Shapland

Question 1

Property of the over-dispersed Poisson model

Answer 1

the fitted incremental claims will exactly equal the fitted incremental claims derived using the standard chain-ladder factors

Question 2

Advantage of the ODP bootstrap model

Answer 2

Although sampling with replacement assumes the residuals are independent and identically distributed, it does not require the residuals to be normally distributed.
This allows the distributional form of the residuals flow through the simulation process. (this is sometimes referred to as a ‘semi-parametric’ bootstrap model since we are not parameterizing the residuals

Question 3

How to include process variance in the future incremental claims

Answer 3

We assume that each future incremental claims follows gamma distribution.
This revised model incorporates process variance and parameter variance in the simulation of the historical and future data

Question 4

Approach 1 for modeling an unpaid loss distribution using incurred data

Answer 4

We run a paid data model in conjunction with the incurred data model.
Then we use the random payment pattern from each iteration of the paid data model to convert the ultimate values from each corresponding incurred model iteration to develop paid losses by AY.
Advantage: it allows us to use the case reserves to help predict the ultimate losses, while still focusing on the payment stream for measuring risk
An improvement to this approach would be the inclusion of correlation between the paid and incurred models

Question 5

Approach 2 for modeling an unpaid loss distribution using incurred data

Answer 5

Apply the ODP bootstrap to the Munich chain-ladder (MCL) model. The MCl uses the inherent relationship/correlation between the paid and incurred losses to predict ultimate losses.
When paid losses are low relative to incurred losses, then future paid loss development tends to be higher than average. When paid losses are high relative to incurred losses, then future paid loss development tends to be lower than average.
2 advantages:
1. it does not require us to model paid losses twice.
2. it explicitly measures the correlation between paid and incurred losses

Question 6

Issue with using the ODP bootstrap

Answer 6

Iterations for the latest few accident years tend to be more variable than what we would expect given the simulations for earlier accident years.
This is due to the fact that MORE age-to-age factors are used to extrapolate the sampled values to develop point estimates for each iteration.

Question 7

How to fix the issue with the ODP bootstrap

Answer 7

Future incremental values can be extrapolated using the BF or Cape Cod method.

Question 8

Two drawbacks of GLM bootstrap

Answer 8

1. The GLM must be solved for each iteration of the bootstrap model, which may slow down the simulation
2. The model is no longer directly explainable to others using age-to-age factors

Question 9

4 benefits of GLM bootstrap

Answer 9

1. Fewer parameters helps avoid over-parametrizing the model
2. Gives us the ability to add parameters for calendar year trends.
3. Gives us the ability to model data shapes other than triangles
4. Allows us to match the model parameters to the statistical features found in the data, and to extrapolate those features

Question 10

How do we produce point estimates using the GLM bootstrap model

Answer 10

Unlike the ODP bootstrap that replicates the chain-ladder model we do not apply age-to-age factors to each sample triangle to produce point estimates.
Instead, we fit the same GLM model underlying the residuals to each sample triangle. Then we use the resulting parameters to produce ultimates and reserve point estimates.
Drawback: the additional time required to fit a GLM to each sample triangle

Question 11

3 options to deal with extreme outcomes

Answer 11

1. identify the extreme iterations and remove them.
2. Recalibrate the model (identify the source of the negative incremental losses and remove it if necessary)
3. Limit incremental losses to zero

Question 12

Should the residuals be adjusted so that their average is zero?

Answer 12

If the average of the residuals is positive, then re-sampling from the residuals will add variability to the resampled incremental losses. I may also cause the resampled incremental losses to have an average greater than the fitted losses. In this case, the residuals should be adjusted.

Question 13

Using an L-year weighted average for the GLM bootstrap

Answer 13

• We use L years of data by excluding the first few diagonals in the triangle (which leaves us with L+1 included diagonals)
• This changes the shape of the triangle to a trapezoid
• The excluded diagonals are given zero weight in the model and fewer calendar year parameters are required.
• When running the bootstrap simulations, we only need to sample residuals for the trapezoid that was used to parametrize the original model. Because the GLM models incremental claims directly and can be parameterized using a trapezoid. Each parameter set is then used to project the sampled triangles to ultimate.

