Math 479 Casualty Actuarial Mathematics

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

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. 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. 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. 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. 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. Expected Loss Ratio (ELR) Method

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

12. Bornhuetter-Ferguson Method (cont.)

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. 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. CAS Exam 6, Fall 2005, #10

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

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