# Non-life insurance mathematics. Nils F. Haavardsson, University of Oslo and DNB Skadeforsikring

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1 Non-life insurance mathematics Nils F. Haavardsson, University of Oslo and DNB Skadeforsikring

2 About the lecturer PhD in risk analysis, Institute for Mathematics, University of Oslo 11 years experience from insurance Senior Actuary in DNB Skadeforsikring (current position) Actuary in Gjensidige Actuary in KLP Actuarial consultant in Avenir 6 years experience from other sectors (energy, research) Quantitative economic risk assessments (consulting, research) 2 years pre-graduate experience (teamleader/instructor in military, transport) Member of Norwegian Actuarial Association 2

3 Overview of this session Module Contents Part in EB* Basic concepts General insurance: an opening look How Monte Carlo simulation is put to work Intro Pricing Portfolio and solvency Risk ceding and reinsurance with risk drivers Risk selection and pricing Intro Enter contracts and their clauses Stochasitc modelling Risk diversification How are random variables samled? Inversion Introduction Computing the reserve When responsibility is limited Dealing with reinsurance Some important concepts of real life (if time) EB*=Computation and modeling in insurance and finance by Erik Bølviken Combined ratio 3

4 Overview Non-life insurance from a financial perspective: for a premium an insurance company commits itself to pay a sum if an event has occured Contract period, a year in property insurance Policy holder signs up for an insurance Policy holder pays premium. Insurance company starts to earn premium retrospective prospective During the duration of the policy, some of the premium is earned, some is unearned How much premium is earned? How much premium is unearned? Is the premium sufficient to cover incurred (retrospective) and future losses (prospective)? 4

5 Why does it work?? Client 1 Client 2 Insurance company Client n-1 Client n Economic risk is transferred from the policyholder to the insurer Due to the law of large numbers (many almost independent clients), the loss of the insurance company is much more predictable than that of an individual Therefore the premium should be based on the expected loss that is transferred from the policyholder to the insurer Much of the course is about computing this expected loss...but first some insurance economics 5

6 How can the result of an insurance company be decomposed? Insurance economics in its most basic form: : + Insurance premium paid by the clients + financial income generated by the premium from the clients - claims paid to the clients - operational costs of the Insurance company = result to be distributed among the owners and the authorities 6

7 Insurance mathematics is fundamental in insurance economics The result drivers of insurance economics: : Result drivers: Risk based pricing, + Insurance premium reinsurance International economy for example interest rate level, + financial income risk profile for example stocks/no stocks risk reducing measures (for example installing burglar alarm), risk selection (client behaviour), change in legislation, weather phenomenons, demographic factors, - claims reinsurance measures to increase operational efficiency, IT-systems, - operational costs wage development = result to be distributed among the owners and the authorities Tax politics 7

8 Insurance economics Risk selection: Object risk which house is most likely to burn down??

9 Insurance economics Risk selection: subject risk.. sloppy client who is always unlucky.

10 Different price elements have different risk drivers Object Geography Subject Standard Electrical system Pipes Roof Construction material The number of wet rooms and the number of kitchens Building year Maintenance level in general Electrical system reviewed? Weather Climate Population density Infrastructure complexity Natural catastrophies Demography Policy holder Age Profession Risk aversion Personality structured or carefree? The number of inhabitants in building Use Inhabited by owner Rental Vacation use (inherited?)

11 Find riskodrivers for every price element Objekt Insurance object Element Fire Water Risk driver Cumule Elektrical system Construction material Flood risk Pipe quality Weather A B C

12 Shows how much of the premium income is spent to cover losses Amounts in NOK 2012 Written gross premium ceded reinsurance premium -270 Change in reserve for unearned gross premium change in reinsurance share of unearned premium 25 Net premium income Amounts in NOK 2012 Paid claims gross Reinsurance share of paid claims gross 120 change in gross claims reserve change in reinsurance part of gross claims reserve 100 Net claims costs -850 Gross Net Incurred losses 1070 ( ) 850 Earned premium 1340 ( ) % 77.6% What does the difference in loss ratio gross and net tell us?

