Shapland

Question 1

ODP model

Answer 1

• uses GLM to model incremental q(w,d) claims
• uses log link fct and ODP for error fct

Question 2

fitted incremental claims using ODP

Answer 2

= fitted incremental claims derived using CL factors

• start with latest diag and divide backwards successively by each DF, obtain fitted cumulative claims then using subtraction get fitted incremental claims
• model is known as ODPB

Question 3

ODPB benefit

Answer 3

-simple link ratio algorithm can be used in place of more complicated GLM while maintaining underlying GLM framework

Question 4

robust GLM: expected incremental formula

Answer 4

Question 5

Bootstrap process

Answer 5

• calc fitted cumulative losses using actual DFs then calc fitted incremental losses
• calc actual increm loss
• calc residuals
• Pearson residuals used since they are calculated consistently with scale parameter ϕ
• sampling with replacement from residuals of data
• sampling can be used to create new sample triangles of increm claims
• sample triangles can be cumulated and DFs can be calculated and applied to calc point estimates for data
• have distribution of point estimates which incorporates process var and parameter var in simulation of hist data

Question 6

sampling with replacement assumes

Answer 6

residuals are independent and identically distributed, does not require them to be normally distributed

Question 7

sampling can be used to create new sample triangles of increm claims -> formula for incremental loss q*

Answer 7

Question 8

Adjustments to unscaled Pearson residuals

Answer 8

• DoF adj factor
• hat matrix adjustment factor

Question 9

DoF adj factor

Answer 9

-DoF adj factor is used to correct for bias in residuals up front aka add more dispersion aka more var. -> scaled Pearson residuals

N: # data cells in triangle p: parameters = 2*AYs -1

Question 10

hat matrix adjustment factor

Answer 10

-hat matrix adjustment factor is considered replacement for and improvement over degrees of freedom factor

Only use diagonal

Question 11

Standardized residuals ensure

Answer 11

that each residual has same variance

Question 12

Negative incremental values if sum of column is positive

Answer 12

Ln(q) for q>0

0 for q=0

-Ln(|q|) for q<0

Question 13

Negative incremental values if column in negative

Answer 13

q+=q-psi

m=m+ + psi

-psi is largest neg in value in triangle (largest ind or sum)

Question 14

Heteroscedasticity

Answer 14

• model errors do not share common variance
• violates assumption that residuals are i.i.d.

Question 15

heteroscedasticity: 3 options

Answer 15

stratified sampling, variance parameters, scale parameters

Question 16

Stratified sampling

Answer 16

group development periods with homogeneous variances/simiilar residual variances

for each simulated incremental loss, only sample residuals from the same age (same group?)-> some groups may lack credibility

Question 17

Calc variance parameters

Answer 17

group, calc std dev of residuals in each of hetero groups, and calc hetero-adj factor for each group -> STANDARDIZED residuals r^H

*this gives residuals constant variance

• sample with replacement among all residuals and divide each residual by adj factor when residuals are resampled

**goes from group3 to group2, divide by group2 hi for qi*

Question 18

Calc scale parameters:

Answer 18

similar but hetero-adj factor is based on scale parameter -> have to look @ unscaled PEARSON residuals r

**use same formula for r^iH and qi*

modify phi so that each hetero group has a different scale parameter when adding future process variance and use hetero-factor to adjust simulated losses similar to variance parameters

Question 19

residual plots

Answer 19

• Tests the assumption residuals are i.i.d.
• DP, AY, or CY
• do residuals exhibit any trends?
• should have random pattern
• do residuals have different variances? -> heteroscedasticity
• if so should group into hetero group and adjust them to common std deviation

Question 20

Standard errors

Answer 20

• should increase over time (oldest to youngest years)
• total reserve std error should be larger than any ind year

Question 21

CoV

Answer 21

• should decrease over time (newest AY could have large because parameter uncertainty could overpower process uncertainty)
• total reserve CoV should be less than any ind year

Question 22

Normality test

Answer 22

• allows comparison of parameter sets and assess skewness
• normality plot: data points should be tightly distributed around diagonal line
• calc test values, results for normal distributed

p-value > 5%

R^2 close to 1

Question 23

Parsimony

Answer 23

model with fewer parameters is preferred as long as goodness of fit is not markedly different

Question 24

options for using multiple models

Answer 24

1. run models with same RVs
2. run models with independent RVs

Question 25

If a ton of outliers?

