1. Munich Chain Ladder Closing the gap between paid and incurred IBNR estimates Dr. Gerhard Quarg Münchener Rück Munich Re Group

2. From Chain Ladder to Munich Chain Ladder

3. MUNICH CHAIN LADDER – Closing the gap between paid and incurred IBNR estimates Triangle of P/I ratios vs. development years

4. MUNICH CHAIN LADDER – Closing the gap between paid and incurred IBNR estimates P/I quadrangle (with separate Chain Ladder estimates)

5. MUNICH CHAIN LADDER – Closing the gap between paid and incurred IBNR estimates Applying Chain Ladder to paid and incurred separately –Interpretation of graphic: A low current P/I ratio yields a low projected ultimate P/I ratio. A high current P/I ratio yields a high projected ultimate P/I ratio. –This is inherent to separate Chain Ladder calculations. Thorough mathematical formulation: For each accident year, the quotient of the ultimate P/I ratio and the average ultimate P/I ratio and the quotient of the current P/I ratio and the corresponding average P/I ratio agree.

6. MUNICH CHAIN LADDER – Closing the gap between paid and incurred IBNR estimates Correlations between paid and incurred data –Explanation: Below-average P/I ratios were succeeded by relatively high paid and/or relatively low incurred development factors. Above-average P/I ratios were succeeded by relatively low paid and/or relatively high incurred development factors. –This can indeed be seen in the data:

7. MUNICH CHAIN LADDER – Closing the gap between paid and incurred IBNR estimates Paid development factors vs. preceding P/I ratios

8. MUNICH CHAIN LADDER – Closing the gap between paid and incurred IBNR estimates Incurred development factors vs. preceding P/I ratios

9. MUNICH CHAIN LADDER – Closing the gap between paid and incurred IBNR estimates Problem: paid dev. factors vs. widely scattered P/I ratios

10. MUNICH CHAIN LADDER – Closing the gap between paid and incurred IBNR estimates Solution (Th. Mack): paid dev. factors vs. I/P ratios

11. MUNICH CHAIN LADDER – Closing the gap between paid and incurred IBNR estimates The residual approach –Problem: high volatility due to not enough data, especially in later development years –Solution: consider all development years together Use residuals to make different development years comparable. Residuals measure deviations from the expected value in multiples of the standard deviation.

12. MUNICH CHAIN LADDER – Closing the gap between paid and incurred IBNR estimates Residuals of paid development factors vs. I/P residuals

13. MUNICH CHAIN LADDER – Closing the gap between paid and incurred IBNR estimates Residuals of incurred dev. factors vs. P/I residuals

14. The stochastic MCL model

15. MUNICH CHAIN LADDER – Closing the gap between paid and incurred IBNR estimates The standard Chain Ladder model –Main assumptions of the standard Chain Ladder model: –These assumptions are designed for the projection of one triangle. –They ignore systematic correlations between paid losses and incurred losses. where and

16. MUNICH CHAIN LADDER – Closing the gap between paid and incurred IBNR estimates Required model features – In order to combine paid and incurred information we need or equivalently where Res ( ) denotes the conditional residual.

17. MUNICH CHAIN LADDER – Closing the gap between paid and incurred IBNR estimates The new model: Munich Chain Ladder –The Munich Chain Ladder assumptions: –Lambda is the slope of the regression line through the origin in the respective residual plot.

18. MUNICH CHAIN LADDER – Closing the gap between paid and incurred IBNR estimates The new model: Munich Chain Ladder –Interpretation of lambda as correlation parameter: –Together, both lambda parameters characterise the interdependency of paid and incurred.

19. MUNICH CHAIN LADDER – Closing the gap between paid and incurred IBNR estimates The new model: Munich Chain Ladder –The Munich Chain Ladder recursion formulas: –Lambda is the slope of the regression line through the origin in the residual plot, sigma and rho are variance parameters and q is the average P/I ratio.

20. Capability and limits of Munich Chain Ladder

21. MUNICH CHAIN LADDER – Closing the gap between paid and incurred IBNR estimates Initial example: ult. P/I ratios (separate CL vs. MCL)

22. MUNICH CHAIN LADDER – Closing the gap between paid and incurred IBNR estimates Triangle of P/I ratios vs. development years

23. MUNICH CHAIN LADDER – Closing the gap between paid and incurred IBNR estimates P/I quadrangle (with separate Chain Ladder estimates)

24. MUNICH CHAIN LADDER – Closing the gap between paid and incurred IBNR estimates P/I quadrangle (with Munich Chain Ladder)

25. MUNICH CHAIN LADDER – Closing the gap between paid and incurred IBNR estimates Another example: ultimate P/I ratios (SCL vs. MCL)

26. MUNICH CHAIN LADDER – Closing the gap between paid and incurred IBNR estimates The remaining gap –Munich Chain Ladder projects ultimate P/I ratios of about “only” 96%. –There is a remaining gap between paid and incurred IBNR estimates. –This is not a failure of Munich Chain Ladder. On the contrary, data suggest a remaining gap after 14 years of development:

27. MUNICH CHAIN LADDER – Closing the gap between paid and incurred IBNR estimates Triangle of P/I ratios vs. development years

28. Thank you for your interest Dr. Gerhard Quarg Münchener Rück Munich Re Group