Verrall Flashcards
stochastic models for CL
- Mack
- over dispersed poisson
- over dispersed negative binomial
- normal approx to NB
MACK
-only on cumulative loss
Bayesian compared to the Mack, the full distribution can be easily calculated & the prediction error can be calculated
ODP
- increm loss
- since neg increm values are possible with reported data, preferable to use paid loss or claim counts
- not obvious it produces CL
ODNB
- 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
normal approx to NB
- allows for neg increm claims
- more parameters
2 areas where expert knowledge is applied
- BF method (row parameters/AY ultimates)
- Insertion of prior knowledge about individual DFs in CL (unlike Bootstrapping)
Bayesian models have 2 important properties
- Can incorporate expert knowledge
- Can be easily implemented
estimate for outstanding losses: CL
estimate for outstanding losses: BF
prediction variance
process variance + estimation variance
prediction error will be [] if less confident in expert opinion
higher
When comparing prediction errors
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
difficulty in calculating the prediction error highlights a few advantages of Bayesian methods
- full predictive distribution can be found using simulation methods
- RMSEP can be obtained directly by calculating the std dev of method
2 cases of intervention in estimation of DFs for CL
- DF changed in some rows due to external information
- DFs = 5yr volume weighted average rather than all of the available data in the triangle
Incorporating Expert opinion about DFs
- 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
if W is large
DF will be pulled closer to CL DF and reserve will closely resemble CL reserve
if W is small
DF will be pulled closer to prior mean and reserve will move away from CL reserve
using BF
- 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
smaller B implies
we are more unsure about M
Bayesian Model for BF (BAYESIAN MEAN RESERVE) -> E[Cij]
formula for Z
beta can control Z
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)
to modify Bayesian framework ->
insert row parameter for each AY and specify low variances
Estimating column parameters (BF RESERVE)
- 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)
fully specify Bayesian model when using 5yr volume weighted average
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
if estimating E[Cij] for multiple periods ie E[Cij] and E[Cij+1] using bayesian credibility
for CL: Di,j-1 will come from latest diagonal and estimated incrementals using CL
for BF: need to do nothing special
estimating incrememtal losses with bayesian credibility -> another way to look at it/calculate
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
if incremental losses are assumed to follow ODNB (for Bayesian using BF) then E[Cij]
E[Cij]=(gamma(i)-1)*sum(Cmj)
how to calculate gamma(i)
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
how to calculate expected ultimate loss when given gammas and the incremental losses follow ODNB
use E[Cij]
**need to start with oldest year first and then go down from there since this process is iterative for newer AYs
estimating reserve for fully stochastic BF and the benefit compared to Bayesian BF based on ODP
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
specify the prior distributions for DFs lambda(i,j) to be used in Bayesian model that will produce CL besides prior knowledge
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
specify a prior distribution for row parameter
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
believe AY should be modeled between CL and BF, how to do this in stochastic framework
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)
describe how bayesian model will behave and how prior distribution will impact model
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
