impacts of frequency contagion on pricing of …...impacts of frequency contagion on pricing of...

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


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 =


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


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%


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%


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%


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%


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%


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%


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%


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.

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