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 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 reserves are more than billion HUF at 2015).
Insurance mathematics II. lecture

3. Unearned premium reserve I.
Example:
we take out a household policy at with HUF yearly premium. We agree yearly payment frequency and we pay total premium in December. What is the realistic P&L situation at ?
2.000 HUF
2.000 HUF
2.000 HUF
HUF
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 2016 and 2017?
If we would book whole premium for 2016 then we would not book any premium for 2017, but we are in risk during 11 months! But we got really the whole premium in 2016.
The solution:
We book whole premium for 2016.
We calculate unearned premium reserve for It means that this part of premium will be the offset of risk in The value of UPR is:
Insurance mathematics II. lecture

5. Unearned premium reserve III.
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

6. Unearned premium reserve IV.
Open problems with this definition:
the lengths of months are not equal (in the example the real UPR is not 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

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.)
Insurance mathematics II. lecture

8. 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;
lx comes 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

9. 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:
Insurance mathematics II. lecture

10. Mortality table example

11. 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).
Insurance mathematics II. lecture

12. 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.
Insurance mathematics II. lecture

13. 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 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

14. 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 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

15. 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).
Insurance mathematics II. lecture

16. 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 two tasks: calculate claims handling reserve with the next formula and calculate so-called ‚initial’ 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

17. 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.
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

18. Insurance mathematics II. lecture
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

19. Insurance mathematics II. lecture
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

20. 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.
Insurance mathematics II. lecture

21. 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.
Insurance mathematics II. lecture

22. Methods of IBNR calculation Grossing Up method I.
Example:
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

23. Methods of IBNR calculation Grossing Up method II.
Example:
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

24. Methods of IBNR calculation Grossing Up method III.
Example:
Is the cumulated triangle complete? No, we have data from earlier years as follows:
Year
Claims until 5th year
Total
Ratio (5th year/Total)
2007
780
830
93,98%
2008
810
890
91,01%
2009
800
860
93,02%
2390
2580
92,64%
Insurance mathematics II. lecture

25. Methods of IBNR calculation Grossing Up method IV.
Example:
Assumption for 2015:
The base of assumption will be 2015 as next table shows:
1. year
2. year
3. year
4. year
5. year
Total
2015
223
534
786
913
942
1017
Ratio
21,9%
52,5%
77,3%
89,8%
92,64%
100%
We assume that the run-off of next years will be equal as 2015.
Insurance mathematics II. lecture

26. Methods of IBNR calculation Grossing Up method V.
Example:
Total
IBNR
1017 75
1152
118
1292
293
1512
718
1751
1367
2571
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

27. 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

28. 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

29. Methods of IBNR calculation Modified Grossing Up methods I.
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

30. Methods of IBNR calculation Modified Grossing Up methods II.
Example:
1. year
2. year
3. year
4. year
5. year
Total
2007
Ratio (%)
170
20,5%
422
50,8%
660
79,5%
750
90,4%
780
94%
830
100%
2008
140
15,7%
435
48,9%
680
76,4%
782
87,9%
810
91%
890
2009
132
15,3%
428
49,8%
670
77,9%
90,7%
803
93%
860
2015
223
21,9%
534
52,5%
786
77,3%
913
89,8%
942
92,7%
1017
Averageratio
18,4%
50,5%
77,8%
89,7%
Insurance mathematics II. lecture

31. Methods of IBNR calculation Modified Grossing Up methods III.
Example:
Year
Total (ultimate) payment
IBNR
2011
1017 75
2012
1153
119
2013
1284
285
2014
1572
778
2015
2090
1706
Total
2964
Insurance mathematics II. lecture

32. 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:
Insurance mathematics II. lecture

33. Methods of IBNR calculation Modified Grossing Up methods V.
Example:
1. year
2. year
3. year
4. year
5. year
Total
2007
Ratio (%)
170
20,5%
422
50,8%
660
79,5%
750
90,4%
780
94%
830
100%
2008
140
15,7%
435
48,9%
680
76,4%
782
87,9%
810
91%
890
2009
132
15,3%
428
49,8%
670
77,9%
90,7%
803
93%
860
2015
223
21,9%
534
52,5%
786
77,3%
913
89,8%
942
92,7%
1017
Minimum ratio
Insurance mathematics II. lecture

