.7160	0.8040	0.8620	0.8940	0.9120
2002	0.2770	0.4850	0.6180	0.7060	0.7820	0.8250	0.8700
2003	0.1690	0.4700	0.6120	0.7240	0.7970	0.8660
2004	0.2000	0.4470	0.5990	0.7080	0.7860
2005	0.1940	0.4230	0.5350	0.6850
2006	0.2060	0.4140	0.5430
2007	0.1800	0.4290
2008	0.2090

PART 2 - Average Disposal Rate Factors

development	12-Ult	24-Ult	36-Ult	48-Ult	60-Ult	72-Ult	84-Ult	96-Ult	108-Ult
Total
simple_5	0.198	0.437	0.582	0.708	0.792	0.851	0.882	0.912	1.0
simple_3	0.198	0.422	0.559	0.705	0.788	0.851	0.882	0.912	1.0
medial_5x1	0.200	0.433	0.585	0.710	0.791	0.862	0.882	0.912	1.0

PART 3 - Selected Disposal Rate Factors

	12-Ult	24-Ult	36-Ult	48-Ult	60-Ult	72-Ult	84-Ult	96-Ult	108-Ult
(All)	0.2000	0.4330	0.5850	0.7100	0.7910	0.8620	0.8820	0.9120	1.0000

Exhibit V Sheet 5

print('Closed Claim Counts')
nb_display(e5_tri['Closed Claim Counts'])
print('Projected Incremental Closed Claim Counts')
nb_display(e5_dr_tri.full_triangle_.cum_to_incr().round(0))

Closed Claim Counts

	12	24	36	48	60	72	84	96
2001	195.00	375.00	510.00	625.00	702.00	752.00	780.00	796.00
2002	199.00	349.00	445.00	508.00	563.00	594.00	626.00
2003	106.00	294.00	383.00	453.00	499.00	542.00
2004	126.00	281.00	377.00	445.00	494.00
2005	114.00	249.00	315.00	403.00
2006	114.00	229.00	300.00
2007	79.00	188.00
2008	127.00

Projected Incremental Closed Claim Counts

	12	24	36	48	60	72	84	96	9999
2001	195.00	180.00	135.00	115.00	77.00	50.00	28.00	16.00	77.00
2002	199.00	150.00	96.00	63.00	55.00	31.00	32.00	24.00	70.00
2003	106.00	188.00	89.00	70.00	46.00	43.00	12.00	18.00	53.00
2004	126.00	155.00	96.00	68.00	49.00	46.00	13.00	19.00	57.00
2005	114.00	135.00	66.00	88.00	52.00	45.00	13.00	19.00	56.00
2006	114.00	115.00	71.00	76.00	49.00	43.00	12.00	18.00	54.00
2007	79.00	109.00	67.00	55.00	36.00	31.00	9.00	13.00	39.00
2008	127.00	140.00	91.00	75.00	49.00	43.00	12.00	18.00	53.00

Exhibit V Sheet 6

print('Paid Claims')
nb_display(e5_tri["Paid Claims"])
print('Incremental Paid CLaims')
nb_display(e5_tri["Paid Claims"].cum_to_incr())
print('Incremental Closed Claim Counts')
nb_display(e5_tri["Closed Claim Counts"].cum_to_incr())
print('Incremental Paid Severities')
nb_display(e5_ipsev)

Paid Claims

	12	24	36	48	60	72	84	96
2001	1,119,962	4,373,268	8,398,345	13,490,793	17,372,233	22,052,662	27,359,691	29,901,361
2002	1,411,957	6,287,005	11,443,820	15,520,552	21,295,572	28,410,418	32,468,911
2003	984,748	6,128,957	10,470,758	14,604,684	21,936,647	23,942,499
2004	1,158,659	5,811,172	10,497,504	15,087,416	18,242,570
2005	1,198,767	5,103,837	9,042,134	15,443,929
2006	1,220,778	4,594,746	8,983,864
2007	796,774	4,233,641
2008	1,445,365

Incremental Paid CLaims

	12	24	36	48	60	72	84	96
2001	1,119,962	3,253,306	4,025,077	5,092,448	3,881,440	4,680,429	5,307,029	2,541,670
2002	1,411,957	4,875,048	5,156,815	4,076,732	5,775,020	7,114,846	4,058,493
2003	984,748	5,144,209	4,341,801	4,133,926	7,331,963	2,005,852
2004	1,158,659	4,652,513	4,686,332	4,589,912	3,155,154
2005	1,198,767	3,905,070	3,938,297	6,401,795
2006	1,220,778	3,373,968	4,389,118
2007	796,774	3,436,867
2008	1,445,365

Incremental Closed Claim Counts

	12	24	36	48	60	72	84	96
2001	195.00	180.00	135.00	115.00	77.00	50.00	28.00	16.00
2002	199.00	150.00	96.00	63.00	55.00	31.00	32.00
2003	106.00	188.00	89.00	70.00	46.00	43.00
2004	126.00	155.00	96.00	68.00	49.00
2005	114.00	135.00	66.00	88.00
2006	114.00	115.00	71.00
2007	79.00	109.00
2008	127.00

Incremental Paid Severities

	12	24	36	48	60	72	84	96
2001	5,743	18,074	29,815	44,282	50,408	93,609	189,537	158,854
2002	7,095	32,500	53,717	64,710	105,000	229,511	126,828
2003	9,290	27,363	48,784	59,056	159,390	46,648
2004	9,196	30,016	48,816	67,499	64,391
2005	10,516	28,926	59,671	72,748
2006	10,709	29,339	61,819
2007	10,086	31,531
2008	11,381

Exhibit V Sheet 7

print('Incremental Paid Severities')
nb_display(e5_ipsev)
print('Annual Change based on Exponential Regression')
nb_display(pd.concat([e5_trend.round(3),e5_trend_ex2001.round(3)]))
print('Goodness of Fit Test of Exponential Regression (R-Squared)')
nb_display(pd.concat([e5_rsq.round(3),e5_rsq_ex2001.round(3)]))

