PIR Book Examples in Aggregate

Python code to recreate exhibits from Pricing Insurance Risk book.

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Table 4.3

Pricing Insurance Risk Table 4.3 on page 76 shows estimated occurrence and aggregate PML points for a simple model of US Hurricane Risk. The table is reproduced in Table 1 using the following code.

import pandas as pd
import numpy as np
import scipy.stats as ss
from aggregate import build

pd.set_option("display.float_format", lambda x: f'{x:,.0f}' if abs(x) > 10 else f'{x:.3f}')

# model parameters
sigma = 2.58
en = 1.74
cv = np.sqrt(np.exp(sigma**2)-1)

# manual severity
sev = ss.lognorm(sigma, scale=np.exp(-sigma**2/2)*5000)

# return periods
Ns = np.array([2, 5, 10, 20, 25, 50, 100, 200, 250, 500, 1000])
Ps = 1 - 1 / Ns
PML = sev.ppf(1 + np.log(Ps) / en)

df = pd.DataFrame({'n': Ns, 'p': Ps, 'Adj p':1 + np.log(Ps) / en,  'Occ PML': PML}).set_index('n')

usw = build(f'''
agg USHurr {en} claims 4000000 xs 0 sev lognorm 5000 cv {cv} poisson
''', bs=20, log2=18, padding=2)

df['Agg PML'] = [usw.q(i) for i in df.p]
df['Sev VaR'] = [sev.isf(1-p) for p in df.p]
df['Est Agg VaR'] = [sev.isf((1-p)/en) for p in df.p]
df['Error'] = df['Est Agg VaR'] / df['Agg PML'] - 1
df.iloc[:, [0, 1, 2, 4, 3, 5, 6]]

Table 1: Estimated occurrence and aggregate PML points for a simple model of US Hurricane risk. Frequency Poisson with mean 1.74. Severity lognormal with mean USD5 billion and \(\sigma=2.58\). Amounts in USD millions at 2-18 price levels. Return period in \(n\) years.

	p	Adj p	Occ PML	Sev VaR	Agg PML	Est Agg VaR	Error
n
2	0.500	0.602	348	179	420	763	0.816
5	0.800	0.872	3,350	1,572	3,940	3,971	0.008
10	0.900	0.939	9,783	4,892	11,080	10,467	-0.055
20	0.950	0.971	23,420	12,490	25,620	24,107	-0.059
25	0.960	0.977	30,196	16,412	32,720	30,873	-0.056
50	0.980	0.988	62,630	35,870	66,180	63,256	-0.044
100	0.990	0.994	121,162	72,473	125,760	121,714	-0.032
200	0.995	0.997	222,461	137,948	228,020	222,931	-0.022
250	0.996	0.998	268,049	167,935	273,900	268,492	-0.020
500	0.998	0.999	466,816	300,935	473,500	467,178	-0.013
1000	0.999	0.999	787,463	520,108	794,820	787,752	-0.009