1. Insurance mathematics II. lecture 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. In the future (from 2016) there will be a change in this viewpoint: it will be just one (two) type(s) of reserve.

2. Insurance mathematics II. lecture Reserving – introduction II. Legal regulation: 8/2001 Ministry of Finance Regulation Requirement: all reserves have to calculate per business lines (not per products) Reserves are important because of measure also (total reserves are about 2.000 billion HUF at 2013).

3. Insurance mathematics II. lecture Unearned premium reserve I. Example: we take out a household policy at 01.12.2014 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 31.12.2014? 2.000 HUF 22.000 HUF 2.000 HUF 22.000 HUF 2.000 HUF 01.12.2014 31.12.2014 30.11.2015

4. Insurance mathematics II. lecture Unearned premium reserve II. Suppose that no any claim related to this policy. What is the real P&L figure in 2014 and 2015? -If we would book whole premium for 2014 then we would not book any premium for 2015, but we are in risk during 11 months! But we got really the whole premium in 2014. The solution: 1.We book whole premium for 2014. 2.We calculate unearned premium reserve for 2014. It means that this part of premium will be the offset of risk in 2015. The value of UPR is:

5. Insurance mathematics II. lecture Unearned premium reserve III. Generalizing: Let it be 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:

6. Insurance mathematics II. lecture Unearned premium reserve IV. 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.

7. Insurance mathematics II. lecture 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.)

8. Insurance mathematics II. lecture 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; - l x comes from mortality table (number of x ages); - n is the unexpired years regarding annuity; - S k is the yearly annuity in the k-th year (with taxes);

9. Insurance mathematics II. lecture 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 limit in the policy related to annuity the formula will change as follows:

10. Insurance mathematics II. lecture 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 unexpected profit or loss. Because of above reasons the insurer usually tends to pay lump sum (typically 50-60% of virtual mathematical reserve).

11. Insurance mathematics II. lecture Mathematical reserve in accidental insurance 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.

12. Insurance mathematics II. lecture 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 totally payed, 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).

13. Insurance mathematics II. lecture 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. 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.

14. Insurance mathematics II. lecture 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).

15. Insurance mathematics II. lecture Outstanding claims reserve and claims handling reserve In separate OS reserve assumption methods the actuaries have no 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 one task: calculate claims handling reserve with the next formula: where CHR – claims handling reserve; CHP – claims handling payment in current year; CP – claims payment in current year; CLR – claims reserve (OS or IBNR).

16. Insurance mathematics II. lecture 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. Run-off triangles Accident year Reporting/payment year ………….. means the total amount of claims which are occurred in i-th year and reported/paid in j-th year

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

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

19. Insurance mathematics II. lecture IBNR reserve IV. The cumulated triangle is complete, if there is no any reporting/paying 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.

20. Insurance mathematics II. lecture 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.

21. Insurance mathematics II. lecture Methods of IBNR calculation Grossing Up method I. Lagging triangle Accident year Development year Example: 2009 2010 2011 2012 2013 1 2 3 4 5 223; 311; 252; 127; 29 254; 378; 249; 153 312; 411; 276 359; 435 384

22. Insurance mathematics II. lecture Methods of IBNR calculation Grossing Up method II. Cumulated triangle Accident year Development year Example: 2009 2010 2011 2012 2013 1 2 3 4 5 223; 534; 786; 913; 942 254; 632; 881; 1034 312; 723; 999 359; 794 384

23. Insurance mathematics II. lecture Methods of IBNR calculation Grossing Up method III. Is the cumulated triangle complete? No, we have data from earlier years as follows: Example: YearClaims until 5th year TotalRatio (5th year/Total) 200578083093,98% 200681089091,01% 200780086093,02% Total2390258092,64%

24. Insurance mathematics II. lecture Methods of IBNR calculation Grossing Up method IV. Assumption for 2009: Example: 1. year2. year3. year4. year5. yearTotal 20092235347869139421017 Ratio21,9%52,5%77,3%89,8%92,64%100% The base of assumption will be 2009 as next table shows: We assume that the run-off of next years will be equal as 2009.

