Verrall

Question 1

Important properties of Bayesian models (2)

Verrall

Answer 1

1. can incorporate expert knowledge

2. easily implemented

Question 2

Main ways expert knowledge can be incorporated in reserve estimates (2)

Answer 2

1. change the LDF in some rows due to external info (BF)

2. limit data informing the LDF selection

Question 3

BF estimated reserve formula

Verrall

Answer 3

estimated reserves = expected loss * % reported * (product of age-to-age factors - 1)

expected loss = m-sub i

> > to make incremental change = expected loss * % reported * (age-to-age factor - 1)

Question 4

Key difference between the CL and BF methods

Verrall

Answer 4

BF incorporates external expert knowledge for the level of each row vs. the CL which is based on the data

Question 5

Stochastic CL reserving methods and what each one estimates (4)

Answer 5

1. Mack’s method
2. ODP
3. over-dispersed negative binomial
4. normal approximation to the negative binomial

*ODP estimates incremental losses, all others can be used to estimate cumulative OR incremental losses

Question 6

Advantage of Mack’s CL method

Verrall

Answer 6

simple - parameter estimates and prediction errors can be obtained with a spreadsheet

Question 7

Disadvantages of Mack’s CL method (2)

Verrall

Answer 7

1. no predictive distribution

2. must estimate additional parameters to calculate variance

Question 8

Expected value and variance of incremental claims using ODP methodology
(Verrall)

Answer 8

E[ incremental claims ] = ultimate loss * % emerged
» E[C-sub ij] = x-sub i * y-sub i

Var( incremental claims ) = mean * dispersion factor

Question 9

Advantage of the ODP model

Verrall

Answer 9

produces reserve estimates that are the same as the CL method

Question 10

Disadvantages of ODP model (2)

Verrall

Answer 10

1. column and row sums of incremental claims must be positive
2. hard to see the connection to the CL method

Question 11

Expected value and variance of incremental AND cumulative claims under the over-dispersed negative binomial model

Answer 11

E[ incremental claims ] = (age-to-age factor - 1) * prior cumulative claims
= (lambda-sub j - 1) * D-sub i, j-1

E[ cumulative claims ] = age-to-age factor * prior cumulative claims
= lambda-sub j * D-sub i, j-1

Var(incremental claims) = Var(cumulative claims) = dispersion factor * age-to-age factor * (age-to-age factor - 1) * prior cumulative claims

*constant dispersion factor

Question 12

Advantage of the over-dispersed negative binomial model

Answer 12

results in reserve estimates that are the same as the CL method, with a clearer connection

Question 13

Disadvantage of the over-dispersed negative binomial model

Answer 13

column sum of incremental claims must be positive

Question 14

Enhancement to the normal approximation of the negative binomial model (over the over-dispersed negative binomial)

Answer 14

alters the variance to allow for negative incremental claims

Question 15

Expected value and variance of incremental AND cumulative claims under the normal approximation to the negative binomial model

Answer 15

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

Question 16

Advantage of the normal approximation to the negative binomial model

Answer 16

allows for negative incremental claims

Question 17

Disadvantage of the normal approximation to the negative binomial model

Answer 17

must estimate additional parameters to calculate variance

Question 18

Advantages of Bayesian methods (2)

Verrall

Answer 18

1. full predictive distribution can be found with simulation methods
2. RMSEP can be obtained directly by calculating the standard deviation of the distribution

Question 19

Expected value and variance for prior distribution for BF method

Answer 19

E[x-sub i] = alpha-i / beta-i = m-i

Var(x-sub i) = alpha-i / beta-i^2 = m-i / beta-i

Question 20

Bayesian credibility model for expected incremental claims

Answer 20

E[ incremental claims ] = Z-sub i,j * E[ CL incremental claims] + (1 - Z-sub i,j) * E[ BF incremental claims]

Z-sub i,j = cumulative % emerged / (beta-i * dispersion factor + cumulative % emerged)

Question 21

Column parameters (gamma-sub i) and expected incremental claims

Answer 21

“reverse CL approach”

gamma-i = 1 + (BF resere for AY i / future development period incremental losses for all prior AYs)

E[ incremental claims] = (gamma-i - 1) * sum(incremental losses from prior AYs in column j)