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Provide one disadvantage of bootstrapping

Shapland They are more complex than other models and more time consuming to create

1

Provide two advantages of bootstrapping

Shapland

1. They allow us to calculate how likely it is that the ultimate value of the claims will exceed a certain amount

2. They are able to reflect the general skewness of Insurance losses

2

Describe how using the over dispersed Poisson model to model incremental claims relates a GLM to the standard chain ladder method

Shapland

If we start with the latest diagonal and divide backwards successively by each age to age factor we obtain fitted cumulative claims. Using subtraction, the fitted cumulative claims can be used to determine the fitted incremental claims. These fitted incremental claims exact match those obtained using the over dispersed Poisson model

3

Briefly describe three important outcomes from this relationship of GLM to the standard chain ladder method using the ODP model to model incremental claims.

Shapland

1. A simple link ratio algorithm can be used in place of the more complicated GLM algorithm while still maintaining an underlying GLM framework

2. The use of age to age factors serves as a bridge to the deterministic framework. This allows models to be more easily explained

3. In general the log link function does not work for negative incremental claims. Using link ratios remedies this problem.

4

Identify the assumptions underlying the residual sampling process. Explain why they are advantageous.

Shapland

The residual sampling process assumes that the residuals are independent and identically distributed. However, it does not require the residuals to be normally distributed. This is an advantage since the distributional form of the residuals will flow through the simulation process.

5

Briefly describe two uses of the degrees of freedom adjustment factor

Shapland

The distribution of reserve point estimates from the sample triangles could be multiplied by the degrees of freedom adjustment factor to allow for over dispersion of the residuals in the sampling process. The Pearson residuals could be multiplied by the degrees of freedom adjustment factor to correct for bias in the residuals

6

Identify a downfall of the degrees of freedom adjustment factor and state how the issue can be remedied

Shapland

The degrees of freedom bias correction does not create standardized residuals. This is important because standardized residuals ensure that each residual has the same variance. In order to calculate the standardized residuals, a hat matrix adjustment factor must be applied to the unscaled Pearson residuals.

7

Discuss the difference between bootstrapping paid data and bootstrapping incurred data

Shapland

Bootstrapping paid data provides a distribution of possible outcomes for total unpaid claims. Bootstrapping incurred data provides a distribution of possible outcomes for IBNR.

8

EXPLAIN how the results of an incurred data model can be converted to a paid data model.

Shapland

To convert the results of an incurred data model to a payment steam we apply payment patterns to the ultimate value of the incurred claims.

9

Explain the benefit of bootstrapping the incurred data triangle (instead of paid)

Shapland

Bootstrapping incurred data leverages the case reserves to better predict the ultimate claims. This improves estimates while still focusing on the payment steam for measuring risk.

10

The over dispersed Poisson model can be generalized by specifying fewer parameters.

Identify four advantages to generalizing the over dispersed Poisson model.

Identify two disadvantages

Shapland

Advantages:

1. use fewer parameters helps avoid over parameterizing 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.

Disadvantages:

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.

11

Identify four stochastic models for the chain-ladder technique. For each model, provide one disad- vantage.

Verrall ⇧ Mack’s model – no predictive distribution ⇧ Over-dispersed Poisson distribution – requires positive incremental values ⇧ Over-dispersed negative binomial distribution – requires positive incremental values ⇧ Normal approximation to the negative binomial model – additional parameters must be estimated in order to calculate the variance

12

a) Provide the formula for the prediction variance. b) Explain the difference between the standard error and the prediction error.

Verrall Part a: Prediction variance = process variance + estimation variance Part b: The standard error considers the uncertainty in parameter estimation, whereas the predic- tion error considers both the uncertainty in parameter estimation and the inherent variabil- ity in the data being forecast

13

Provide two advantages of Bayesian methods.

Verrall ⇧ The full predictive distribution can be found using simulation methods ⇧ The prediction error can be obtained directly by calculating the standard deviation of the predictive distribution

14

a) Provide a disadvantage to including calendar year trends in the over-dispersed Poisson model.

b) Explain how this problem can be remedied.

Shapland

Part a:By including calendar year trends, the system of equations underlying the GLM no longer has a unique solution

Part b: To deal with this issue, we start with a model with one lambda parameter, one parameter and one parameter. We then add and remove parameters as needed

15

The over-dispersed Poisson model assumes that residuals are identically distributed with mean zero. a) Explain why the average of the residuals may be less than zero in practice.

Shapland

If the magnitude of losses is higher for an accident year that shows higher development than the weighted average, then the average of the residuals will be negative

16

The over-dispersed Poisson model assumes that residuals are identically distributed with mean zero. b) Explain why the average of the residuals may be greater than zero in practice.

