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TRANSCRIPT
Impacts of Frequency Contagion
on Pricing of Catastrophe Excess of
Loss Reinsurance for Australian
Natural Perils
Dr Will Gardner
This presentation has been prepared for the 2016 General Insurance Seminar.The Institute Council wishes it to be understood that opinions put forward herein are not necessarily
those of the Institute and the Council is not responsible for those opinions.
Agenda
• Australian Natural Perils
• Frequency Influences
• Poisson Contagion
• Convolution
• Cat XoL Pricing
Photo: Pinterest3
Insurance Council of Australia Disaster List
• Events from 1967 to present
• Ten Years of Catastrophe
– Available to public
• ICA Catastrophe Database
– Available to members
– Recently updated
• Great starting point
4
Additional Event Research• Older events that had been intentionally removed from
ICA list
• Missing events– Wikipedia and other online disaster lists
– Various academic research and publications
– Historic newspapers
– Publicly reported insurance losses
• Added roughly an additional 200 events between 1967 and 2016
5
What was the actual event? Sub-perils?• 1974 Brisbane Floods – Tropical Cyclone Wanda
• Flood, wind
• 2007 Newcastle and Hunter Valley Severe Storm – East Coast Low• Flood, wind, surge
• 2010 Melbourne Hail – Severe Convective Storm• Hail, flood, wind, tornado
• 2010 Perth Hail – Severe Convective Storm• Hail, flood, wind
• 2011 Melbourne Christmas Day Storm – Severe Convective Storm• Hail, wind, flood, tornado
6
Peril versus Sub-Peril
7
Sub-Peril
Wind Flood Hail Storm
Surge
…
Pe
ril
Tropical Cyclone Yes Yes No Yes
Severe Convective Storm Yes Flash Yes No
Low Pressure System Yes Yes Maybe Yes
…
Peril types
42 10%
14939%
98
17%
64
28%
6 6%
Count Loss
1989 Newcastle NSW $4,223m
1994 Ellalong NSW $150m
1968 Meckering WA $74m
1974 Cyclone Tracy $5,330m
1974 Cyclone Wanda $3,447m
1973 Cyclone Madge $1,944m
2011 SEQ Floods $3,084m
2007 Newcastle Storm $2,270m
2015 East Coast NSW $950m
1999 Sydney Hailstorm $5,599m
1985 Brisbane Hailstorm $2,689m
1990 Sydney Hailstorm $1,690m
1983 Ash Wednesday $2,341m
2009 Black Saturday $1,650m
2003 Canberra Fires $860m
Bushfire
Cyclone
Earthquake
Severe Convective Storm
Low Pressure System
Losses rescaled to 1/1/2016
8
Australia + New Zealand = One Event?
• East Coast Low impacts
Australia
• Same “event” impacts
New Zealand
• No significant events in
ICA/ICNZ records but
appears possible
9
Key Takeaway #1
• Define the event by the
underlying Peril
Then
• Identify the damage and
loss due to each Sub-peril
10
Source: Youtube
Influence of Global Weather Systems
• Different systems
– El Nino Southern Oscillation
– Indian Ocean Dipole
– Southern Annular Mode
– Etc.
• Impact on catastrophe modelling
– Make the event frequency non-Poisson
11
Simeon Denis Poisson
• Poisson statistical distribution– One event at a time
– Events independent
– Probability of an event is proportional to time
• Australian events don’t appear to be Poisson
Portrait: Francois Seraphin Delpech
12
Poisson
13
PUB
Non-Poisson
14
PUB
ENSO effect on Atlantic storm frequency
• Neelin, J.D, “Climate
Change and Climate
Modelling”, 2011
• Correlation coefficient,
goodness of fit = 0.40
El NinoLa Nina
15
Indian Ocean Dipole (IOD)
• Sustained changes in the difference between Sea
Surface Temperatures (SST) of the tropical western and
eastern Indian Ocean
• According to BOM, the IOD is a “key driver of the
Australian climate”
16
IOD
17
Southern Oscillation Index (SOI)
• Irregularly periodical variations in winds and sea surface
temperature over the tropical eastern Pacific Ocean
• Difference in surface air pressure between Tahiti and
Darwin
Figure : NIWA
18
SOI
19
Southern Annular Mode (SAM)
• Describes the north-south movement
of the westerly wind belt that circles
Antarctica
• Positive phase = high pressure over
southern Australia
• Negative phase = low pressure over
southern AustraliaImage: IPCC
20
SAM
21
Nino 3.4
• Periodical variation in Sea Surface Temperature (SST) in
particular regions of the equatorial Pacific Ocean
• Nino 3.4 covers region from 5° N – 5° S and 120° – 170° W
Image: NOAA
22
Nino 3.4
23
Multivariate ENSO Index (MEI)
• Used to characterise the intensity of ENSO events
• Based on sea-level pressure (P), zonal (U) and
meridional (V) components of the surface wind, sea
surface temperature (S), surface air temperature (A),
and total cloudiness fraction of the sky (C)
24
MEI
25
Best Fits by Peril
26
Key Takeaway #2
• Global weather systems
influence the frequency of
Australian natural perils
although relationship with
individual indices does not
capture all of the variability
27Source: thechive.com
• Base Poisson Frequency = 6.5
• Mixing/Smear/Contagion
• 3.5 or 6.5 or 9.5 equal chance
• “Contagioned” distribution
Average = 6.5
Average = 9.5
Average = 3.5
Average = 6.5
28
+
=
+
Poisson and Contagioned Poisson
29
Pr 𝑁 = 𝑛 =𝑒−𝜆𝜆𝑛
𝑛!
