Reserving – introduction I .

1. Reserving – introduction I. One of the most important function of actuaries. Every year-end closure there is a requirement to calculate reserves for future liabilities and grant the realistic P&L. Now there is a lot of type of reserves with special calculation rules. From 2016 – because of Solvency II. - there was a change in this viewpoint: it will be new types of reserves. Insurance mathematics II. lecture

2. Reserving – introduction II. Legal regulation: 43/2015 Regulation of Government Requirement: all reserves have to calculate per business lines (not per products) Reserves are important because of measure also (total SII reserves are more than 2.000 billion HUF at 2017). Insurance mathematics II. lecture

3. Unearned premium reserve I. Example: we take out a household policy at 01.12.2017 with 24.000 HUF yearly premium. We agree yearly payment frequency and we pay total premium in December. What is the realistic P&L situation at 31.12.2017? 30.11.2018 01.12.2017 31.12.2017 2.000 HUF 2.000 HUF 2.000 HUF 22.000 HUF 22.000 HUF Insurance mathematics II. lecture

4. Unearned premium reserve II. Suppose that no any claim related to this policy. What is the real P&L figure in 2017 and 2018? If we would book whole premium for 2017 then we would not book any premium for 2018, but we are in risk during 11 months! But we got really the whole premium in 2017. The solution: We book whole premium for 2017. We calculate unearned premium reserve for 2017. It means that this part of premium will be the offset of risk in 2018. The value of UPR is: Insurance mathematics II. lecture

5. Unearned premium reserve III. Insurance mathematics II. lecture

6. Unearned premium reserve IV. Generalizing: We note d the premium due to payment frequency; T the duration of the payment which continues into next year; K the duration till date of year-closure, then: Insurance mathematics II. lecture

7. Unearned premium reserve V. Open problems with this definition: the lengths of months are not equal (in the example the real UPR is not 21.962 HUF ?); the risk can be not equal in the whole period (example: fleets); based on Hungarian regulation it should not reserve UPR more then 1 year; difference between Hungarian and international regulation. The base of UPR is the yearly premium in international standard, but the payment regarding frequency in Hungarian regulation. Insurance mathematics II. lecture

8. Mathematical reserve Now we are dealing with mathematical reserve in non-life insurance, i.e. related to liability insurance and accident insurance. (Apart from these types there are mathematical reserve regarding life insurance and health insurance also.) Insurance mathematics II. lecture

9. Mathematical reserve in liability insurance I. This type is used generally if the insurer has to pay annuity based on liability insurance (typically in Hungary MTPL). The total future annuity payments are estimated with the next formula: where: • x is an age of annuitant; • lxcomes from mortality table (number of x ages); • n is the unexpired years regarding annuity; • Sk is the yearly annuity in the k-th year (with taxes); Insurance mathematics II. lecture

10. Mathematical reserve in liability insurance II. • i is the technical interest (the yield which the insurer will reach till end of annuity with guarantee; now based on the regulation the maximum of technical interest is 0 according to liability insurance); • d,b are cost factors; typically one of them is 0. If there is an L limitin the policy related to annuity the formula will change as follows: Insurance mathematics II. lecture

11. Mortality table example

12. Mathematical reserve in liability insurance III. The formula is uncomplicated, but the estimation of parameters is not easy: - in the most cases the insurer knows just the initial annuity. In the long run it can be a lot of change regarding the health status of annuitant, inflation, etc.; • the mortality of annuitant is different comparing the total mortality rate (but there are no separate mortality table of annuitant). It can cause unexpectedprofit or loss. Because of above reasons the insurer usually tends to pay lump sum (typically 50-60% of virtual mathematical reserve). Insurance mathematics II. lecture

13. Mathematical reserve in accidentalinsurance The formula is similar as in liability insurance, but the change of annuity is not so frequent – because usually the measure of annuity is exact in the policy. Insurance mathematics II. lecture

14. Claims reserve I. Reasons: • lag in reporting of claims; • lag in payment of claims. There are two types of claims reserve: • If the insurer has known the claims but the claims are not (or not totally) paid, there are Outstanding Claims Reserve (OS Reserve); • If the insurer has not yet known the claims, it can be used IBNR reserve (incurred but not reported). Insurance mathematics II. lecture