Incremental Paid Severities

	12	24	36	48	60	72	84	96
2001	5,743	18,074	29,815	44,282	50,408	93,609	189,537	158,854
2002	7,095	32,500	53,717	64,710	105,000	229,511	126,828
2003	9,290	27,363	48,784	59,056	159,390	46,648
2004	9,196	30,016	48,816	67,499	64,391
2005	10,516	28,926	59,671	72,748
2006	10,709	29,339	61,819
2007	10,086	31,531
2008	11,381

Annual Change based on Exponential Regression

development	12	24	36	48	60	72	84	96
all_years	0.088	0.056	0.120	0.109	0.122	-0.294	-0.331	NaN
latest_6	0.038	0.001	0.120	0.109	0.122	-0.294	-0.331	NaN
latest_4	0.018	0.016	0.095	0.050	0.122	-0.294	-0.331	NaN
all_years_ex_2001	0.064	0.001	0.049	0.050	-0.217	-0.797	NaN	NaN

Goodness of Fit Test of Exponential Regression (R-Squared)

development	12	24	36	48	60	72	84	96
all_years	0.787	0.349	0.644	0.722	0.084	0.19	1.0	NaN
latest_6	0.704	0.000	0.644	0.722	0.084	0.19	1.0	NaN
latest_4	0.208	0.306	0.856	0.518	0.084	0.19	1.0	NaN
all_years_ex_2001	0.737	0.000	0.478	0.518	0.290	1.00	NaN	NaN

Exhibit V Sheet 8

print('Incremental Paid Severities')
nb_display(e5_ipsev)
print('Trended Incremental Paid Severities Assuming 5% Annual Trend')
nb_display(e5_ipsev.trend(0.05))
print('Average Trended Incremental Paid Severities')
nb_display(combine_tri(e5_sevs).round(0).to_frame())
print('Selected Incremental Paid Severities')
nb_display(e5_sevs_sel[e5_sevs_sel.development<=60])

Incremental Paid Severities

	12	24	36	48	60	72	84	96
2001	5,743	18,074	29,815	44,282	50,408	93,609	189,537	158,854
2002	7,095	32,500	53,717	64,710	105,000	229,511	126,828
2003	9,290	27,363	48,784	59,056	159,390	46,648
2004	9,196	30,016	48,816	67,499	64,391
2005	10,516	28,926	59,671	72,748
2006	10,709	29,339	61,819
2007	10,086	31,531
2008	11,381

Trended Incremental Paid Severities Assuming 5% Annual Trend

	12	24	36	48	60	72	84	96
2001	8,082	25,432	41,953	62,309	70,930	131,717	266,697	223,524
2002	9,508	43,554	71,986	86,718	140,711	307,567	169,962
2003	11,857	34,923	62,262	75,372	203,427	59,536
2004	11,177	36,485	59,336	82,045	78,268
2005	12,173	33,486	69,077	84,215
2006	11,806	32,346	68,155
2007	10,590	33,107
2008	11,381

Average Trended Incremental Paid Severities

development	12	24	36	48	60	72	84	96
Total
simple_5	11426.0	34069.0	66163.0	78132.0	123334.0	166273.0	218329.0	223524.0
simple_3	11259.0	32980.0	65523.0	80544.0	140802.0	166273.0	218329.0	223524.0
medial_5x1	11455.0	33839.0	66498.0	80544.0	109489.0	131717.0	218329.0	223524.0

Selected Incremental Paid Severities

	12	24	36	48	60
2008	11,259	32,980	65,523	80,544	140,802

Exhibit V Sheet 9

e5_s9 = pd.DataFrame(
    data = [[
        e5_incr_ccc[e5_incr_ccc.development >= 60].sum().sum(), 
        e5_incr_ccc[e5_incr_ccc.development >= 72].sum().sum()
    ]],
    index = ['Total Closed Claim Counts'],
    columns = ['Age 60 & Older','Age 72 & Older']
)
e5_s9.loc['Total Trended Paid Claims'] = [
        e5_incr_paid_trended[e5_incr_paid_trended.development >= 60].sum().sum(), 
        e5_incr_paid_trended[e5_incr_paid_trended.development >= 72].sum().sum()
    ]
e5_s9.loc['Estimated Trended Tail Severity'] = e5_s9.iloc[1] / e5_s9.iloc[0]
e5_s9.loc['Estimated Incremental Trended Tail Severity'] = e5_sevs_sel.iloc[:,:,:,4:6].values.flatten()
e5_s9.loc['Selected Trended Paid Severity'] = e5_ipsev_full.values[0,0,-1,4:6]
print('Incremental Closed Claim Counts')
nb_display(e5_incr_ccc[e5_incr_ccc.development >= 60][e5_incr_ccc.origin <= '2004'])
print('Incremental Paid CLaims')
nb_display(e5_incr_paid[e5_incr_paid.development >= 60][e5_incr_paid.origin <= '2004'])
print('Trended Incremental Paid Severities')
nb_display(e5_incr_paid_trended[e5_incr_paid_trended.development >= 60][e5_incr_paid_trended.origin <= '2004'])
nb_display(e5_s9)

Incremental Closed Claim Counts

	60	72	84	96
2001	77.00	50.00	28.00	16.00
2002	55.00	31.00	32.00
2003	46.00	43.00
2004	49.00

Incremental Paid CLaims

	60	72	84	96
2001	3,881,440	4,680,429	5,307,029	2,541,670
2002	5,775,020	7,114,846	4,058,493
2003	7,331,963	2,005,852
2004	3,155,154

Trended Incremental Paid Severities

	60	72	84	96
2001	5,461,576	6,585,834	7,467,523	3,576,385
2002	7,739,079	9,534,574	5,438,769
2003	9,357,649	2,560,032
2004	3,835,109

