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Severity Distributions for GLMs: Gamma or Lognormal?. Presented by Luyang Fu, Grange Mutual Richard Moncher, Bristol West 2004 CAS Spring Meeting Colorado Springs, Colorado May 18, 2004. Session Outline. Introduction Distribution Assumptions Simulation Method Simulation Results

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severity distributions for glms gamma or lognormal

Severity Distributions for GLMs: Gamma or Lognormal?

Presented by

Luyang Fu, Grange Mutual

Richard Moncher, Bristol West

2004 CAS Spring Meeting

Colorado Springs, Colorado

May 18, 2004

session outline
Session Outline
  • Introduction
  • Distribution Assumptions
  • Simulation Method
  • Simulation Results
  • Conclusions
introduction
Introduction
  • Common characteristics of loss distributions
  • Typical GLM forms in actuarial practice
  • Lognormal and Gamma are most widely-used distributions in size of loss (severity) analysis
  • Lognormal or Gamma?
distribution characteristics of insurance losses
Distribution Characteristics of Insurance Losses
  • Non-negative
  • Positively skewed
  • Variance is positively correlated with mean.
  • Normal is not appropriate:

negative,

symmetric,

constant variance

advantages of glms
Advantages of GLMs
  • Exponential Distribution Selections:

Poisson, Gamma, Binomial, Inverse Gaussian, Negative Binomial, etc.

Lognormal is not in exponential family.

  • Link Function Selections:

Identity, Log, Logit, Power, Probit, etc.

typical glm forms in actuarial practice
Typical GLM Forms in Actuarial Practice
  • Severity:

Log link, Gamma Distribution

  • Frequency:

Log link, Poisson Distribution

  • Retention (Renewal):

Logit link, Binomial Distribution

gamma or lognormal
Gamma or Lognormal?
  • Gamma and lognormal are the two most popular selections of loss distributions
  • On CAS website (www.casact.org), we found 31 papers by searching “Lognormal” and 37 papers by searching “Gamma”
lognormal is one of most widely used loss distributions
Lognormal Is One of Most Widely-Used Loss Distributions

Proceedings of the Casualty Actuarial Society

  • Ratemaking and Reinsurance

Wacek, Michael G.(1997)

Bear, Robert A.; Nemlick, Kenneth J. (1990)

Hayne, Roger M. (1985)

Mack, Thomas (1984)

Ter Berg, Peter (1980)

Benckert, Lars-Gunnar (1962)

lognormal is one of most widely used loss distributions9
Lognormal Is One of Most Widely-Used Loss Distributions

Proceedings of the Casualty Actuarial Society

  • Reserving and Reinsurance

Kreps, Rodney E. (1997)

Ramsay, Colin M.; Usabel, Miguel A. (1997)

Doray, Louis G. (1996)

Levi, Charles; Partratm, Christian  (1991)

Hertig, Joakim (1985)

lognormal is one of most widely used loss distributions10
Lognormal Is One of Most Widely-Used Loss Distributions
  • In actuarial practice

Increased Limit Factors

Excess of Loss Calculations

Weather Load Quantile

Loss Reserve Variability

gamma or lognormal11
Gamma or Lognormal?
  • Desirable Features of Gamma and Lognormal Distributions:

1. Non-negative

2. Positively skewed

3. Variance is proportional to the mean-squared (Constant Coefficient of Variation)

gamma or lognormal12
Gammaor Lognormal?

Advantages of Lognormal:

  • Easy to understand (related to normal distribution)
  • Consistent with other actuarial procedures, such as increased limits ratemaking
  • Fits data with large skewness well

Disadvantage of Lognormal:

  • Not in exponential family, and GLM coefficients need volatility adjustment
gamma or lognormal13
Gamma or Lognormal?
  • Under what conditions are the severity distribution assumptions important?
  • If severity distribution is unknown, which distribution yields most accurate and stable results (i.e., minimized estimation bias and standard error)?
classical distribution assumptions
Classical Distribution Assumptions
  • Normal

Constant Variance

  • Gamma

Constant Coefficient of Variation

classical distribution assumptions15
Classical Distribution Assumptions
  • Lognormal

Constant Coefficient of Variation

does normal necessarily imply constant variance
Does Normal Necessarily Imply Constant Variance?
  • Normal

Constant Coefficient of Variation:

Variance function is like Gamma

  • Normal

Variance proportional to mean:

Variance function is like Poisson

does gamma necessarily imply constant coefficient of variation
Does Gamma Necessarily Imply Constant Coefficient of Variation?
  • Gamma

Variance is proportional to mean:

Variance function is like Poisson.

distribution assumptions
Distribution Assumptions
  • One of two parameters is constant
  • Which one is selected as constant should be based on data
  • Classical assumptions are most-widely used distribution forms, and generally fit data better
  • Can we assume none of them are constant?

