# Mathematical Concepts in the Insurance Industry. Felix Rosenbaum, Risk Management, SCIPP Seminar April 2011

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1 Mathematical Concepts in the Insurance Industry Felix Rosenbaum, Risk Management, SCIPP Seminar April 2011

2 Table of Contents / Agenda n Introduction to Swiss Re n Insurance Math n Nat Cat Modelling n Natural Catastrophes on the rise? 2

3 Introduction to Swiss Re 3

4 Swiss Re n Re-Insurance Company n Founded 19th December 1863 n 2010 Premiums earned: 19'652 Million USD n 2010 net income: 863 Million USD n Combined Ratio: P&C 93.9% and L&H 88.7% n Number of employees: 10'362 4

5 Insurance Math n An introduction 5

6 Example: Insure your friend's car How much would you charge to insure your friends car? n Simple questions: How often per year does he have an accident? (=f) n How much money does it usually cost to repair his car? (=X) The average loss per year: S = f*x n You also want to make some money, so add a profit percentage (=p). Price = f*l*(1+p) 6

7 Example: Insure your friend's car n That was easy! n But, how much money do need to keep aside (=reserves) to pay your friend, in case he has an accident? n If you insure one car only, you will have to have reserves up to the maximum possible loss, in other words, the value of the car. 7

8 More cars... n But now we want to insure many cars. n The yearly loss now is (X is the loss, N the number of losses): n It is obvious, that S will not be the same for every year, but has a distribution. The challenge is to find distributions for X~F(x) and N~P. 8

9 Loss distribution n A often used distribution for the loss is Pareto 9

10 Pareto n The alpha depends on the type of risk. F(x) alpha = 2 alpha = 1 loss 10

11 Frequency n Very commonly used is the Poisson distribution n Poisson works fine if events are rare and independent. 11

12 The Result n We now have a distribution for the loss size and loss number to represent S. n The aggregated cdf is usually calculated with Monte Carlo methods: - draw the number of losses per year - draw the loss amounts and add them up. n Ordered by loss amount of the year one can calculate the aggregated CDF. n The average of these outcomes returns the expected loss. 12

13 Aggregated CDF Probability Agg Loss 13

14 More things to consider But there is more to think of... n Long term/short term claims n Capital costs n Liquidity n Profit margin n Brokerage n Recovery n Internal costs n Taxes 14

15 And the reserves? n How much money do we have to reserve now? n To hold the MPL for all contracts would be way too expensive! n Therefore we hold reserves cover two 99% shortfall years: The shortfall is defined as: shf(s) = <S S>Q(99%)> n We calculate distribution of the losses versus the capital we hold for the whole Swiss Re group. n There is a possibility that we go bankrupt! Otherwise we would be way too expensive. 15

16 Research areas n Correlations! For example Pandemic will not only trigger many life insurances, but the stock market will go down, too! n Avoid surprises! Swiss Re is constantly looking at possible emerging risks as climate change, nano-particles, cell phone radiation etc. 16

17 Nat Cat Modelling n Hurricanes 17

18 Natural Catastrophes n Swiss Re develops own models for natural catastrophes. n This covers models for flood, hail, winter storms, earthquakes, bushfires, and tropical cyclones. 18

19 Principles of Nat Cat Modelling Four basic sets of data are needed to be fed into a loss model: n Hazard: Where, how often and with what intensity do events occur? n Vulnerability: What is the extent of damage at a given event intensity? n Value distribution: Where are the various types of insured objects located and how high is their value? n Insurance conditions: What proportion of the loss is insured? 19

20 Natural Catastrophes - Hurricanes n There is a lot of historical data about hurricanes: n Data is recorded since 1891 of more than 1'000 hurricanes. 20

21 Natural Catastrophes - Hurricanes n These historical storms can be used as a basis for hurricane modelling. n Basic Formula to calculate the loss of hurricane wind speed * vulnerability * insured value * insurance conditions = loss n These points consist of many sections, for example... - wind speed at a distance x from the storm - construction type - age of building - elevation (storm surge) 21

22 Hurricanes n The historical storms are not granular enough... n... therefore storms have to be invented. 22

23 Hurricanes n How can you create 'daughter storms'? à Random Walk 23

24 Hurricanes n Find border conditions to limit the random walk so that the resulting tracks are realistic. 24

25 Hurricanes n How much can a track deviate from the mother storm? n How much can the wind speed vary? n Landfall n Maximum Intensity 25

26 Hurricanes - LFC 26

27 Hurricanes - LFC 27

28 Natural Catastrophes on the rise? 28

29 Insured catastrophe losses Ocean Drive, FL, 1926 Ocean Drive, FL, 2000 USD bn, at 2009 prices n increased insurance penetration n more values n more values in high - risk areas n higher vulnerability n climate change (storm,flood) 1999: Winter storm Lothar 1992: Hurricane Andrew 1994: Northridge EQ 2004: Hurricanes Ivan, Charley, Frances 2001: Attack on WTC : Hurricanes Katrina, Rita, Wilma 2008: Hurricanes Ike, Gustav Earthquake/tsunami Weather- related Nat C ats Man- made disasters 2009: EQs Chil New Zealand 29 Source: Swiss Re, sigma No 1/2010; 1/2011, Figure 3

30 Number of events per year An event has a loss and victim threshold. 30

31 Number of victims per year 31

32 Natural Catastrophes - Hurricanes 32

33 Thank you

34 Legal notice 2011 Swiss Re. All rights reserved. You are not permitted to create any modifications or derivatives of this presentation or to use it for commercial or other public purposes without the prior written permission of Swiss Re. Although all the information used was taken from reliable sources, Swiss Re does not accept any responsibility for the accuracy or comprehensiveness of the details given. All liability for the accuracy and completeness thereof or for any damage resulting from the use of the information contained in this presentation is expressly excluded. Under no circumstances shall Swiss Re or its Group companies be liable for any financial and/or consequential loss relating to this presentation. 34

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