Verrall

Question 1

stochastic models for CL

Answer 1

1. Mack
2. over dispersed poisson
3. over dispersed negative binomial
4. normal approx to NB

Question 2

MACK

Answer 2

-only on cumulative loss

Bayesian compared to the Mack, the full distribution can be easily calculated & the prediction error can be calculated

Question 3

ODP

Answer 3

• increm loss
• since neg increm values are possible with reported data, preferable to use paid loss or claim counts
• not obvious it produces CL

Question 4

ODNB

Answer 4

• same for increm and cumulative losses
• if link ratio is less than 1 or id column sums of incremental loss are positive, produce negative variance
• expected value for each increm cell is equivalent to CL estimate AKA form of mean is the same as CL

Question 5

normal approx to NB

Answer 5

• allows for neg increm claims
• more parameters

Question 6

2 areas where expert knowledge is applied

Answer 6

• BF method (row parameters/AY ultimates)
• Insertion of prior knowledge about individual DFs in CL (unlike Bootstrapping)

Question 7

Bayesian models have 2 important properties

Answer 7

1. Can incorporate expert knowledge
2. Can be easily implemented

Question 8

estimate for outstanding losses: CL

Answer 8

Question 9

estimate for outstanding losses: BF

Answer 9

Question 10

prediction variance

Answer 10

process variance + estimation variance

Question 11

prediction error will be [] if less confident in expert opinion

Answer 11

higher

Question 12

When comparing prediction errors

Answer 12

it’s best to think of the prediction error as a percentage of the prediction, since the reserve estimate itself may vary greatly from model to model

Question 13

difficulty in calculating the prediction error highlights a few advantages of Bayesian methods

Answer 13

• full predictive distribution can be found using simulation methods
• RMSEP can be obtained directly by calculating the std dev of method

Question 14

2 cases of intervention in estimation of DFs for CL

Answer 14

• DF changed in some rows due to external information
• DFs = 5yr volume weighted average rather than all of the available data in the triangle

Question 15

Incorporating Expert opinion about DFs

Answer 15

• means and variance of prior distributions of DFs reflect expert opinion
• lamda has mean and var W
• mean is opinion
• W depends on strength of opinion

Question 16

if W is large

Answer 16

DF will be pulled closer to CL DF and reserve will closely resemble CL reserve

Question 17

if W is small

Answer 17

DF will be pulled closer to prior mean and reserve will move away from CL reserve

Question 18

using BF

Answer 18

• BF assumes expert opinion about level of each row xi from ODP, need to specify prior distribution for xi
• uses Gamma

E[xi] = alpha/beta = M

Var(xi) = alpha/beta^2 = M/beta

-for given choice of M, variance can be altered by changing beta

Question 19

smaller B implies

Answer 19

we are more unsure about M

Question 20

Bayesian Model for BF (BAYESIAN MEAN RESERVE) -> E[Cij]

Answer 20

Question 21

formula for Z

Answer 21

Question 22

beta can control Z

Answer 22

so large beta aka more conf., more weight to BF

-mean of incremental claims is credibility formula where Z controls trade-off between prior mean (BF) and data (CL)

Question 23

to modify Bayesian framework ->

Answer 23

insert row parameter for each AY and specify low variances

Question 24

Estimating column parameters (BF RESERVE)

Answer 24

• To account for all variability, we also need to estimate the column parameters (yj )
• use estimates from traditional CL
• or define prior dist for column parameters and estimate column parameters first
• Once we define improper prior distributions (i.e. large variances) for the column parameters and estimate them, we obtain an over-dispersed negative binomial with mean

E[Cij]=(gamma(i)-1)*sum(Cmj)

Question 25

fully specify Bayesian model when using 5yr volume weighted average

Answer 25

E[Inc Loss or Cum Loss] = ... Var(Inc Loss or Cum Loss) = ... lambda (i,j) = lamda (j) for most recent 5 CY diagonals lambda (i,j) = lamda\* (j) for all diagonals prior to latest 5

Question 26

if estimating E[Cij] for multiple periods ie E[Cij] and E[Cij+1] using bayesian credibility

Answer 26

for CL: Di,j-1 will come from latest diagonal and estimated incrementals using CL for BF: need to do nothing special

Question 27

estimating incrememtal losses with bayesian credibility -\> another way to look at it/calculate

Answer 27

calc E[Cij] for CL and BF seperately E[Cij]=Cum loss to date \* (DF-1) E[Cij]=M\*(%rptd@t+1 - %rptd@t) then calc Z and credibility weight estimate

Question 28

if incremental losses are assumed to follow ODNB (for Bayesian using BF) then E[Cij]

Answer 28

E[Cij]=(gamma(i)-1)\*sum(Cmj)

Question 29

how to calculate gamma(i)

Answer 29

use E[Cij]=(gamma(i)-1)\*sum(Cmj) and given column parameters lambda ie calculate expected incremental loss using CL (fill out remainder of triangle) then solve for gamma by using those and the E[Cij] formula above

Question 30

how to calculate expected ultimate loss when given gammas and the incremental losses follow ODNB

Answer 30

use E[Cij] \*\*need to start with oldest year first and then go down from there since this process is iterative for newer AYs

Question 31

estimating reserve for fully stochastic BF and the benefit compared to Bayesian BF based on ODP

Answer 31

this is the same as incremental loss following ODNB with gamma values \*fully stochastic model in both row and column parameters where as Baysian BF based on ODP uses static column factors, LDFs, calculated from the loss triangle

Question 32

specify the prior distributions for DFs lambda(i,j) to be used in Bayesian model that will produce CL besides prior knowledge

Answer 32

need to set a prior distribution for each AY and development period for non knowledge DFs, set prior distribution to have mean = volume weighted LDF and large variance for knowledge DFs, set prior distribution to have mean = knowledge and small variance \*setting variances -\> anything relatively large/small compared to mean should be fine

Question 33

specify a prior distribution for row parameter

Answer 33

x(i)~gamma(alpha, beta) know M=alpha/beta from given info pick small variance -\> do this by using small CoV variance =(CoV\*mean)^2 then solve for alpha and beta based on variance and M

Question 34

believe AY should be modeled between CL and BF, how to do this in stochastic framework

Answer 34

use xi~gamma(alpha,beta) or gamma(mean, var) select larger variance for a priori estimate ^ set variance to modify beta and reflect credibility weighting between CL and BF larger var -\> closer to CL estimate smaller var -\> closer to BF estimate credibility weighting of CL: Z=%paid/(%paid + psi\*beta)

Question 35

describe how bayesian model will behave and how prior distribution will impact model

Answer 35

lambdas have prior distributions with a mean of volume weighted LDFs and large variance -\> large variance indicates the model will reproduce the CL results from these development factors for AYs development periods (), prior distribution is set to mean of 1.2 and small variance -\> this will pull posterior distribution toward the 1.2 factor instead of what the data alone indicates