Pr 𝑁 = 𝑛 = 𝑒−𝜆𝑢 𝜆𝑢 𝑛
𝑛!𝑓 𝑢 𝑑𝑢
0
0 < 𝑢 <
Linear fit to Contagion
30
𝑓 𝑢 = 1
2𝑦
1− 𝑦 < 𝑢 < 1 + 𝑦
Pr 𝑁 = 𝑛 = 𝑒−𝜆𝑢 𝜆𝑢 𝑛
𝑛!
1
2𝑦𝑑𝑢
1+y
1−𝑦
= 𝑒−𝑧𝑧𝑛𝑑𝑧 − 𝑒−𝑧𝑧𝑛𝑑𝑧
𝜆(1+𝑦)
0
𝜆(1+𝑦)
0
𝜆𝑛!
1
2𝑦
=𝛾 𝑥 + 1, 𝜆 1 + 𝑦 − 𝛾 (𝑥 + 1, 𝜆 1− 𝑦
𝜆𝑛!
1
2𝑦
y
Quadratic fit to Contagion
31
𝑓 𝑢 = 8 1 + 𝑦 − 𝑢
9𝑦2
1− 𝑦/2 < 𝑢 < 1 + 𝑦
yPr 𝑁 = 𝑛 =
𝑒−𝜆𝑢 𝜆𝑢 𝑛
𝑛!
8 1 + 𝑦 − 𝑢
9𝑦2𝑑𝑢
1+y
1−𝑦2
=
8(1 + 𝑦)
9𝑦² 𝛾 𝑥 + 1, 𝜆 1 + 𝑦 − 𝛾 (𝑥 + 1, 𝜆 1−
𝑦2
𝜆𝑛!
− 8
9𝑦² 𝛾 𝑥 + 2, 𝜆 1 + 𝑦 − 𝛾 (𝑥 + 2, 𝜆 1 −
𝑦2
𝜆²𝑛!
Gamma fit to Contagion
32
i.e. a Negative Binomial𝐸(𝑢) = 1.0
𝑉𝑎𝑟(𝑢) = contagion
𝑓 𝑢 =𝑢
1𝜃−1 𝑒−𝑢𝜃
Γ 1𝜃 𝜃
1𝜃
𝜃 =
Pr 𝑁 = 𝑛 = 𝑒−𝜆𝑢 𝜆𝑢 𝑛
𝑛!
0
𝑢
1𝜃−1 𝑒−𝑢𝜃
Γ 1𝜃 𝜃
1𝜃
𝑑𝑢
= 𝑛 +
1𝜃 − 1
𝑛
𝜃𝜆
𝜃𝜆 + 1 𝑛
1
𝜃𝜆 + 1
1𝜃
Practical Calculations
33
N
Poisson
without
Contagion
Using Equation
(5)
Using 10
Bands
Using 100
Bands
0 0.082085 0.198652 0.196598 0.198632
1 0.205212 0.191914 0.194009 0.191936
2 0.256516 0.175070 0.175219 0.175071
3 0.213763 0.146995 0.147098 0.146996
4 0.133602 0.111901 0.111974 0.111902
5 0.066801 0.076808 0.076808 0.076808
6 0.027834 0.047563 0.047502 0.047563
7 0.009941 0.026674 0.026587 0.026673
8 0.003106 0.013619 0.013537 0.013618
9 0.000863 0.006366 0.006305 0.006365
10 0.000216 0.002739 0.002701 0.002739
Key Takeaway #3
• The impact of global
weather influences on
peril frequency may be
easily quantified for use
in loss modelling
34
Excess of Loss Reinsurance
X = 0 when G < A
= G-A when A < G < A+L
= L when G > A+L
where
G is the Gross event loss net of deductibles and inuring reinsurance;
A is the layer attachment per occurrence; and
L is the layer limit per occurrence.
