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

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

Example of Fitted Incremental Data Fitted Points 1 2 3 4 5 6 7 8 9 1989 100 86 73 . . . 1990 107 92 . . . 1991 . . 61 1992 . . 1993 . 1994 1995 1996 1997 Assume: CY trend = 7% Then, 100 x 0.84 x 1.076 = 61 DY decay = 0.8 All AY on same level

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

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

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

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

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

Calendar Year Trend Example Original Paid Data (Cumulative) 1 2 3 4 5 6 7 8 9 10 1989 1,000 2,913 5,156 7,005 8,613 9,928 11,010 11,606 11,914 12,067 1990 1,063 3,039 5,300 7,155 8,633 9,916 11,009 11,545 11,796 1991 1,123 3,288 5,806 7,993 9,820 11,287 12,504 13,149 1992 1,190 3,542 6,181 8,483 10,424 11,961 13,205 1993 1,258 3,505 6,082 8,274 10,013 11,502 1994 1,307 3,744 6,634 8,935 10,952 1995 1,362 3,953 6,811 9,024 1996 1,416 3,888 6,710 1997 1,446 4,304 1998 1,473 LDF's 1 - 2 2 - 3 3 - 4 4 - 5 5 - 6 6 - 7 7 - 8 8 - 9 9 - 10 1989 2.913 1.770 1.359 1.230 1.153 1.109 1.054 1.026 1.013 1990 2.859 1.744 1.350 1.207 1.149 1.110 1.049 1.022 1991 2.929 1.766 1.377 1.229 1.149 1.108 1.052 1992 2.977 1.745 1.372 1.229 1.147 1.104 1993 2.787 1.735 1.360 1.210 1.149 1994 2.865 1.772 1.347 1.226 1995 2.903 1.723 1.325 1996 2.746 1.726 1997 2.977 Tail All yr wtd 2.882 1.746 1.355 1.222 1.149 1.108 1.051 1.024 1.013 1.013 Cum 11.720 4.066 2.328 1.718 1.406 1.224 1.105 1.051 1.026 1.013 Accident Yr 1998 1997 1996 1995 1994 1993 1992 1991 1990 1989 Ult Loss 17,267 17,500 15,622 15,505 15,402 14,074 14,588 13,816 12,102 12,223

Calendar Year Trend Example CY Trend From To Observed Model Fit Cum 1989 1990 6.3% 6.0% 1.060 1990 1991 5.6% 6.0% 1.124 1991 1992 6.0% 6.0% 1.191 1992 1993 5.7% 6.0% 1.262 1993 1994 3.9% 4.0% 1.313 1994 1995 4.2% 4.0% 1.365 1995 1996 4.0% 4.0% 1.420 1996 1997 2.1% 2.0% 1.449 1997 1998 1.9% 2.0% 1.477 1998 1999 2.0% 1.507 1999 2000 2.0% 1.537 2000 2001 2.0% 1.568 2001 2002 2.0% 1.599 2002 2003 2.0% 1.631 2003 2004 2.0% 1.664 2004 2005 2.0% 1.697 2005 2006 2.0% 1.731 2006 2007 2.0% 1.766 2007 2008 2.0% 1.801 2008 2009 2.0% 1.837 2009 2010 2.0% 1.874 2010 2011 2.0% 1.911 2011 2012 2.0% 1.950 2012 2013 2.0% 1.989 2013 2014 2.0% 2.028

Calendar Year Trend Example Observed Decay: Yr \ From.. 1 - 2 2 - 3 3 - 4 4 - 5 5 - 6 6 - 7 7 - 8 8 - 9 9 - 10 1989 1.80 1.11 0.78 0.82 0.79 0.79 0.53 0.51 0.49 1990 1.76 1.08 0.78 0.77 0.83 0.82 0.48 0.46 1991 1.82 1.10 0.84 0.80 0.77 0.81 0.52 1992 1.87 1.08 0.84 0.81 0.78 0.79 1993 1.72 1.10 0.82 0.78 0.84 1994 1.79 1.14 0.78 0.86 1995 1.83 1.08 0.76 1996 1.71 1.12 1997 1.94 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10 10-11 11-12 Avg 1.80 1.10 0.80 0.81 0.80 0.80 0.51 0.48 0.49 Selection 1.80 1.10 0.80 0.80 0.80 0.80 0.50 0.50 0.50 0.50 0.50 Product 1.80 1.980 1.584 1.267 1.014 0.811 0.406 0.203 0.101 0.051 0.025