	Age 60 & Older	Age 72 & Older
Total Closed Claim Counts	4.270000e+02	2.000000e+02
Total Trended Paid Claims	6.155653e+07	3.516312e+07
Estimated Trended Tail Severity	1.441605e+05	1.758156e+05
Estimated Incremental Trended Tail Severity	1.408017e+05	1.662731e+05
Selected Trended Paid Severity	1.408017e+05	1.758156e+05

Exhibit V Sheet 10

print('Incremental Paid Severities')
nb_display(e5_ipsev)
print('Selected Incremental Paid Severities')
nb_display(e5_ipsev_full[e5_ipsev_full.origin == '2008'])
print('Incremental Paid Severities Adjusted to Cost Level of Accident Year Assuming 5% Annual Trend Rate')
nb_display(e5_ipsev_full_complete)

Incremental Paid Severities

	12	24	36	48	60	72	84	96
2001	5,743	18,074	29,815	44,282	50,408	93,609	189,537	158,854
2002	7,095	32,500	53,717	64,710	105,000	229,511	126,828
2003	9,290	27,363	48,784	59,056	159,390	46,648
2004	9,196	30,016	48,816	67,499	64,391
2005	10,516	28,926	59,671	72,748
2006	10,709	29,339	61,819
2007	10,086	31,531
2008	11,381

Selected Incremental Paid Severities

	12	24	36	48	60	72	84	96	9999
2008	11,259	32,980	65,523	80,544	140,802	175,816	175,816	175,816	175,816

Incremental Paid Severities Adjusted to Cost Level of Accident Year Assuming 5% Annual Trend Rate

	12	24	36	48	60	72	84	96	9999
2001	5,743	18,074	29,815	44,282	50,408	93,609	189,537	158,854	124,949
2002	7,095	32,500	53,717	64,710	105,000	229,511	126,828	131,196	131,196
2003	9,290	27,363	48,784	59,056	159,390	46,648	137,756	137,756	137,756
2004	9,196	30,016	48,816	67,499	64,391	144,644	144,644	144,644	144,644
2005	10,516	28,926	59,671	72,748	121,630	151,876	151,876	151,876	151,876
2006	10,709	29,339	61,819	73,056	127,711	159,470	159,470	159,470	159,470
2007	10,086	31,531	62,403	76,709	134,097	167,443	167,443	167,443	167,443
2008	11,381	32,980	65,523	80,544	140,802	175,816	175,816	175,816	175,816

Exhibit V Sheet 11

print('Projected Incremental Closed Claim Counts')
nb_display(e5_s11_ccc.round(0))
print('Incremental Paid Severities Adjusted to Cost Level of Accident Year Assuming 5% Annual Trend Rate')
nb_display(e5_ipsev_full_complete)
print('Projected Incremental Paid Claims')
nb_display(e5_s11_ip)
print('Projected Cumulative Paid Claims')
nb_display(e5_s11_ip.incr_to_cum())

Projected Incremental Closed Claim Counts

	12	24	36	48	60	72	84	96	9999
2001	195.00	180.00	135.00	115.00	77.00	50.00	28.00	16.00	77.00
2002	199.00	150.00	96.00	63.00	55.00	31.00	32.00	24.00	70.00
2003	106.00	188.00	89.00	70.00	46.00	43.00	12.00	18.00	53.00
2004	126.00	155.00	96.00	68.00	49.00	46.00	13.00	19.00	57.00
2005	114.00	135.00	66.00	88.00	52.00	45.00	13.00	19.00	56.00
2006	114.00	115.00	71.00	76.00	49.00	43.00	12.00	18.00	54.00
2007	79.00	109.00	67.00	55.00	36.00	31.00	9.00	13.00	39.00
2008	127.00	140.00	91.00	75.00	49.00	43.00	12.00	18.00	53.00

Incremental Paid Severities Adjusted to Cost Level of Accident Year Assuming 5% Annual Trend Rate

	12	24	36	48	60	72	84	96	9999
2001	5,743	18,074	29,815	44,282	50,408	93,609	189,537	158,854	124,949
2002	7,095	32,500	53,717	64,710	105,000	229,511	126,828	131,196	131,196
2003	9,290	27,363	48,784	59,056	159,390	46,648	137,756	137,756	137,756
2004	9,196	30,016	48,816	67,499	64,391	144,644	144,644	144,644	144,644
2005	10,516	28,926	59,671	72,748	121,630	151,876	151,876	151,876	151,876
2006	10,709	29,339	61,819	73,056	127,711	159,470	159,470	159,470	159,470
2007	10,086	31,531	62,403	76,709	134,097	167,443	167,443	167,443	167,443
2008	11,381	32,980	65,523	80,544	140,802	175,816	175,816	175,816	175,816

Projected Incremental Paid Claims

	12	24	36	48	60	72	84	96	9999
2001	1,119,962	3,253,306	4,025,077	5,092,448	3,881,440	4,680,429	5,307,029	2,541,670	9,596,072
2002	1,411,957	4,875,048	5,156,815	4,076,732	5,775,020	7,114,846	4,058,493	3,121,653	9,156,850
2003	984,748	5,144,209	4,341,801	4,133,926	7,331,963	2,005,852	1,672,908	2,509,362	7,360,795
2004	1,158,659	4,652,513	4,686,332	4,589,912	3,155,154	6,628,548	1,867,197	2,800,795	8,215,665
2005	1,198,767	3,905,070	3,938,297	6,401,795	6,298,114	6,893,395	1,941,801	2,912,702	8,543,926
2006	1,220,778	3,373,968	4,389,118	5,560,412	6,298,790	6,894,135	1,942,010	2,913,015	8,544,843
2007	796,774	3,436,867	4,178,868	4,224,415	4,785,383	5,237,685	1,475,404	2,213,106	6,491,779
2008	1,445,365	4,625,496	5,994,996	6,060,338	6,865,102	7,513,974	2,116,612	3,174,918	9,313,094

