Combined analysis of paid and incurred losses B. Posthuma E.A. Cator

1. Posthuma Partners Combined analysis of paid and incurred lossesB. Posthuma E.A. Cator Washington, September 2008

2. Modern accounting (IFRS), capital management and global regulation rules put more stringent demands on loss reserving • Existing loss reserving methods are struggling to provide an adequate, yet sufficiently flexible, solution • Our combined analysis provides an excellent tool for modern risk and capital management

3. Principles of modern Financial Economics are: • expected present value of future cash flows, and • their variances • Markowitz (1952), Sharpe (1964) and Modigliani, Miller (1958) already applied economic theory to business administration

4. International Financial Reporting Standards (IFRS) are fully based on two basic principles. • Fair Value =expected present value plus margin for riskor market value • Actuarial prudence and solvency analyses have to be in agreement with financial economic theory

5. This requires: • stochastic loss reserving on a continuous time basis, including discounting • adequate assessment of percentile ranges • flexibility in aggregating various datasets for branches Other prerequisites: • projections of expired and risk in force • adjustments in time to incurred loss properly modeled

6. The datasets used often consist of two loss triangles, for paid and reported incurred, together with a measure for exposure • Many methods and models have been developed for analyzing a single loss triangle • We are able to model the loss triangles for paid and reported incurred simultaneously, and show that this leads to a more accurate analysis of the loss reserve

7. (1) • Loss period l • Development period k • Y incremental paid losses • Y incremental incurred losses lk (2) lk

8. Start by supposing that all losses are independent and normally distributed Now note: as all claims are settled eventually, cumulative paid and incurred losses for a given loss period must be equal

9. (1) (2) lk lk S S (1) (2) • Therefore we condition the incrementallosses Y and Y on the event that • Y = Y • This conditioning preserves normalityAlso, conditioning can be used to predict future losses given the observed losses lk lk k k

10. Advantages of using the normal distribution: • projections for expired insurance risk as well as risk in force are readily available • the formats of data such as time units for loss period, development period or other aggregations are easily handled • aggregation of neighboring cells strengthens the assumption of normality • flexible projections for discounted values naturally exists • negative incrementals in incurred are not uncommon

11. (1) (2) lk lk • Now we need a parametric model formeans and variances of Y and Y

12. (i) (i) , lk l • Means and variances: • EY = mlPki = 1,2 • where • m=W eXb • and Wl is the exposure. • var(Y)= s mP i = 1,2 ~ (i) (i ) 2 , lk i l k

13. S S S S ~ ~ (2) (1) (2) (1) • Development curves sum to 1 : • P = P = P = P = 1 k k k k k k k k

14. A four parameter family is constructed for development curves • {f(x;b,g,m,s):b,g,s > 0, m 0} • with useful properties: • integrate to 1 • can be negative if m >1 • first and second primitive explicit • direct control over boundary behavior

15. Ample flexibility, including negative values

16. A simulation experiment was conducted: • using a real dataset of partially observed paid and incurred loss tables = loss triangles • estimating the parameters (b,g,m,s)using maximum likelihood • generating 6000 pairs of complete tables from our model with the estimated parameters • predicting the reserve R on the basis of the observed part for each of the 6000 samples • The combined analysis is compared to using • the single paid triangle only

17. single combined Variance of the combined analysis is 3x smaller!

18. Conclusion: • Our combined analysis of paid and incurred proves to be a flexible and accurate tool for loss reserving, providing results for simulated and real data that are superior to existing methods