Answer 25

• model is poor fit
• GLMB: new parameters chosen or error dist changed
• ODPB: L-year weighted avg

Question 26

Heteroecthesious data

Answer 26

• ODPB requires symmetrical shape and homoecthesious data (similar exposures)
• can relax symmetrical shape by using L-yr wght avg or excl diags
• common Heteroecthesious data triangles
• partial first development period data
• partial last calendar period data

Question 27

matrix notation and how to solve for parameters

Answer 27

model specification:

Y=X*A

Y=[ln(q(1,1);ln(q(2,1);…..]

A=[alpha1; alpha2; …; beta2; ….; gamma2; 2*gamma2; ….]

X= 0 and 1 matrix

*alpha is AY parameter

model with fitted parameters and soln matrix W:

W=X*A

W=[ln(m(1,1);ln(m(2,1);…..]

*solve for parameters in A that minimize squared difference between vectors Y (actual log loss) and W (expected log loss)

*use iteratively weighted least squares to solve for A

Question 28

how can CY trend be incorporated in matrix/model

Answer 28

include CY parameter, gamma_k in model specification of 0 for diagonal, 1 for second diagonal and so on

*fitted parameter will be CY trend

Question 29

benefit and disadvantage to using single AY parameter ie alpha

Answer 29

advantage = using fewer parameters helps avoid overfitting and can help improve the model performance for future losses

disadvantage = if there are signifcatnt differences in level by AY, model cant directly reflect those differences and may fit data worse

Question 30

Standardized Pearson residuals and scale parameter process

Answer 30

cumulative losses

back out LDFs

fit incremental losses

calc unscaled pearson residuals: r=(q-m)/sqrt(m^z)

calc scale parameter: phi = sum(r^2)/(N-p) where N is number of data points

calc hat matrix triangle

calc f^H triangle -> sqrt(1/(1-H))

standardized pearson residuals: r^H=r*f^H

Question 31

how process variance can be incorporated in simulation of incremental losses

Answer 31

projected cumulative loss = use latest diag from simulated and use CDFs from simulation

calc projectected incremental loss

simulate future incrementals from Gamma distribution to add process variance

q(w,d) ~ gamma(mean=m(w,d), var= phi*m(w,d))

Question 32

process of adjusting for negative column sum

Answer 32

calc adjusted incremental losses: q⁺=q-psi
calc fitted adjusted losses with simplified GLM -> calc LDFs from adjusted cumulative losses and divide backwards to get fitted adjusted cumulative losses

calc fitted adj incremental losses: m⁺

calc fitted incrementals: m=m⁺ + psi

calc residuals as normal

Question 33

multiple models: same random variables

calc unpaid loss for each iteration

Answer 33

incremental loss = sum(weight(model) * inc loss(model))

Question 34

multiple models: weighted mixture of models AKA independent RVs

calc unpaid loss for each iteration

Answer 34

random values from [0,1] for each accident year and iteration to mix models

*make table of selected models for each AY and iteration

if random value < model weight #1, pick model 1

if random value > model weight #1, pick model 2

model weights will be given by AY

*make table of best estimate of unpaid losses

Question 35

problem with differing levels of variability of residuals at different development periods impacts appropriateness of model

Answer 35

if earlier periods have greater standardized Pearson residual variability than later periods (heteroscedasticity), then during simulation residuals could be sampled from later periods to simulate losses at earlier development

when this happens, there will be less variability in results than there should be for earlier periods and model is not appropriate

Question 36

if they ask you to calculate point estimate of projected unpaid loss before process variance is added

Answer 36

use n=ln[m]=alpha +Σbeta

to get estimates of projected incrementals ie the points outside of triangle

then add these together to get unpaid estimate

these point estimates are also m^projected(w,d)

if incorporating process variance: q^sim(w,d)=gamma(m^projected, phi*m^projected)

AKA simulated future incrementals

Question 37

if they ask you to estimate expected incremental loss

Answer 37

use n=ln[m]=alpha +Σbeta

to calculate incremental loss triangle m(w,d)

Question 38

simulated incremental losses

Answer 38

q*(w,d)=r*sqrt(m^z)+m

where r* is simulated residuals

Question 39

2 advantages and disadvantages of using GLM framework compared to simplified GLM

Answer 39

ADV: GLM can use fewer paramers than simplified to avoid overparameterizing

GLM is more flexible and we can add different parameters such as CY trend parameters

DISADV: GLM framework requires us to re-fit GLM for each iteration of sampled data -> can slow down process

model isn’t directly explainable to others using LDFs like simplified GLM is