13 Skadeprosent B/K villa, content and cabin ,0 % 90,0 % 80,0 % 70,0 % 60,0 % 50,0 % Difference of 33 pp in 6 years!?! Yearly premium of 9 billion 40,0 % % Skade% 75,2 86,9 79,2 85,9 74,0 67,8 56,4 55,5 60,6 64,5 73,1 73,8 88,6 74,8 The graph presents loss ratio for all insurance companies for villa, content and cabin The graph illustrates the delay in the price adjustments and the need for reinsurance The graph illustrates the effect of claims frequency (frost 2010) Source: FNO.no

14 Amounts in NOK 2012 Sales costs -85 Insurance related operational costs -200 received provision for ceded reinsurance 45 Insurance related operational costs -240 Sales costs: Provisions, sales offices, marketing, back-office sale Insurance related operation costs: management, accounting, actuary, house rent, HR, IT etc.up to 2012 also claims settling costs NB:these were transferred to claims in 2012 Received provision reinsurance: - Normally it constitutes 20% to 25% of ceded premium.. - NB: Cost income in the table why? - Why do the companies receive this provision?

15 Cost ratio (percent) Shows how much of the premium income is spent to cover operational costs Amounts in NOK 2012 Written gross premium ceded reinsurance premium -270 Change in reserve for unearned gross premium change in reinsurance share of unearned premium 25 Net premium income Amounts in NOK 2012 Sales costs -85 Insurance related operational costs -200 received provision for ceded reinsurance 45 Insurance related operational costs -240 Gross Net Operational costs 285 ( ) 240 Earned premium 1340 ( ) 1095 Cost ratio (percent) 21.3% 21.9% What does the difference in cost ratio gross and net tell?

16 Cost ratio (percent) Cost ratio What is causing the reduction in cost ratio to 20%?Where are the companies heading? Source: fno.no Results in non-life insurance: includes all non-life insurance companies in Norway

17 Combined ratio Shows how much of the premium income that is spent to cover claims and operational costs Gross Net 79.9% 77.6% Cost ratio 21.3% 21.9% Combined ratio 101.1% 99.5% Combined ratio above 100 % implies that the insurance operations are not profitable What do the combined ratio gross and net express for the example company? Long term CR for insurance companies in Norway are between 90% and 95% What key ratio is most problematic for the example company?

18 Key parameters for non-life insurance in Norway The graph shows loss ratio (left axis), result degree (total revenue minus total costs, right axis), cost ratio (right axis) andincome from investments in percent of premium (right axis) for the period The result degree and the loss ratio vary a lot. The loss ratio seems to be the most important driver for profitability in non-life insurance The cost ratio and income from investments in percent of premium are decreasing during the period. Result degree Cost ratio Income from investments as percent of premium 18

19 Outline of the course Models treated Curriculum Duration in lectures Basic concepts and introduction EB 1.2, 2.3.1, 2.3.2, 3.2, How is claim frequency modelled? Poisson, Compound Poisson, Poisson regression, negative binomial model EB 8.2, 8.3, How is claim reserving modelled? Delay modelling, chain ladder EB 8.5, Note 1-2 How is claim size modelled? Gamma distribution, log-normal distribution, Pareto distribution, Weibull distribution EB How is pricing done? EB 10 1 Solvency EB 10, Note 1-2 Credibility theory Buhlmann Straub EB 10 1 Reinsurance EB 10 1 Repetition 1 19

20 Course literature Curriculum: Chapter 1.2, 2.3.1, 2.3.2, 2.5, 3.2, 3.3 in EB Chapter 8,9,10 in EB Note on Chain Ladder Lecture notes by NFH The following book will be used (EB): Computation and Modelling in Insurance and Finance, Erik Bølviken, Cambridge University Press (2013) Additions to the list above may occur during the course Final curriculum will be posted on the course web site in due time 20

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