35
Aggregate Claims
36
𝐶 = 𝑋1 + 𝑋2 + … + 𝑋𝑁
𝐸(𝐶) = 𝐸(𝑁) 𝐸(𝑋)
𝑉𝑎𝑟(𝐶) = 𝑉𝑎𝑟(𝑁)𝐸(𝑋)2 + 𝐸(𝑁)𝑉𝑎𝑟(𝑋)
Poisson and Contagioned Poisson
37
𝑉𝑎𝑟(𝐶) = 𝑉𝑎𝑟(𝑁)𝐸(𝑋)2 + 𝐸(𝑁)𝑉𝑎𝑟(𝑋)
= 𝐸(𝑁)𝐸(𝑋2)
𝑉𝑎𝑟(𝐶) = 𝑣2 ∗ 𝐸 𝑁 2 𝐸 𝑋 2 + 𝐸(𝑁)𝑉𝑎𝑟(𝑋)
= 𝐸 𝑁 𝐸 𝑋2 + 𝑣2𝐸 𝑁 2𝐸 𝑋 2 − 𝐸 𝑁 𝐸(𝑋)2
= Variance of Compound Poisson + 𝑣2 −1
𝐸 𝑁 𝐸(𝐶)2
Poisson : (Compound Poisson)
Contagioned Poisson :
Where 𝑣 is the coefficient of variation of the Contagioned frequency
Low layer “Frequency” covers
• Protect against earnings
volatility
• May involve complicated
Drop-down triggers
• Influence of contagion will
vary by occurrence
• Need to be modelled
individually
38
6-10 years
3-6 years
2-3 years
1-2 years
0.5-1 year
Below First Second Third Fourth
Event Occurrence
Mo
de
lled
Re
turn
Pe
rio
ds
Simulation Error
• 1/6 chance of attachment
• Prob[4 or more events]= 0.00281%
(Assuming Poisson)
• Number of hits on 1 million
simulations = 28
• Simulation error of ±19%
39
6-10 years
3-6 years
2-3 years
1-2 years
0.5-1 year
Below First Second Third Fourth
Closed form Approximations
• Normal Approximation
• Normal Power Approximation
• Power Transformation
40
Panjer’s Recursion
41
𝑃𝑟 𝑁 = 𝑛 = 𝑎 +𝑏
𝑛 Pr(𝑁 = 𝑛 − 1)
Requires..
Event Loss Recursion
42
𝑔 𝐶 = Pr 𝑁 = 𝑛 𝑓𝑛∗(𝐶)
∞
𝑛=0
𝑓𝑛∗ 𝐶 is the 𝑛-fold convolution probability that, if there are exactly n
claims, the total claim cost will be exactly 𝐶.
