Shapland and Leong

Question 1

Alphas, betas in row 3, age 4 for mean of incremental losses

Answer 1

a3 + b2 + b3 + b4

Question 2

Alphas, betas in row 2, age 2 for mean of incremental losses

Answer 2

a2 + b2

Question 3

Variance of incremental losses for (w,d) in ODP bootstrap

Answer 3

Var(q(w,d)) = Om

O is dispersion factor

Question 4

Residual for ODP bootstrap

Answer 4

r = (q - m)/ sqrt(m^z)

Question 5

Error distributions that result from variance assumption (GLM)

Answer 5

Var(q(w,d) = Om^z
z=0, Normal
z=1, Poisson
z=2, Gamma
z=3, Inverse Gaussian

Question 6

Assumptions needed so GLM result equals CL result

Answer 6

Variance proportional to mean
Poisson errors
Parameter for each row and column (except first column)

Question 7

5 Bootstrap Steps

Answer 7

1. Create sampled triangle from residuals and means
2. Determine alphas and betas
3. Calculate mean and variance of each cell
4. Draw losses from gamma distribution to add process variance
5. Sum bottom half of triangle to get simulated unpaid loss

Question 8

ODP Bootstrap - Dispersion factor

Answer 8

O = sum(r^2)/(n-p)
n = # data points
p = 2m - 1

Question 9

Standardized residual

Answer 9

Residual times hat matrix adjustment

Question 10

Verrall adjustment to residuals

Answer 10

Multiplied by sqrt[n/(n-p)] for degrees of freedom
n = # data points
p = 2m - 1

Question 11

Three options to deal with negative incrementals in bootstrap

Answer 11

1. Model -ln[-q(w,d)]
2. Add constant to each loss so sum of losses in each column is positive, then subtract constant from results
3. If using simplified GLM, can use CL

Question 12

Simulating loss when mean is negative

Answer 12

Simulate positive value, then shift the mean Gamma(-m,-Om) + 2m; keeps distribution skewed to right

Question 13

Stratified sampling (for heteroskedasticity)

Answer 13

Split triangle into groups

When sampling, only sample from residuals in same group

Question 14

Hetero-Adjustment

Answer 14

Find SD of residuals in each group
Scale residuals by h, so that the SD is same for each group [h = max(SD of groups)/SD of the group]
For whole triangle, sample from these residuals
When applying residual, divide by h to recognize size of residuals in group

Question 15

Steps to take if negative values lead to extreme results

Answer 15

1. Remove outliers
2. Recalibrate model
3. Limit incremental losses below at zero
4. Understand driver of negative development

Question 16

Parametric bootstrapping

Answer 16

Once residuals have been calculated, fit a distribution to these residuals and sample from distribution

Question 17

Heteroecthesious Data

Answer 17

Data where exposures are not same:

1. Valuation of losses before year is fully earned - model will still forecast full year
2. Last diagonal is valued between normal schedule - we don’t have a model for it

Question 18

Correlating multiple lines of business in bootstrap

Answer 18

Location mapping – take sampled residual from same location in triangle from each business
Re-sorting

Question 19

Bootstrap box and whisker

Answer 19

Box is between 25th and 75th percentiles (IQ range)

Any residuals exceeding 75th percentile by 1.5x IQ range should be investigated

Question 20

Bootstrap diagnostics

Answer 20

Mean, standard error, COV, %iles, Min/Max
Standard error should be higher for more recent years
CoV should be highest for older years
Check min/max for reasonability

Question 21

Future research possibilities for bootstrap model

Answer 21

Test ODP bootstrap against additional data sets
Expand model to include Munich chain ladder, claim counts and severity
Research other risk measures
Use in Solvency II
Research correlation matrix