Projected Cumulative Paid Claims

	12	24	36	48	60	72	84	96	9999
2001	1,119,962	4,373,268	8,398,345	13,490,793	17,372,233	22,052,662	27,359,691	29,901,361	39,497,433
2002	1,411,957	6,287,005	11,443,820	15,520,552	21,295,572	28,410,418	32,468,911	35,590,564	44,747,415
2003	984,748	6,128,957	10,470,758	14,604,684	21,936,647	23,942,499	25,615,407	28,124,769	35,485,564
2004	1,158,659	5,811,172	10,497,504	15,087,416	18,242,570	24,871,118	26,738,314	29,539,109	37,754,774
2005	1,198,767	5,103,837	9,042,134	15,443,929	21,742,043	28,635,438	30,577,240	33,489,942	42,033,868
2006	1,220,778	4,594,746	8,983,864	14,544,276	20,843,065	27,737,200	29,679,210	32,592,225	41,137,068
2007	796,774	4,233,641	8,412,509	12,636,923	17,422,307	22,659,992	24,135,396	26,348,502	32,840,281
2008	1,445,365	6,070,861	12,065,857	18,126,195	24,991,297	32,505,270	34,621,883	37,796,801	47,109,895

Exhibit V Sheet 12

summary_exh(
    e5_tri['Reported Claims']/1000,
    e5_tri['Paid Claims']/1000,
    e5_s11_ip.incr_to_cum()/1000
)

	Reported	Paid	Ult Claims	Case Outstanding	IBNR	Unpaid
2001-01-01	35592.0	29901.0	39497.0	5691.0	3905.0	9596.0
2002-01-01	36330.0	32469.0	44747.0	3861.0	8417.0	12278.0
2003-01-01	31900.0	23942.0	35486.0	7958.0	3586.0	11544.0
2004-01-01	39716.0	18243.0	37755.0	21473.0	-1961.0	19512.0
2005-01-01	32667.0	15444.0	42034.0	17223.0	9367.0	26590.0
2006-01-01	27774.0	8984.0	41137.0	18790.0	13363.0	32153.0
2007-01-01	16246.0	4234.0	32840.0	12012.0	16594.0	28606.0
2008-01-01	8216.0	1445.0	47110.0	6771.0	38894.0	45665.0

e5_dr_selected.disposal_rate_

	12-Ult	24-Ult	36-Ult	48-Ult	60-Ult	72-Ult	84-Ult	96-Ult	108-Ult
(All)	0.2000	0.4330	0.5850	0.7100	0.7910	0.8620	0.8820	0.9120	1.0000

#Exhibit V Sheet 1
assert np.all(e5_ccc_selected.cdf_.round(3).values == np.array([4.769, 2.199, 1.682, 1.390, 1.256, 1.174, 1.123, 1.100]))
#Exhibit V Sheet 2
assert np.all(e5_rcc_selected.cdf_.round(3).values == np.array([0.753, 0.746, 0.818, 0.897, 0.932, 0.968, 1.007, 1.000]))
#Exhibit V Sheet 3
assert np.all(e5_s3['Selected Ult CC'].values==np.array([873., 720., 626., 629., 588., 553, 438., 609.]))
#Exhibit V Sheet 4
assert np.all(e5_dr_selected.disposal_rate_.values == np.array([0.200, 0.433, 0.585, 0.710, 0.791, 0.862, 0.882, 0.912, 1.000]))
#Exhibit V Sheet 5
lhs = (e5_dr_tri.full_triangle_.cum_to_incr()-e5_tri['Closed Claim Counts'].cum_to_incr()).values.flatten()
rhs = np.array([
                                                        77.,  
                                                24.,    70.,  
                                        12.,    18.,    54.,  
                                46.,    13.,    19.,    57.,  
                        52.,    45.,    13.,    19.,    56.,  
                76.,    49.,    43.,    12.,    18.,    54.,  
        67.,    55.,    36.,    31.,    9.,     13.,    39.,  
140.,   91.,    75.,    49.,    43.,    12.,    18.,    53.
])
assert np.all(abs(lhs[~np.isnan(lhs)] - rhs) < 1)
#Exhibit V Sheet 7
assert np.all(e5_trend.loc['latest_4'].values[:7].round(3) == np.array([0.018, 0.016, 0.095, 0.050, 0.122, -0.294, -0.331]))
assert np.all(e5_rsq.loc['latest_6'].values[:7].round(3) == np.array([0.704, 0.000, 0.644, 0.722, 0.084, 0.190, 1.000]))
#Exhibit V Sheet 8
assert np.all(e5_sevs_sel.values[...,:5].round(0) == np.array([11259.,  32980.,  65523.,  80544., 140802.]))
#Exhibit V Sheet 9
assert np.all(e5_s9.iloc[2].values.round(0) == np.array([144160., 175816.]))
#Exhibit V Sheet 10
lhs = e5_ipsev_full_detrended[e5_ipsev_full_detrended.valuation > e5_ipsev.valuation_date].round(0).values.flatten()
rhs = np.array([
                                                        124949.,
                                                131196.,131196.,
                                        137756.,137756.,137756.,
                                144644.,144644.,144644.,144644.,
                        121630.,151876.,151876.,151876.,151876.,
                73056.,	127711.,159470.,159470.,159470.,159470.,
        62403.,	76709.,	134097.,167443.,167443.,167443.,167443.,
32980., 65523.,	80544.,	140802.,175816.,175816.,175816.,175816.,
])
assert np.all(lhs[~np.isnan(lhs)] == rhs)
#Exhibit V Sheet 11
assert np.allclose(
    e5_s11_ip.incr_to_cum().iloc[:,:,:,-1].values.round(0).flatten(),
    np.array([39497433., 44743308., 35510981., 37766027., 42007442., 41113459., 32859080., 47109641.]),
    rtol = .001
)

