1. 1 Math 479 Casualty Actuarial Mathematics Fall 2014 University of Illinois at Urbana-Champaign Professor Rick Gorvett Session 5: Loss Reserving II September 9, 2014

2. 2 Agenda Review of basic loss development technique and essential metrics / quantities Other loss reserving methods –Expected loss ratio –Bornhuetter-Ferguson Diagnostics Reserving issues

3. 3 Review of LDF Technique General approach of loss development techniques –Loss development triangle (cumulative data) –Age-to-age factors (link ratios) Approaches to selecting factors – e.g., –Straight average –Average of last three years –(Middle) 3 of 5 years –Weighted average (by dollars) –To-ultimate loss development factors –For each (accident) year: (actual loss) × (LDF) = estimated ultimate loss –Reserve = (ultimate loss) – (to-date paid loss) –IBNR (broad def ’ n.) = (ult. loss) – (case incurred loss)

4. 4 Review of LDF Technique (cont.) Possible bases: –Accident Year (AY) –Policy Year (PY) –Report Year (RY) Possible types of data: –Paid losses (with or without (A)LAE) –Case incurred losses (with or without (A)LAE) –Closed claim counts –Reported / incurred claim counts –Paid loss severity –Incurred loss severity

5. 5 Review of LDF Technique (cont.) Data relationships –Cumulative incurred loss = cumulative paid loss + “ snapshot ” case reserve –Cumulative reported / incurred claim count = cumulative paid claim count + “ snapshot ” open claim count –Paid severity = cumulative paid loss ÷ cumulative paid claim count –Reported / incurred severity = cumulative incurred loss ÷ cumulative reported claim count

6. 6 Metrics and Quantities of Interest Paid loss + case reserve = (case) incurred loss Ultimate AY or PY loss = paid loss + case reserve + IBNR IBNR (Incurred But Not Reported) loss = ultimate loss – (case) incurred loss Total loss reserve = ultimate loss – paid loss Total loss reserve = case reserve + IBNR Loss = frequency × average severity

7. 7 Other Basic Loss Reserving Techniques Expected loss ratio (ELR) method –Loss ratio = loss ÷ premium –Ultimate loss = premium × ELR –ELR per best estimate (e.g., original loss provision in premium per ratemaking process) Bornhuetter-Ferguson method –A “middle-ground” between the LDF and ELR methods

8. 8 Expected Loss Ratio (ELR) Method

9. 9 Bornhuetter-Ferguson (B-F) Method Merges the LDF and ELR techniques Steps (e.g., for case-incurred B-F): (1) Determine the to-ultimate LDF (2) Determine the proportion of ultimate loss unreported to-date = 1 – {1 / (1)} (3) Earned premium × ELR = ( “ a priori ” ) expected ultimate loss (4) Expected unreported loss = (2) × (3) (5) B-F ultimate loss = (4) + (case incurred loss to- date) See example

10. 10 B-F Method: Example of Use of LDF “Unpaid” –Suppose the 12-ult. Paid loss LDF (loss development factor) is 5.000 –What proportion of the ultimate loss is expected to have been paid @ 12 months? “Unreported” –Suppose the 24-ult. Incurred LDF is 2.500 –What proportion of the ultimate loss is expected to have been reported @ 24 months?

11. 11 Bornhuetter-Ferguson Method “A priori” Or “Initial Expected”

12. 12 Bornhuetter-Ferguson Method (cont.)

13. 13 Diagnostics Relationship of projected ultimate L/R to ELR LDF / payout patterns compared with industry or other sources –“ Internal vs external ” Paid-to-incurred (or –to-reserve) ratios Closed-to-reported claim count ratios Average paid claim Reserve as percentage of ultimate loss

14. 14 Loss Reserving Issues Considering multiple results –LDF vs ELR vs B-F –Paid vs incurred vs frequency-severity Point estimates vs reserve ranges Loss reserve discounting –Regulatory considerations –Payment patterns –Interest rates

15. 15 CAS Exam 6, Fall 2005, #10

16. 16 CAS Exam 6, Fall 2005, #10 (cont.)

17. 17 Next Time Loss Reserving III Loss reserving principles CAS Statement of Loss Reserving Principles Accounting issues Blanchard study note