Verrall Flashcards
Important properties of Bayesian models (2)
Verrall
- can incorporate expert knowledge
- easily implemented
Main ways expert knowledge can be incorporated in reserve estimates (2)
- change the LDF in some rows due to external info (BF)
- limit data informing the LDF selection
BF estimated reserve formula
Verrall
estimated reserves = Mean * % Unpaid
expected incremental paid = M_i * y_i
Key difference between the CL and BF methods
Verrall
BF incorporates external expert knowledge for the level of each row vs. the CL which is based on the data
Stochastic CL reserving methods and what each one estimates (4)
- Mack’s method
- ODP
- over-dispersed negative binomial
- normal approximation to the negative binomial
*ODP estimates incremental losses, all others can be used to estimate cumulative OR incremental losses
Advantage of Mack’s CL method
Verrall
simple - parameter estimates and prediction errors can be obtained with a spreadsheet
Disadvantages of Mack’s CL method (2)
Verrall
- no predictive distribution
- must estimate additional parameters to calculate variance
Expected value and variance of incremental claims using ODP methodology
(Verrall)
E[ incremental claims ] = ultimate loss * % emerged
» E[C-sub ij] = x-sub i * y-sub i
Var( incremental claims ) = mean * dispersion factor
Advantage of the ODP model
Verrall
produces reserve estimates that are the same as the CL method
Disadvantages of ODP model (2)
Verrall
- column and row sums of incremental claims must be positive
- hard to see the connection to the CL method
Expected value and variance of incremental claims under the over-dispersed negative binomial model
E[C_i,j] = (lambda - 1) * D_i,j-1
Var[C_i,j] = Var[D_i,j] = phi * lambda * E[C_i,j]
C = incremental
D = cumulative
lambda = LDF
Advantage of the over-dispersed negative binomial model
results are the same as ODP = CL
method looks like chainladder so easier to explain
Disadvantage of the over-dispersed negative binomial model
column sum of incremental claims must be positive
Enhancement to the normal approximation of the negative binomial model (over the over-dispersed negative binomial)
alters the variance to allow for negative incremental claims
Expected value and variance of incremental AND cumulative claims under the normal approximation to the negative binomial model
E[ incremental claims ] and E[ cumulative claims] are the same as the over-dispersed negative binomial model
Var(incremental claims) = Var(cumulative claims) = dispersion factor * prior cumulative claims
*dispersion factor for each column
Advantage of the normal approximation to the negative binomial model
allows for negative incremental claims
Disadvantage of the normal approximation to the negative binomial model
must estimate additional parameters to calculate variance
Advantages of Bayesian methods (2)
Verrall
- full predictive distribution can be found with simulation methods
- RMSEP can be obtained directly by calculating the standard deviation of the distribution
Expected value and variance for prior distribution for BF method
E[x-sub i] = alpha-i / beta-i = m-i
Var(x-sub i) = alpha-i / beta-i^2 = m-i / beta-i
Bayesian credibility model for expected incremental claims
E[C_i,j] = CL * Z_i + BF * (1 - Z_i)
where
Z_i,j = p_j-1 / (B_i * phi + p_j-1)
p_j-1 is the expected % paid to date
B_i is the beta from the prior distribution
Column parameters (gamma-sub i) and expected incremental claims
-reverse CL approach for CL parameterization
-iterative x_i * q^i for ODP with stochastic column parameters
(q^i is an index not an exponent)
E[C_ij] = (gamma_i - 1) * Sum(C_mj)
