Shapland

Question 1

Technique to address model risk

Answer 1

weight multiple models

Question 2

Error distributions (4)

Answer 2

1. normal, z = 0
2. Poisson, z = 1
3. Gamma, z = 2
4. inverse Gaussian, z = 3

Question 3

Variance of incremental claims

Shapland

Answer 3

var(q(w,d)) = phi * m^z

m is fitted incremental loss

Question 4

Linear predictor GLM parameters (4)

Answer 4

1. c = constant level parameter
2. alpha = AY adjustments to constant level parameter
3. beta = development period parameter
4. gamma = CY trend parameter

Question 5

Consequences of ODP fitted incremental claims = CL incremental claims (3)
(Shapland)

Answer 5

1. simple link ratio algorithm can be used w/in GLM framework
2. use of age-to-age factors allows the model to be easily explained
3. allows for negative incremental claims (which would be a problem to model)

Question 6

Unscaled (aka normalized) residual

Shapland

Answer 6

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

q = actual incremental losses
m = fitted incremental losses
z = 1 for ODP

Question 7

Scale parameter formula

Answer 7

sum(r^2) / (N - p)

where N = # data cells and p = # parameters

**should always be calculated from unscaled residuals

Question 8

Number of parameters

Shapland

Answer 8

= 2 * # AYs - 1

or with additional development period/adjustment parameters:
= # AYs
+ (# development period parameters - 1)
+ (# hetero adjustment groups - 1)

Question 9

Scaled residuals and what scaling accounts for

Answer 9

scaled residual = unscaled residual * DOF adj. factor

DOF adj. factor = sqrt ( N / (N - p))

> > corrects for bias/over-dispersion in residuals

Question 10

Standardized residuals and what standardization accounts for

Answer 10

standardized residuals = unscaled residual * hat matrix adjustment factor

hat matrix adjustment factor = sqrt ( 1 / (1 - ith position on diagonal of hat matrix) )

> > ensures residuals have constant variance

Question 11

Scale parameter approximation using standardized residuals

Answer 11

scale parameter = sum (squared standardized residuals) / N

Question 12

How to incorporate process variance into incremental claims estimates

Answer 12

assume each future incremental value is the mean and each variance(fitted incremental value) the variance of a gamma distribution and simulate future incremental losses

Question 13

Outcomes when modeling paid vs. incurred data

Answer 13

paid data - outcomes represent total unpaid

incurred data - outcomes represent IBNR

Question 14

Common problem with ODP bootstrap model

Shapland

Answer 14

most recent AYs have more variance than expected b/c more age-to-age factors are used to extrapolate the sample values

> > correct for this by using BF/CC method

Question 15

Limitations of the ODP bootstrap model (2)

Answer 15

1. does not account for CY effects
2. tends to over-parameterize the model

Question 16

Drawbacks to the GLM bootstrap model (2)

Answer 16

1. GLM must be solved with every iteration, which slows down simulation
2. model is no longer directly explainable with age-to-age factors

Question 17

Benefits of the GLM bootstrap model (4)

Answer 17

flexibility

1. fewer parameters help to avoid over-parameterization
2. ability to add CY trend parameter
3. flexibility to model triangles with incomplete data
4. allows for matching model parameters to the statistical features of the data

Question 18

ODP vs. GLM bootstrap sampling with trapezoidal data OR when using an L-yr weighted average of LDFs

Answer 18

ODP - models cumulative claims instead of incremental, sample from trapezoid to entire triangle

GLM - models incremental claims directly, sample from trapezoid to trapezoid

Question 19

ODP (3) vs. GLM bootstrap methods for handling missing data

Answer 19

ODP -

1. estimate missing value from surrounding values
2. exclude missing value from LDF calculations (and no corresponding residual)
3. if on latest diagonal, estimate from surrounding or use 2nd latest diagonal value

GLM - simply reduce N

Question 20

Options for handling outliers in the ODP bootstrap model (2) and GLM bootstrap model

Answer 20

ODP -

1. exclude and treat as a missing value
2. exclude residual but still sample that cell; this removes extreme residual but keeps variability; use sample value to calculate LDFs and ultimate

GLM - treat like a missing value

Question 21

Methods for handling heteroscedasticity under ODP (3) and GLM bootstrap models

Answer 21

ODP -

1. stratified sampling
2. variance parameter adjustment
3. scale parameter adjustment

