Shapland Flashcards

1
Q

Technique to address model risk

A

weight multiple models

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2
Q

Error distributions (4)

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

Variance of incremental claims

Shapland

A

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

m is fitted incremental loss

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4
Q

Linear predictor GLM parameters (4)

A
  1. c = constant level parameter
  2. alpha = AY adjustments to constant level parameter
  3. beta = development period parameter
  4. gamma = CY trend parameter
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5
Q

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

A
  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)
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6
Q

Unscaled (aka normalized) residual

Shapland

A

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

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

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7
Q

Scale parameter formula

A

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

where N = # data cells and p = # parameters

**should always be calculated from unscaled residuals

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8
Q

Number of parameters

Shapland

A

= 2 * # AYs - 1

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

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9
Q

Scaled residuals and what scaling accounts for

A

scaled residual = unscaled residual * DOF adj. factor

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

> > corrects for bias/over-dispersion in residuals

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10
Q

Standardized residuals and what standardization accounts for

A

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

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11
Q

Scale parameter approximation using standardized residuals

A

scale parameter = sum (squared standardized residuals) / N

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12
Q

How to incorporate process variance into incremental claims estimates

A

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

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13
Q

Outcomes when modeling paid vs. incurred data

A

paid data - outcomes represent total unpaid

incurred data - outcomes represent IBNR

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14
Q

Common problem with ODP bootstrap model

Shapland

A

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

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15
Q

Limitations of the ODP bootstrap model (2)

A
  1. does not account for CY effects
  2. tends to over-parameterize the model
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16
Q

Drawbacks to the GLM bootstrap model (2)

A
  1. GLM must be solved with every iteration, which slows down simulation
  2. model is no longer directly explainable with age-to-age factors
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17
Q

Benefits of the GLM bootstrap model (4)

A

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
18
Q

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

A

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

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

19
Q

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

A

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

20
Q

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

A

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

21
Q

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

A

ODP -

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

GLM - adding/removing parameters can reduce heteroscedasticity

22
Q

Advantage and disadvantage of stratified sampling

A

advantage - simple and straightforward to implement

disadvantage - small number of residuals limits variability

23
Q

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

A

variance parameter adjustment&raquo_space; standardized residuals

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

24
Q

Variance parameter adjustment factor

A

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

25
Q

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

A

both alter the original distribution of residuals

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

26
Q

Number of parameters for variance and scale parameter adjustments for heteroscedasticity

A

group adjustment factors are considered new parameters (impacts p)

27
Q

Scale parameter adjustment factors

A

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

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

28
Q

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

A

reduce (scale) projected payments

29
Q

Handling partial last calendar period data under the ODP bootstrap model

A

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

30
Q

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

A

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

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

31
Q

Tail factors in the GLM bootstrap model

A

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

32
Q

Purposes of diagnostic tests (3)

A
  1. test model assumptions
  2. gauge quality of fit
  3. guide adjustment of model parameters
33
Q

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

A

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

  • alternative: can graph relative standard deviations to visualize groupings
34
Q

Tests for normality (4)

A
  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

35
Q

When to add/remove parameters

A

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

36
Q

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

A

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

37
Q

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

A
  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)
38
Q

Methods for combining multiple models (2)

A
  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
39
Q

Correlation processes for aggregating LOB results (2)

A
  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
40
Q

How to solve for GLM parameters (Shapland)

A

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

41
Q

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

A
  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)