Yes, but it will increase the number of parameters and reduce the degrees of freedom

why simulation
Why Simulation?
  • The distributions of GLM coefficients and predicted values are unknown in the case of small samples
  • Statistical analysis based on asymptotic distributions is not reliable
  • In an individual regression, we don’t know if the difference between predicted value and observed value is from random variation or systematic bias
simulation assumptions
Simulation Assumptions
  • 32 Severity Observations for Two Class Variables
  • 8 Age Groups
  • 4 Vehicle-Use Groups
  • Data Source: Private Passenger Auto Collision used in Mildenhall (1999) and McCullagh and Nelder (1989)
simulation assumptions21
Simulation Assumptions
  • Individual Losses Have Constant Coefficient of Variation
  • Multiplicative Relationship Between Severities and Rating Variables
  • Known “True” Base Severities & Relativities
  • Known CVs for the Severity Distribution
simulation procedures
Simulation Procedures
  • Generate individual losses based on lognormal and gamma distributions and calculate 32 claim severities
  • Fit three regressions: GLM with Gamma, GLM with Normal, and GLM with log-transformed severity
  • Repeat Steps 1-2 one thousand times, and generate sampling distributions of GLM coefficients and predicted values
performance measurements
Performance Measurements
  • Weighted Absolute Bias, which measures the systematic bias (accuracy):
  • Weighted Standard Error, which measures random variation (stability):
adjustments for log transformed regressions
Adjustments for Log-Transformed Regressions
  • GLMs with Gamma and Normal
  • Log-transformed Regression

is called the “Volatility Adjustment Factor”

simulation results
Simulation Results
  • Data Generated
  • Regression Results
  • Residual Diagnostics
data generated
Data Generated

Reporting on Two Different Classes:

  • Classification I - Age 17-20 and Pleasure Use, with 21 observations.
  • Classification II - Age 40-49 and Short Drive to Work, with 970 observations.
data generated gamma severity for age 17 20 and pleasure use with coefficient of variation 3 0
Data Generated: Gamma Severity for Age 17-20 and Pleasure Use with Coefficient of Variation 3.0
data generated gamma severity for age 40 49 and dtw short use with coefficient of variation 3 0
Data Generated: Gamma Severity for Age 40-49 and DTW Short Use with Coefficient of Variation 3.0
data generated lognormal severity for age 17 20 and pleasure use with coefficient of variation 3 0
Data Generated: Lognormal Severity for Age 17-20 and Pleasure Use with Coefficient of Variation 3.0
data generated lognormal severity for age 40 49 and dtw short use with coefficient of variation 3 0
Data Generated: Lognormal Severity for Age 40-49 and DTW Short Use with Coefficient of Variation 3.0
regression results

CV

wab

wse

G-G

G-L

G-N

G-G

G-L

G-N

1.0

0.180

0.240

0.221

8.170

8.177

8.568

2.0

0.475

0.852

0.509

16.498

16.514

17.239

3.0

0.860

1.808

1.139

25.223

25.097

26.986

Regression Results

Overall Unbiasedness and Stability of Predicted Severities for Gamma Loss

regression results32

CV

wab

wse

L-G

L-L

L-N

L-G

L-L

L-N

1.0

0.151

0.202

0.175

8.309

8.284

8.754

2.0

0.498

0.844

0.604

16.426

16.113

17.721

3.0

0.720

1.589

1.006

24.328

23.214

27.608

Regression Results

Overall Unbiasedness and Stability of Predicted Severities for Lognormal Loss

slide33
Regression Results: Predicted Severities for Gamma Loss with Coefficient of Variation 3.0 for Age 17-20 and Pleasure Use
slide34
Regression Results: Predicted Severities for Gamma Loss with Coefficient of Variation 3.0 for Age 40-49 and DTW Short Use
slide35
Regression Results: Predicted Severities for Lognormal Loss with Coefficient of Variation 3.0 for Age 17-20 and Pleasure Use
slide36
Regression Results: Predicted Severities for Lognormal Loss with Coefficient of Variation 3.0 for Age 40-49 and DTW Short Use
slide38
Residual Diagnostics: Predicted Severities vs Standardized Residuals for Gamma Loss with Coefficient of Variation 3.0
residual diagnostics standardized residuals for lognormal loss with coefficient of variation 3 0
Residual Diagnostics: Standardized Residuals for Lognormal Loss with Coefficient of Variation 3.0
slide40
Residual Diagnostics: Predicted Severities vs Standardized Residuals for Lognormal Loss with Coefficient of Variation 3.0
slide41
Residual Diagnostics: Standardized Residuals for Gamma Loss with Coefficient of Variation 1.0 Based on Individual Data
slide42

Residual Diagnostics: Predicted Severities vs Standardized Residuals for Gamma Loss with Coefficient of Variation 1.0 Based on Individual Data

slide43
Residual Diagnostics: Standardized Residuals for Lognormal Loss with Coefficient of Variation 1.0 Based on Individual Data
slide44

Residual Diagnostics: Predicted Severities vs Standardized Residuals for Lognormal Loss with Coefficient of Variation 1.0 Based on Individual Data

conclusions
Conclusions
  • When the gamma distribution is “true”, the G-G model is dominant in both unbiasedness and stability (except the G-L model is slightly more stable in the case of large volatility).
conclusions46
Conclusions
  • When the lognormal distribution is “true”, the L-L model is dominant in terms of stability.
conclusions47
Conclusions
  • GLMs with a normal distribution never dominate based on any criteria, and they have the worst weighted standard error.
conclusions48
Conclusions
  • GLMs with a gamma distribution are dominant in terms of unbiasedness, no matter whether the “true” distribution is gamma or lognormal.
conclusions49
Conclusions
  • In general, GLMs with a gamma distribution are recommended because they perform slightly better than the log-transformed model.
conclusions50
Conclusions
  • When the data is not volatile, the distribution selection for GLMs may not be as important because all distribution assumptions yield small biases and standard errors.
conclusions51
Conclusions
  • When the data is very volatile, the log-transformed regression is recommended because it provides the most stable estimation.
conclusions52
Conclusions
  • When the log-transformed model is used, the classification relativities should be adjusted by a volatility-adjustment factor. Without the adjustment, the relativities could be undervalued.
conclusions53
Conclusions
  • Residual plots may work well to examine the distribution assumptions on individual data, but not necessarily on summarized/average data.
questions answers
Questions & Answers
  • Questions?
  • Thank You!