15. Claims reserve II. There are two different possible approach: • separate assumption for OS and IBNR reserve or • together estimating with statistical methods. Now in Hungary it is used the separate approach generally, but because of Solvency II. in the future the second approach will come into view also. The measure of lag is characteristic for products, for example the CASCO and accident products have usually higher speed run-off, and MTPL and other liability products have usually slower run-off. Insurance mathematics II. lecture

16. Claims reserve III. For assumption it can consider inflation and the yield of reserves also. It is interesting and generally unanswerable question how is the most useful splitting of portfolio for the assumption: The target is to find homogenous groups of risks. It can be per products or per business lines or sometimes in one product there is useful to further splitting (for example, in MTPL splitting between annuities and non-annuities, or splitting big claims and non-big claims). Insurance mathematics II. lecture

17. Outstanding claims reserve and claims handling reserve In separate OS reserve assumption methods the actuaries have not a lot of tasks. The claim experts have experience how much can be the ‘best estimate’ of the different claim event. The actuaries have just two tasks: calculate claims handling reserve with the next formula and calculate so-called ‚initial’ (when the insurer does not know anything for the claim) claims reserve. where CHR – claims handling reserve; CHP – claims handling payment in current year; CP – claims payment in current year; CLR – claims reserve (OS or IBNR). Insurance mathematics II. lecture

18. IBNR reserve I. There are a lot of different algorithm to evaluate IBNR reserve. Before the detailed description of these methods it can be useful to define several basic ideas. Reporting/payment year ………….. Run-off triangles Accident year means the total amount of the claims which are occurred in i-th year and reported/paid in j-th year ………….. Insurance mathematics II. lecture

19. IBNR reserve II. Development year ………….. Accident year Lagging triangles ………….. means the total amount of claims which are occurred in i-th year and reported/paid in(i+j-1)-th year Insurance mathematics II. lecture

20. IBNR reserve III. Development year ………….. Accident year Cumulated triangles ………….. means the total amount of claims which are occurred in i-th year and reported/paid till (i+j-1)-th year Insurance mathematics II. lecture

21. IBNR reserve IV. The cumulated triangle is complete, if there is no any reporting/paying claims event after n-th year (difficult to say). It is possible to make run-off triangles for number of claims also. Hungarian regulation requires for IBNR calculation just using run-off triangles. For assumption the cumulated triangle will be the basic usually. Insurance mathematics II. lecture

22. IBNR reserve V. Denote the claims which are reported/paid after n-th year regarding claims occurred in i-th year. Our target is estimating the next formula (for each i): Our best estimate is as follows: Our problem is that in practice usually we do not know covariance and common distribution of claims. That is why we simplify in the methods which we are using for calculation of IBNR. Insurance mathematics II. lecture

23. Methods of IBNR calculation Grossing Up method I. Example: 1 2 3 4 5 Development year 2011 223; 311; 252; 127; 29 Accident year 2012 254; 378; 249; 153 2013 312; 411; 276 Lagging triangle 2014 359; 435 2015 384 Insurance mathematics II. lecture

24. Methods of IBNR calculation Grossing Up method II. Example: 1 2 3 4 5 Development year 2011 223; 534; 786; 913; 942 Accident year 2012 254; 632; 881; 1034 2013 312; 723; 999 Cumulated triangle 2014 359; 794 2015 384 Insurance mathematics II. lecture

25. Methods of IBNR calculation Grossing Up method III. Example: Is the cumulated triangle complete? No, we have data from earlier years as follows: Insurance mathematics II. lecture

26. Methods of IBNR calculation Grossing Up method IV. Example: Assumption for 2015: The base of assumption will be 2015 as next table shows: We assume that the run-off of next years will be equal as 2015. Insurance mathematics II. lecture

27. Methods of IBNR calculation Grossing Up method V. Example: 1 2 3 4 5 2011 223; 534; 786; 913; 942 Occurring year 2012 254; 632; 881; 1034 2013 312; 723; 999 2014 359; 794 2015 384 Insurance mathematics II. lecture

28. Methods of IBNR calculation Grossing Up method VI. Generalizing: 1. Based on earlier year we estimate If we have no any data from earlier year we can use data from similar products or OS reserves. 2. Further factors: …. 3. Ultimate payment estimation: …. Insurance mathematics II. lecture

