Taylor Flashcards
Notation
Y_k,j = incremental losses for AY i and development period j
X_k,j = SUM(Y_k,i) = cumulative losses
f_j = all year volume wtd avg age-to-age factor from development period j to j + 1
d = CY
K AYs/CYs
J development periods
BF Method
A priori loss = P_k * ELR_k
Cape Cod Method
A priori loss = P_k * [SUM(P_i * % emerged_i) * (Culm Loss + R_i) / P_i] / SUM(P_i * % emerged_i)
Uses the same ELR for each AY = wtd average of each AY’s CL LR with weights = Premium * % Emerged
Exponential Dispersion Family (EDF)
ln(pdf) = (y * theta - b(theta)) / a(phi) + c(y,phi)
theta is a location parameter = canonical parameter
phi is a dispersion parameter = scale parameter
b(theta) = cumulant function, determining the shape
exp(c(y,phi)) = normalizing factor
a(phi) = phi / w -> assume w =1
E(Y) = b’(theta) = mu
Var(Y) = a(phi) * b’‘(theta) = a(phi) * V(mu)
where V(mu) is the variance function that depends on the mean
Tweedie Sub-Family
V(mu) = mu^p where p<=0 or p>=1
mu = [(1 - p) * theta]^(1/(1-p)) for p <> 1 (when p = 1, mu = exp(theta) - Poisson)
Assuming a(phi) = phi, Var(Y) = phi * mu^p
p = 0 -> Normal
p = 1 -> ODP
p = 2 -> Gamma
p = 3 -> Inverse Gaussian
Tail heaviness of Tweedie distributions increases as p increases
If a model based on a specific p generates more widely dispersed residuals than are consistent with that model, then an increase in p might be warranted
Over-Dispersed Poisson Sub-Family
p = 1
b(theta) = exp(theta) = mu
a(phi) = phi
Useful when little is known of the subject distribution
Selection of a GLM
- Selection of a cumulant function b(theta), controlling the model’s assumed error distribution
- Selection of an index p, controlling the relationship between the model’s mean and variance
- Selection of the covariates
- Selection of a link function, controlling the relationship between the mean mu_i and the associated covariates
Parameters of a GLM are often estimated using maximum likelihood estimation (MLE)
Scaled Deviance
D = 2 * SUM[loglikelihood(saturated) - loglikelihood(actual)]
Saturated model includes a parameter for every observation so that Estimated = Actual
When this difference is small, then the fitted values are close to the actual value - want to minimize
Unscaled Deviance
Common to minimize unscaled deviance so we don’t need the parameter phi
D* = 2 * SUM[loglikelihood(saturated) - loglikelihood(actual)]
phi can be estimated by D* / (n - p)
Standardized Pearson Residuals
R_i = (Actual - Estimated) / SD_i
Should exhibit unbiasedness (revolve around y=0) and homoscedasticity (constant variance) when plotted against covariates
Will reproduce any non-normality that exists in the observations (ex. skewed loss data
Standardized Deviance Residuals
Much closer to normal than the Pearson residuals, better for model assessment
R_i = sgn(Actual - Estimated) * SQRT(d_i / phi)
sgn produces -1 if negative, 0 if 0, and 1 if positive
d_i is the contribution of the ith observation to the unscaled deviance
Non-Parametric Mack Model
M1: AYs are stochastically independent
M2: The cumulative losses form a Markov chain (means that X_k,j is only dependent on X_k,j-1)
M3a: E(X_k,j+1|X_j,k) = f_j * X_k,j
M3b: Var(X_k,j+1|X_k,j) = sigma_j^2 * X_k,j
Result 1: The conventional CL estimators of f_j are unbiased AND minimum variance estimators among estimators that are unbiased linear combinations of the f_k,j
Result 2: The conventional CL reserve estimate is unbiased
Parametric Mack Models
Assigns distributions to the incremental losses Y_k,j
EDF Mack Model
M1: AYs are stochastically independent
M2: The cumulative losses form a Markov chain (means that X_k,j is only dependent on X_k,j-1)
M3a: E(X_k,j+1|X_j,k) = f_j * X_k,j
M3b: Y_k,j+1 | X_k,j ~ EDF
Tweedie Mack Model
M1: AYs are stochastically independent
M2: The cumulative losses form a Markov chain (means that X_k,j is only dependent on X_k,j-1)
M3a: E(X_k,j+1|X_j,k) = f_j * X_k,j
M3b: Y_k,j+1 | X_k,j ~ Tweedie
ODP Mack Model
M1: AYs are stochastically independent
M2: The cumulative losses form a Markov chain (means that X_k,j is only dependent on X_k,j-1)
M3a: E(X_k,j+1|X_j,k) = f_j * X_k,j
M3b: Y_k,j+1 | X_k,j ~ ODP
Theorem 3.1
The EDF Mack model results in the following assuming the data array is a triangle:
- If M3b holds, then the maximum likelihood estimators (MLEs) of the f_i are the conventional CL estimators (which are unbiased)
- If we are in the special case of the ODP Mack model AND the dispersion parameters are just column dependent, then the conventional CL estimators are minimum variance unbiased estimators (MVUEs). In addition, the cumulative loss estimates and the reserve estimates are also MVUEs
The estimators here are the minimum variance out of ALL unbiased estimators, not just out of the linear combinations of the f_k,j
EDF Cross-Classified Models
The incremental loss random variables are stochastically independent
Y_k,j ~ EDF
E(Y_k,j) = alpha_k * beta_j
SUM(beta_j) = 1
Includes explicit row and column parameters
Theorem 3.2
For an ODP cross classified model with the dispersion parameter identical for all cells, the MLE fitted values and forecasts are the same as those given by the conventional CL method
Theorem 3.3
If ODP cross classified model assumptions apply and the fitted values/forecasts are corrected for bias, then they are MVUEs
Forecasts from the ODP Mack and ODP cross-classified models are identical and the same as those from the conventional CL method despite the different formulations
ODP Cross-Classified Model Algorithm
Iteratively solve for the next beta and alpha parameter, starting with alpha_1 and beta_J
- Set alpha_1 = latest cumulative loss
- Determine other alphas using alpha_k = Cumulative Loss_k / (1 - SUM(remaining beta_j))
- Determine other betas using beta_j = SUM(column k Inc Loss) / SUM(column k alpha_k)
alpha_1 represents the ultimate loss for AY 1
Each beta_j is % incremental emergence
GLM representation of ODP Mack Model
f_k,j - 1|X_k,j ~ ODP(f_j - 1, phi_j / X_k,j)
Identity link function
Weights underlying the variance are the cumulative losses
GLM representation of ODP Cross-Classified Model
Y_k,j ~ ODP(alpha_k * beta_j, phi)
Log link function -> mu_i,j = exp(ln(alpha_i) + ln(beta_j)) = alpha_i * beta_j
Weights underlying the variance are 1
Leads to parameter redundancy/aliasing
