Verrall

Question 1

State 2 examples of situations where results may need to be adjusted

Answer 1

1. A change in payment pattern due to a change in company policy
2. Legislature has enacted benefit limitations that restrict potential for loss development

Question 2

State 2 important properties of Bayesian models

Answer 2

1. Can incorporate expert knowledge
2. Can be easily implemented

Question 3

Name the 2 areas where expert knowledge is applied in Verrall

Answer 3

1. Expected losses in BF method
2. Insertion of prior knowledge about individual development factors in the C-L method.

Question 4

Describe Mack stochastic model for the Chain Ladder method

Answer 4

E(Loss_AY,k) = LDF_k * Loss_AY,k-1
LDF = lambda
Loss = D

V(Loss_AY, k) = sj^2 * Loss_AY,k-1

Question 5

Name 2 advantages of the mack stochastic model for CL

Answer 5

1. Easy to implement
2. Parameter estimates and prediction errors (reserve ranges) can be calculated in a spreadsheet

Question 6

Name 2 disadvantages of the mack stochastic model for CL

Answer 6

1. Since a distribution is not specified, there is no predictive distribution.
2. Separate parameters for the variance must also be estimated apart from LDFs.

Question 7

Describe the ODP model for incremental losses (GLM approach)

Answer 7

E(Loss_AY,k) = exp(c+ai+bj) = mij
Loss_AY,k = Cij

Question 8

Describe the ODP model for the CL method (Row-Column approach)

Answer 8

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

Question 9

Name 3 advantages of ODP model

Answer 9

1. Does not necessarily break down if there are some negatives incremental values
2. Gives the same reserve estimate as CL method
3. More stable than log-normal model of Kremer

Question 10

Name 1 disadvantage of ODP model

Answer 10

Connection to the chain ladder is not immediately apparent

Question 11

Describe the ODNB distribution model for
a) INCREMENTAL losses
b) CUMULATIVE losses

Answer 11

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)

Question 12

Name 2 advantages of ODNB distribution model

Answer 12

1. Does not necessarily break down if there are some negative incremental losses
2. Gives the same reserve estimate and has the same form as the CL method

Question 13

Name 1 disadvantage of ODNB distribution model

Answer 13

Cannot handle negative development (column sums of incremental losses must be positive). Otherwise, would produce negative variance.

Question 14

Describe the Normal Approximation to the NB Model for
a) incremental losses
b) cumulative losses

Answer 14

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

Question 15

Name 1 advantage of the Normal Approx Model

Answer 15

Allows the possibility of negative incremental losses

Question 16

Name 1 disadvantage of the Normal Approx Model

Answer 16

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.

Question 17

Calculate the Prediction Error of a Reserve

Answer 17

Prediction Error = Root men square error of prediction (RMSEP)

Prediction variance = process variance + estimation variance = MSEP

Prediction Error = Prediction variance^0.5 = RMSEP

Question 18

Briefly explain the difference between prediction error and standard error

Answer 18

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)

Question 19

2 advantages of Bayesian methods for calculating prediction error

Answer 19

1. Full predictive distribution can be found using simulation methods
2. RMSEP can be obtained directly by calculating the standard deviation of the distribution

Question 20

Describe 2 common ways to incorporate expert opinion about LDFs

Answer 20

1. A development factor is override in some rows due to external information
2. Development factors are based on a 5y volume weighted average rather than all of the available data in the triangle

Question 21

Describe the Bayesian Model for the BF method formulas

Answer 21

xi follows Gamma(ai, bi)

E(xi) = Mi = ai/bi
V(xi) = Mi/bi = ai/bi^2

Question 22

Briefly explain what it means in Bayesian Model for BF method to select larger value of bi

Answer 22

Choosing a larger value of bi implies we are more certain about the value of Mi.

Question 23

Describe the Credibility-Weighted Bayesian Model for the BF Method formulas

Answer 23

See image

Question 24

Explain the impact of beta on the variance of the prior distribution

Answer 24

1. Large variances (small betas) for prior distribution mean parameter estimates are not significantly influenced.
   Thus, results will be close to CL method.
2. Small variances (large betas) for prior distribution mean we are confident in the parameters.
   Thus, results will be close to BF method.

Question 25

Describe the Fully Stochastic BF model

Answer 25

See image vi = 1 + (BF Reserves for AY i / IncLosses for Prior Ays in Future Dev Periods for AY i) v1 = 1.00

Question 26

Describe 2 options to estimate the column parameters (yj) in the BF Model. Which one is preferred?

Answer 26

1. Use plug-in estimates from traditional CL method its no variability 2. 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.

Question 27

List the steps to define a fully stochastic version of the BF method

Answer 27

1. Estimate column parameters 2. Incorporate prior information into the distributions for the parameters xi 3. Use xi to determine vi 4. Calculate expected future incremental losses using vi 5. Sum of expected future incremental losses will be the BF reserve

Question 28

Explain how the bayesian model for the BF method can be interpreted as a trade-off between standard chain ladder method and BF method.

Answer 28

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).

Question 29

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.

Answer 29

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.

Question 30

Discuss how you can adjust for judgmental selection in link ratios when not confident. Describe the effect of this change on simulated results.

Answer 30

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.

Question 31

Describe an advantage of Bayesian approach over bootstrapping.

Answer 31

Bayesian approach can incorporate expert opinion in parameter selection.

Question 32

Describe an advantage of Bayesian approach over Mack method.

Answer 32

Bayesian approach gives full predictive distribution of loss reserves instead of just the first 2 moments.

Question 33

Discuss a modification to the Bayesian framework for the chain-ladder method so that it applies to the BF method.

Answer 33

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.