GLM - adding/removing parameters can reduce heteroscedasticity

Question 22

Advantage and disadvantage of stratified sampling

Answer 22

advantage - simple and straightforward to implement

disadvantage - small number of residuals limits variability

Question 23

Type of residuals used in variance parameter adjustment and scale parameter adjustments for heteroscedasticity

Answer 23

variance parameter adjustment&raquo_space; standardized residuals

scale parameter adjustment&raquo_space; use unscaled residuals to calculated adjustment factors but adjust standardized residuals

Question 24

Variance parameter adjustment factor

Answer 24

= standard deviation (all residuals) / standard deviation (residuals in group i)

Question 25

Disadvantage of variance and scale parameter adjustments and how to correct them

Answer 25

both alter the original distribution of residuals

> > adjust back to normal after re-sampling by dividing sampled residuals by destination group adjustment factor

Question 26

Number of parameters for variance and scale parameter adjustments for heteroscedasticity

Answer 26

group adjustment factors are considered new parameters (impacts p)

Question 27

Scale parameter adjustment factors

Answer 27

= sqrt ( overall scale parameter ) / sqrt ( scale parameter for group i )

scale parameter = [ (N / (N - p)) * sum (squared unscaled residuals) ] / N

Question 28

Handling partial 1st development period data under the ODP bootstrap model

Answer 28

reduce (scale) projected payments

Question 29

Handling partial last calendar period data under the ODP bootstrap model

Answer 29

annualize exposures when calculating LDFs and reduce (scale) projected payments

Question 30

Tail factor decay model and standard deviation rule of thumb for the ODP bootstrap model

Answer 30

under decay model: next age-to-ultimate factor = 1 + decay percentage * (prior age-to-ultimate - 1)

standard deviation (tail factor) <= .5 * (tail factor - 1)

Question 31

Tail factors in the GLM bootstrap model

Answer 31

implicit assumption that last development/CY period parameters applies incrementally until the effect is negligible (get a tail factor without needing to specify one)

Question 32

Purposes of diagnostic tests (3)

Answer 32

1. test model assumptions
2. gauge quality of fit
3. guide adjustment of model parameters

Question 33

Test for whether residuals are i.i.d. (and alternative visualization)

Answer 33

plot residuals against development period/AY/CY and look for a random pattern

• alternative: can graph relative standard deviations to visualize groupings

Question 34

Tests for normality (4)

Answer 34

1. plot of normal inverse against residuals - want residuals tightly distributed around diagonal
2. p-value > 5%
3. R^2 value close to 1.00
4. small AIC or BIC values (which penalize for additional parameters)

**residuals required to be i.i.d., but not normal

Question 35

When to add/remove parameters

Answer 35

add: if residuals are not randomly scattered around the 0 line
remove: if not statistically significant

Question 36

Relationship between standard error, age, and CoV and individual years vs. total for AY and CY

Answer 36

For AY
SE is larger for recent years because there are more unpaid losses; decreases with age (CY is opposite)

CoV smaller for recent years (except most recent) due to large unpaid losses; variability is offset
CoV all years < CoV individual (diversification)

most recent CoV high due to parameter uncertainty

Question 37

Reasons for CoV to increase in most recent years (2)

Answer 37

1. parameter uncertainty increases as age decreases, so with additional parameters, parameter variance may overpower process uncertainty
2. model may be over-estimating variability in most recent years (use BF/CC)

Question 38

Methods for combining multiple models (2)

Answer 38

1. run models with the same random variables (sample residuals from the same position) - each model gets a weight for every iteration (correlated results)
2. run models with independent random variables - for each iteration only use 1 model where model selected is determined by weights

Question 39

Correlation processes for aggregating LOB results (2)

Answer 39

1. location mapping - selects residuals from same location in all triangles and preserves correlation
2. re-sorting - specify desired correlation by re-sorting residuals until the rank correlation matches the desired correlation

Question 40

How to solve for GLM parameters (Shapland)

Answer 40

Solve for the parameters that minimize the squared difference between ln(actual incremental losses) and ln(expected incremental losses).

Question 41

Handling negative incremental values for the GLM bootstrap model (2)

Answer 41

1. If sum of incremental losses in column > 0 use the modified log link function
2. If sum of incremental losses in the column < 0 shift the entire loss triangle by the amount of the largest negative value. After running the GLM reverse the adjustment on the fitted incremental losses (before calculating residuals)