A5 - A10

Question 0

Provide two advantages of bootstrapping

Answer 0

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

Question 1

Provide one disadvantage of bootstrapping

Answer 1

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

Question 2

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

Answer 2

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

Question 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.

Answer 3

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.

Question 4

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

Answer 4

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.

Question 5

Briefly describe two uses of the degrees of freedom adjustment factor

Answer 5

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

Question 6

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

Answer 6

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.

Question 7

Discuss the difference between bootstrapping paid data and bootstrapping incurred data

Answer 7

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.

Question 8

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

Answer 8

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.

Question 9

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

Answer 9

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.

Question 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

Answer 10

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.

Question 11

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

Answer 11

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

Question 12

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

Answer 12

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

Question 13

Provide two advantages of Bayesian methods.

Answer 13

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

Question 14

a) Provide a disadvantage to including calendar year trends in the over-dispersed Poisson model.
b) Explain how this problem can be remedied.

Answer 14

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

Question 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.

Answer 15

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

Question 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.

Answer 16

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

Question 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.

Answer 17

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

Question 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.

Answer 18

Shapland

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

Question 19

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

Answer 19

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

Question 20

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

Answer 20

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

Question 21

Explain the difference between homoscedastic residuals and heteroscedastic residuals.

Answer 21

Shapland

Homoscedasticity – residuals are independent and identically distributed.

Heteroscedasticity – residuals are independent, but NOT identically distributed

Question 22

a) Define heteroecthesious data.
b) Briefly describe two types of heteroecthesious data.

Answer 22

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

Question 23

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

Answer 23

Shapland

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

Question 24

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

Answer 24

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)

Question 25

Briefly describe the interaction between heteroscedasticity and credibility.

Answer 25

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

Question 26

Describe two options when adjusting residuals for heteroscedasticity.

Answer 26

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

Question 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

Answer 27

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)

Question 28

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

Answer 28

Marshall

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

Question 29

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

Answer 29

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

Question 30

a) Briefly describe systemic risk.
b) Briefly describe two sources of systemic risk and provide an example for each source.

Answer 30

Marshall

Part a: Systemic risk represents risks that are common across valuation classes or claim groups

Part b:

• Internal systemic risk – risks internal to the insurance liability valuation/modeling process (i.e. model parameter risk)
• External systemic risk – risks external to the insurance liability valuation/modeling process (i.e. changes in building costs)

Question 31

a) Briefly describe independent risk.
b) Briefly describe two sources of independent risk.

Answer 31

Marshall

Part a:

• Independent risk represents risks that occur due to the randomness inherent in the insurance process

Part b:

• Parameter risk – represents the extent to which the randomness associated with the insur- ance process affects the ability to select appropriate parameters in the valuation models
• Process risk – represents the pure e↵ect of the randomness associated with the insurance process

Question 32

a) Identify two sources of risk that can be fully analyzed using modeling techniques such as bootstrapping or a stochastic chain-ladder model.

Answer 32

Marshall

Part a: Independent risk and historical external systemic risk

Question 33

Identify two sources of risk that cannot be fully analyzed using modeling techniques such as bootstrapping or a stochastic chain-ladder model.

Answer 33

Marshall

Internal systemic risk and future external systemic risk

Question 34

Briefly explain why traditional modeling techniques cannot capture all sources of uncertainty.

Answer 34

Marshall

Since models fit past data, they are only able to remove past episodes of external systemic risk. They are not able to capture future external systemic risk

Question 35

Briefly describe three sources of internal systemic risk.

Answer 35

Marshall

Specification error – the error that arises because the model cannot perfectly model the insurance process

Parameter selection error – the error that arises because the model cannot adequately measure all predictors of future claim costs or trends in these predictors

Data error – the error that arises due to the lack of credible data. Data error can also refer to an inadequate knowledge of the portfolio being analyzed, including pricing, underwriting and claims management processes

Question 36

Fully describe the balanced scorecard approach for assessing internal systemic risk.

