Impacts of Frequency Contagion on Pricing of Catastrophe Excess of Loss Reinsurance for Australian Natural Perils

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2 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.

3 Agenda Australian Natural Perils Frequency Influences Poisson Contagion Convolution Cat XoL Pricing Photo: Pinterest 3

4 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

5 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

6 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

7 Peril Peril versus Sub-Peril Sub-Peril Wind Flood Hail Storm Surge Tropical Cyclone Yes Yes No Yes Severe Convective Storm Yes Flash Yes No Low Pressure System Yes Yes Maybe Yes 7

8 Peril types 6% 28% 17% Losses rescaled to 1/1/2016 Earthquake 1989 Newcastle NSW $4,223m 1994 Ellalong NSW $150m 1968 Meckering WA $74m Cyclone 1974 Cyclone Tracy $5,330m 1974 Cyclone Wanda $3,447m 1973 Cyclone Madge $1,944m Low Pressure System 2011 SEQ Floods $3,084m 2007 Newcastle Storm $2,270m 2015 East Coast NSW $950m % Severe Convective Storm 1999 Sydney Hailstorm $5,599m 1985 Brisbane Hailstorm $2,689m 1990 Sydney Hailstorm $1,690m 42 10% Count Loss Bushfire 1983 Ash Wednesday $2,341m 2009 Black Saturday $1,650m 2003 Canberra Fires $860m 8

9 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

10 Define the event by the underlying Peril Key Takeaway #1 Then Identify the damage and loss due to each Sub-peril Source: Youtube 10

11 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

12 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 Simeon Denis Poisson Portrait: Francois Seraphin Delpech 12

13 Poisson PUB 13

14 Non-Poisson PUB 14

15 ENSO effect on Atlantic storm frequency Neelin, J.D, Climate Change and Climate Modelling, 2011 Correlation coefficient, goodness of fit = 0.40 La Nina El Nino 15

16 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

17 IOD 17

18 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

19 SOI 19

20 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 Australia Image: IPCC 20

21 SAM 21

22 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 W Image: NOAA 22

23 Nino

24 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

25 MEI 25

26 Best Fits by Peril 26

27 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 Source: thechive.com 27

28 Average = 6.5 Base Poisson Frequency = Average = 9.5 Average = 3.5 Mixing/Smear/Contagion 3.5 or 6.5 or 9.5 equal chance = Average = 6.5 Contagioned distribution 28

29 Poisson and Contagioned Poisson Pr N = n = e λ λ n n! Pr N = n = 0 e λu λu n n! f u du 0 < u < 29

30 Linear fit to Contagion y Pr N = n = 1+y 1 y e λu λu n n! 1 2y du 1 f u = 2y 1 y < u < 1 + y = λ(1+y) 0 e z z n dz λn! λ(1+y) 0 e z z n dz γ x + 1, λ 1 + y γ (x + 1, λ 1 y = λn! 1 2y 1 2y 30

31 Quadratic fit to Contagion y Pr N = n = 1+y 1 y 2 e λu λu n n! y u 9y 2 du f u = y u 9y 2 = 8(1 + y) 9y² 8 9y² γ x + 1, λ 1 + y γ (x + 1, λ 1 y 2 λn! γ x + 2, λ 1 + y γ (x + 2, λ 1 y 2 λ²n! 1 y/2 < u < 1 + y 31

32 Gamma fit to Contagion Pr N = n = 0 e λu λu n n! u 1 θ 1 e u θ Γ 1 θ θ1 θ du f u = u 1 θ 1 e u θ θ = Γ 1 θ θ1 θ E(u) = 1.0 Var(u) = contagion = n + 1 θ 1 n θλ θλ + 1 i.e. a Negative Binomial n 1 θλ θ 32

33 N Practical Calculations Poisson without Contagion Using Equation (5) Using 10 Bands Using 100 Bands

34 Key Takeaway #3 The impact of global weather influences on peril frequency may be easily quantified for use in loss modelling 34

35 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

36 Aggregate Claims C = X 1 + X X N E(C) = E(N) E(X) Var(C) = Var(N)E(X) 2 + E(N)Var(X) 36

37 Poisson and Contagioned Poisson Poisson : (Compound Poisson) Var(C) = Var(N)E(X) 2 + E(N)Var(X) = E(N)E(X 2 ) Contagioned Poisson : Var(C) = v 2 E N 2 E X 2 + E(N)Var(X) = E N E X 2 + v 2 E N 2 E X 2 E N E(X) 2 = Variance of Compound Poisson + v 2 1 E N E(C) 2 Where v is the coefficient of variation of the Contagioned frequency 37

38 Modelled Return Periods Low layer Frequency covers 6-10 years 3-6 years 2-3 years 1-2 years year Below First Second Third Fourth Event Occurrence Protect against earnings volatility May involve complicated Drop-down triggers Influence of contagion will vary by occurrence Need to be modelled individually 38

39 Simulation Error 6-10 years 3-6 years 2-3 years 1-2 years year Below First Second Third Fourth 1/6 chance of attachment Prob[4 or more events]= % (Assuming Poisson) Number of hits on 1 million simulations = 28 Simulation error of ±19% 39

40 Closed form Approximations Normal Approximation Normal Power Approximation Power Transformation 40

41 Panjer s Recursion Requires.. Pr N = n = a + b n Pr (N = n 1) 41

42 f n C f(x) Event Loss Recursion g C = Pr N = n f n (C) n=0 is the n-fold convolution probability that, if there are exactly n claims, the total claim cost will be exactly C. b f n C = f x f n 1 (C x) x=0 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 42

43 Calculation Algorithm 1. 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 f n c ; 8. Complete loops 6, 5 and 4; and 9. Write results to output file 43

44 Key Takeaway #4 A recursive process can be used to determine accurate metrics of loss for excess of loss covers in a timely manner Source: Darren Pateman 44

45 Run time Use of Graphics Processors Excel VBA Visual Basic C# C++ C++/CUDA Run time (seconds) Speed increase of 1862 times Source: MSI Simultaneous processes 45

46 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 year, 1-2 years, 2-3 years, 3-6 years, 6-10 years Multiple events 1st, 2nd, 3rd and 4th events 46

47 Occurrence PML Expected Loss using Poisson 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% year 78.5% 44.5% 18.8% 6.2% Below First Second Third Fourth Return Period (Years) Expected Loss (using event loss recursion on GPU) 47

48 Adding Contagion to Poisson 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% 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% year 75.8% 43.4% 19.9% 7.6% Below First Second Third Fourth Expected Loss Poisson Expected Loss Poisson with ±50% contagion 48

49 Expected Loss with Contagion ± 50% 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% 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% year 96.5% 97.5% 105.9% 123.1% Below First Second Third Fourth Expected Loss Proportion of Poisson 49

50 Expected Loss with Contagion ± 80% 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% 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% year 90.7% 94.9% 115.0% 155.1% Below First Second Third Fourth Expected Loss Proportion of Poisson 50

51 Frequency variability due to global weather patterns can increase technical prices of catastrophe excess of loss for events beyond first loss Key Takeaway #5 Photo: Kirkstall Bridge Inn 51

52 Photo: FAIRFAX ARCHIVES 52

53 Thank you 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. 53