25. Insurance mathematics II. lecture Methods of IBNR calculation Grossing Up method V. Occurring year Example: 2009 2010 2011 2012 2013 1 2 3 4 5 223; 534; 786; 913; 942 254; 632; 881; 1034 312; 723; 999 359; 794 384 TotalIBNR 101775 1152118 1292293 1512718 17511367 Total2571

26. Insurance mathematics II. lecture 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: ….

27. Insurance mathematics II. lecture 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.

28. Insurance mathematics II. lecture Methods of IBNR calculation Modified Grossing Up methods I. 1.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:

29. Insurance mathematics II. lecture Methods of IBNR calculation Modified Grossing Up methods II. Example: 1. year2. year3. year4. year5. yearTotal 2005 Ratio (%) 170 20,5% 422 50,8% 660 79,5% 750 90,4% 780 94% 830 100% 2006 Ratio (%) 140 15,7% 435 48,9% 680 76,4% 782 87,9% 810 91% 890 100% 2007 Ratio (%) 132 15,3% 428 49,8% 670 77,9% 780 90,7% 803 93% 860 100% 2009 Ratio (%) 223 21,9% 534 52,5% 786 77,3% 913 89,8% 942 92,7% 1017 100% Average ratio 18,4%50,5%77,8%89,7%92,7%100%

30. Insurance mathematics II. lecture Methods of IBNR calculation Modified Grossing Up methods III. Example: YearTotal (ultimate) payment IBNR 2009101775 20101153119 20111284285 20121572778 201320901706 Total2964

31. Insurance mathematics II. lecture Methods of IBNR calculation Modified Grossing Up methods IV. 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:

32. Insurance mathematics II. lecture Methods of IBNR calculation Modified Grossing Up methods V. Example: 1. year2. year3. year4. year5. yearTotal 2005 Ratio (%) 170 20,5% 422 50,8% 660 79,5% 750 90,4% 780 94% 830 100% 2006 Ratio (%) 140 15,7% 435 48,9% 680 76,4% 782 87,9% 810 91% 890 100% 2007 Ratio (%) 132 15,3% 428 49,8% 670 77,9% 780 90,7% 803 93% 860 100% 2009 Ratio (%) 223 21,9% 534 52,5% 786 77,3% 913 89,8% 942 92,7% 1017 100% Minimum ratio 15,3%48,9%76,4%87,9%91%100%

33. Insurance mathematics II. lecture Methods of IBNR calculation Modified Grossing Up methods VI. Example: YearTotal (ultimate) payment IBNR 2009103593 20101177143 20111308309 20121625831 201325022118 Total3493

34. Insurance mathematics II. lecture Methods of IBNR calculation Modified Grossing Up methods VII. 3. version: We estimateas 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.

35. Insurance mathematics II. lecture Methods of IBNR calculation Modified Grossing Up methods VIII. Example: 1. year2. year3. year4. year5. yearTotal 2009 Ratio (%) 223 21,9% 534 52,5% 786 77,3% 913 89,8% 942 92,6% 1017 100% 2010 Ratio (%) 254 22,1% 632 54,9% 881 76,5% 1034 89,8% 1152 100% 2011 Ratio (%) 312 24% 723 55,6% 999 76,9% 1299 100% 2012 Ratio (%) 359 24,6% 794 54,3% 1461 100% 2013 Ratio (%) 384 23,1% 1659 100%

36. Insurance mathematics II. lecture Methods of IBNR calculation Modified Grossing Up methods IX. Example: YearTotal (ultimate) payment IBNR 2009101775 20101152118 20111299300 20121461667 201316591275 Total2434

37. Insurance mathematics II. lecture Methods of IBNR calculation Modified Grossing Up methods X. 4. version: We estimateas 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.

