Non-life insurance mathematics

1. Non-lifeinsurancemathematics Nils F. Haavardsson, University of Oslo and DNB Skadeforsikring

2. PhD in risk analysis, Institute for Mathematics, University of Oslo 10 yearsexperience from insurance • Actuary in DNB Skadeforsikring (currentposition) • Actuary in Gjensidige • Actuary in KLP • Actuarialconsultant in Avenir 6 yearsexperience from othersectors (energy, research) • Quantitativeeconomic risk assessments (consulting, research) MemberofNorwegianActuarial Association nilsfri@math.uio.no Aboutthelecturer

3. Overviewofthissession Non-lifeinsurance from a financialperspective – result elements and result drivers The balancesheetof a non-lifeinsurancecompany Premium Income Losses Loss ratio Costs, cost ratio and combined ratio

4. Overview Non-lifeinsurance from a financialperspective: for a premium an insurancecompanycommitsitself to pay a sum if an event has occured Contractperiod Result elements retrospective prospective The balancesheet Policy holder signs up for an insurance During thedurationofthe policy, someof thepremium is earned, some is unearned • How muchpremium is earned? • How muchpremium is unearned? • Is theunearnedpremiumsufficient? Premium Income Premium reserve, prospective Policy holder payspremium. Insurance company starts to earnpremium Losses Loss ratio Costs

5. Overview Contractperiod retrospective prospective Sign up Result elements Premium earning starts • Is theunearnedpremiumsufficient? Premium reserve, prospective • Risk premium • = Likelihoodofevent * Economicconsequenceofevent • Risk premiumexpressesclaimscost per policy • Risk premium is used to pricetheportfolio • Sincepricing is prospective, wetypicallywant to knowhowwelldidwe do? (i.e., howuncertainareour risk premiumestimates) • The uncertaintyofthe risk premium is closelyrelated to thepremium reserve risk The balancesheet Likelihood ofevent Premium Income Consequence ofevent Losses Loss ratio Distribution of requiredpremium expressespremium reserve risk Risk premium Costs Expectedrequiredpremium Likelihood ofevent Consequence ofevent

6. Overview Non-lifeinsurance from a financialperspective: for a premium an insurancecompanycommitsitself to pay a sum if an event has occured Contractperiod Result elements retrospective prospective The balancesheet Sign up Premium reserve, prospective Premium earning starts Premium Income During thedurationofthe policy, claimsmight or might not occur: • How do wemeasurethenumber and sizeofunknownclaims? • How do weknowifthe reserves onknownclaimsaresufficient? Claims reserve, retrospective Losses Accident date Reporting date Claims payments Claimsclose Claims reopening Claims payments Claimsclose Loss ratio Distribution of claimssettling time. Costs Time

7. Whydoes it work?? Client 1 Client 2 Result elements Insurance company The balancesheet Client n-1 Client n Premium Income Losses • Economic risk is transferred from the policyholder to theinsurer • Due to thelawoflargenumbers (manyalmostindependentclients), the loss oftheinsurancecompany is much more predictablethanthatof an individual • Thereforethepremiumshould be basedontheexpected loss that • is transferred from the policyholder to theinsurer Loss ratio Costs Muchofthecourse is aboutcomputingthisexpected loss ...but first someinsuranceeconomics

8. How cantheresultof an insurancecompany be decomposed? Result elements Insurance economics in its most basic form: The balancesheet Premium Income Losses Loss ratio Costs

9. Insurance mathematics is fundamental in insuranceeconomics The result drivers ofinsuranceeconomics: Result elements The balancesheet Premium Income Losses Loss ratio Costs

10. Insurance economics • Risk selection: Object risk Result elements The balancesheet Premium Income Losses Loss ratio Costs ……whichhouse is most likely to burndown??

11. Insurance economics • Risk selection: subject risk Result elements The balancesheet Premium Income Losses Loss ratio Costs ………..”sloppy” clientwho is alwaysunlucky….

12. What is a balancesheet? • The balancesheetcontainsresult and balance: Result elements The resultpresents thedevelopment in balancebetweentwopoint in time. The result 2012 shows thevaluecreation in theperiodbetween 31.12.11 and 31.12.12. The balancesheet Premium Income Losses Loss ratio Costs

13. What is a balancesheet? • The balancesheetcontainsresult and balance: • Questions: - How large wastheequity 31.12.11? - What is there to sayaboutthefinancialyield? - Is thecompanyfinancially solid? Result elements The balance presents thefinancialstateofthecompany at a given point in time, for example 31.12.12. Equity = theresidualbetweenassets and debt. The change in equitybetweentwobalanceperiods = theannualresult. The balancesheet Premium Income Losses Loss ratio Costs

