Clark

Question 1

Loglogistic Distribution

Answer 1

Question 2

Weibull Distribution

Answer 2

Question 3

Clark Assumes Variance is Proportional to Mean-Estimate the ratio

Answer 3

Question 4

In Clark, what is the log likelihood formula?

Answer 4

Question 5

Clark’s LDF Method-Estimate Ultimate Losses

Answer 5

Question 6

Clark’s Cape Cod-Estimate Ultimate Losses

Answer 6

Question 7

Clark-Estimate Calendar Year Development

Answer 7

Question 8

Assumptions in Clark Model

Answer 8

Question 9

Calculate Residuals for Clark’s Model

Answer 9

Question 10

Provide three advantages of using parameterized curves to describe loss emergence patterns.

Answer 10

• Estimation is simple since we only have to estimaet two parameters
• We can use data from triangles that do NOT have evenly spaced evaluation data
• The final pattern is smooth and does not follow random movements in the historical age-to-age factors

Question 11

In a stochastic framework, explain why the Cape Cod method is preferred over the LDF method when few data points exist.

Answer 11

The Cape Cod method requires the estimation of fewer parameters. Since the LDF method requires a method for each AY, as well as the parameters for the growth curve, it tends to be over-paramterized when few data points exist

Question 12

Briefly describe the two components of the variance of the actual loss emergence.

Answer 12

• Process variance-the random variation in the actual loss emergence
• Parameter variance-the uncertainty in the estimate

Question 13

Provide two advantages of using the over-dispersed Poisson distribution to model the actual loss emergence.

Answer 13

• Inclusion of scaling factors allows us to match the first and second moments of any distribution. Thus, there is high flexibility.
• Maximum likelihood estimation produces the LDF and Cape Cod estimates of ultimate losses. Thus, the results can be presented in a familiar format.

Question 14

Fully describe the key assumptions underlying the model outlines in Clark.

Answer 14

Assumption 1: Incremental losses are independent and identially distributed.

• “Independence” means that one period does not affect the surrounding periods
• “Identically distributed” assumes that the emergence pattern is the same for all accident years, which is clearly over-simplified.

Assumption 2: The variance/mean parameter is fixed and known.

-Technically, the variance should be estimated simultaneously with the other model paremeters, with the variance around its estimate included in the covariance matrix. However, doing so results in messy mathematics. For convenience and simplicity, we assume the variance is fixed and known.

Assumption 3: Variance estimates are based on an approximation to the Rao-Cramer lower bound

• The estimate of variance based on the information matrix is only exact when we are using linear functions.
• Since our model is non-linear, the variance estimate is a Rao-Cramer lower bound (ie, the variance estimate is as low as it possibly can be)

Question 15

Briefly descrbie three graphical tests that can be used to validate Clark’s model assumptions

Answer 15

Plot the normalized residuals using the following:

• Increment age-if residuals are randomly scattered around zero with a roughly constant variance, we can assume the growth curve is appropriate.
• Expected loss in each increment age-if residuals are randomly scattered around zero with a roughly constant variance, we can assume the variance/mean ratio is constant.
• Calendar year-if residuals are randomly scattered around zero with a roughly constant variance, we can assume that there are no calendar year effects

Question 16

Briefly explain why it might be necessary to truncate LDFs when using growth curves

Answer 16

For curves with heavy tails (such as loglogistic), it may be necessary to truncate the LDF at a finite point in time to reduce reliance on the extrapolation

Question 17

Compare and contrast the process and paremeter variances of the Cape Cod method and the LDF method

Answer 17

• Process variance-the Cape Cod method can produce a higher or lower process variance than the LDF method.
• Parameter variance-the Cape Cod method produces a lower paremeter variance than the LDF method since it requires fewer paremeters and incorporates information from the exposure base.

Question 18

An actuary used maximum likelihood to paremeterize a reserving model. Due to management discretion, the carried reserves differ from the maximum likelihood estimate.

-Explain why it may or may NOT be appropriate to use the coefficient of variation in the model to describe the carried reserve.

Answer 18

• Since the standard deviation in the MLE model is directly tied to the maximum likelihood estimate, it may not be appropriate for the carried reserves.
• Since the final carried reserve is a selection based on a number of factors, it stands to reason that the standard deviation should also be a selection. The output from the MLE model is a reasonable basis for that selection.