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Adv Res.ppt

Advantages and Limitations of Applying Regression Based

Reserving Methods to Reinsurance Pricing

Thomas Passante, FCAS, MAAASwiss Re New Markets

CAS Seminar on ReinsuranceJune 16, 2000

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Adv Res.ppt

Regression Model for In-Period Loss Payments

Pij = AYi ( ) e j

ij

Log Transformed:

ln Pij = ln AYi + ( ) + j + ln ij

k

i j

kCY

yr1

k

i j

CY k

yr1

ln

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Adv Res.ppt

Fitted Points 1 2 3 4 5 6 7 8 9

1989 100 86 73 . . .1990 107 92 . . .1991 . . 61 1992 . .1993 .1994199519961997

Assume: CY trend = 7% Then, 100 x 0.84 x 1.076 = 61 DY decay = 0.8All AY on same level

Example of Fitted Incremental Data

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Adv Res.ppt

A Regression Based Model: Why?

Competitive Advantage

• Additional Information

• Better Understanding

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Adv Res.ppt

Some Short Term Limitations:

• Time Investment

– Learning how

– Actually applying the techniques

• Ability to Explain

– Colleagues

– Clients

• Intuition takes time to develop

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Adv Res.ppt

Regression Modeling - Two Main Advantages:

• Obtain a distribution around a point estimate

• Isolate Calendar Year trends

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Adv Res.ppt

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

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Adv Res.ppt

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

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Adv Res.ppt

Calendar Year Trend ExampleOriginal Paid Data (Cumulative)

1 2 3 4 5 6 7 8 9 101989 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 1.0131990 2.859 1.744 1.350 1.207 1.149 1.110 1.049 1.0221991 2.929 1.766 1.377 1.229 1.149 1.108 1.0521992 2.977 1.745 1.372 1.229 1.147 1.1041993 2.787 1.735 1.360 1.210 1.1491994 2.865 1.772 1.347 1.2261995 2.903 1.723 1.3251996 2.746 1.7261997 2.977

TailAll yr wtd 2.882 1.746 1.355 1.222 1.149 1.108 1.051 1.024 1.013 1.013Cum 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 1989Ult Loss 17,267 17,500 15,622 15,505 15,402 14,074 14,588 13,816 12,102 12,223

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Adv Res.ppt

Calendar Year Trend ExampleCY Trend

From To Observed Model Fit Cum1989 1990 6.3% 6.0% 1.060 1990 1991 5.6% 6.0% 1.1241991 1992 6.0% 6.0% 1.1911992 1993 5.7% 6.0% 1.2621993 1994 3.9% 4.0% 1.3131994 1995 4.2% 4.0% 1.3651995 1996 4.0% 4.0% 1.4201996 1997 2.1% 2.0% 1.4491997 1998 1.9% 2.0% 1.4771998 1999 2.0% 1.5071999 2000 2.0% 1.5372000 2001 2.0% 1.5682001 2002 2.0% 1.5992002 2003 2.0% 1.6312003 2004 2.0% 1.6642004 2005 2.0% 1.6972005 2006 2.0% 1.7312006 2007 2.0% 1.7662007 2008 2.0% 1.8012008 2009 2.0% 1.8372009 2010 2.0% 1.8742010 2011 2.0% 1.9112011 2012 2.0% 1.9502012 2013 2.0% 1.9892013 2014 2.0% 2.028

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Adv Res.ppt

Calendar Year Trend ExampleObserved 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

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Adv Res.ppt

Fitted Calendar Year Trends

6%

4%

2%

2%

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Adv Res.ppt

Fitted Decay Parameters

0.8 0.51.11.8

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Adv Res.ppt

Fitted Points1 2 3 4 5 6 7 8 9 10

1989 997.5 1903.3 2219.2 1881.9 1595.8 1327.7 1104.7 574.4 293.0 149.41990 1057.4 2017.4 2352.3 1994.8 1659.7 1380.8 1148.9 585.9 298.8 152.41991 1120.8 2138.5 2493.5 2074.6 1726.1 1436.1 1171.8 597.6 304.8 155.41992 1188.1 2266.8 2593.2 2157.6 1795.1 1464.8 1195.3 609.6 310.9 158.61993 1259.3 2357.5 2697.0 2243.9 1831.0 1494.1 1219.2 621.8 317.1 161.71994 1309.7 2451.8 2804.8 2288.7 1867.6 1524.0 1243.6 634.2 323.5 165.01995 1362.1 2549.8 2860.9 2334.5 1905.0 1554.5 1268.4 646.9 329.9 168.31996 1416.6 2600.8 2918.1 2381.2 1943.1 1585.5 1293.8 659.8 336.5 171.61997 1444.9 2652.9 2976.5 2428.8 1981.9 1617.3 1319.7 673.0 343.2 175.11998 1473.8 2705.9 3036.0 2477.4 2021.6 1649.6 1346.1 686.5 350.1 178.6

Errors (Fitted - Actual)

1 2 3 4 5 6 7 8 9 101989 -2.5 -10.1 -23.6 33.0 -12.6 13.3 22.6 -22.0 -14.6 -4.11990 -5.6 41.8 90.6 140.6 181.7 97.4 55.8 50.2 47.71991 -1.7 -27.1 -24.4 -112.4 -100.9 -31.1 -45.3 -47.31992 -1.8 -85.1 -45.9 -144.7 -145.3 -72.4 -48.81993 1.6 109.9 120.7 51.7 91.7 5.31994 3.0 14.4 -84.9 -12.6 -149.11995 0.5 -41.6 3.4 121.01996 0.5 128.5 96.51997 -0.9 -205.41998 0.5

Avg Error -0.7 -8.3 16.5 10.9 -22.4 2.5 -3.9 -6.4 16.6 -4.1Sum 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

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Adv Res.ppt

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

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Adv Res.ppt

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

Z 17

Adv Res.ppt

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

Z 18

Adv Res.ppt

LPT’s vs. Prospective Contracts• LPT’s:

– 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

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Adv Res.ppt

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

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Adv Res.ppt

Prospective Contracts

P(L)

Conditional Loss distribution given claim count Co

L

P(C)

Claim count as % of payroll

C

Co

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Adv Res.ppt

Prospective ContractsConditional Loss given Co

Conditional Loss given C1

Conditional Loss given C2

P(L)

L|Co L|C1

L|C2

L

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Adv Res.ppt

Prospective ContractsConditional Loss given Co

Conditional Loss given C1

Conditional Loss given C2

L

P(L)

L|Co L|C1

L|C2

P(X)

Unconditionalloss distribution

X

Unconditional Loss distribution

Z 23

Adv Res.ppt

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)

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Adv Res.ppt

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

Z 25

Adv Res.ppt

Regression Modeling - Limitations• Positive incremental data issue:

Two possible solutions?

1 Add a constant value to the entire triangle such that all values are now positive

2 Smoothing the data

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Adv Res.ppt

Regression Modeling - Limitations• Solution #1: Adding A Constant

– NOT RECOMMENDED.

– Log (x2+c) / Log (x1+c) does

not equal Log(x2) / Log (x1)

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Adv Res.ppt

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….

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Adv Res.ppt

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?

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Adv Res.ppt

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)

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Adv Res.ppt

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

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Adv Res.ppt

Regression Modeling - Limitations

In Period Loss Payments

Development Year

Accident Year

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Adv Res.ppt

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

Z 33

Adv Res.ppt

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)

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Adv Res.ppt

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

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Adv Res.ppt

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