Clark Flashcards
Expected Loss Emergence
- 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/LDFx
- x is the time in months from the avg accident date to the evaluation date
Loglogistic Function
- G(x | ω, Θ) = xω/(xω +Θω)
- LDFx = 1 + Θω + x-ω
Weibull Function
- G(x|ω,Θ) = 1- exp(-(x/Θ)ω)
*
Paramterized Curves Assumption
- A strictly increasing pattern is assumed
- If negative development is expected (salvage) use different models
Advantages of Paramterized Curves
- 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
Methods for Estimation
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
LDF Method
- µAY;x,y = ULTAY * [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
Cape Cod Method
- PremiumAY * 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
Process Variance
- 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)ΣAY,tn * (cAY,t - µAY,t)2/µAY,t
- n = #data points, p=#paramters, cAY,t= actual increment loss emergence, µAY,t=expected increment loss emerg
- Assume actual loss emergence follows over-dispersed Poisson distribtuion with a scaling factor of σ2
Over-Dispersed Poisson Distribution
- 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
Parameter Variance
- 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
The likelihood function
- loglikelihood for an overdispersed Poisson distribution needs to be maximized
- l = Σici * ln(µi) - µi
- Advantage of the maximum loglikelihood function is that it works in the presence of negative or zero incremental losses
MLE LDF Method
- ULTi = Σt (ci,t) / Σt[G(xt) - G(xt-1)]
- The MLE estimate for each ULTi is equivalent to the “LDF Ultimate”
- ci,t = actual loss in AY i, development period t
- xt-1 = beginning age for development period t
- xt = ending age for development period t
MLE Cape Cod Method
- ELR = Σt (ci,t) / (Σi,t Pi* [G(xt) - G(xt-1)])
- ci,t = actual loss in AY i, development period t
- xt-1 = beginning age for development period t
- xt = ending age for development period t
- Pi = premium for AY i
- The MLE estimate for the ELR is equivalent to the Cape Cod Ultimate
Variance of the Reserves
- In general CC Method has a lower total variance
- Process Variance of R = σ2 * ΣµAY;x,y
- Paramter Variance of R = too complicated
Key Assumptions of the Model (Clark)
- 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
Residual calculation
- rAY;x,y = (cAY;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
Test Constant ELR Assumption in CC model
- Graph the ultimate loss ratios by AY
- If an increasing or decreasing pattern exists assumption doesnt hold
Variance of the prospective losses
- 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
Calendar Year Development
- 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
Variability in discounted reserves
- 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
Triangles vs Tabular
- The MLE model works best when using a tabular format
- We need a consistent aggregation of losses evaluated at more than one date
Reason for using the Curves
- 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)
Variance in Discounted Reserves
- Rd = ΣAYΣk=1 to y-x UltAY *vk-1/2 * (G(x+k) - G(x+k-1))
- Var(Rd) =σ2ΣAYΣk=1 to y-x UltAY *v2k-1 * (G(x+k) - G(x+k-1))
- Discounted reserve and process variance
- v= 1/(1+i)