Taylor

Question 1

Calculate the Expected Value and Variance for the Tweedie distribution.

Answer 1

E(Y) = u = ((1-p)theta)^(1/(1-p))
V(Y) = phiu^p

p = 0 if Normal (u = theta)
p = 1 if ODP (u = exp(theta))
p = 2 if Gamma (u = -1/theta)
p = 3 if Inverse Gaussian (-2*theta)^-0.5

Question 2

3 selections to specify a GLM

Answer 2

1. Error distribution (must be EDF), including index p
2. Explanatory variables (Xis)
3. Link function h() (ex: identity, log)

Question 3

Calculate GLM deviance

Answer 3

D* = 2*(ll_saturated - ll_model)

Saturated model has a parameter for each observation so the model completely fits the observations.

Question 4

Calculate Standardized Pearson Residuals.
How do you use them to validate model?

Answer 4

RiP = (Yi - Yihat)/sihat

Standardized Pearson residuals should be random around zero (unbiased) and have uniform dispersion (homoscedasticity)

Question 5

Calculate Standardized Deviance Residuals

Answer 5

RiD = sgn(Yi - Yihat)*sqrt(di/phi)

di is the Contribution of observation Yi to deviance D*

Standardized deviance residuals should be normally distributed.

Question 6

Provide 1 solution if a model based on specific p generates more widely dispersed residuals than are consistent with that model.

Answer 6

Increase p

Question 7

Estimate scale parameter (phi) from Deviance

Answer 7

phihat = D*/(n-p)
n = # observations
p = # parameters

Question 8

Provide a solution if residual plot exhibits heteroscedasticity (non-homogeneous variance around 0)

Answer 8

Use weights

Observations should be assigned weights that are inversely proportional to the variance of the residuals.

We can remove the influence of outliers by assigning them weights of 0 in the model.

Question 9

Summarize the Non-Parametric Mack Model Assumptions

Answer 9

1. Accident years are stochastically independent.
2. For each AY k, cumulative losses Xkj form a Markov Chain
3. For each AY k and development period j
   E(Xk,j+1 given Xk,j) = fj*Xkj (fj is the LDF)
   V(Xk,j+1 given Xk,j) = sj^2 * Xk,j

Question 10

Summarize the 2 results from the Mack Model

Answer 10

1. The conventional chain ladder estimators (LDF = fi) are
   a) Unbiased
   b) Minimum variance among estimators that are unbiased linear combinations of the age-to-age factors
2. The conventional chain ladder reserve estimator, Rk, is unbiased

Question 11

Summarize the 2 parametric Mack Model Assumptions

Answer 11

Same assumptions as Mack Model, except:
1. Yk,j+1 given Xk,j follows EDF(theta_kj, phi_kj ; alb,c)
2. Variance assumption is removed - Variance is driven by the selected EDF distribution

Question 12

Briefly explain Theorem 3.1 regarding EDF Mack Model

Answer 12

a. If the original Mack variance assumption also holds, then the MLEs of the fj parameters are the chain ladder LDF estimators, fjhat, and these are unbiased estimators.

b. If the model is restricted to the ODP Mack model and if the dispersion parameters are just column dependent, phi_k,j = phi_j, then the weighted average chain ladder LDFs, fjhat, are MVUEs.

c. For the same conditions as (b), the cumulative loss estimates, That_k,j and the reserve estimates, Rhat_k, also MVUEs.

Question 13

Briefly explain the implication of Theorem 3.1

Answer 13

This theorem means that the conventional chain ladder estimates and forecasts are optimal estimators (both MLE and MVUEs).

This is a stronger implication than the original Mack model.

It shows the LDF estimators, fhatj, are minimum variance for all unbiased estimators, not just of the linear combinations of the age-to-age factors.

Question 14

Explain the 2 Cross-Classified Model assumptions

Answer 14

1. Random variables Y_k,j are stochastically independent.
2. For each accident year k and development period j:
   Y_k,j+1 given X_k,j follows EDF(theta, phi; a,b,c)
   E(Y_k,j) = ak*bj
   sum(bj) = 1

Question 15

Briefly explain theorem 3.2 regarding Cross-Classified Model

Answer 15

For the Cross-Classified, if the following assumptions also hold:
1. Y_k,j is restricted to an ODP distribution
2. The dispersion parameters are identical for all cells: phi_k,j = phi

The, the MLE fitted values Y_k,j are the same as those from the conventional chain ladder method.

Question 16

Briefly explain theorem 3.3 regarding Cross-Classified Model

Answer 16

If the theorem 3.2 assumptions hold AND the fitted values Yhat_k,j and reserve estimates Rhat_k are corrected for bias, then they are MVUEs of Y_k,j and Rk.

Question 17

Briefly explain the results of Taylor’s theorems

Answer 17

Theorems 2 and 3 are similar to theorem 1 about ODP mack model and mean:
1. Forecasts from ODP Mack and ODP Cross-Classified models are identical and the same as those from the chain ladder method despite the different formulations.
2. We can get forecasts for the ODP cross-classified model without considering the model directly and working as if the model was an ODP Mack model.

Question 18

Describe Y, X and b matrices in ODP Mack Model GLM representation.

Answer 18

See image

Question 19

Describe the Y, X and b matrices in the ODP cross-classified model GLM representation (4x4 triangle)

Answer 19

See image

Question 20

Calculate ak, bj and forecasts using Cross-Classified Model Parameters.

Answer 20

See image

Question 21

Calculate re-normalized ODP Cross-Classified Model parameters

Answer 21

ak_norm = ak*sum(bj)
bj_norm = bj/sum(bj)