Answer 36

Marshall

A balanced scorecard is developed to objectively assess the model specification against a set of criteria. For each of the sources of internal systemic risk, risk indicators are developed and scored against the criteria. The scores are aggregated for each valuation class and mapped to a quantitative measure (CoV) of the variation arising from internal systemic risk. Despite the focus on objectivity, some subjective decisions must be made. These include the risk indicators, the measurement and scoring criteria, the weight given to each risk indicator and the CoVs that map to each score

Question 37

Briefly describe three external systemic risk categories.

Answer 37

Marshall

1. Claim management process change risk – the uncertainty associated with changes in claim reporting, payment, estimation, etc.
2. Expense risk – the uncertainty associated with the cost of managing the run-off of the insurance liabilities or the cost of maintaining the unexpired risk until the date of loss
3. Event risk – the uncertainty associated with claim costs arising from events (i.e. cats), either natural or man-made

Question 38

For the external systemic risk category; ‘Claim management process change risk’, provide an approach for assessing the risk.

Answer 38

Marshall

Claim management process change risk

• Analysis of past experience should help identify past systemic episodes impacted by the claim management process.
• Discussions with claim managers can help identify the process changes that contributed to those systemic episodes, as well as the potential for any future changes to the process
• Sensitivity testing of key valuation assumptions is useful in the assessment of CoVs for this risk category

Question 39

For the external systemic risk category; ‘Expense Risk’, provide an approach for assessing the risk.

Answer 39

Marshall

Expense risk

• An actuary should spend time with product and claim management to better under- stand the key drivers of policy maintenance and claim handling expenses
• Similar to claim management process change risk, an analysis of past expense levels can help identify past systemic episodes

Question 40

For the external systemic risk category; ‘Event risk’, provide an approach for assessing the risk.

Answer 40

Marshall

Event risk

• For outstanding claim liabilities, discussions with claim management should help set expectations on final claim cost outcomes. The range of development patterns on past events may influence the view on uncertainty as well
• For premium liabilities, past experience for event claims, output from proprietary cat models, and output from models for perils not covered by cat models can help quantify liabilities

Question 41

Fully describe the correlation effects within AND between ‘Independent Risk’

Answer 41

Marshall

Independent risk – assumed to be uncorrelated with any other source of uncertainty and between valuation classes

Question 42

Fully describe the correlation effects within AND between ‘Internal systemic risk’

Answer 42

Marshall

Internal systemic risk – assumed to be uncorrelated with independent risk, and with each source of external systemic risk. However, internal systemic risk tends to be correlated between valuation classes

Question 43

Fully describe the correlation effects within AND between ‘External systemic risk’

Answer 43

Marshall

External systemic risk – assumed to be uncorrelated with independent risk and internal systemic risk. However, correlation may exist between risks that belong to similar external systemic risk categories. If correlation occurs, correlated risk categories can be aggregated into broader categories that are not correlated

Question 44

Identify two distributions used to calculate risk margins. For each distribution, explain any differences in the calculated risk margins.

Answer 44

Marshall

Normal distribution – produces a higher risk margin at lower probabilities of adequacy, irrespective of the size of the consolidated CoV

Lognormal distribution – produces a higher risk margin at higher probabilities of adequacy as long as the consolidated CoV is not too high. For extremely high CoVs, the risk margin can actually reduce as a percentage of the CoV

Question 45

For Independent risk, describe any internal benchmarking that should occur.

Answer 45

Marshall

• Portfolio size – the larger the portfolio, the lower the volatility arising from random effects
• Length of claim run-off – the longer a portfolio takes to run-off, the more time there is for random effects to have an impact

Question 46

For Internal systemic risk, describe any internal benchmarking that should occur.

Answer 46

Marshall

• Classes with homogeneous claim groups should have similar CoVs
• Long-tailed portfolios should have higher CoVs than short-tailed portfolios

Question 47

For External systemic risk, describe any internal benchmarking that should occur.

Answer 47

Marshall

Long-tailed portfolios should have higher CoVs than short-tailed portfolios in most cases (exceptions include event risk and liability risk for home classes)

Question 48

Briefly describe a situation in which external benchmarking is useful.