Exhibit VI Analysis

#loading data and assumptions
e6_tri = e2_tri[e2_tri.origin >= '2001'][e2_tri.development <= 96]
e6_disp_assumptions = {}
e6_disp_assumptions['simple_3'] = {'n_periods':3, 'average':'simple'}
e6_disp_assumptions['simple_2'] = {'n_periods':2, 'average':'simple'}
e6_disp_assumptions['medial_5x1'] = {'n_periods':5, 'average':'simple','drop_high':1, 'drop_low':1}
e6_sev_assumptions = {}
e6_sev_assumptions['simple_3'] = {'n_periods':3}
e6_sev_assumptions['simple_2'] = {'n_periods':2}
e6_sev_assumptions['medial_5x1'] = {'n_periods':5,'drop_high':1, 'drop_low':1}

#calculating disposal rate and complete the closed count triangle
e6_rcc_ult = e2_rcc_cl.ultimate_[e2_rcc_cl.ultimate_.origin >= '2001']
e6_drs = average_dr(e6_tri['Closed Claim Counts'],e6_rcc_ult,e6_disp_assumptions)
e6_dr_selected = e6_drs['simple_2']
e6_dr_selected.disposal_rate_ = e6_dr_selected.disposal_rate_.round(3)
e6_dr_tri = e6_dr_selected.transform(e6_tri['Closed Claim Counts'],sample_weight = e6_rcc_ult)

#calculate and select incremental paid severity
e6_ipsev = e6_tri["Paid Claims"].cum_to_incr() / e6_tri["Closed Claim Counts"].cum_to_incr() * 1000
e6_ipsev_adj = e6_ipsev.trend(0.05) * xyz_tort_adjustment.fit(e6_ipsev).olf_.values
e6_sevs = average_sev(e6_ipsev_adj,e6_sev_assumptions)
e6_sevs_sel = e6_sevs['simple_2'].copy()
e6_sevs_sel.origin = ['2008']

#calculate and select tail paid severity
e6_incr_ccc = e6_tri["Closed Claim Counts"].cum_to_incr()
e6_incr_paid = e6_tri["Paid Claims"].cum_to_incr() * 1000
e6_incr_paid_trended = e6_ipsev_adj * e6_incr_ccc

e6_ipsev_full = e6_ipsev_adj.copy()
#extending the severity triangle to 108 months
e6_ipsev_full = cl.concat((e6_ipsev_full,e6_ipsev_full.latest_diagonal.rename("development",[9999])),axis=3)
#setting the before 72 months to selected incremental
e6_ipsev_full.iloc[:,:,:,:6] = e6_sevs_sel.iloc[:,:,:,:6]
#setting the after 86 months to selected tail
e6_ipsev_full.iloc[:,:,:,6:] = (
    e6_incr_paid_trended[e6_incr_paid_trended.development >= 84].sum().sum()
    /
    e6_incr_ccc[e6_incr_ccc.development >= 84].sum().sum() 
)
#setting the valuation_date to the future
e6_ipsev_full.valuation_date = pd.to_datetime(cl.options.ULT_VAL)
#trending back from latest accident year
e6_ipsev_full_detrended = e6_ipsev_full.trend(1/1.05-1,start='2008-12-31',end='2001-01-01') / xyz_tort_adjustment.fit(e6_ipsev).olf_.values
#compositing the actual incremental severities with dtrended selected
e6_ipsev_full_complete = e6_ipsev[e6_ipsev.valuation <= e6_ipsev.valuation_date] + e6_ipsev_full_detrended[e6_ipsev_full_detrended.valuation > e6_ipsev.valuation_date]

#complete the incremental paid triangle
e6_s7_ccc = e6_dr_tri.full_triangle_.cum_to_incr()
e6_s7_ip = e6_s7_ccc * e6_ipsev_full_complete / 1000

/home/docs/checkouts/readthedocs.org/user_builds/chainladder-python/envs/experimental/lib/python3.11/site-packages/chainladder/utils/triangle_weight.py:367: UserWarning: Some exclusions have been ignored. At least 1 (use preserve = ...) link ratio(s) is required for development estimation.
  warnings.warn(warning)
/home/docs/checkouts/readthedocs.org/user_builds/chainladder-python/envs/experimental/lib/python3.11/site-packages/chainladder/utils/triangle_weight.py:367: UserWarning: Some exclusions have been ignored. At least 1 (use preserve = ...) link ratio(s) is required for development estimation.
  warnings.warn(warning)

Exhibit VI Sheet 1

print('Part 1 - Disposal Rate Triangle')
nb_display(cl.DisposalRate().fit_transform(e6_tri['Closed Claim Counts'],sample_weight = e6_rcc_ult).disposal_rate_tri.round(3))

print('PART 2 - Average Disposal Rate Factors')
nb_display(combine_disposal(e6_drs).round(3).to_frame())

print('PART 3 - Selected Disposal Rate Factors')
nb_display(e6_dr_selected.disposal_rate_)

Part 1 - Disposal Rate Triangle

	12-Ult	24-Ult	36-Ult	48-Ult	60-Ult	72-Ult	84-Ult	96-Ult
2001	0.2090	0.4680	0.6430	0.7510	0.8420	0.9330	0.9840	0.9940
2002	0.1310	0.3910	0.5410	0.7010	0.8540	0.9420	0.9800
2003	0.1110	0.3770	0.5770	0.7750	0.9240	0.9620
2004	0.1040	0.3750	0.6370	0.8190	0.8970
2005	0.1230	0.4660	0.6930	0.8100
2006	0.1830	0.5400	0.7150
2007	0.2510	0.6040
2008	0.2360

PART 2 - Average Disposal Rate Factors

development	12-Ult	24-Ult	36-Ult	48-Ult	60-Ult	72-Ult	84-Ult	96-Ult	108-Ult
Total
simple_3	0.223	0.537	0.682	0.801	0.892	0.945	0.982	0.994	1.0
simple_2	0.244	0.572	0.704	0.814	0.910	0.952	0.982	0.994	1.0
medial_5x1	0.180	0.461	0.636	0.778	0.875	0.942	0.982	0.994	1.0

PART 3 - Selected Disposal Rate Factors

	12-Ult	24-Ult	36-Ult	48-Ult	60-Ult	72-Ult	84-Ult	96-Ult	108-Ult
(All)	0.2440	0.5720	0.7040	0.8140	0.9100	0.9520	0.9820	0.9940	1.0000

