Total Variance
Parameter Variance + Process Variance
Process Variance
Ratio of Variance:Mean = (Sum of ChiSq Errors) / (n-p)
ChiSq error = (actual - expected)^2 / expected
n
number of data points in triangle
p (general)
number of parameters
1 + #parameters in growth curve
p for LDF method
AYs + 2
Ults, theta, w
p for Cape Cod method
3
ELR, theta, w
p for expected loss emergence
parameters in G(x) + #AYs
Loglogistic emergence curve G(x|theta, w)
x^w / (x^w + theta^w)
Weibull emergence curve G(w|theta, w)
1 - exp(-(x/theta)^w)
LDF method likelihood function
Calculate expected Incremental triangle
MLE term = Act Inc * ln(Exp Inc) - Exp Inc
l = Sum( MLE terms)
The best fitting parameters will maximize l
Ulthat = sum(Cit) / sum(G(xt) - G(xt-1))
Cape Cod method likelihood function
Calculate expected Incremental triangle
MLE term = Act Inc * ln(Exp Inc) - Exp Inc
l = Sum( MLE terms)
The best fitting parameters will maximize l
ELRhat = sum(Cit) / sum[ Premi * (G(xt) - G(xt-1)) ]
Key assumptions of the model
- Incremental losses are independent and iid (test with residual analysis)
- Variance/mean scale parameter sigma^2 is fixed and known
- Variance estimates are based on an approximation of the Rao-Cramer lower bound (makes information matrix more exact)
Cape Cod Reserve
ELR * OLP * (1-G(x))
where x is the average age for the AY
ELR always done before truncation = sum of losses to date / sum of used up prem
Cape Cod Parameter Variance from information matrix
Var(ELR) * Premium^2
normalized residual
(actual - expected) / sqrt(sigma^2*expected)
Clark usually does incremental values
Variance of Reserves (LDF Method) general steps
Set up table of AY | Avg Age | Cum Loss | G(x) | CDF (to trunc point if relevant) | Loss at Ult (trunc) | Estimated Reserve
Calculate expected incrementals, then get Chi-Sq error (act-exp)^2/exp
sigma^2 = Chi-Sq error / (n-p)
Process Var = sigma^2*Reserve
Total Var = Process + Param Var
Variance of Reserves (Cape Cod Method) general steps
Set up tale AY | Avg Age | G(x) | Cum Loss | Prem | Used Prem = G(x)*Prem
ELR = Sum(Losses to Date)/Sum(Used Prem)
Calculate expected ult losses as Full Prem * ELR
%Unpaid is limited at truncation point if applicable
Calculate expected incrementals, then get Chi-Sq error (act-exp)^2/exp
sigma^2 = Chi-Sq error / (n-p)
Process Var = sigma^2*Reserve
Total Var = Process + Param Var
Coefficient of Variance
StdDev(Reserve) / Reserve
