Actuarial Mathematics in Reinsurance Ing. Mgr. Tomáš Oravec December 14th, 2020 About VIG (Re) Introduction to Reinsurance Basic Reinsurance Contracts Basic Pricing Approaches Capital Modelling … Actuarial Mathematics in Reinsurance: Table of Contents 2 About VIG (Re) Actuarial Mathematics in Reinsurance Vienna Insurance Group The Leading Insurance Group in Austria & CEE Premiums written EUR 10.4bn (+7.7%) Profit before taxes EUR 521.6mn (+7.4%) Combined Ratio 95.4% (-0.6ppt) Return on Equity 10.5% Solvency ratio 210% S&P Rating A+ with stable outlook Market capitalization ~ € 2.2bn (03/31/2020) More than 25,000 Employees Profit before taxes € 201.2 mn PBT increased by 21.8% (-€55.9mn) • PBT decrease driven by goodwill impairments and lower financial result due to last year’s changed consolidation method of the nonprofit housing societies • Net profit decreased to €126.3 mn (-16.3%) • Earnings per share of €1.97 Gross written premium € 5,577.4 mn GWP up by 2.4% (+€130.7mn) • Strongest contribution by markets coming from Austria (+€100mn) and Poland (+€46m) • Other property business increasing by €180mn (+7.0%) • CEE share of 54.6% Vienna Insurance Group Highlights H1 2020 Solid 6M 2020 results despite adverse effects of COVID-19 Combined Ratio 95.5% Improved by 0.9% due to better claims experience • Claims ratio down to 63.8% • Cost ratio stable at 31.7% • CoR of <100% and improvements in all segments • Stable combined ratio compared to 95.4% at YE 2019 Solvency Ratio 183% YE 2019: 210% • Own funds: €7.1 mn • SCR: €3.9 mn Part of Vienna Insurance Group. VIG Re VIG Re Briefly Rating A+ with stable outlook, unaltered since 2008, confirmed as of 15 October 2020. VIG Re was founded in 2008. Today leading reinsurer in the CEE, it is providing risk and capital management solutions to insurance companies of Vienna Insurance Group and outside of VIG. VIG Re: Offices and Underwriting Territories Today, VIG Re is providing made-to-measure reinsurance to 440+ insurance companies in 39 countries. VIG Re Key Figures 2019 0% 5% 10% 15% 20% 201220102009 20172011 2013 2014 2015 2016 2018 2019 16.6% 13.6% 16.7% 17.9% 19.1% 14.3% 15.3% 15.5% 16.3% 15.3% 15.1% ROE 2019 (before tax) Portfolio Split 2019 90,6% 4,5% 4,9% P&C Life Health in EUR mn 2009 2010  2011 2012 2013 2014 2015 2016 2017 2018 2019 Gross written premium 257.2 273.8 274.8 450.3 390.9 405.1 357.7 383.1 423.6 456.9 527.5 Net earned premium 157.9 176.3 170.0 324.7 247.9 233.7 205.9 222.4 256.0 226.0 309.6 Combined ratio* 95.7% 96.1% 95.8% 94.6% 97.6% 97.6% 95.7% 93.2% 94.8% 92.5% 96.0% Profit before tax 14.4 18.6 21.1 23.6 17.9 19.7 22.3 22.3 23.6 26.2 26.2 *for Non-Life VIG Re Key Performance Indicators 2019 External business getting more important each year. 450 50 500 0 350 550 150 100 200 250 300 400 2015 456.9450.3 2014201320122009 2010 2011 2016 2017 2018 2019 357.7 257.2 273.8 405.1 274.8 390.9 383.1 423.6 527.5 EUR mn External Business Group P&C Business Group Life Business 50% 55% 60% 65% 70% 75% 80% 85% 90% 95% 100% 30 5 0 35 10 20 15 25 14.4 20172010 18.6 20192011 EUR mn 20152009 20162012 2013 2014 2018 21.1 23.6 17.9 19.7 22.3 22.3 23.6 26.2 26.2 Underwriting Result*Combined Ratio Investment Income * Including Other Income/Expenditure Gross Written Premium Profitability VIG Re Management Report H1 2020 Financial Performance H1 2020 P&L (in EUR mn) H1 2020 Actual Gross Written Premium 309.5 Net Earned Premium 184.1 Underwriting Result 7.8 Investment Income* 0.3 Other Expenditures** 2.1 Profit Before Tax 10.2 Profit After Tax 7.9 Key Ratios H1 2020 Actual Combined Ratio 96.9% Admin Cost Ratio 2.0% * Ordinary income 4.2 mn, Extraordinary income EUR -3.1 mn, Interest on Sub. debt EUR -0.7 mn, Expenditures EUR -0.1 mn ** FX effects Investment income impaired due to market volatilities, income impaired by FX effects. Solid UW results, with 2ppt Covid-19 impact on combined ratio. 