Exhibit VI Sheet 2

print('Closed Claim Counts')
nb_display(e6_tri['Closed Claim Counts'])
print('Projected Incremental Closed Claim Counts')
nb_display(e6_dr_tri.full_triangle_.cum_to_incr().round(0))

Closed Claim Counts

	12	24	36	48	60	72	84	96
2001	304	681	936	1,092	1,225	1,357	1,432	1,446
2002	203	607	841	1,089	1,327	1,464	1,523
2003	181	614	941	1,263	1,507	1,568
2004	235	848	1,442	1,852	2,029
2005	295	1,119	1,664	1,946
2006	307	906	1,201
2007	329	791
2008	276

Projected Incremental Closed Claim Counts

	12	24	36	48	60	72	84	96	9999
2001	304.00	377.00	255.00	156.00	133.00	132.00	75.00	14.00	9.00
2002	203.00	404.00	234.00	248.00	238.00	137.00	59.00	21.00	10.00
2003	181.00	433.00	327.00	322.00	244.00	61.00	39.00	16.00	8.00
2004	235.00	613.00	594.00	410.00	177.00	109.00	78.00	31.00	16.00
2005	295.00	824.00	545.00	282.00	235.00	103.00	74.00	29.00	15.00
2006	307.00	599.00	295.00	177.00	155.00	68.00	48.00	19.00	10.00
2007	329.00	462.00	160.00	133.00	116.00	51.00	36.00	15.00	7.00
2008	276.00	388.00	156.00	130.00	114.00	50.00	36.00	14.00	7.00

Exhibit VI Sheet 3

print('Paid Claims')
nb_display(e6_tri["Paid Claims"])
print('Incremental Paid CLaims')
nb_display(e6_tri["Paid Claims"].cum_to_incr())
print('Incremental Closed Claim Counts')
nb_display(e6_tri["Closed Claim Counts"].cum_to_incr())
print('Incremental Paid Severities')
nb_display(e6_ipsev)

Paid Claims

	12	24	36	48	60	72	84	96
2001	1,539	5,952	12,319	18,609	24,387	31,090	37,070	38,520
2002	2,318	7,932	13,823	22,096	31,945	40,629	44,438
2003	1,743	6,240	12,683	22,893	34,505	39,320
2004	2,221	9,898	25,950	43,439	52,811
2005	3,043	12,219	27,073	40,026
2006	3,531	11,778	22,819
2007	3,529	11,865
2008	3,409

Incremental Paid CLaims

	12	24	36	48	60	72	84	96
2001	1,539	4,413	6,367	6,289	5,778	6,703	5,980	1,450
2002	2,318	5,614	5,891	8,273	9,850	8,683	3,809
2003	1,743	4,497	6,443	10,209	11,612	4,815
2004	2,221	7,677	16,052	17,489	9,372
2005	3,043	9,176	14,854	12,953
2006	3,531	8,247	11,041
2007	3,529	8,336
2008	3,409

Incremental Closed Claim Counts

	12	24	36	48	60	72	84	96
2001	304.00	377.00	255.00	156.00	133.00	132.00	75.00	14.00
2002	203.00	404.00	234.00	248.00	238.00	137.00	59.00
2003	181.00	433.00	327.00	322.00	244.00	61.00
2004	235.00	613.00	594.00	410.00	177.00
2005	295.00	824.00	545.00	282.00
2006	307.00	599.00	295.00
2007	329.00	462.00
2008	276.00

Incremental Paid Severities

	12	24	36	48	60	72	84	96
2001	5,064	11,705	24,969	40,317	43,445	50,781	79,730	103,551
2002	11,417	13,896	25,175	33,359	41,386	63,382	64,556
2003	9,631	10,386	19,703	31,706	47,592	78,942
2004	9,452	12,524	27,023	42,657	52,947
2005	10,315	11,136	27,255	45,934
2006	11,502	13,768	37,427
2007	10,726	18,043
2008	12,351

Exhibit VI Sheet 4

print('Incremental Paid Severities')
nb_display(e6_ipsev)
print('Trended Incremental Paid Severities Assuming 5% Annual Trend and Adjusted for Tort Reform')
nb_display(e6_ipsev_adj)
print('Average Trended Incremental Paid Severities')
nb_display(combine_tri(e6_sevs).round(0).to_frame())
print('Selected Incremental Paid Severities')
nb_display(e6_sevs_sel[e6_sevs_sel.development<=72])

Incremental Paid Severities

	12	24	36	48	60	72	84	96
2001	5,064	11,705	24,969	40,317	43,445	50,781	79,730	103,551
2002	11,417	13,896	25,175	33,359	41,386	63,382	64,556
2003	9,631	10,386	19,703	31,706	47,592	78,942
2004	9,452	12,524	27,023	42,657	52,947
2005	10,315	11,136	27,255	45,934
2006	11,502	13,768	37,427
2007	10,726	18,043
2008	12,351

Trended Incremental Paid Severities Assuming 5% Annual Trend and Adjusted for Tort Reform

	12	24	36	48	60	72	84	96
2001	4,774	11,035	23,540	38,009	40,958	47,874	75,166	97,623
2002	10,251	12,477	22,604	29,952	37,159	56,908	57,963
2003	8,236	8,881	16,848	27,112	40,696	67,504
2004	7,698	10,199	22,007	34,739	43,119
2005	8,000	8,637	21,139	35,627
2006	9,511	11,384	30,947
2007	11,262	18,945
2008	12,351

Average Trended Incremental Paid Severities

development	12	24	36	48	60	72	84	96
Total
simple_3	11041.0	12989.0	24698.0	32493.0	40325.0	57429.0	66564.0	97623.0
simple_2	11807.0	15165.0	26043.0	35183.0	41908.0	62206.0	66564.0	97623.0
medial_5x1	9591.0	10155.0	21917.0	33439.0	40827.0	56908.0	66564.0	97623.0