Fitted Calendar Year Trends 6% 4% 2%

Fitted Decay Parameters 1.8 1.1 0.8 0.5

Calendar Year Trend Example Fitted Points 1 2 3 4 5 6 7 8 9 10 1989 997.5 1903.3 2219.2 1881.9 1595.8 1327.7 1104.7 574.4 293.0 149.4 1990 1057.4 2017.4 2352.3 1994.8 1659.7 1380.8 1148.9 585.9 298.8 152.4 1991 1120.8 2138.5 2493.5 2074.6 1726.1 1436.1 1171.8 597.6 304.8 155.4 1992 1188.1 2266.8 2593.2 2157.6 1795.1 1464.8 1195.3 609.6 310.9 158.6 1993 1259.3 2357.5 2697.0 2243.9 1831.0 1494.1 1219.2 621.8 317.1 161.7 1994 1309.7 2451.8 2804.8 2288.7 1867.6 1524.0 1243.6 634.2 323.5 165.0 1995 1362.1 2549.8 2860.9 2334.5 1905.0 1554.5 1268.4 646.9 329.9 168.3 1996 1416.6 2600.8 2918.1 2381.2 1943.1 1585.5 1293.8 659.8 336.5 171.6 1997 1444.9 2652.9 2976.5 2428.8 1981.9 1617.3 1319.7 673.0 343.2 175.1 1998 1473.8 2705.9 3036.0 2477.4 2021.6 1649.6 1346.1 686.5 350.1 178.6 Errors (Fitted - Actual) 1989 -2.5 -10.1 -23.6 33.0 -12.6 13.3 22.6 -22.0 -14.6 -4.1 1990 -5.6 41.8 90.6 140.6 181.7 97.4 55.8 50.2 47.7 1991 -1.7 -27.1 -24.4 -112.4 -100.9 -31.1 -45.3 -47.3 1992 -1.8 -85.1 -45.9 -144.7 -145.3 -72.4 -48.8 1993 1.6 109.9 120.7 51.7 91.7 5.3 1994 3.0 14.4 -84.9 -12.6 -149.1 1995 0.5 -41.6 3.4 121.0 1996 0.5 128.5 96.5 1997 -0.9 -205.4 1998 0.5 Avg Error -0.7 -8.3 16.5 10.9 -22.4 2.5 -3.9 -6.4 16.6 -4.1 Sum Error -6.6 -74.8 132.4 76.6 -134.5 12.5 -15.7 -19.1 33.2 -4.1 sum = 0.0

Calendar Year Trend Example 11 12 13 14 15 16 17 18 19 20 Reserve 76.2 38.9 19.8 10.1 5.2 2.6 1.3 0.7 0.3 0.2 155.3 77.7 39.6 20.2 10.3 5.3 2.7 1.4 0.7 0.4 0.2 310.8 79.3 40.4 20.6 10.5 5.4 2.7 1.4 0.7 0.4 0.2 621.8 80.9 41.2 21.0 10.7 5.5 2.8 1.4 0.7 0.4 0.2 1,243.9 82.5 42.1 21.5 10.9 5.6 2.8 1.5 0.7 0.4 0.2 2,487.9 84.1 42.9 21.9 11.2 5.7 2.9 1.5 0.8 0.4 0.2 4,061.7 85.8 43.8 22.3 11.4 5.8 3.0 1.5 0.8 0.4 0.2 6,047.8 87.5 44.6 22.8 11.6 5.9 3.0 1.5 0.8 0.4 0.2 8,550.0 89.3 45.5 23.2 11.8 6.0 3.1 1.6 0.8 0.4 0.2 11,697.5 91.1 46.4 23.7 12.1 6.2 3.1 1.6 0.8 0.4 0.2 14,637.3

Calendar Year Trend Example Ultimates Paid Reserves* Regression Model LDF Method To Date Regression Model LDF Method 12,223 12,223 12,067 155 156 12,107 12,102 11,796 311 306 13,771 13,816 13,149 622 667 14,449 14,588 13,205 1,244 1,384 13,390 14,074 11,502 2,488 2,572 15,013 15,402 10,952 4,062 4,450 15,072 15,505 9,024 6,048 6,481 15,260 15,622 6,710 8,550 8,912 16,002 17,500 4,304 11,698 13,196 16,111 17,267 1,473 14,637 15,793 143,997 148,099 94,183 49,814 53,916 * In this case, LDF Method produces 8.24% higher reserves

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

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

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

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

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

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

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)

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

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

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

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 -100 appeared somewhere earlier on in the triangle….

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?

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)

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

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

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

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)

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

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