1. When can accident years be regarded as development years? Glen Barnett, Ben Zehnwirth, Eugene Dubossarsky Speaker: Dr Glen Barnett Senior Research Statistician, Insureware

2. Outline of talk What do we mean by “the chain ladder”? The basic “transpose-invariance” result Demonstrating the result (outline of a simple proof) What does the result tell us? i) Structure: accident years vs development years ii) Number of parameters iii) Cross-classification structure and ordering What lessons are there for other ratio methods?

3. What do we mean by “the chain ladder”? In its standard form: loss development technique for cumulative & incurred arrays use volume-weighted average ratios (where “volume” is previous column) gives factor, b j = “sum of column”/“sum of previous” where sum is over observations present in both.

4. Chain ladder for incremental arrays You can think of the chain ladder as a way to forecast incremental arrays as well: - take an incremental array (say incremental paid) 1 cumulate across 2 compute ratios 3 forecast 4 difference back to incrementals ①② ③ ④ ⊝  ⃝

5. The basic result Produce two tables: 1) forecast an incremental array with the chain ladder (using the 4 steps) 2) transpose the array (interchange accident and development years), then forecast that with the chain ladder and transpose back Tables 1 and 2 are identical

6. Another way to think about it 1) forecast an incremental array with the chain ladder 2) take an incremental array and apply new steps: 1 cumulate down 2 compute ratios running down 3 forecast down 4 difference (up) back to incrementals Tables 1 and 2 are identical ① ② ③ ④⊝  ⃝

7. Demonstrating the result (i)show that an incremental forecast from the CL is the same as finding sums of incrementals in each region (A,B,C) and computing BC/A ** (ii) Note that the result is the same for a transposed array B C ij p ˆ A = B.C/A ij p ˆ ** Replace any future values in shaded regions with their CL forecasts

8. Step (i) - (a) first, show for next diagonal, C.L. ratio = (A+B)/A; previous cumulative = C  Forecast cumulative = (A+B)C/A  Forecast incremental = (A+B)C/A – C = B.C/A B C ij p ˆ A = B.C/A ij p ˆ

9. = B.C/A ij p ˆ B C A Replace any future values in shaded regions with their CL forecasts Step (i) - (b) for later diagonals, Note that forecast values already “follow the ratio”, so adding them in to A and B leaves B/A unchanged. Also, the next forecast is based on the previous cumulative forecast, so C also contains the incremental forecasts  Future forecast incremental = B.C/A

10. One advantage of this fact Easy in Excel to forecast incrementals directly Single formula can be pasted to each forecast cell Ratios can be computed from the last forecast row (aside)

11. Step (ii) Plainly BC/A = CB/A; So the calculation is the same for the transposed array (B and C merely interchange their roles). = B.C/A ij p ˆ B C A A B C = C.B/A ij p ˆ

12. What does the result tell us? We call this property “transpose invariance” (more strictly, “transpose-forecast commutativity) i) Structure: accident years vs development years does not differentiate between accident & development year directions – chain ladder treats them identically

13. Of course we know that development years are quite different from accident years! adjusted for trend in other direction raw data

14. What does this result tell us? ii) It also tells us that there are in fact parameters in both accident and development directions; (This has been a source of argument – e.g. Mack vs Renshaw & Verrall) we’re aware of the parameter corresponding to the column effect (the “B” effect) – it’s the “ratio” but we usually condition on (i.e. ignore) the row effect (the “C” effect); it’s a degree of freedom that the model has to fit the data – i.e. a parameter

15. What does this result tell us?  There are parameters in both accident and development directions: s  s triangle has 2s–1 parameters for the mean (Overparameterization)

16. iii) cross classification structure (two-way-ANOVA-like) – no account of ordering In a cross-classification structure, you can interchange the row labels (or the column labels) with no impact - but we know that order matters! Can easily tell which of these has had its labels scrambled

17. iii) cross classification structure (ctd) – in fact there’s abundant information in nearby acci yrs e.g. – in same dev. yr: If you left out a point, how would you guess what it was? - observations at same delay very informative.

18. iii) cross classification structure (ctd) – information in different dev. yrs: - nearby delays also informative (smooth trends) (could leave out whole development & still guess where it was)

19. iii) cross classification structure (ctd) information in nearby accident and development yrs… need to use more of that information!

20. Are there lessons for other ratio methods? Mack (93) and Murphy (94) are able to write a model that includes several ratio methods as wtd regression - chain ladder uses one particular set of weights  Conditionally on previous cumulative, can write a number of other methods as weighted chain ladder:

21. Are there lessons for other ratio methods? Example: “average development factor”: fix the development year (i.e. hold it constant) at j Let y i be the cumulative in accident year i, and let x i be the previous cumulative. b = 1/n  (y i /x i ) =  (w i y i ) /  (w i x i ), where w i = 1/x i Hence ave. devel. factor is a weighted chain ladder

22. Are there lessons for other ratio methods? …for many methods, a weighted version of this result still holds (weighted) two-way cross-classification structure - many of the problems carry over!

23. Are there lessons for other ratio methods? - Still parameters in both directions. 2s-1 parameters

24. Are there lessons for other ratio methods? - Still ignores location information in nearby accident and development years

25. s  s triangle: ratio methods use 2s–1 parameters for mean How many parameters needed to describe this:

26. Can describe shape of curve with say 2-3 parameters Can describe stable accident year level with 1 (many arrays are similarly simple) for this triangle, ratio methods use 20 parameters (and wastes those: ratios don’t predict the next increment for that array)

27. Effects of overparameterisation - fitting noise rather than signal - high parameter uncertainty - unstable forecasts (small change in data – large change in prediction I.e. projects and amplifies noise into the future)

28. Conclusions “Transpose invariance” is an important feature – serious implications for the chain ladder; – many of the lessons apply more generally

29. Lessons be aware of the structure of loss data – don’t ignore what you know need to be aware of the specific structure in a triangle – does the model succinctly describe the main features in the data? (diagnostics, model validation, parsimony)