1. Advantages and Limitations of Applying Regression Based Reserving Methods to Reinsurance Pricing Thomas Passante, FCAS, MAAA Swiss Re New Markets CAS Seminar on Reinsurance June 16, 2000

2. Regression Model for In-Period Loss Payments
Pij = AYi ( ) e j ij
Log Transformed:
ln Pij = ln AYi + ( ) + j + ln ij i + j Õ CY g e k k =
yr1 i + j å g ln CY e k k =
yr1

3. Example of Fitted Incremental Data
Fitted Points
1994
1995
1996
1997
Assume:
CY trend = 7% Then, 100 x 0.84 x = 61
DY decay = 0.8
All AY on same level

4. A Regression Based Model: Why?
Competitive Advantage
Additional Information
Better Understanding

5. Some Short Term Limitations:
Time Investment
Learning how
Actually applying the techniques
Ability to Explain
Colleagues
Clients
Intuition takes time to develop

6. Regression Modeling - Two Main Advantages:
Obtain a distribution around a point estimate
Isolate Calendar Year trends

7. Distribution Around a Point Estimate
Capital Requirement =
F( …. , volatility , … )
Volatility measured by: standard deviation, downside result, low percentile result, other…
Directly affects Return on Equity (ROE) and therefore influences decision making process

8. Calendar Year Trend
Situations may exist where calendar year trends are identifiable using regression techniques, but not with traditional techniques
May use information to your advantage

9. Calendar Year Trend Example
Original Paid Data (Cumulative)
, , , , , , , , , ,067
, , , , , , , , ,796
, , , , , , , ,149
, , , , , , ,205
, , , , , ,502
, , , , ,952
, , , ,024
, , ,710
, ,304
,473
LDF's
Tail
All yr wtd
Cum
Accident Yr
Ult Loss 17, , , , , , , , , ,223

10. Calendar Year Trend Example
CY Trend
From To Observed Model Fit Cum
% 6.0%
% 6.0% 1.124
% 6.0% 1.191
% 6.0% 1.262
% 4.0% 1.313
% 4.0% 1.365
% 4.0% 1.420
% 2.0% 1.449
% 2.0% 1.477
% 1.507
% 1.537
% 1.568
% 1.599
% 1.631
% 1.664
% 1.697
% 1.731
% 1.766
% 1.801
% 1.837
% 1.874
% 1.911
% 1.950
% 1.989
% 2.028

11. Calendar Year Trend Example
Observed Decay:
Yr \ From..
Avg
Selection
Product

12. Fitted Calendar Year Trends6% 4% 2%

13. Fitted Decay Parameters
1.8
1.1
0.8
0.5

14. Calendar Year Trend Example
Fitted Points
Errors (Fitted - Actual)
Avg Error
Sum Error sum = 0.0

15. Calendar Year Trend Example
Reserve
,243.9
,487.9
,061.7
,047.8
,550.0
,697.5
,637.3

16. Calendar Year Trend Example
Ultimates Paid Reserves*
Regression Model LDF Method To Date Regression Model LDF Method
12, , ,
12, , ,
13, , ,
14, , , , ,384
13, , , , ,572
15, , , , ,450
15, , , , ,481
15, , , , ,912
16, , , , ,196
16, , , , ,793
143, , , , ,916
* In this case, LDF Method produces 8.24% higher reserves

17. Regression Modeling - Limitations
Need a good exposure base
Ultimate Claim Count is a preferred measure
Poorer fits with other measures
Loss Portfolio Transfers vs. Prospective Contracts

18. LPT’s vs. Prospective Contracts
Often can estimate claim count very well
Prospective:
Estimate future claim count using past relationship to some other exposure base (one more easily predictable/verifiable for next year)
Use other exposure base

19. Prospective Contract: Example
Corporate Client, Workers’ Compensation
Payroll may be easy to predict next year (budget item, and is auditable)
Estimate distribution (mean, volatility, etc.) of claim count as a percentage of payroll
Incorporate this volatility into estimate for prospective exposure base

20. Prospective Contracts
P(C)
Claim count as % of payroll C Co
P(L)
Conditional Loss distribution given claim count Co L

21. Prospective Contracts
Conditional Loss given Co
Conditional Loss given C1
Conditional Loss given C2
P(L)
L|Co
L|C1
L|C2 L

22. Prospective Contracts
Conditional Loss given Co
Conditional Loss given C1
Conditional Loss given C2 L
P(L)
L|Co
L|C1
L|C2
P(X)
Unconditional
loss distribution X
Unconditional Loss distribution

23. Regression Modeling - Limitations
Not always good for excess/ reinsurance data
Zero’s in early development periods
cannot model log (data)
Varying effect of threshold for excess data year by year
In traditional methods there are ways to adjust (eg., Pinto Gogol)

24. Regression Modeling - Limitations
Can only use positive incremental data
Again, issue with log (data)
Rarely can we model incurred data
Usually not a problem with paid data, although issues may occur with recoveries appearing at later maturities

25. Regression Modeling - Limitations
Positive incremental data issue: Two possible solutions?
Add a constant value to the entire triangle such that all values are now positive
Smoothing the data

26. Regression Modeling - Limitations
Solution #1: Adding A Constant
NOT RECOMMENDED.
Log (x2+c) / Log (x1+c) does not equal Log(x2) / Log (x1)

27. Regression Modeling - Limitations
Solution #1: Example
Suppose the following incremental data: 50, 40, 32, 25.6, ...
Decay in natural process is 0.8.
However, suppose a value of appeared somewhere earlier on in the triangle….

28. Regression Modeling - Limitations
Solution #1: Example (continued…)
Now we model: 150, 140, 132, 125.6, …,
Decay parameters are no longer constant: .933, .943, .952,…
Now modeling a different process
What do you select for future decay?

29. Regression Modeling - Limitations
Solution #2: Smoothing the data
Our preferred solution
However, must be aware of:
Higher goodness of fit statistics
Lower volatility estimate (must adjust for this in the end)

30. Regression Modeling - Limitations
Difficult to interpret results
Difficult to separate out the effects of three dimensions (Development year, Accident year, and Calendar Year)
Three dimensions are difficult to interpret / visualize

31. Regression Modeling - Limitations
In Period Loss Payments
Development Year
Accident Year

32. Regression Modeling - Limitations
Tail behavior
Cannot just run regression out to infinity!
Still need external sources to determine length of payout
May determine number of future years to pay out using industry tail information and decay parameter

33. Regression Modeling - Limitations
Often not good for situations where difficulty exists using traditional development based methods
Long tailed latent claims
Other poor data sets (regression models can sometimes be as poor as loss development models)

34. Regression Modeling - Conclusions
Often difficult to use when traditional methods fail
However, does provide some very useful additional information which is not provided by other more traditional techniques

35. Regression Modeling - Conclusions (continued)
Still continue to analyze using traditional methods, but use additional information from regression techniques, especially when:
results make sense
output can be explained