Selected Incremental Paid Severities

	12	24	36	48	60	72
2008	11,807	15,165	26,043	35,183	41,908	62,206

Exhibit VI Sheet 5

print('Incremental Closed Claim Counts')
nb_display(e6_incr_ccc[e6_incr_ccc.development >= 72][e6_incr_ccc.origin <= '2003'])
print('Incremental Paid CLaims')
nb_display(e6_incr_paid[e6_incr_paid.development >= 72][e6_incr_paid.origin <= '2003'])
print('Trended Incremental Paid Severities')
nb_display(e6_incr_paid_trended[e6_incr_paid_trended.development >= 72][e6_incr_paid_trended.origin <= '2003'])
e6_s5 = pd.DataFrame(
    data = [[
        e6_incr_ccc[e6_incr_ccc.development >= 72].sum().sum(), 
        e6_incr_ccc[e6_incr_ccc.development >= 84].sum().sum()
    ]],
    index = ['Total Closed Claim Counts'],
    columns = ['Age 72 & Older','Age 84 & Older']
)
e6_s5.loc['Total Trended Paid Claims'] = [
        e6_incr_paid_trended[e6_incr_paid_trended.development >= 72].sum().sum(), 
        e6_incr_paid_trended[e6_incr_paid_trended.development >= 84].sum().sum()
    ]
e6_s5.loc['Estimated Trended Tail Severity'] = e6_s5.iloc[1] / e6_s5.iloc[0]
e6_s5.loc['Estimated Incremental Trended Tail Severity'] = e6_sevs_sel.iloc[:,:,:,5:7].values.flatten()
e6_s5.loc['Selected Trended Paid Severity'] = e6_ipsev_full.values[0,0,-1,5:7]
nb_display(e6_s5)

Incremental Closed Claim Counts

	72	84	96
2001	132.00	75.00	14.00
2002	137.00	59.00
2003	61.00

Incremental Paid CLaims

	72	84	96
2001	6,703,092	5,979,750	1,449,714
2002	8,683,334	3,808,804
2003	4,815,462

Trended Incremental Paid Severities

	72	84	96
2001	6,319,389	5,637,453	1,366,728
2002	7,796,454	3,419,788
2003	4,117,743

	Age 72 & Older	Age 84 & Older
Total Closed Claim Counts	4.780000e+02	1.480000e+02
Total Trended Paid Claims	2.865756e+07	1.042397e+07
Estimated Trended Tail Severity	5.995304e+04	7.043223e+04
Estimated Incremental Trended Tail Severity	6.220620e+04	6.656428e+04
Selected Trended Paid Severity	6.220620e+04	7.043223e+04

Exhibit VI Sheet 6

print('Incremental Paid Severities')
nb_display(e6_ipsev)
print('Selected Incremental Paid Severities')
nb_display(e6_ipsev_full[e6_ipsev_full.origin == '2008'])
print('Incremental Paid Severities Adjusted to Cost Level of Accident Year Assuming 5% Annual Trend Rate and Tort Reform Adjustment')
nb_display(e6_ipsev_full_complete)

Incremental Paid Severities

	12	24	36	48	60	72	84	96
2001	5,064	11,705	24,969	40,317	43,445	50,781	79,730	103,551
2002	11,417	13,896	25,175	33,359	41,386	63,382	64,556
2003	9,631	10,386	19,703	31,706	47,592	78,942
2004	9,452	12,524	27,023	42,657	52,947
2005	10,315	11,136	27,255	45,934
2006	11,502	13,768	37,427
2007	10,726	18,043
2008	12,351

Selected Incremental Paid Severities

	12	24	36	48	60	72	84	96	9999
2008	11,807	15,165	26,043	35,183	41,908	62,206	70,432	70,432	70,432

Incremental Paid Severities Adjusted to Cost Level of Accident Year Assuming 5% Annual Trend Rate and Tort Reform Adjustment

	12	24	36	48	60	72	84	96	9999
2001	5,064	11,705	24,969	40,317	43,445	50,781	79,730	103,551	74,709
2002	11,417	13,896	25,175	33,359	41,386	63,382	64,556	78,444	78,444
2003	9,631	10,386	19,703	31,706	47,592	78,942	82,366	82,366	82,366
2004	9,452	12,524	27,023	42,657	52,947	76,384	86,485	86,485	86,485
2005	10,315	11,136	27,255	45,934	54,032	80,203	90,809	90,809	90,809
2006	11,502	13,768	37,427	42,549	50,682	75,230	85,179	85,179	85,179
2007	10,726	18,043	24,803	33,508	39,912	59,244	67,078	67,078	67,078
2008	12,351	15,165	26,043	35,183	41,908	62,206	70,432	70,432	70,432

Exhibit VI Sheet 7

print('Projected Incremental Closed Claim Counts')
nb_display(e6_s7_ccc.round(0))
print('Incremental Paid Severities Adjusted to Cost Level of Accident Year Assuming 5% Annual Trend Rate and Tort Reform Adjustment')
nb_display(e6_ipsev_full_complete)
print('Projected Incremental Paid Claims')
nb_display(e6_s7_ip)
print('Projected Cumulative Paid Claims')
nb_display(e6_s7_ip.incr_to_cum())

Projected Incremental Closed Claim Counts

	12	24	36	48	60	72	84	96	9999
2001	304.00	377.00	255.00	156.00	133.00	132.00	75.00	14.00	9.00
2002	203.00	404.00	234.00	248.00	238.00	137.00	59.00	21.00	10.00
2003	181.00	433.00	327.00	322.00	244.00	61.00	39.00	16.00	8.00
2004	235.00	613.00	594.00	410.00	177.00	109.00	78.00	31.00	16.00
2005	295.00	824.00	545.00	282.00	235.00	103.00	74.00	29.00	15.00
2006	307.00	599.00	295.00	177.00	155.00	68.00	48.00	19.00	10.00
2007	329.00	462.00	160.00	133.00	116.00	51.00	36.00	15.00	7.00
2008	276.00	388.00	156.00	130.00	114.00	50.00	36.00	14.00	7.00