14. The result as presented in theannual report The annualreportpresentation (left) and thesummarizedresult (below) areconnected: Result elements The balancesheet Premium Income Losses Loss ratio The following slides arebasedonthesummarizedresult Costs

15. Premium income Result elements • Writtenpremium : Sum ofpremium for policiescommenced (new business or renewals) in theperiod. The entirepremium for theagreementperiod is included. Thus it is possible to accountpremiumbelonging to the ”next” reportingyear. • Change in reserve for unearnedpremium: This includesaccountedwrittenpremium in whichtheagreementperiodspansthenextreportingyear. • Writtenpremium +- change in reserve for unearnedpremium = gross earnedpremium • Cededreinsurancepremium and change in reinsuranceshareofunearnedpremiumarethereinsurersshareoftheentriesabove The balancesheet Premium Income Losses Loss ratio Costs

16. Premium income • The relationshipbetweenwrittenpremium, unearnedpremium and paidpremium: Result elements The balancesheet Premium Income Losses Loss ratio • The yearlypremium for theentireagreementperiod is 12 000. The maturity is September 1st. Numberofinstallments is 1. • The premium is earnedwith 1/12 per month, i.e., 1000 NOK in theexample • Premium arrears = clientreceivable Costs

17. Premium income • Cededreinsurancepremium – premium to thereinsurer: Result elements The balancesheet Premium Income Losses • 19% ofthewritten gross premium is ceded to thereinsurer in theexample. Whypayreinsurancepremium? Loss ratio Costs

18. Premium income • Catastropheevents: - The Kielland-accident(1980) - Scandinavian Star (1990) - The tsunami(2004) • Natural catastrophes: - Dagmar and Berit (2011) - 23.000 claims– 1,7 billion • Large singular claims or eventsthatyieldhighclaimsfrequency(frost 2010) Result elements The balancesheet Premium Income Losses Picture above (text in Norwegian): Last year (2010) wasthecoldestsince 1941. Onlytwoyears have beencolderthan 2010 and only parts ofNorthernNorwayescapedthecold. Loss ratio Costs ”Recordhigh losses from villa fires The insurancecompanies’ losses from villa fire were sky high at 3.3 billion NOK in the first ninemonthsoftheyear (2010). This is a 31% increasecomparedwith the same period last year, according to statistics from FNO”

19. Whydoes it work?? Insurance company 1 Insurance company2 Reinsurance company Result elements Insurance companyn-1 The balancesheet Insurance companyn Premium Income • Economic risk is transferred from theinsurancecompany to thereinsurer • Due to thelawof large numbers, the loss ofthereinsurancecompany is much more predictablethanthatof a smallerinsurancecompany • Thereforethepremiumshould be basedontheexpected loss that • is transferred from the policyholder to theinsurer • Note thatthefinancialyieldofthereinsurerwillexceedthefinancialyieldoftheinsurancecompany, sincethereinsurerdoes not payoutuntiltheclaim is settled Losses Loss ratio Accident date Reporting date Claims payments Claimsclose Claims reopening Claims payments Claimsclose Costs The insurancecompany starts payingouthere The reinsurancecompany refundstheinsurance companyhere Extrafinancialyield for thereinsurer

20. Premium income • Earningofpremiumadjustmentstake 2 years in non-lifeinsurance: Result elements The balancesheet Premium Income Losses • Assumesthatpremiumadjustment is implementedJanuary 1st. • Assumesthattheportfolio’smaturitypattern is evenlydistributed during theyear Loss ratio Costs

21. Losses (claimincidents) Result elements The balancesheet • Loss disbursements: payments in thereportingperiod. Can be incurred losses in thereportingyear and incurred losses from previousyears. • Change in gross loss reserve. Consistsof reserves for reported losses and incurred losses thatare not reported (yet) • Loss disbursements +- change in loss reserve = gross accrued losses • Reinsurance shareof loss disbursements and change in reinsuranceshareof gross loss reserve arethereinsurer’sshareoftheentriesabove Premium Income Losses Loss ratio Costs

22. Losses • The loss reserve is an importantmeasure in a non-lifeinsurancecompany – 30-40% ofthebalance • The companyaccounts as incometheshareofthepremiumincomethat covers risks during theperiod (earnedpremium). The incurred losses areaccounted as costs in the same period. Incurred losses thatare not disbursedareincluded in the loss reserve • Whichclaimsareincurredbut not disbursed? - claimsthatarereported to thecompanybut not settled (RBNS- reported but not settled) - claimsthatareincurredbut not reported to thecompany (IBNR – incured but not reported) • Examplesofwhat RBNS and IBNR can be? Result elements The balancesheet Premium Income Losses Loss ratio Costs