34. Methods of IBNR calculation Modified Grossing Up methods VI.
Example:
Year
Total (ultimate) payment
IBNR
2011
1035 93
2012
1177
143
2013
1308
309
2014
1625
831
2015
2502
2118
Total
3493
Insurance mathematics II. lecture

35. Methods of IBNR calculation Modified Grossing Up methods VII.
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

36. Methods of IBNR calculation Modified Grossing Up methods VIII.
Example:
1. year
2. year
3. year
4. year
5. year
Total
2011
Ratio (%)
223
21,9%
534
52,5%
786
77,3%
913
89,8%
942
92,6%
1017
100%
2012
254
22,1%
632
54,9%
881
76,5%
1034
1152
2013
312
24%
723
55,6%
999
76,9%
1299
2014
359
24,6%
794
54,3%
1461
2015
384
23,1%
1659
Insurance mathematics II. lecture

37. Methods of IBNR calculation Modified Grossing Up methods IX.
Example:
Year
Total (ultimate) payment
IBNR
2011
1017 75
2012
1152
118
2013
1299
300
2014
1461
667
2015
1659
1275
Total
2434
Insurance mathematics II. lecture

38. Methods of IBNR calculation Modified Grossing Up methods X.
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

39. Methods of IBNR calculation Modified Grossing Up methods XI.
Example:
1. year
2. year
3. year
4. year
5. year
Total
2011
Ratio (%)
223
21,9%
534
52,5%
786
77,3%
913
89,8%
942
92,6%
1017
100%
2012
254
22,1%
632
54,9%
881
76,5%
1034
1152
2013
312
24%
723
55,6%
999
1306
2014
359
24,6%
794
1512
2015
384
1753
Insurance mathematics II. lecture

40. Methods of IBNR calculation Modified Grossing Up methods XII.
Example:
Year
Total (ultimate) payment
IBNR
2011
1017 75
2012
1152
118
2013
1306
307
2014
1512
718
2015
1753
1369
Total
2587
Insurance mathematics II. lecture

41. 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 reserve estimation:
Insurance mathematics II. lecture

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

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

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

45. Methods of IBNR calculation Chain ladder method II.
Example:
Year
Total (ultimate) payment
IBNR
2011
1017 75
2012
1152
118
2013
1300
301
2014
1458
664
2015
1648
1264
Total
2420
Insurance mathematics II. lecture

46. 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.
Insurance mathematics II. lecture

47. 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

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

49. 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

50. 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
Insurance mathematics II. lecture

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

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

53. 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: …
Insurance mathematics II. lecture

54. 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:
Insurance mathematics II. lecture

55. Methods of IBNR calculation Separation method VII.
Example:
Occurring year
2013
375; 250; 188
2014
273; 341
2015
324
Insurance mathematics II. lecture

56. Methods of IBNR calculation Separation method VIII.
We have information about the number of claims also:
Insurance mathematics II. lecture

57. Methods of IBNR calculation Separation method IX.
Than we can calculate the missing part of the triangle:
The IBNR reserve will be the sum of these elements:
Insurance mathematics II. lecture

58. Methods of IBNR calculation Separation method X.
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.
Insurance mathematics II. lecture

59. 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:
Insurance mathematics II. lecture

60. Insurance mathematics II. lecture
Bonus reserve I.
Insurance mathematics II. lecture

61. Insurance mathematics II. lecture
Bonus reserve II.
Insurance mathematics II. lecture

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

63. Insurance mathematics II. lecture
Other reserves I.
Bonus reserve for life insurance
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).
Insurance mathematics II. lecture

64. Insurance mathematics II. lecture
Other reserves II.
Cancellation reserve
Based on own experience it has to be reserved the expected cancelled part of premium in the future. It is important to take into consideration the unearned premium reserve (it is prohibited to reserve for unearned part of premium) and in life insurance the saving part of the premium (it is prohibited to reserve for the saving part of the premium).
Example:
If we have HUF written premium for a policy which has 1 month delay at year-end, and based on our experience 5% the probability that premiums delaying 1 month won’t be paid then the cancellation reserve will be ·5%=500 HUF. But if there is UPR related to this policy then we have just ( )·5%=375 HUF cancellation reserve. (If the UPR more than HUF then it is prohibited to reserve.)
Insurance mathematics II. lecture

65. Insurance mathematics II. lecture
Other reserves III.
Insurance mathematics II. lecture