Answer 48

Marshall

External benchmarking is beneficial when little information is available for analytics

Question 49

Briefly describe general hindsight analysis.

Answer 49

Marshall

General hindsight analysis compares past estimates of liabilities against the latest view of the equivalent liabilities. Any movement/variation can be converted to a coe cient of variation reflecting actual past volatility. This coe cient of variation can then be used to help estimate current liabilities

Question 50

Describe mechanical hindsight analysis and give an example of how it can be used to analyze independent risk.

Answer 50

Marshall

Mechanical hindsight analysis applies a mechanical approach to estimating liabilities by systematically removing the most recent claims experience.

For example:

• Apply the chain-ladder method to a triangle of cumulative claims
• Use an objective approach to calculate the development factors (volume-weighted av- erage)
• The outstanding claim payment derived using all of the data up to the valuation date is the ‘current’ estimate
• Remove diagonals one at a time and apply the same method to derive the outstanding claims payments at past valuation dates
• Compare each of the past estimates with the current estimate

To analyze independent risk, mechanical hindsight analysis should be applied to periods with stable experience

Question 51

When Bayesian model results are closer to CL then BE, this implies what about the variance of the prior distribution?

Answer 51

Verrall

Variance of the prior distribution is large if Bayesian model results are closer to CL then BF

Question 52

Discuss how the Beta value of a prior gamma distribution influences the trade off between chain ladder and BF estimates in a Bayesian model.

Answer 52

Verrall

For a gamma distribution, a large beta results in small variance. This means that we have strong confidence in the expert opinion regarding the a priori ultimate loss. Thus, more weight will be given to BF estimate in the Bayesian model

Question 53

Describe one advantage that a Bayesian approach has over bootstrapping, and one advantage that a Bayesian approach has over the Mack method.

Answer 53

Compared to bootstrapping, the Bayesian approach can incorporate expert knowledge into the selection of link ratios. Compared to the Mack method, the full distribution can be easily calculated AND the prediction error can be calculated. Verrall

Question 54

Explain why modelling paid losses or claim counts may be preferable to modeling reported losses when using an over dispersed Poisson model.

Answer 54

Verrall An over dispersed Poisson model requires positive incremental values. Since negative incremental values stress possible when using reported losses, it’s preferable to use paid losses or claim counts

Question 55

What is the principle of parsimony?

Answer 55

Shapland and Leong Model with less parameters is preferred.

Question 56

For each of the plots below, determine whether the model is light tailed, heavy tailed, or biased
high:

Answer 56

C - Based on the U shape, this model is biased high

(Meyers)

Question 57

For each of the plots below, determine whether the model is light tailed, heavy tailed, or biased high:

Answer 57

A - Based on the backward S shape, this model is heavy tailed

B - Based on the S shape, this model is light tailed

(Meyers)

Question 58

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.

Answer 58

Meyers

• The Mack model multiplies the age-to-age factors by the last observed loss. These observed losses act as fixed level parameters. A model that treats the level of the accident year as random will predict more risk. One such model is the Leveled Chain Ladder model
• The Mack model assumes that the loss amounts for different accident years are independent. A model that allows for correlation between accident years will predict more risk. One such model is the Correlated Chain Ladder model

Question 59

Briefly describe two consequences of including a payment year trend in a model.

Answer 59

Meyers

• The model should be based on incremental paid loss amounts rather than cumulative paid loss amounts. This is because cumulative losses include settled claims which do not change over time
• Incremental paid loss amounts tend to be skewed to the right and are occasionally negative. We need a loss distribution that allows for these features

Question 60

Describe an alternative model for paid losses that outperforms the CIT and LIT models.

(Meyers)

Answer 60

The claims settlement rate model is used to model cumulative losses instead of incremental losses. Thus, it removes the payment year trend and skew normal distribution. It uses a claims settlement rate parameter to account for any speedup in the claims settlement rate. This model passes the K-S test for uniformity

Question 61

Briefly describe why model risk can be thought of as a special type of parameter risk.

(Meyers)

Answer 61

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