Meyers

Question 1

Diagram of Models Tested

Answer 1

Question 2

Kolmogorov-Smirnov (KS) Test

Answer 2

D=max |pi-ei|

Reject the null hypothesis (predicted percentiles are normal) if D>136/n^.5

Question 3

LIght Tailed Distributions

Answer 3

Question 4

Heavy Tailed Distributions

Answer 4

Question 5

Biased High/Upward

Answer 5

Question 6

Diagnostics

What to look for in the p-p plots

Answer 6

 Slope at the right tail & left tail

 Shallow Slope indicates the model has a light tail

 Steep Slope indicates that model has a heavy tail

 Predicted Percentiles are accepted as uniform unless they fail the KS Test at the 5th percentile

Question 7

Two metrics to test stability in the book of business

Answer 7

• Consistency in net earned premium
• Consistency in the ratio (Net Premium/Direct Premium)

Question 8

Mack (Incurred)

Answer 8

• Fits a mean & standard deviation to the unpaid losses for each AY
• Overall:
  
  • Light left tail
  • Very light right tail
  • Fails KS test
• Need to look for a predictive distribution that has fatter tails

Question 9

Leveled Chain Ladder (Incurred) - LCL

Answer 9

• Add “column” correlation - Treat the value at the last diagonal for the row as a random variable
• u(w,d) = alpha(w) + b(d)
• Has more variance than Mack
• Overall:
  
  • Definite light tails
  • Fails KS test
• Improvement over Mack but still forecasts tails that are too light

Question 10

Correlated Chain Ladder (Incurred) - CCL

Answer 10

• Continue to use “column” correlation
• Add a parameter p to create correlation of losses by AY
• u(w,d) = alpha(w) + b(d) +p(ln(C(w-1,d) - u(w-1,d))
• More variance than Mack
• Overall:
  
  • Very slight light tails on both sides
  • Passes KS test
• Meyers concludes this model is good enough

Question 11

ODP Bootstrap (Paid)

Answer 11

• Overall:
  
  • Very shallow in the left tail
  • Looks biased high
  • Fails KS test

Question 12

Mack (Paid)

Answer 12

• Overall:
  
  • Slight shallow right tail
  • Very shallow left
  • Biased high
  • Fails KS test
• Two plausible explainations:
  
  • Insurance loss environment has experienced changes
  • Other models can validate

Question 13

Correlated Chain Ladder (Paid) - CCL

Answer 13

• Continue to use “column” correlation
• Add a parameter p to create correlation of losses by AY
• Overall:
  
  • Biased High
  • Fails KS test

Question 14

1Correlated Incremental Trend (Paid) - CIT

Answer 14

• Add “column” correlation - Treat the value at the last diagonal for the row as a random variable
• Add a parameter p to create correlation of losses by AY
• Add CY trend parameter - t
  
  • Applies to incremental losses
• Overall:
  
  • Biased high
  • Fails KS test

Question 15

2 properties of incremental losses

Answer 15

• Skewed right
• Occasionally has negative values

Question 16

Tighter variance parameters on which 2 variables for the CIT

Answer 16

• t - CY trend
• sigma (d)

Question 17

Leveled Incremental Trend (Paid) - LIT

Answer 17

• Add “column” correlation - Treat the value at the last diagonal for the row as a random variable
• Add CY trend parameter - t
• Overall:
  
  • Very light left tail
  • Biased high
  • Fails KS test

Question 18

Changing Settlement Rate (Paid) - CSR

Answer 18

• Add “column” correlation - Treat the value at the last diagonal for the row as a random variable
• Add parameter “y” that allows for recognition of a speed up in claim payments
• Overall:
  
  • Fits well
  • Removed high bias
  • No indication of light tails
• Meyers says this is the best model for paid data

Question 19

Variance (Meyers)

Answer 19

=E(Process Variance) + Var (Hypothetical Means)

=EVPV + VHM

= Process Risk + Paramter Risk (CF)

Question 20

Larger component of total risk - Parameter or Process Risk

Answer 20

Parameter Risk

Question 21

Describe three tests for uniformity for n predicted percentiles

Answer 21

Question 22

Describe two ways to increase the variability of the predictive distribution produced by the Mack model on incurred losses. For each one, identify a model that accomplishes this goal.

Answer 22

Question 23

Briefly describe two formulations for the skew normal distribution.

Answer 23

• One formulation produces the skew normal distribution by expressing it as a mixed truncated normal-normal distribution
• Another formulation produces the skew normal distribution by expressing it as a mixed lognormal-normal distribution

Question 24

Briefly describe why model risk can be thought of as a special type of parameter risk.

Answer 24

Model risk is the risk that we did not select the right model. In a sense, we can think of model risk as a special case of parameter risk because the possible models can be thought of as “known unknowns” similar to the rest of the parameters in the model

Question 25

Briefly describe how to test for the existence of model risk.

Answer 25

To test for model risk, we formulate a model that is a weighted average of the various candidate models, where the weights are parameters. If the posterior distribution of the weights assigned to each model has significant variability, then model risk exists

Question 26

Standard Deviation over time (Meyers)

Answer 26

* Cumulative - As time increases your standard deviation should decrease * Highest variability at early ages * Incremental - As time increases your standard deviation should increase

Question 27

An actuary used Bayesian MCMC processes to simulate losses from a lognormal distribution with unknown parameters μ and sigma a) Briefly describe how the posterior distributions of the parameters can be used to determine the total loss volatility.

Answer 27

⇧ Sample from the posterior distributions for each parameter to create parameter sets. Simulate a random loss from a lognormal distribution for each parameter set. The variability in the simulated losses represents the total volatility of the losses

Question 28

An actuary used Bayesian MCMC processes to simulate losses from a lognormal distribution with unknown parameters μ and sigma. b) Briefly describe how the posterior distributions of the parameters can be used to determine the parameter risk portion of the total loss volatility.

Answer 28

Sample from the posterior distributions for each parameter to create parameter sets. Calculate the expected value of losses for each parameter set using the lognormal distribution. The variability in the expected values represents the parameter risk