Shapland

If the magnitude of losses is lower for an accident year that shows higher development than the weighted average, then the average of the residuals will be positive

17

The over-dispersed Poisson model assumes that residuals are identically distributed with mean zero. c) Discuss the arguments for and against adjusting the residuals to an overall mean of zero.

Shapland

Argument for adjusting the residuals • If the average of the residuals is positive, then re-sampling from the residuals will add variability to the resampled incremental losses. It may also cause the resampled incremental losses to have an average greater than the fitted loss

Argument against adjusting the residuals • The non-zero average of the residuals is a characteristic of the data set

18

The over-dispersed Poisson model assumes that residuals are identically distributed with mean zero. d) If the decision is made to adjust the residuals to an overall mean of zero, explain the process for doing so.

Shapland

We can add a single constant to all residuals such that the sum of the shifted residuals is zero

19

Briefly describe three approaches to managing missing values in the loss triangle.

Shapland

  1. Estimate the missing value using surrounding values
  2. Exclude the missing value
  3. If the missing value lies on the last diagonal, we can use the value in the second to last diagonal to construct the fitted triangle

20

Briefly describe three approaches to managing outliers in the loss triangle.

Shapland

  1. If these values occur on the first row of the triangle where data may be sparse, we can delete the row and run the model on a smaller triangle
  2. Exclude the outliers completely
  3. Exclude the outliers when calculating the age-to-age factors and the residuals, but re-sample the corresponding incremental when simulating triangles

21

Explain the difference between homoscedastic residuals and heteroscedastic residuals.

Shapland

Homoscedasticity – residuals are independent and identically distributed.

Heteroscedasticity – residuals are independent, but NOT identically distributed

22

a) Define heteroecthesious data.

b) Briefly describe two types of heteroecthesious data.

Shapland

Part a: Heteroecthesious data refers to incomplete or uneven exposures at interim evaluation dates

Part b: 

  • Partial first development period data – occurs when the first development column has a different exposure period than the rest of the columns
  • Partial last calendar period data – occurs when the latest diagonal only has a six-month development period

23

In order to assess the quality of a stochastic model, various diagnostic tests should be run. Identify three purposes of using diagnostic tools.

Shapland

  1. Test various assumptions in the model
  2. Gauge the quality of the model fit
  3. Guide the adjustment of model parameters

24

Describe the process for determining if a model is over-parameterized.

Shapland

To test whether or not a model is over-parameterized, use the following steps:

  1. Start with the basic model which includes one parameter for accident, development and calendar periods
  2. Use trial and error to find a good fit to the data (i.e. add and remove parameters until a good fit is found)
  3. Run all of the standard diagnostics (normality plots, box-whisker plots, p-values, etc.) and compare them to the model with more parameters
  4. If the diagnostics are comparable, then the model with less parameters is preferred (principle of parsimony)

25

Briefly describe the interaction between heteroscedasticity and credibility.

Shapland

Since there are fewer residuals for older development periods, credibility decreases in the tail of the triangle. It’s important NOT to overreact to “apparent” heteroscedasticity in older development years

26

Describe two options when adjusting residuals for heteroscedasticity.

Shapland

  1. Stratified sampling
    • Group development periods with homogeneous variances
    • Sample with replacement from the residuals in each group separately
  2. Variance parameters
    • Group development periods with homogeneous variances
    • Calculate the standard deviation of the residuals in each of the “hetero” groups
    • Calculate the hetero-adjustment factor for each group
    • Multiply all residuals in each group by the hetero-adjustment factor for that group
    • All groups now have the same standard deviation, and we can sample with replacement from among ALL residuals

27

An actuary is estimating ultimate loss ratios by accident year using a bootstrap model.

a) Briefly describe how the actuary can estimate the complete variability in the loss ratio.

b) Briefly describe how the actuary can estimate the future variability in the loss ratio

Shapland

Part a: He can estimate the complete variability in the loss ratio by using all simulated values to estimate the ultimate loss ratio by accident year (rather than just using the values beyond the end of the historical triangle)

Part b: He can estimate the future variability in the loss ratio by using only the future simulated values to estimate the ultimate loss ratio (i.e. add the estimated unpaid losses to the actual cumulative losses to date)

28

Explain why qualitative approaches are preferred over quantitative approaches when populating a correlation matrix.

Marshall

Quantitative techniques require a significant amount of data, time and cost to produce credible and intuitive results

29

Briefly describe the bolt-on approach to determining risk margins.

Marshall

A bolt-on approach occurs when separate analyses are completed to develop a central esti- mate of insurance liabilities and/or estimate risk margins. It is called a “bolt-on” approach because it does not involve a single unified distribution of the entire distribution of possible future claim costs