62% 14% 6% 3% 3% 2% 4% 5%0% Property MTPL MOD GTPL Marine PA Life Health Other VIG Re Portfolio Split 2019 Non-CEE markets: accounting for 40% of GWP, with short-tail lines continuing to dominate the portfolio. GWP by Underwriting Territory GWP by Line of Business 35% 11% 9% 6% 6% 6% 5% 3% 2% 18% Austria Germany Czech Republic Italy Poland Turkey Slovakia France Romania Other VIG Re VIG Business in 2019 CAGR: Compound annual growth rate 100 0 150 50 200 250 300 350 400 450 500 EUR mn 2009 2010 2011 2012 2013 2014 2015 2016 2017 2018 2019 +3% CAGR GWP by Business Segments P&C Health Life GWP by Underwriting Territory 51% 12% 7% 5% 5% 3% 3% 2% 2% 11% Austria Czech Republic Turkey Slovakia Poland Romania Serbia Hungary Croatia Other Austria and Czech Republic: most important contributors. 28% 17% 8% 8% 7% 5% 4% 4% 2% 18% Germany Italy France Poland Austria Japan Slovakia Switzerland Czech Republic Other VIG Re Third Party Business 2019 CAGR: Compound annual growth rate GWP by Business Segments GWP by Underwriting Territory P&C external business almost doubled during the last 2 years driven by opening of the German & French branches. 100 120 0 60 20 140 40 80 160 180 200 2013 EUR mn 2009 2010 2011 2012 2014 2015 2016 2017 2018 2019 +28% CAGR P&C Health Life Introduction to Reinsurance Actuarial Mathematics in Reinsurance Introduction to Reinsurance Definition of Reinsurance Reinsurance is insurance that is purchased by an insurance company Reinsurance is the transfer of part of the hazards or risks that a direct insurer assumes by way of insurance contract or legal provision on behalf of an insured, to a second insurance carrier, the reinsurer, who has no direct contractual relationship with the insured Introduction to Reinsurance Origins First professional company Cologne Re (founded in 1846) as consequence of the catastrophic fire in Hamburg in 1842 • 1846: Cologne Re • 1853: Aachen Re • 1857: Frankfurt Re • 1863: Swiss Re • 1880: Munich Re • 1886: Frankona Re Today about 200 professional reinsurers worldwide and large number of direct insurers writing also reinsurance Introduction to Reinsurance Risk Sharing Policies (Risks) Direct Insurer (Cedent) Reinsurers Retrocessionaires cession retro- cession Introduction to Reinsurance Purpose • Protection against large single losses and catastrophes • Reduces volatility of the financial results • Reduction of solvency capital requirement • More underwriting capacity for smaller players • Know-how transfer Introduction to Reinsurance Why Reinsurance? (1/4) Introduction to Reinsurance Why Reinsurance? (2/4) Introduction to Reinsurance Why Reinsurance? (3/4) Introduction to Reinsurance Why Reinsurance? (4/4) Introduction to Reinsurance Reinsurance Market Subjects • Insurance Companies (reinsurance buyers – risk cedants) • Brokers (intermediaries; main roles: risk modelling, contract structuring, negotiating with reinsured(s) and reinsurer(s), administration of treaties) • Reinsurers (providing capacity) • Supervisory Authorities (EIOPA, Czech National Bank; main roles: enhance reliability, stability and control of the (re)insurance market) Introduction to Reinsurance Top 10 (Re)insurance Brokers Source: https://www.atlas-mag.net/en/article/top-10-brokers-in-insurance-and-reinsurance Introduction to Reinsurance Top 10 Reinsurers Source: https://www.atlas-mag.net/en/article/top-50-global-reinsurers-ranking-of-2019 Basic Reinsurance Contracts Actuarial Mathematics in Reinsurance Basic Reinsurance Contracts Types of Reinsurance I Facultative: each risk considered individually, obligation neither to cede nor to accept Obligatory: group of homogenous risks automatically ceded to the reinsurance treaty Combination: either Fac-Obligatory or Obl-Facultative, rare cases Basic Reinsurance Contracts Types of Reinsurance II Proportional • Quota Share • Surplus Non-Proportional (Excess of Loss) • Risk XL • Cat XL (Event XL) • Stop Loss • Aggregate XL Basic Reinsurance Contracts Proportional: Quota Share • ceded loss calculated as: • ratio used also for premium (q) • commission fixed / sliding scale / profit commission (c) • loss corridor / loss participation / loss ratio cap Basic Reinsurance Contracts Proportional: Quota Share (Example) • Sum Insured SI = 5,000,000 • Treaty Capacity 8,000,000 • Gross Individual Loss X = 1,000,000 • q = 40% • c = 20% • Gross Premium P = 2,000 Reinsurer's share: Reinsured's result: XR = q * X = 400,000 (P-PR)-(X-XR)+c*PR = -598,640 PR = q * P = 800 Basic Reinsurance Contracts Proportional: Surplus • ceded loss calculated as: • ratio (1 – a/S) used also for premium • commission mainly fixed for Basic Reinsurance Contracts Proportional: Surplus (Example) • Sum Insured a = 5,000,000 • Treaty Capacity 8,000,000 • Retention S = 2,000,000 and limit 6,000,000 (i.e. 3 lines) • Ceded percentage = max(1-a/S; 0) = 60% • c = 20% • Premium P = 2,000 Reinsurer's share: Reinsured's result: XR = s * X = 600,000 (P-PR)-(X-XR)+c* PR = - 398,960 PR = s * P = 1,200 Basic Reinsurance Contracts Non-Proportional: Risk XL • ceded loss calculated as: • premium mostly as % of GNPI (sometimes flat premium) • reinstatements free / paid / combination for Basic Reinsurance Contracts Non-Proportional: Risk XL (Example) • Sum Insured = 5,000,000 • Total Capacity = 8,000,000 with Retention 2,000,000 … layer of 6m XS 2m • Gross Individual Loss X = 3,000,000 • r = 8% (Rate on Premium) • Gross Premium P = 2,000 Reinsurer's share: Reinsured's result: XR = max(0; X-D) = 1,000,000 (P-PR)-(X-XR) = - 1,998,160 PR = r * P = 160 Basic Reinsurance Contracts Non-Proportional: Cat XL • ceded loss calculated as: • premium as % of GNPI • reinstatements free / paid / combination Basic Reinsurance Contracts Non-Proportional: Cat XL Basic Reinsurance Contracts Non-Proportional: Stop Loss LoB 2 LoB 1 LoB 3 Ceded to XL Ceded to surplus Ceded to XL and QS Retained Retained Retained Retained Loss Ratio Ceded Loss Ratio Basic Reinsurance Contracts Reinsurance Program: Example (in CZK) Line of Business Type of reinsurance Layer Retention Limit Rate Reinstatements Personal Accident XL 1 2 000 000 18 000 000 0,65% 1-3@100% Personal Accident XL 2 20 000 000 30 000 000 0,16% 1@100% Motor Liability XL 1 30 780 000 23 220 000 2,10% 1-15@free Motor Liability XL 2 54 000 000 81 000 000 0,80% 1-15@free Motor Liability XL 3 135 000 000 Unlimited 0,21% unlimited@free Marine Hull QS - 100 000 000 100 000 000 - General Liability XL 1 10 000 000 10 000 000 6,50% 1-30@free General Liability XL 2 20 000 000 30 000 000 3,60% 1-20@free General Liability XL 3 50 000 000 50 000 000 1,50% 1-10@free General Liability XL 4 100 000 000 200 000 000 0,80% 1-5@free Fire Surplus - 100 000 000 1 500 000 000 - Fire QS - 50 000 000 50 000 000 - Fire XL - 30 000 000 70 000 000 0,39% 1@free; 2-3@50% Nat Cat XL 1 270 000 000 630 000 000 1,20% 2@100% Nat Cat XL 2 900 000 000 1 100 000 000 0,90% 1@100% Nat Cat XL 3 900 000 000 1 500 000 000 0,30% 1@100% Nat Cat RPP - fixed price, covering reinstatement premium L1-L2 Basic Reinsurance Contracts How much Capacity to Buy? (Nat Cat) Solvency 2 (Partial) Internal Model? X SII Standard Formula? Basic Reinsurance Contracts How much Capacity to Buy? (Complex Example) • Maximise capacity to gain competitive advantage • Price for „empty capacity“ • Unlimited cover for MTPL Line of Business Type of reinsurance Layer Retention Limit Rate Reinstatements Fire Surplus - 100 000 000 1 500 000 000 - Fire QS - 50 000 000 50 000 000 - Fire XL - 30 000 000 70 000 000 0,39% 1@free; 2-3@50% •important to know the hierarchy (inurance) •e.g. SI = 400.000.000 CZK, gross loss = 200.000.000 •Surplus cover applied as first -> Then QS cover -> Finally XL on net retention -> Ceded losses to the contracts? •loss ceded to surplus (in mio.) = ((400-100)/400) * 200 = 150 •loss ceded to QS (in mio.) = (200 – 150) * 0,5 = 25 •loss ceded to XL (in mio.) = Max (0; (200 – 150 – 25) – 30) = 0 •net loss (in mio.) = 25 Basic Reinsurance Contracts How to Structure Reinsurance Program? • Define capacity • What is maximum retention (e.g. group rules) • Targets of the company might be: ➢ Minimise