23. Losses • The loss reserve is important in severalcontexts: - it is used to periodizethebalancesheet, ie., to assess a ”correct” result - it is used by theproduct managers in pricing and whenworkingwith terms - it is reported to thereinsurers • - it is used by the fund managers • - last but not leastimportanttheclaimssettlingunit.......butthe loss reserve is basedonjudgement and computations. ”Deviations” arecalleddevelopmentresult (avviklingsresultat in Norwegian) Result elements The balancesheet Premium Income Losses Loss ratio Costs

24. Losses • Developmentresult: Result elements The balancesheet Premium Income Losses Loss ratio Costs

25. Losses • Developmentresult: can be delopmentprofit or development loss. • Computation 31.12.12: Result elements The balancesheet Premium Income Losses Loss ratio Costs

26. Loss ratio • Shows howmuchofthepremiumincome is spent to cover losses Result elements The balancesheet Premium Income Losses Loss ratio Costs • Whatdoesthedifference in loss ratio gross and net tell us?

27. Loss ratio • The claimseverity and theclaimfrequency– twokey drivers for the loss ratio • Numberofclaims: - 1 claimincidentmay hit several covers. Ex. The incident fire may hit both villa and contents • - Wemaycountthenumberofincurredclaims or thenumberofreportedclaims • - 0-claims may or may not be included • Numberof policy years: - The total amountof time all activepolicies have been in force in theperiod - A company has twoclients. 1 wasactive 1/1 and 1 entered 1/9. Numberof policy years = 1+8/12 = 1,67 Result elements The balancesheet Premium Income Losses Loss ratio Costs Pure premium = Claimfrequency x claimseverity

28. Loss ratio • The graph presents loss ratio for all insurancecompanies for villa, content and cabin • The graphillustratesthedelay in thepriceadjustments and theneed for reinsurance • The graphillustratestheeffectofclaimsfrequency (frost 2010) • Source: FNO.no • Differenceof 33 pp in 6 years !?! • Yearlypremiumof 9 billion Loss ratio villa, content and cabin 1998-2011 Result elements The balancesheet Premium Income Losses Loss ratio% Loss ratio Costs

29. Costs Result elements The balancesheet • Sales costs: Provisions, sales offices, marketing, back-office sale • Insurance relatedoperationcosts: management, accounting, actuary, house rent, HR, IT etc.Up to 2012 alsoclaimssettlingcosts– NB:theseweretransferred to claims in 2012 • Receivedprovisionreinsurance: - Normally it constitutes 20% to 25% ofcededpremium.. - NB: ”Costincome” in thetable – why? • - Why do thecompaniesreceivethisprovision? Premium Income Losses Loss ratio Costs

30. Cost ratio (percent) • Shows howmuchofthepremiumincome is spent to cover operationalcosts Result elements The balancesheet Premium Income Losses Loss ratio Costs • Whatdoesthedifference in cost ratio gross and net tell?

31. Cost ratio (percent) Cost ratio Result elements The balancesheet Premium Income Losses Loss ratio • What is causingthereduction in cost ratio to 20%?Where arethecompanies heading? • Source: fno.no– Results in non-lifeinsurance: includes all non-lifeinsurancecompanies in Norway Costs

32. Combined ratio • Shows howmuchofthepremiumincomethat is spent to cover claims and operationalcosts Result elements The balancesheet Premium Income Losses • Combined ratio above100 % impliesthattheinsuranceoperationsare not profitable • What do thecombined ratio gross and netexpress for theexamplecompany? • Long term CR for insurancecompanies in Norwayarebetween 90% and 95% • Whatkey ratio is most problematic for theexamplecompany? Loss ratio Costs

33. Key parameters for non-lifeinsurance in Norway • The graph shows loss ratio (leftaxis), resultdegree (total revenue minus total costs, right axis), cost ratio (right axis) andincome from investments in percentofpremium (right axis) for theperiod1998-2011 • The resultdegree and the loss ratio vary a lot. • The loss ratio seems to be the most important driver for profitability in non-lifeinsurance • The cost ratio and income from investments in percentofpremiumaredecreasing during theperiod. Result elements The balancesheet Premium Income Loss ratio Losses Resultdegree Loss ratio Cost ratio Income from investments as percentofpremium Costs

34. Outlineofthecourse

35. Course literature Curriculum: Chapter 8(except 8.5),9,10 by Professor Erik Bølviken(UIO) Note by Patrick Dahl (Stockholm University, Sweden) Lecture notes by NFH Additionalliterature (for deeperunderstanding and personal development): Non-lifeinsurancepricingwithgeneralized linear models (2010) Esbjörn Ohlsson and Björn Johansson Stochasticclaimsreservingmethods in insurance (2008) Mario Wüthrich and Michael Merz Generalized Linear models (1989) John Nelder and Peter McCullagh