29. Methods of IBNR calculation Grossing Up method VII. 4. Reserve assumption: The above calculation is the basic version, but there are some modified possibility of this method. Insurance mathematics II. lecture

30. Methods of IBNR calculation Modified Grossing Up methodsI. • version: If we have data related to earlier year splitting per year then we can calculate more exact the d factors. • Let and the other experience d factors. Then the ultimate used factors: Insurance mathematics II. lecture

31. Methods of IBNR calculation Modified Grossing Up methodsII. Example: Insurance mathematics II. lecture

32. Methods of IBNR calculation Modified Grossing Up methodsIII. Example: Insurance mathematics II. lecture

33. Methods of IBNR calculation Modified Grossing Up methodsIV. 2. version: If we have data related to earlier year splitting per year then we can calculate more exact the d factors. Let and the other experience d factors. Then the ultimate used factors: Insurance mathematics II. lecture

34. Methods of IBNR calculation Modified Grossing Up methodsV. Example: Insurance mathematics II. lecture

35. Methods of IBNR calculation Modified Grossing Up methodsVI. Example: Insurance mathematics II. lecture

36. Methods of IBNR calculation Modified Grossing Up methodsVII. 3. version: We estimate as in the 1. version. After it we judge ultimate payment for 2.year: With this result we define the d factors and estimate ultimate payment as follows: After it we continue this process till each d factors and payments will be calculated. Insurance mathematics II. lecture

37. Methods of IBNR calculation Modified Grossing Up methodsVIII. Example: Insurance mathematics II. lecture

38. Methods of IBNR calculation Modified Grossing Up methodsIX. Example: Insurance mathematics II. lecture

39. Methods of IBNR calculation Modified Grossing Up methodsX. 4. version: We estimate as in 1. version. After it we judge ultimate payment for 2.year: With this result we define the d factors and estimate ultimate payment as follows: After it we continue this process till each d factors and payments will be calculated. Insurance mathematics II. lecture

40. Methods of IBNR calculation Modified Grossing Up methodsXI. Example: Insurance mathematics II. lecture

41. Methods of IBNR calculation Modified Grossing Up methodsXII. Example: Insurance mathematics II. lecture

42. Methods of IBNR calculation Link ratio methods I. We suppose that ratio does not depend on significantly for i 1. Determining is similar then in the Grossing Up method. (with experience of earlier years or OS reserve). Other factors will be defined as function of actual . With choosing different function will be defined the different version of link ratio method. 2. Ultimate payment and IBNR reserveestimation: Insurance mathematics II. lecture

43. Methods of IBNR calculation Link ratio methodsII. 2. Ultimate payment and IBNR reserveestimation continued: … … Basic version: 1. modification: Insurance mathematics II. lecture

44. Methods of IBNR calculation Link ratio methodsIII. 2. modification: 3. modification: In 3. modification with special α factors we will get the most popular ‘chain-ladder’ method. Insurance mathematics II. lecture

45. Methods of IBNR calculation Chain ladder method I. This is the most popular process for IBNR estimation. Insurance mathematics II. lecture

46. Methods of IBNR calculation Chain ladder method II. Example: Insurance mathematics II. lecture

47. Methods of IBNR calculation Naive loss ratio method In the next methods we are using premium data also (not just claim data). We suppose that the ultimate loss of i-th year will be the -th part of the premium. Then the reserve can be calculated as follows: , where signsthe earned premium of i-th year. The disadvantage of this method is that IBNR reserve is independent of actual claim data.Starting company without any own claim data can use this method. Insurance mathematics II. lecture

48. Methods of IBNR calculation Bornhuetter-Ferguson method I. This method combines the naive loss ratio and grossing up (or link ratio) methods. 1. We calculate the ultimate loss payment with naive claim ratio methods: 2. For calculating development factors we are using grossing up (or link ratio method): Insurance mathematics II. lecture

49. Methods of IBNR calculation Bornhuetter-Ferguson method II. 3. The reserves will be estimated as follows: …. Insurance mathematics II. lecture

50. Methods of IBNR calculation Separation method I. In this method we do not use cumulated triangles, but we are using lagging triangles. The used triangles have to be complete. We suppose that where signs the number of claims in the i-th year (known), signs an inflation, c is the average claim amount. There are two types of this method: - arithmetic; - geometric. Now we consider detailed the arithmetic version. Insurance mathematics II. lecture