price ➢ Maximise mean net underwriting result ➢ Minimise volatility of net underwriting result ➢ Maximise return on capital ➢ Other (financial, cash-flow, etc.) Basic Reinsurance Contracts Timeline of Renewal 1.1. xxxx • May – September: data preparation by insurance companies and submissions to brokers, suitability assessed by brokers / internally • September – December: data sent to reinsurers who provide their quotes > decision about final terms (FOT) and wordings > placement • December 31st – all treaties need to be placed (rare exceptions possible) Basic Reinsurance Contracts Decisions about Final Terms Final terms are subject to further negotiations: • Not all requested reinsurers (wanted to) quote • Offer of whole package VS “cherry-picking” • Leader(s) and followers, differential terms Layer Limit Retention Swiss Re R+V Markel Amlin Novae Scor Aspen Final? 1 23 220 000 30 780 000 1,30% 1,10% 0,89% 1,30% 2,00% 1,70% 1,10% 1,1% - 1,3% 2 81 000 000 54 000 000 0,65% 0,72% 0,59% 0,85% 0,79% 0,60% 0,99% 0,65% - 0,72% 3 Unlimited 135 000 000 0,26% 0,19% 0,15% 0,21% 0,10% 0,30% 0,26% 0,21% - 0,26% Quoted Rate (as % of GNPI) Layer Limit Retention Swiss Re R+V Markel Amlin Novae Scor Aspen 1 23 220 000 30 780 000 30% 25% 10% 15% 20% 30% 25% 2 81 000 000 54 000 000 30% 25% 10% 15% 20% 30% 25% 3 Unlimited 135 000 000 30% 25% 5% 15% 5% 30% 35% Written line (indication) Total 105% 95% 115% Basic Reinsurance Contracts Reinsurance Market Cycles Basic Pricing Approaches Actuarial Mathematics in Reinsurance Basic Reinsurance Contracts Components of Reinsurance Price Basic Reinsurance Contracts Pricing Techniques Basic Reinsurance Contracts Pricing Techniques Basic Reinsurance Contracts Pricing Techniques – Split of Claims Basic Reinsurance Contracts Profit Loading (Premium Risk and Reserve Risk Loading) Premium Risk (including Nat Cat) = risk that we do not collect enough premium to cover losses: 1 in 200 years scenario (0,5% probability) Reserve Risk = risk that reserves are not sufficient for the next accounting year (again 1 in 200 years, i.e. 0,5% probability) Premium Risk Year 1 Year NYear 3Year 2 Reserve Risk Basic Reinsurance Contracts Pricing Techniques – Experience Rating Basic Reinsurance Contracts Burning Cost – Data Basic Reinsurance Contracts Burning Cost – Example Basic Reinsurance Contracts Burning Cost – Example Basic Reinsurance Contracts Burning Cost – Example Basic Reinsurance Contracts Burning Cost – Example Basic Reinsurance Contracts Burning Cost – Example Basic Reinsurance Contracts Burning Cost L1 – Drawbacks Basic Reinsurance Contracts Burning Cost L2 – Drawbacks Basic Reinsurance Contracts Burning Cost – Real Example Basic Reinsurance Contracts Probabilistic Approach – Possible Remedy for BC Drawbacks Basic Reinsurance Contracts Probabilistic Approach – Severity Model Basic Reinsurance Contracts Probabilistic Approach – Severity Model Basic Reinsurance Contracts Probabilistic Approach – Frequency Model Basic Reinsurance Contracts Probabilistic Approach – Frequency Model Basic Reinsurance Contracts Simulations – Methodology 1) Select number of simulations: Each simulation represents 1 treaty year, i.e. by 500 simulations we simulate 500 years (500 different outcomes of 2020 treaty year) 2) Generate number of claims based on the frequency model: in each year (simulation) different numbers of claims are generated 3) Generate severity of each claim based on the severity model: for each simulated claim in previous step a different claim size is generated 4) Apply reinsurance structure: within each of 500 years as-if reinsurance structure is applied and we can see the expected average ceded loss (reinsurer´s result), their volatility and many other statistics Basic Reinsurance Contracts Simulations – Example Basic Reinsurance Contracts Simulations – Number of Generated Claims per Year Basic Reinsurance Contracts Simulations – Generated Severity and Applied Reinsurance Basic Reinsurance Contracts Simulations – CDFs for Ceded