Incremental Paid Severities Adjusted to Cost Level of Accident Year Assuming 5% Annual Trend Rate and Tort Reform Adjustment

	12	24	36	48	60	72	84	96	9999
2001	5,064	11,705	24,969	40,317	43,445	50,781	79,730	103,551	74,709
2002	11,417	13,896	25,175	33,359	41,386	63,382	64,556	78,444	78,444
2003	9,631	10,386	19,703	31,706	47,592	78,942	82,366	82,366	82,366
2004	9,452	12,524	27,023	42,657	52,947	76,384	86,485	86,485	86,485
2005	10,315	11,136	27,255	45,934	54,032	80,203	90,809	90,809	90,809
2006	11,502	13,768	37,427	42,549	50,682	75,230	85,179	85,179	85,179
2007	10,726	18,043	24,803	33,508	39,912	59,244	67,078	67,078	67,078
2008	12,351	15,165	26,043	35,183	41,908	62,206	70,432	70,432	70,432

Projected Incremental Paid Claims

	12	24	36	48	60	72	84	96	9999
2001	1,539	4,413	6,367	6,289	5,778	6,703	5,980	1,450	672
2002	2,318	5,614	5,891	8,273	9,850	8,683	3,809	1,621	811
2003	1,743	4,497	6,443	10,209	11,612	4,815	3,224	1,290	645
2004	2,221	7,677	16,052	17,489	9,372	8,324	6,732	2,693	1,346
2005	3,043	9,176	14,854	12,953	12,716	8,258	6,678	2,671	1,336
2006	3,531	8,247	11,041	7,552	7,851	5,098	4,123	1,649	825
2007	3,529	8,336	3,960	4,458	4,634	3,010	2,434	974	487
2008	3,409	5,890	4,071	4,583	4,764	3,094	2,502	1,001	500

Projected Cumulative Paid Claims

	12	24	36	48	60	72	84	96	9999
2001	1,539	5,952	12,319	18,609	24,387	31,090	37,070	38,520	39,192
2002	2,318	7,932	13,823	22,096	31,945	40,629	44,438	46,059	46,869
2003	1,743	6,240	12,683	22,893	34,505	39,320	42,545	43,834	44,479
2004	2,221	9,898	25,950	43,439	52,811	61,135	67,867	70,560	71,906
2005	3,043	12,219	27,073	40,026	52,742	61,000	67,678	70,350	71,685
2006	3,531	11,778	22,819	30,371	38,222	43,321	47,444	49,093	49,918
2007	3,529	11,865	15,825	20,283	24,917	27,926	30,360	31,334	31,821
2008	3,409	9,299	13,370	17,954	22,718	25,812	28,314	29,315	29,816

Exhibit VI Sheet 8

summary_exh(
    e6_tri['Reported Claims'],
    e6_tri['Paid Claims'],
    e6_s7_ip.incr_to_cum()
)

	Reported	Paid	Ult Claims	Case Outstanding	IBNR	Unpaid
2001-01-01	38798.0	38520.0	39192.0	278.0	394.0	672.0
2002-01-01	48169.0	44438.0	46869.0	3731.0	-1300.0	2431.0
2003-01-01	44373.0	39320.0	44479.0	5053.0	106.0	5159.0
2004-01-01	70288.0	52811.0	71906.0	17477.0	1618.0	19095.0
2005-01-01	70655.0	40026.0	71685.0	30629.0	1030.0	31659.0
2006-01-01	48804.0	22819.0	49918.0	25985.0	1114.0	27099.0
2007-01-01	31732.0	11865.0	31821.0	19867.0	89.0	19956.0
2008-01-01	18632.0	3409.0	29816.0	15223.0	11184.0	26407.0

#Exhibit VI Sheet 1
assert np.all(e6_dr_selected.disposal_rate_.values == np.array([0.244, 0.572, 0.704, 0.814, 0.910, 0.952, 0.982, 0.994, 1.000]))
#Exhibit VI Sheet 2
lhs = (e6_dr_tri.full_triangle_.cum_to_incr()-e6_tri['Closed Claim Counts'].cum_to_incr()).values.flatten()
rhs = np.array([
                                                        9.,  
                                                21.,    10.,  
                                        39.,    16.,    8.,  
                                109.,   78.,    31.,    16.,  
                        235.,   103.,   74.,    29.,    15.,  
                177.,   155.,   68.,    48.,    19.,    10.,  
        160.,   133.,   116.,   51.,    36.,    15.,    7.,  
389.,   156.,   130.,   114.,   50.,    36.,    14.,    7.
])
assert np.all(abs(lhs[~np.isnan(lhs)] - rhs) < 1)
#Exhibit VI Sheet 4
assert np.all(e6_sevs_sel.values[...,:6].round(0) == np.array([11807.,15165.,26043.,35183.,41908.,62206.]))
#Exhibit VI Sheet 5
assert np.all(e6_s5.iloc[4].values.round(0) == np.array([62206.,70432.]))
#Exhibit VI Sheet 6
lhs = e6_ipsev_full_detrended[e6_ipsev_full_detrended.valuation > e6_ipsev.valuation_date].values.flatten()
rhs = np.array([
                                                        74709., 
                                                78444., 78444., 
                                        82367., 82367., 82367., 
                                76384., 86485., 86485., 86485., 
                        54032., 80203., 90809., 90809., 90809., 
                42549., 50682., 75230., 85179., 85179., 85179., 
        24803., 33508., 39912., 59244., 67079., 67079., 67079., 
15165., 26043., 35183., 41908., 62206., 70432., 70432., 70432., 
])
assert np.allclose(lhs[~np.isnan(lhs)], rhs, atol=1)
#Exhibit VI Sheet 7
assert np.allclose(
    e6_s7_ip.incr_to_cum().iloc[:,:,:,-1].values.round(0).flatten(),
    np.array([39192.,46869.,44479.,71906.,71684.,49913.,31805.,29828.]),
    rtol = .001
)