Question 14

Using an L-year weighted average for the ODP bootstrap

Answer 14

• We calculate L year average factors instead of all year factors
• We exclude the first few diagonals when calculating residuals
• We still sample residuals for the entire triangle when running bootstrap. Because the ODP bootstrap requires cumulative values in order to calculate link ratios. Once we have cumulative values for each sample triangle, we use L-year average factors to project the sample triangles to ultimate

Question 15

What does missing values affect

Answer 15

1. Loss development factors
2. fitted triangle (if the missing value lies on the last diagonal)
3. Residuals
4. Degree of freedom

Question 16

Dealing with missing values for ODP bootstrap

Answer 16

1. Estimate the missing value using surrounding values
2. Exclude the missing value when calculating the loss development factors. No corresponding residual will be calculated for the missing value. Similar to the L-year weighted average, sample for the entire triangle. Once the sample triangles are calculated, we should exclude the cells corresponding to the missing values from the projection process
3. If the missing value lies on the last diagonal, we can either estimate the value OR we can use the value in the second to the last diagonal to contract the fitted triangle

Question 17

Dealing with missing values for GLM bootstrap

Answer 17

The missing data simply reduced the number of observation used in the model.
Similar to ODP, we could use any one of the 3 method to estimate the missing data

Question 18

Managing outlies for ODP bootstrap

Answer 18

1. Exclude the outliers completely (proceed in the same manner as a missing value)
2. Exclude the outliers when calculating the age-to-age factors and the residuals (similar to missing values), BUT include the outlier cells during the sample triangle projection process. (remove the extreme impact of the incremental cell by excluding the outlier during the fitting process while still including some non-extreme variability by including the cell in the sample triangle projections)

Question 19

3 options when excluding outliers to calculate age-to-age factors

Answer 19

1. Exclude in the numerator
2. Exclude in the denominator
3. Exclude in the numerator and denominator

Question 20

Managing outliers for GLM bootstrap

Answer 20

Outliers are treated similarly to missing data.
If the data is not considered representative of real variability, the outliers should be excluded and the model should be parameterized without it

Question 21

What do we do if there are a significant number of outliers

Answer 21

1. Might indicate that the model is a poor fit to the data
2. For GLM, new parameters could be chosen OR the distribution of the error could be changed.
3. For ODP, an L-year weighted average could be used to provide a better model fit.

Question 22

3 options to adjust for heteroscedasticity

Answer 22

1. Stratified sampling
2. Calculating variance parameters
3. Calculating scale parameters

Question 23

Describe stratified sampling

Answer 23

1. Group development periods with homogeneous variances
2. Sample with replacement from the residuals in each group separately

Question 24

Advantage of stratified sampling

Answer 24

It’s straightforward and easy to implement

Question 25

Disadvantage of stratified sampling

Answer 25

Some groups may only have a few residuals in them, which limits the amount of variability in the possible outcomes

Question 26

What is heteroecthesious data

Answer 26

incomplete or uneven exposures at interim evaluation dates

Question 27

Describe partial first development period data

Answer 27

Occurs when the first development column has a different exposure period than the rest of the columns.
This is NOT a problem for parameterizing the ODP bootstrap model since the Pearson residuals use the square root of the fitted value to make them all exposure independent

Question 28

How to adjust for the partial first development period data

Answer 28

In a deterministic analysis (not bootstrapping), the most recent accident needs to be adjusted to remove exposures beyond the evaluation date. We can reduce the projected future payments by half to remove the exposures from 6/30 to 12/31.
During ODP bootstrap simulation process, we do the same thing. Once the projected future values have been reduced by half, we simulate the process variance as usual.
Alternatively, we can reduce the future values by half AFTER simulating the process variance

