Clark

Question 1

Expected Loss Emergence

Answer 1

• Model will estiamte the expected amount of loss to emerge based on
  
  • An estimate of the ultimate loss by year
  • An estiamte of the patter of loss emergence
• G(x) = 1/LDF_x
  
  • x is the time in months from the avg accident date to the evaluation date

Question 2

Loglogistic Function

Answer 2

• G(x | ω, Θ) = x^ω/(x^ω +Θ^ω)
• LDF_x = 1 + Θ^ω + x^-ω

Question 3

Weibull Function

Answer 3

• G(x|ω,Θ) = 1- exp(-(x/Θ)^ω)
  *

Question 4

Paramterized Curves Assumption

Answer 4

• A strictly increasing pattern is assumed
• If negative development is expected (salvage) use different models

Question 5

Advantages of Paramterized Curves

Answer 5

• Estimation is simple since we only have to estimate two paramters
• We can use data that is not from a triangle with evenly spaced evaluation data. Such as when latest diagonal is only nine months from second latest diagonal.
• The final pattern is not smooth and does not follow random movements in the historical age-to-age factors

Question 6

Methods for Estimation

Answer 6

LDF Method- Assumes the loss amount in each AY is independent from all other years

Cape Cod Method- Assumes that there is a known relationship between expected ultimate losses across accident years, where the relationship is identified by an exposure base

Question 7

LDF Method

Answer 7

• µ_AY;x,y = ULT_AY * [G(y|ω,Θ)-G(x|ω,Θ)]
• The LDF method requres an estimation of a number of paramters, it tends to be overparamterized when few data points exist

Question 8

Cape Cod Method

Answer 8

• Premium_AY * ELR * [G(y|ω,Θ)-G(x|ω,Θ)]
• The CC Method is preferred since data is summarized into a loss triangle with relatively few data points

Question 9

Process Variance

Answer 9

• The “random” amount
• Cape Cod method can have higher or lower process variance than LDF method
• Assume that the loss in any period has a constant ratio of variance/mean
• variance/mean = σ²≈ 1/(n-p)Σ_AY,tⁿ * (c_AY,t - µ_AY,t)²/µ_AY,t
• n = #data points, p=#paramters, c_AY,t= actual increment loss emergence, µ_AY,t=expected increment loss emerg
• Assume actual loss emergence follows over-dispersed Poisson distribtuion with a scaling factor of σ²

Question 10

Over-Dispersed Poisson Distribution

Answer 10

• E[c] = λσ² = µ
• Var[c] = λσ⁴ = µσ²
• Key Advantages
  
  • Inclusion of scaling factors allow us to match the first and second moments of any distribution, making highly flexible
  • Maximum likelihood estimation produces the LDF and Cape Cod estimates of ultimate losses, so the results can be presented in a familiar format

Question 11

Parameter Variance

Answer 11

• The uncertainty in the estimator, also known as the estimation error
• Cape Cod Method has a smaller parameter variance
• We need the covariance matrix to calculate the parameter variance
• VERY COMPLEX

Question 12

The likelihood function

Answer 12

• loglikelihood for an overdispersed Poisson distribution needs to be maximized
• l = Σ_ic_i * ln(µ_i) - µ_i
• Advantage of the maximum loglikelihood function is that it works in the presence of negative or zero incremental losses

Question 13

MLE LDF Method

Answer 13

• ULT_i = Σ_t (c_i,t) / Σ_t[G(x_t) - G(x_t-1)]
• The MLE estimate for each ULT_i is equivalent to the “LDF Ultimate”
• c_i,t = actual loss in AY i, development period t
• x_t-1 = beginning age for development period t
• x_t = ending age for development period t

Question 14

MLE Cape Cod Method

Answer 14

• ELR = Σ_t (c_i,t) / (Σ_i,t P_i* [G(x_t) - G(x_t-1)])
• c_i,t = actual loss in AY i, development period t
• x_t-1 = beginning age for development period t
• x_t = ending age for development period t
• P_i = premium for AY i
• The MLE estimate for the ELR is equivalent to the Cape Cod Ultimate

Question 15

Variance of the Reserves

Answer 15

• In general CC Method has a lower total variance
• Process Variance of R = σ² * Σµ_AY;x,y
• Paramter Variance of R = too complicated

Question 16

Key Assumptions of the Model (Clark)

Answer 16

• Assumption 1: Incremental losses are independent and identically distributed
  
  • Independence means one period doesnt affect another. Do residual analysis. Positive correlation if loss inflation. Negative correlation if large settlement affects later periods
  • Identically distributed assumes emergence patter for all AYs. Different risk and mix of business would have been written in each historical period
• Assumption 2: The variance/mean scale parameter σ² is fixed and known
  
  • Technically, σ² should be estimated with other paramters but this is messy math
• Assumption 3: Variance estimates are based on approximation of the Rao-Cramer lower bound
  
  • Estimate of variance based on info matrix is only exact when using linear functions
  • Our model is non linear so use Rao-Cramer

Question 17

Residual calculation

Answer 17

• r_AY;x,y = (c_AY;x,y - µ^(hat)_AY;x,y)/sqroot(σ² *µ^(hat)_AY;x,y)
• Plot the residuals against a number of things
  
  • Increment age, expected loss in increment (tp test if var/mean is constant), AY, CY(diagonal effects)
• Want residuals to be randomly scattered around the zero line

Question 18

Test Constant ELR Assumption in CC model

Answer 18

• Graph the ultimate loss ratios by AY
• If an increasing or decreasing pattern exists assumption doesnt hold

Question 19

Variance of the prospective losses

Answer 19

• Uses the CC Method
• If we have an estimate of future year premium we can calculate expected loss becasuse we have max likelihood ELR
• Process variance is calculated as usual

Question 20

Calendar Year Development

Answer 20

• Instead of estimating IBNR for AY, we can estimate development for next CY period
• For next 12 months take difference in growth functions at two evalulation ages and multiply by estimated ult losses
• Process and paramter variance calculated as usual
• Major reason for calculating the 12 month development is that the estimate is testable w/in a short timeframe
• One yr later we can compare it to actual development

Question 21

Variability in discounted reserves

Answer 21

• Uses the same payout pattern and model parameters that were used with undiscounted reserves
• The coefficient of variation for discounted reserves is lower since the tail of the payout curve has the greates paramter variance and also receives the deepest discount

Question 22

Triangles vs Tabular

Answer 22

• The MLE model works best when using a tabular format
• We need a consistent aggregation of losses evaluated at more than one date

Question 23

Reason for using the Curves

Answer 23

• Weibull and loglogistic go smoothly from 0% to 100%
• Closely match the empirical data
• First and seconds derivatives are calculable
• Method is not limited to these forms; other curves could be used
• Main conclusion is that paramter variance is generally larger than the process variance (need for complete data outweighs the need for more sophisticated models)

Question 24

Variance in Discounted Reserves

Answer 24

• R_d = Σ_AYΣ_{k=1 to y-x} Ult_AY *v^k-1/2 * (G(x+k) - G(x+k-1))
• Var(R_d) =σ²Σ_AYΣ_{k=1 to y-x} Ult_AY *v^2k-1 * (G(x+k) - G(x+k-1))
• Discounted reserve and process variance
• v= 1/(1+i)