38. Insurance mathematics II. lecture Methods of IBNR calculation Modified Grossing Up methods XI. Example: 1. year2. year3. year4. year5. yearTotal 2009 Ratio (%) 223 21,9% 534 52,5% 786 77,3% 913 89,8% 942 92,6% 1017 100% 2010 Ratio (%) 254 22,1% 632 54,9% 881 76,5% 1034 89,8% 1152 100% 2011 Ratio (%) 312 24% 723 55,6% 999 76,5% 1306 100% 2012 Ratio (%) 359 24,6% 794 52,5% 1512 100% 2013 Ratio (%) 384 21,9% 1753 100%

39. Insurance mathematics II. lecture Methods of IBNR calculation Modified Grossing Up methods XII. Example: YearTotal (ultimate) payment IBNR 2009101775 20101152118 20111306307 20121512718 201317531369 Total2587

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

41. Insurance mathematics II. lecture Methods of IBNR calculation Link ratio methods II. 2. Ultimate payment and IBNR reserve estimation: …… Basic version: 1. modification:

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

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

44. Insurance mathematics II. lecture Methods of IBNR calculation Chain ladder method II. Example: YearTotal (ultimate) payment IBNR 2009101775 20101152118 20111300301 20121458664 201316481264 Total2420

45. Insurance mathematics II. lecture 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 signs the 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.

46. Insurance mathematics II. lecture 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):

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

48. Insurance mathematics II. lecture 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.

49. Insurance mathematics II. lecture Methods of IBNR calculation Separation method II. We suppose that It means that signs the expected proportion of claims development year j without inflation effect. In this case is the inflation factor according to first year. Then we define the next formula: And we use the lagging triangle for these new elements as follows

50. Insurance mathematics II. lecture Methods of IBNR calculation Separation method III.... ……….. After we define diagonal sums as follows: Accident year Development year...

51. Insurance mathematics II. lecture Methods of IBNR calculation Separation method IV. ….

52. Insurance mathematics II. lecture Methods of IBNR calculation Separation method V. From the last equation we will get the first assumptions: After that we could calculate recursively the next factors as follows: …

53. Insurance mathematics II. lecture Methods of IBNR calculation Separation method VI. Then we will assume the future inflations: After we fill the remaining part of the lagging triangle: The assumption for the IBNR reserve will be the next:

54. Insurance mathematics II. lecture Methods of IBNR calculation Separation method VII. The geometric separation method will be calculated similar, just the beginning assumption is different: In the estimator formulas we use the products of diagonal instead of sums of diagonal.

55. Insurance mathematics II. lecture Methods of IBNR calculation Which method do will use in practice? There are no one ‘true’ method which is the most useful in each case (products, business lines). What can the actuary do in practice? The most useful possibility is calculating IBNR reserve with methods as much as possible, back-testing the results, and year by year fining the method. There is important to consider result of earlier IBNR as follows:

56. Insurance mathematics II. lecture Bonus reserve I.

57. Insurance mathematics II. lecture Bonus reserve II.

58. Insurance mathematics II. lecture Bonus reserve III. If the premium refund is affected when the policy is claim-free then the formula will be quieter as follows: Then we will get for bonus reserve:

59. Insurance mathematics II. lecture Other reserves I. This reserve is affected just in life insurance (it has to be calculated in case of plus yield return to policyholder). Equalization reserve This reserve can be calculated for those business line which are profitable (it has to be calculated, but the measure of reserve can be 0). If the result of business line is negative then the reserve has to be reduced with the measure of negative result. Large claim reserve This reserve has to be calculated in case of huge risks (for example: nuclear power station). Bonus reserve for life insurance

60. Insurance mathematics II. lecture Other reserves II. Cancellation reserve

61. Insurance mathematics II. lecture Solvency capital and security capital I. The reserves are calculated generally based on the expected value. For the worse scenario the insurer has to have solvency capital which is a part of own fund. Security capital This capital need for the fundamental operation of insurer. If the own fund is lower then the security capital the authority will act immediately. Solvency capital

62. Insurance mathematics II. lecture Solvency capital and security capital II.