Verrall Flashcards
State 2 examples of situations where results may need to be adjusted
- A change in payment pattern due to a change in company policy
- Legislature has enacted benefit limitations that restrict potential for loss development
State 2 important properties of Bayesian models
- Can incorporate expert knowledge
- Can be easily implemented
Nem the 2 areas where expert knowledge is applied in Verrall
- Expected losses in BF method
- Insertion of prior knowledge about individual development factors in the C-L method.
Describe Mack stochastic model for the Chain Ladder method
E(Loss_AY,k) = LDF_k * Loss_AY,k-1
LDF = lambda
Loss = D
V(Loss_AY, k) = sj^2 * Loss_AY,k-1
Name 2 advantages of the mack stochastic model for CL
- Easy to implement
- Parameter estimates and prediction errors (reserve ranges) can be calculated in a spreadsheet
Name 2 disadvantages of the mack stochastic model for CL
- Since a distribution is not specified, there is no predictive distribution.
- Separate parameters for the variance must also be estimated apart from LDFs.
Describe the ODP model for incremental losses (GLM approach)
E(Loss_AY,k) = exp(c+ai+bj) = mij
Loss_AY,k = Cij
Describe the ODP model for the CL method (Row-Column approach)
E(Cij) = xi * yj
E(Loss_AY,k) = RowFactor_AY * ColFactor_k
xi = expected ultimate loss for accident year I up to last development period of the triangle
yj = % of ultimate loss emerging in development period j
Name 3 advantages of ODP model
- Does not necessarily break down if there are some negatives incremental values
- Gives the same reserve estimate as CL method
- More stable than log-normal model of Kremer
Name 1 disadvantage of ODP model
Connection to the chain ladder is not immediately apparent
Describe the ODNB distribution model for
a) INCREMENTAL losses
b) CUMULATIVE losses
a) E(Loss_AY,k) = (LDF_k - 1)*Loss_AY,k-1
LDF_k = lambda_j
Loss_AY,k-1 = Di,j-1
Loss_AY,k = Cij
b) E(Loss_AY,k) = LDF_k * Loss_AY,k-1
V(Cij) = dispersionlambda_j(lambda_j - 1)*Dij-1
dispersion = psi
Note: the reserve estimates are the same as the CL method (All LDFs must be greater than 1, no overall negative development, or variance would be negative)
Name 2 advantages of ODNB distribution model
- Does not necessarily break down if there are some negative incremental losses
- Gives the same reserve estimate and has the same form as the CL method
Name 1 disadvantage of ODNB distribution model
Cannot handle negative development (column sums of incremental losses must be positive). Otherwise, would produce negative variance.
Describe the Normal Approximation to the NB Model for
a) incremental losses
b) cumulative losses
a) E(Loss_AY,k) = (LDF_k - 1)*Loss_AY,k-1
b) E(Loss_AY,k) = LDF_k * Loss_AY,k-1
V(Loss_AY,k) = dispersion*Loss_AY,k-1
Name 1 advantage of the Normal Approx Model
Allows the possibility of negative incremental losses
Name 1 disadvantage of the Normal Approx Model
Constant dispersion parameter psi is replaced by column-specific parameter psi_j. This is disadvantageous since additional parameters must be estimated in order to calculate the variance.
Calculate the Prediction Error of a Reserve
Prediction Error = Root men square error of prediction (RMSEP)
Prediction variance = process variance + estimation variance = MSEP
Prediction Error = Prediction variance^0.5 = RMSEP
Briefly explain the difference between prediction error and standard error
standard error = estimation variance^0.5
Standard error only accounts for parameter estimation error
Prediction error is concerned with the variability of the forecast and accounts for both:
1. Uncertainty in parameter estimation (estimation variance)
2. Variability in data being forecast (process variance)
2 advantages of Bayesian methods for calculating prediction error
- Full predictive distribution can be found using simulation methods
- RMSEP can be obtained directly by calculating the standard deviation of the distribution
Describe 2 common ways to incorporate expert opinion about LDFs
- A development factor is override in some rows due to external information
- Development factors are based on a 5y volume weighted average rather than all of the available data in the triangle
Describe the Bayesian Model for the BF method formulas
xi follows Gamma(ai, bi)
E(xi) = Mi = ai/bi
V(xi) = Mi/bi = ai/bi^2
Briefly explain what it means in Bayesian Model for BF method to select larger value of bi
Choosing a larger value of bi implies we are more certain about the value of Mi.
Describe the Credibility-Weighted Bayesian Model for the BF Method formulas
See image
Explain the impact of beta on the variance of the prior distribution
- Large variances (small betas) for prior distribution mean parameter estimates are not significantly influenced.
Thus, results will be close to CL method. - Small variances (large betas) for prior distribution mean we are confident in the parameters.
Thus, results will be close to BF method.
Describe the Fully Stochastic BF model
See image
vi = 1 + (BF Reserves for AY i / IncLosses for Prior Ays in Future Dev Periods for AY i)
v1 = 1.00
Describe 2 options to estimate the column parameters (yj) in the BF Model.
Which one is preferred?
- Use plug-in estimates from traditional CL method its no variability
- Define prior distributions for the column parameters, and estimate the column parameters first, before applying prior distributions for the row parameters and estimating these.
Second option is preferred since it allows us to consider the fact that column parameters have been estimated when calculating the prediction errors, predictive distribution, etc. In other words, the second option provides a fully stochastic version of the BF method.
List the steps to define a fully stochastic version of the BF method
- Estimate column parameters
- Incorporate prior information into the distributions for the parameters xi
- Use xi to determine vi
- Calculate expected future incremental losses using vi
- Sum of expected future incremental losses will be the BF reserve
Explain how the bayesian model for the BF method can be interpreted as a trade-off between standard chain ladder method and BF method.
The model is credibility-weighting between the incremental unpaid loss estimates from the data (CL) and the prior mean (BF) methods.
The level of credibility is set by adjusting the parameter beta(AY).
Explain whether the actuary should select high or low variance for the prior distributions of the link ratios to produce results that closely resemble deterministic chain-ladder.
Use a high variance for the prior distribution (weak priors).
If we are not confident in our prior estimates, a large variance will place more weight on the chain-ladder estimate.
Discuss how you can adjust for judgmental selection in link ratios when not confident.
Describe the effect of this change on simulated results.
Set the prior distribution of LDF to have a mean of the actuary’s selection, but use a large variance to reflect the lower confidence in the selection.
This means the model will still place weight on the historical data for the period development.
It will pull the model’s LDF parameter closer to the actuary’s selection. Because a larger variance is used for the prior distribution, the prediction error will still be high.
Describe an advantage of Bayesian approach over bootstrapping.
Bayesian approach can incorporate expert opinion in parameter selection.
Describe an advantage of Bayesian approach over Mack method.
Bayesian approach gives full predictive distribution of loss reserves instead of just the first 2 moments.
Discuss a modification to the Bayesian framework for the chain-ladder method so that it applies to the BF method.
We can use row parameters of expected losses for each accident year with a prior distribution for each row parameter. A smaller variance for the row parameters draws the model results closer to the BF method result.
