Z for link ratio method
slope / c
where c = avg ult LR / avg undeveloped LR
When to use losses vs loss ratio for least squares method
use LR when significant premium growth; otherwise loss is fine
solution when intercept is negative
link ratio method (LDF = ybar / xbar)
solution when slope is negative
use budgeted loss, aka ELR method
VHM
( E[%pd] * sigma(Ult) )^2
EVPV
sigma(%pd)^2 * ( sigma(Ult)^2 + E[Ult]^2 )
Z
VHM / (EVPV + VHM)
Bayesian Ult
Z * PdLosses / E[%pd] + (1-Z)*E[Ult]
Caseload Effect setup
E[X|Y=y] = d*y + x0
calculate parameters that define development ratio for modified link ratio method
Caseload Bayesian Ult: assume LDFs vary by caseload
Same setup as basic Bayesian method for Z
Create system of equations; set up two scenarios, as given in the problem
Calculate parameters that satisfy: E[X|Y=y] = d*y + x0
Calculate cred-weighted Ult:
yhat = Z(x - x0)/d + (1-Z)E[Y]
Where E[Y] uses baseline assumptions
Best Linear Approx of Bayesian Loss Estimate
(x-E[X])*Cov(X,Y)/Var(X) + E[Y]
(RepLosses - E[Rep Losses]) * Cov/Var(X) + E[Ult Losses]
Advantage of Least Squares
more flexible than link ratio, BF, and budgeted loss
credibility weighting of link ratio and budgeted loss
produces more reasonable results when the data has random, severe year-to-year fluctuations (like in a small book of business)
Adjustments to data for Least Squares
if large exposure change, divide losses by premium
adjust for inflation! incurred loss data should be on a constant-dollar basis
Hugh White’s question - reported losses(x) come in higher than expected
- Budgeted Loss Method (fixed prior case) – The ultimate loss estimate is fixed, so we decrease the loss reserve estimate by the same amount as the unexpected increase in reported losses. This method
treats the increased loss as losses coming in faster than expected. - BF Method – The ultimate loss estimate increases by the amount losses were greater than expected.
The loss reserve is unchanged. The BF method treats the unexpected increased loss as a random fluctuation (e.g. a large loss). - Link Ratio Method (fixed reporting case) – The ultimate loss estimate increases in proportion to
the excess losses by applying the LDF, so we increase the loss reserve estimate. This method assumes that a fixed percentage of ultimate losses is reported, so if reported losses increases, the ultimate loss estimate will increase proportionally.
What is the best linear approximation to the Bayesian estimate
Least Squares - lowest MSE
Key assumption of Least Squares
Steady distribution of random variables X and Y; there should not be a systemic shift in the book of business
