Meyers Flashcards
Diagram of Models Tested
Kolmogorov-Smirnov (KS) Test
D=max |pi-ei|
Reject the null hypothesis (predicted percentiles are normal) if D>136/n.5
LIght Tailed Distributions
Heavy Tailed Distributions
Biased High/Upward
Diagnostics
What to look for in the p-p plots
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
Two metrics to test stability in the book of business
- Consistency in net earned premium
- Consistency in the ratio (Net Premium/Direct Premium)
Mack (Incurred)
- 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
Leveled Chain Ladder (Incurred) - LCL
- 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
Correlated Chain Ladder (Incurred) - CCL
- 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
ODP Bootstrap (Paid)
- Overall:
- Very shallow in the left tail
- Looks biased high
- Fails KS test
Mack (Paid)
- 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
Correlated Chain Ladder (Paid) - CCL
- Continue to use “column” correlation
- Add a parameter p to create correlation of losses by AY
- Overall:
- Biased High
- Fails KS test
1Correlated Incremental Trend (Paid) - CIT
- 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
2 properties of incremental losses
- Skewed right
- Occasionally has negative values
Tighter variance parameters on which 2 variables for the CIT
- t - CY trend
- sigma (d)
Leveled Incremental Trend (Paid) - LIT
- 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
Changing Settlement Rate (Paid) - CSR
- 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
Variance (Meyers)
=E(Process Variance) + Var (Hypothetical Means)
=EVPV + VHM
= Process Risk + Paramter Risk (CF)
Larger component of total risk - Parameter or Process Risk
Parameter Risk
Describe three tests for uniformity for n predicted percentiles
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.
Briefly describe two formulations for the skew normal distribution.
- 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
Briefly describe why model risk can be thought of as a special type of parameter risk.
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
Briefly describe how to test for the existence of model risk.
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
Standard Deviation over time (Meyers)
- Cumulative - As time increases your standard deviation should decrease
- Highest variability at early ages
- Incremental - As time increases your standard deviation should increase
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.
⇧ 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
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.
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