Losses to L1 and L2 Basic Reinsurance Contracts Simulations – Ceded Losses to L1 and L2 Basic Reinsurance Contracts Simulations – Real Example from Broker Basic Reinsurance Contracts Expected Reinsurer Deficit (ERD) Test Basic Reinsurance Contracts Property FLS Exposure Rating – Limited Data Availability • Risk profiles (number of risks, PML/SI, premium per band) • Exposure (historical and estimated GNPI) • Per Risk Large Claims (gross or ceded basis depending on treaty type) • Per Event Large Claims (if covered and experienced, gross or ceded basis depending on treaty type) • Aggregate claims (required for proportional and aggregate contracts) • The largest risks Basic Reinsurance Contracts Property FLS Exposure Rating – Exposure Curve Parameters Selection Basic Reinsurance Contracts Property Exposure Rating – Freq/Severity Model Capital Modelling Actuarial Mathematics in Reinsurance Capital Modelling Partial Internal Model under Solvency II • After the renewal we merge all models and create one huge stochastic model (incl. natural catastrophes on cross-country level) • This serves for calculation of SCR (expected from 2020) • Retrocession optimization • Assessment of business profitability • Creation of business plan Capital Modelling Scope of Partial Internal Model in VIG Re Capital Modelling Reserve Risk Process RI contracts segmentation into homogenous risk groups (SII LoBs, accounting basis, country of cedant, etc.) Best Estimates (claims TP) and one-year standard deviations Simulation of ultimate claims 80 Retrocession Structure Interest Rate curve Fixed Costs ANCA: Adjustment of natural catastrophe models results Capital Modelling Data Preparation: Checking appropriateness, completeness and accuracy of figures from cedent renewal information (e.g., exposure, risk profiles, run-off triangulation, etc.) MS Excel Quotation Tool: Inflation, Ultimate Claims calculation, etc. Treaties Simulations: simulation of external gross exposure, application of RI covers, VIG Re gross exposure ariSE Contract Specification: Extraction of necessary information from cedent’s demand/broker’s submission (e.g., RI treaty parameters, options to be quoted, retention, limit, etc.) PACE Portfolio Accumulation for Control and Evaluation: Results of Natural Catastrophes models (ELT, YLT) Claims Severity and Frequency distribution parameters estimation Pricing Tools Unmodelled Business 81 Capital Modelling Pricing of NL & NSLT Health Reinsurance & Capital Modelling Pricing of NL Reinsurance: Simple Example Capital Modelling Pricing of NL Reinsurance: Complex Example Capital Modelling Aggregated MetaRisk Model 85 Correlated YLT Active Reinsurance Contracts VIG Re Retrocession Capital Modelling YE 2019 Model: Reinsurance Programs & Contracts P&C and NSLT Health RI Programs Facultative – capital add-on Obligatory - modelled 86 Portfolio Program Contract Number of RI programs 2017 2018 2019 2020 Non-Proportional Facultative: 24 53 163 206 Proportional Facultative: 12 57 108 114 Non-Proportional Obligatory: 435 511 650 778 Proportional Obligatory: 135 154 205 221 Total 606 775 1 126 1 319 Capital Modelling Parameterisation of MetaRisk Model each contract model consists of one, but usually more loss causes each loss cause needs to be parameterised (frequency, severity, payout patterns) for Nat Cat treaties we need external modelling complex retrocession 87 Number of Loss Causes 2017 2018 2019 2020 Large Loss Causes 551 1 105 1 567 1 885 Attritional/Aggregate Loss Causes 368 323 323 502 Nat Cat Loss Causes 85 472 472 728 Total Number of Loss Causes 1 004 1 900 2 362 3 115 YE19 Composition of SCR Capital Modelling 88 Capital Modelling YE19 Standard vs. Partial Internal Model SCR 89 156,3% 52,0%PIM Net SCR/SF Net SCR PIM Gross SCR/SF Gross SCR YE19 Underwriting Risks Composition Capital Modelling 90 December 14th, 2020 Thank You! www.vig-re.com t.oravec@vig-re.com tomas.oravec@mail.muni.cz Disclaimer ©2020 VIG Re. 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