Code
import pandas as pd
from aggregate import build, qd, __version__
from style import wprint, GT, refuse, figures
figures()
print(__version__)1.43.1Using reinstatements
Paid reinstatements on an occurrence layer, with a ceded premium that varies with the annual recovery
reinsurance
variable feature
P&L
An occurrence layer \(y\) xs \(a\) recovers \(R = \sum_i \min((X_i - a)^+, y)\) in a year from the claims \(X_1, \dots, X_N\). Property catastrophe and per-risk treaties rarely give that cover without limit. Each loss to the layer uses up part of its limit, and the cedant reinstates the limit by paying a reinstatement premium, pro rata to the amount reinstated but not to the time left in the year. With base premium \(B\), rate on line \(r = B/y\), and \(m\) reinstatements priced at multiples \(\alpha_1, \dots, \alpha_m\) of \(r\), the recovery and the ceded premium are \[ \min(R, (m+1)y) \qquad\text{and}\qquad B + r\sum_{j=1}^{m} \alpha_j \min\bigl((R - (j-1)y)^+, y\bigr). \] Both are nondecreasing functions of \(R\), so the ceded premium is random and is largest in the years when the recovery is largest.
The reinstatements clause decorates one occurrence layer and needs a base premium clause, rol, deposit, or rate, to fix \(r\). The schedule is written in treaty language, with free for a multiplier of zero and counts as digits or the words one to five, or as an explicit list of multipliers. The form no reinstatements gives one annual limit and no reinstatement premium; omitting the clause leaves the cover free and unlimited. A pnl or xpnl books the random premium (see Using pnl and xpnl); a plain agg has no premium context and ignores the clause with a warning. Reinstatements are the occurrence counterpart of the variable features on aggregate covers: swing rating, the sliding scale, the profit commission, and the loss corridor.
occurrence net of 1000 xs 1000 rol 0.45 reinstatements 1 at 100%
1000 xs 1000 rol 0.45 reinstatements 1 free and 2 at 100% # four annual limits
1000 xs 1000 rol 0.45 reinstatements [0 1 1] # the same schedule
1000 xs 1000 deposit 450 reinstatements two at 50%
1000 xs 1000 rol 0.45 no reinstatements # one annual limitA reinstated layer on the base book
The variable feature pages share one base book: premium 10,000 at a 70% loss ratio, a policy limit of 2,000, lognormal severity with mean 150 and CV 2.5, gamma mixed claim counts with mixing CV 0.25, and expenses of 25% of premium. The layer 1000 xs 1000 is a working layer, with a rate on line of 45% and one reinstatement at 100%. About one claim a year exceeds 1,000, and 0.27 a year reach the policy limit and exhaust the layer.
Code
book = '''agg Book
10000 premium at 0.7 lr
2000 xs 0
sev lognorm 150 cv 2.5 {occurrence}
mixed gamma 0.25 {aggregate}'''
program = '''xpnl {name}
inherit premium
less
''' + book + '''
less
0.25 premium expenses'''
layer = 'occurrence net of 1000 xs 1000 rol 0.45 {} as Layer'
x = build(program.format(name='Reinstated', aggregate='',
occurrence=layer.format('reinstatements 1 at 100%')))
qd(x)
e = x.engine
print(f'claims a year over 1,000 {e.n * e.sevs[0].sf(1000):.4f}, '
f'at 2,000 {e.n * e.sevs[0].sf(1999.99):.4f}')PnL('Reinstated': 2 groups, 5 legs -> 'margin')
EX SD CV Skew P01 Median P99
Step Side
Gross Consideration 10000 0 0 10000 10000 10000
Obligation -9500 2704.6 -0.2847 -0.67669 -17080 -9200 -4640
Margin 500 2704.6 5.4092 -0.67669 -7080 800 5360
Layer Consideration -635.95 194.51 -0.30587 -0.38086 -900 -549 -450
Obligation 508.15 602.68 1.186 1.0039 0 220 2000
Margin -127.8 427.51 -3.3453 1.5169 -450 -329 1100
Net 372.2 2450.8 6.5847 -0.6246 -6420 620 4930
All Consideration 9364.1 194.51 0.020772 -0.38086 9100 9451 9550
Obligation -8991.9 2358 -0.26223 -0.64943 -15580 -8740 -4620
Margin 372.2 2450.8 6.5847 -0.6246 -6420 620 4930
Impact -127.8 427.51 -3.3453 1.5169 -450 -329 1100
claims a year over 1,000 0.9925, at 2,000 0.2705The Layer step shows the effect of the clause. The ceded premium has mean 635.95 and standard deviation (SD) 194.51. It runs from the base premium of 450 in a year with no recovery to 900 once the first limit is used up and reinstated. The recovery has mean 508.15, and its 99th percentile, 2,000, is the cap of two limits. Net of the layer, the cedant’s mean result falls from 500 to 372.20 and its SD from 2,704.60 to 2,450.82.
Refused programs
A reinstatements clause needs a base premium on its layer, decorates an occurrence layer only, and requires that layer to be the only occurrence layer.
Code
reinst = 'rol 0.45 reinstatements 1 at 100%'
refuse(program.format(name='R1', aggregate='',
occurrence='occurrence net of 1000 xs 1000 '
'reinstatements 1 at 100%'),
program.format(name='R2', occurrence='',
aggregate=f'aggregate net of 4000 xs 8000 {reinst}'),
program.format(name='R3', aggregate='',
occurrence='occurrence net of 500 xs 500 rol 0.9 and '
f'1000 xs 1000 {reinst}'))DecL: a 'reinstatements' clause needs a base premium clause (deposit / rol /
rate) on the same layer; the base rate on line r = base_premium / limit would
otherwise be undefined ([reins-premium]).
DecL: a 'reinstatements' clause decorates an occurrence layer, not aggregate
reinsurance.
DecL: 'reinstatements' require a single occurrence layer; found 2
([reins-single-layer]). A reinstated layer cannot share the occurrence tier with
other layers (the engine ceiling is the 2-D (L, R) joint).