Question 29

Describe the partial last calendar period data

Answer 29

occurs when the latest diagonal only has a 6 months development period

Question 30

How to adjust for the partial last calendar period data

Answer 30

In a deterministic analysis, we can exclude the latest diagonal when calculating age-to-age factors, interpolate those factors for the exposures in the latest diagonal, and use the interpolated factors to project the future values.
When parameterizing the ODP bootstrap model, we annualize the exposures in the last diagonal to make them consistent with the rest of the triangle. The fitted triangles is calculated based on this annualized triangle to obtain residuals
During the ODP bootstrap simulation process, age-to-age factors are calculated from the annualized sample triangles and interpolated. Then, the latest diagonal in the sample triangle is adjusted back to a six month period. The cumulative values are then multiplied by the interpolated age-to-age factors to project future values. We must reduce the future values for the latest accident year by half

Question 31

Exposure adjustments under the ODP bootstrap model

Answer 31

We divide the claim data by earned exposure for each AY. this normally improved the fit of the model
The simulation process is then run on the adjusted data.
After the process variance step is completed, we multiply the results by the earned exposures to restate them in terms of total values

Question 32

Exposure adjustments under the GLM bootstrap model

Answer 32

Similar to the ODP, the GLM model is fit to the exposure adjusted losses.
Main difference: exposure adjusted losses with HIGHER exposures are assumed to have LOWER variance when fitting the GLM.
Exposure adjustments could allow fewer AY parameters for the GLM bootstrap model

Question 33

Selecting tail factors for ODP bootstrap model

Answer 33

Tail factor can be extrapolated
The tail factor standard deviation is 50% or less of the tail factor -1

Question 34

Selecting tail factors for GLM bootstrap model

Answer 34

Assume that the final development period will continue to apply incrementally until its effect on the future incremental claims is negligible

Question 35

Diagnostic tool 1 - Residual graphs

Answer 35

Testing the assumption that residuals are independent and identically distributed
We can graph the residuals by development period, accident period or calendar period or against the fitted incremental losses

Question 36

Trends in residual graphs

Answer 36

We should be able to draw a relatively flat line through the residuals.
Residuals should appear random

Question 37

Adjusting Heteroscedasticity in residual graphs

Answer 37

We should group residuals into hetero groups and adjust them to a common standard deviation.
TO help visualize HOW the residuals should be grouped, we can graph relative standard deviations and look for natural groupings

Question 38

Diagnostic tool 2 - Normality test

Answer 38

Although the ODP model does not require residuals to be normally distributed, it’s still helpful to compare residuals against a normal distribution
This allows us to compare parameter sets and assess the skewness of the residuals.
This test uses both graphs AND calculated test values

Question 39

Describe normality plots

Answer 39

If the data points tightly distributed around the diagonal line, then the residuals are assumed to be normally distributed

Question 40

Describe calculated test values for testing normality

Answer 40

1. P-value: P-value should be large (greater than 5%). typically based on the Shapiro test for normality
2. R^2 - R^2 should be close to 1
3. AIC & BIC, these adjust for the number of parameters used in the model. They should be small

Question 41

How to identify outliers

Answer 41

Use a box-whisker plot.
The values beyond the whiskers (the largest values within 3 times the inter-quartile range) are considered outliers

Question 42

Describe the principle of parsimony

Answer 42

A model with fewer parameters is preferred as long as the goodness of fit is not markedly different

Question 43

How to find the optimal mix of parameters in the GLM bootstrap model

Answer 43

1. Start with a basic GLM model which includes one parameter for accident, development, and calendar period
2. Check the residual plots. If it doesn’t look right, we add more parameters.
   The implied development patterns for the GLM should look like a smoothed version of the ODP bootstrap chain-ladder development pattern