𝑓(𝑥) is the distribution of losses to the layer after the occurrence attachment
and limit have been applied
𝑏 is the maximum value of net occurrence loss to layer 𝑋 from all events
𝑓𝑛∗ 𝐶 = 𝑓 𝑥 𝑓 𝑛−1 ∗(𝐶 − 𝑥)
𝑏
𝑥=0
Calculation Algorithm1. Define program variables;
2. Read ELT frequencies into bins, adjusting each event loss for layer attachment
and limit;
3. Rescale total frequency of all events to 1.0;
4. Loop through number of events 𝑛 from 1 to a predefined maximum event
count;
5. Loop through the values of 𝐶 from 0 to 𝑛 times the maximum net layer event
loss;
6. Loop through the values of 𝑋 from 0 to the maximum net layer event loss;
7. Add the product of 𝑓 𝑥 and 𝑓 𝑛−1 ∗ 𝑐 − 𝑥 to 𝑓𝑛∗ 𝑐 ;
8. Complete loops 6, 5 and 4; and
9. Write results to output file43
Key Takeaway #4
• A recursive process
can be used to
determine accurate
metrics of loss for
excess of loss covers in
a timely manner
44
Source: Darren Pateman
Use of Graphics Processors
45
0.1
1
10
100
1000
1 10 100 1000 10000 100000R
un
tim
e
Simultaneous processes
Speed increase of 1862 times
Source: MSI
654.53
640.74
152.11
85.02
5.51
0 60 120 180 240 300 360 420 480 540 600 660 720
Excel VBA
Visual Basic
C#
C++
C++/CUDA
Run time (seconds)
Impact of Frequency Contagion• Technical pricing, ignoring market conditions and company
losses
• Simulation of low pressure system losses (including sub-perils)
• “Average” Industry Exposure portfolio
• Artificial reinsurance programme– Multiple layers
• 0.5-1 year, 1-2 years, 2-3 years, 3-6 years, 6-10 years
– Multiple events• 1st, 2nd, 3rd and 4th events
46
Expected Loss using Poisson
47
0.5 1 2 3 6 10 20
Oc
cu
rre
nc
e P
ML
Return Period (Years)
6-10 years 12.3% 0.8% 0.0% 0.0%
3-6 years 20.5% 2.2% 0.2% 0.0%
2-3 years 34.1% 6.6% 0.9% 0.1%
1-2 years 53.2% 17.2% 3.8% 0.6%
0.5-1 year 78.5% 44.5% 18.8% 6.2%
Below First Second Third Fourth
Expected Loss
(using event loss recursion on GPU)
Adding Contagion to Poisson
48
6-10 years 12.3% 0.8% 0.0% 0.0%
3-6 years 20.5% 2.2% 0.2% 0.0%
2-3 years 34.1% 6.6% 0.9% 0.1%
1-2 years 53.2% 17.2% 3.8% 0.6%
0.5-1 year 78.5% 44.5% 18.8% 6.2%
Below First Second Third Fourth
6-10 years 12.2% 0.8% 0.0% 0.0%
3-6 years 20.3% 2.4% 0.2% 0.0%
2-3 years 33.6% 6.9% 1.0% 0.1%
1-2 years 51.8% 17.6% 4.5% 0.9%
0.5-1 year 75.8% 43.4% 19.9% 7.6%
Below First Second Third Fourth
Expected Loss
Poisson with ±50% contagion
Expected Loss
Poisson
Expected Loss with Contagion ± 50%
49
6-10 years 12.2% 0.8% 0.0% 0.0%
3-6 years 20.3% 2.4% 0.2% 0.0%
2-3 years 33.6% 6.9% 1.0% 0.1%
1-2 years 51.8% 17.6% 4.5% 0.9%
0.5-1 year 75.8% 43.4% 19.9% 7.6%
Below First Second Third Fourth
6-10 years 99.4% 108.5% 127.7% 158.6%
3-6 years 99.0% 107.4% 125.5% 155.0%
2-3 years 98.3% 105.3% 121.7% 148.9%
1-2 years 97.4% 102.4% 116.3% 140.6%
0.5-1 year 96.5% 97.5% 105.9% 123.1%
Below First Second Third Fourth
Expected Loss Proportion of Poisson
Expected Loss with Contagion ± 80%
50
6-10 years 12.1% 0.9% 0.1% 0.0%
3-6 years 20.0% 2.6% 0.3% 0.0%
2-3 years 32.6% 7.5% 1.3% 0.2%
1-2 years 49.6% 18.3% 5.4% 1.3%
0.5-1 year 71.2% 42.2% 21.6% 9.6%
Below First Second Third Fourth
6-10 years 98.4% 121.9% 170.2% 254.8%
3-6 years 97.4% 119.1% 164.5% 243.9%
2-3 years 95.6% 113.8% 154.2% 225.6%
1-2 years 93.3% 106.6% 140.3% 201.6%
0.5-1 year 90.7% 94.9% 115.0% 155.1%
Below First Second Third Fourth
Expected Loss Proportion of Poisson
Key Takeaway #5
• Frequency variability due
to global weather
patterns can increase
technical prices of
catastrophe excess of
loss for events beyond first
lossPhoto: Kirkstall Bridge Inn
51
Photo: FAIRFAX ARCHIVES 52
53
Dr Will [email protected]
Thank you
This presentation has been prepared for the 2016 General Insurance Seminar.The Institute Council wishes it to be understood that opinions put forward herein are not necessarily
those of the Institute and the Council is not responsible for those opinions.