What are the 2 objectives in creating a formal model of loss reserving
- Describe loss emergence in simple mathematical terms as a guide to selecting amount for carried reserves
- Provide a means of estimating the range of possible outcomes around the “expected” reserve.
Write down the formulas of the growth function of a weibull and loglogistic curve
Write
True or false: We can use these curves when we expect salvage recoveries.
False, those curves have a strictly increasing pattern. If we expect salvage recoveries, we should use different models
What are the advantages to use parameterized curves to describe the emergence pattern?
- Estimation is simple since we only have to estimate 2 parameters
- We can use data that is not from a triangle with evenly spaced evaluation data - such as the case in which the latest diagonal is only nine months from the second latest diagonal
- The final pattern is smooth and does not follow random movements in the historical age-to-age factors
Write down the formulas for expected emergence with LDF method and Cape Cod method
Write
In general, the Cape Cod method is preferred than the LDF method. Why?
- Data is summarized into a triangle with relatively few data points. Since the LDF method requires an estimation of a number of parameters, it tends to be over-parameterized when few data points exist
- Cape Cod method has a smaller parameter variance than LDF method due to the additional information given by the exposure base and the fewer parameters –> The process variance can be lower or higher than LDF method. Overall, the total variance is lower when using Cape Cod
Define Parameter variance and process variance
Process Variance is the variance due to randomness in the insurance process
Parameter variance is the variance in the estimate of the parameters
Write down how to calculate the constant variance to mean ratio
Provide 2 advantages of using the over-dispersed Poisson distribution to model the actual loss emergence
- Inclusion of scaling factors allows us to match the first and the second moments of any distribution, allowing high flexibility
- Maximum likelihood estimation produces the LDF and Cape Cod estimates of ultimate losses, so the results can be presented in a familiar format
Write down the formula of the loglikelihood that we need to maximize with over-dispersed Poisson
How we calculate the process variance?
Sigma^2 * Reserves
What is the 1st assumption underlying the model outlined in Clark?
- Incremental losses are independent and identically distributed (iid)
Independence means that one period does not affect the surrounding periods
- -> Can be tested using residual analysis
- -> Positive correlation could exist if all periods are equally impacted by a change in loss inflation
- -> Negative correlation could exist if a large settlement in one period replaces a stream of payments in later periods
Identically distributed assumes that the emergence pattern is the same for all accident years, which is clearly over-simplified
–> Different risks and a different mix of business would have been written in each historical period, each subject to different claims handling and settlement practices
What is the 2nd assumption underlying the model outlined by clark?
The variance/mean scale parameter sigma^2 is fixed and known
What is the 3rd assumption underlying the model outlined in clark?1
Variance estimates are based on an approximation to the Rao-Cramer Lower bound
How we calculate normalized residuals?
Write
Describe 3 graphical tests that can be used to validate Clark’s model assumptions
- Increment age : If residuals are randomly scattered around 0 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 0 with a roughly constant variance, we can assume the variance/mean ratio is constant
- Calendar year : If residuals are randomly scattered around 0 with a roughly constant variance, we can assume that there is no calendar year effects
Explain why it might be necessary to truncate LDFs when using growth curves
For curves with “heavy” tails (loglogistic), it may be necessary to truncate the LDF at a finite point in time to reduce reliance on the extrapolation.
Name a major reason why we calculate the 12 month development
The estimate is testable within a short time frame. One year later, we can compare it to the actual development and see if it was within the forecast range.
If we selected a carried reserve other than the maximum likelihood estimate, can we still use the coefficient of variation from the model? (2 answers)
- Sine 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.
What is the main conclusion of the paper?
The parameter variance is generally larger than the process variance, implying that our need for more complete data (such as the exposure information in the Cape cod method) outweighs the need for more sophisticated models
Be able to calculate the process variance of discounted reserves