Question 44

When reviewing the estimated unpaid model results

Answer 44

The standard error should increase when moving from the oldest years to the most recent years (because the standard errors follows the magnitude of the results)
The total standard error should be larger than any individual error
The coefficient of variance should generally decrease when moving from the oldest years to the most recent years. (because the older AYs have fewer payments remaining, which causes all of the variability to be reflected in the coefficient)
The total coefficient of variation should be smaller than any individual year’s coefficient of variation
The standard error or coefficient of variance for all years combined will be LESS than the sum of standard error or coefficient of variation for individual years. Because accident years are assumed to be independent

Question 45

Why the coefficient of variation may rise in the most recent years

Answer 45

1. With an increasing number of parameters in the model, parameter uncertainty increases when moving from the oldest years to the most recent years. This parameter uncertainty may overpower the process uncertainty, causing an increase in variability
2. The model may simply be overestimating the variability in the most recent years. In this case, the BF or Cape Cod models may need to be used in place of the CL method.

Question 46

Two methods for combining the results from multiple models

Answer 46

1. Run models with the same random variables. Once all the models have been run, the incremental values for each model are weighted together (for each iteration by AY)
2. Run models with independent random variables. once all the models have been run, the weights are used to select a model (for each iteration by AY) by randomly sampling the specified percentage of iterations from each model. The result is a weighted mixture of models

Question 47

How can we use a smoothed results after fitting the distribution

Answer 47

1. Access the quality of the fit
2. Parameterize a DFA (dynamic financial analysis) model
3. Estimate extreme values
4. Estimate TVaR

Question 48

Benefit of using smoothed results

Answer 48

Some of the random noise is prevented from distorting the calculations of specific metrics

Question 49

Reviewing estimated cash flow results

Answer 49

For AY, standard errors increase and CoV decrease as we move from older to more recent years.
For CY, standard errors decrease and CoV increase as we move from older the more recent years.

Question 50

How to simulate correlated variables

Answer 50

Using a multivariate distribution whose parameters and correlations have been specified.
However, we don’t know the distribution of each BU

Question 51

2 correlation process for the Bootstrap model

Answer 51

1. Location mapping
2. Re-sorting

Question 52

Describe location mapping

Answer 52

• Pick a BU
• For each iteration, sample a residual and then note where it belonged in the original residual triangle
• each of the segments is then sampled using the residuals at the same locations for their respective residuals triangles.
  This preserves the correlation of the original residuals in the sampling process

Question 53

Pros and Cons of location mapping

Answer 53

Benefit: it can be easily implemented in a spreadsheet and it does not require us to estimate a correlation matrix.
Cons: it requires all of the business segments to come with residual triangles that are the same size and have no missing values for tress testing purposes

Question 54

Describe re-sorting

Answer 54

To cause correlation among BU in a bootstrap model, the residuals are re-sorted until the rank correlation between each business matches the desired correlation.
P-values can be calculated for each correlation coefficient to test its significances

Question 55

Benefits of re-sorting

Answer 55

Residual triangles may have different shapes/sizes, different correlation assumptions may be employed AND different correlation algorithms may have beneficial impacts on the aggregate distribution

Question 56

Cons of re-sorting

Answer 56

need to specify a desired correlation matrix

Question 57

Advantages of the GLM framework

Answer 57

1. Can tailor the model to the statistical features of the data
2. Can use fewer parameters to avoid over-parameterization
3. Can model data that’s not in a loss triangle

Question 58

Disadvantages of the GLM Framework

Answer 58

1. Simulation is slower because the GLM must be solved for in each iteration
2. Can’t directly explain the model using LDFs

Question 59

Advantages of the ODP bootstrap

Answer 59

1. Can use the simpler LDF method and the model will still be based on the GLM framework
2. Using LDFs makes the model more easily explainable to others
3. The GLM uses a log-link and may not work with negative incremental, but the simplified GLM will still get a solution

Question 60

Disadvantages of the ODP bootstrap

Answer 60

1. Unable to adjust for calendar-year effects
2. Requires many parameters and can over-fit the data