Define Least Squares Method
A credibility weighted expectation of link ratio method (a=0) and budgeted loss method (b=0)
Assumptions using Least Squares Method
LSM assumes a steady distribution of random variables X and Y
It’s inappropriate if there is a systematic shift in the book of business
Formula for b and a in Least Square method
Problem with Least Squares Method
can result in weird numbers (a<0 or b<0)
Solutions to resolve weird results from Least Squares Method
(a<0): this will cause the estimate of developed losses (y) to be negative for small values of x. -> use the link ratio method instead
(b<0): this causes the estimate of y to decrease as x increases -> use budgeted loss method instead
For Least Squares method, what are the problems if a<0 or b<0? what are some of the possible corrections?
(a<0): estimate of developed losses (y) to be negative for small values of x. -> use the link ratio method instead
(b<0): estimate of y to decrease as x increases -> use budgeted loss method instead
Pros of Least Square Method
More flexible than link ratio, budgeted loss, and BF methods
Credibility weighting of the link ratio and budgeted loss estimates. Gives more/less credibility to the loss experience as appropriate
Hugh White’s Question -
Given the reported losses come in higher than expected, how will the different methods estimate different changes to outstanding loss reserve -
Budgeted Loss Method - decrease loss reserve estimate
BF Method - loss reserve is unchanged
Link Ratio Method - increase loss reserve estimate
What’s the purpose of Least-Squares Method
Least-Squares method fits a regression line through the data to estimate developed losses (or loss ratios, as below)
Estimates for developed losses for Link Ratio, Budgeted Loss, and Least Squares
In what special cases is Least Squares Method equal to Link Ratio or budgeted loss method
If future losses are totally uncorrelated to undeveloped losses, then b = 0 and L(x) =a
this will result in budgeted loss method
If the regression lien fits the origin, a = 0 and L(x)=bx
this will result in CL
How is the Least Squared Method more flexible than BF method
BF - Ult losses fixed b=1 so BF-Ult always equal to incurred losses + expected reserve
Least Squares allows b to vary and adjust this proportion of incurred losses to be weighted
Also LS method allows for negative development
what situations will result in problems for estimating parameters for the Least Squares Method
- Significant changes to the nature o the loss experience in the book of business
- Normal sampling error will lead to variance in a and b estimates (aka, noise instead of signal in loss data itself)
When is the Least Squares method appropriate to use? When is it inappropriate?
- appropriate is we have a series of years of data where we can assume stable distributions for Y and X
— we assume fluctuations are driven by random chance - inappropraite if yoy changes are due to systematic changes in the book of business (mix-of-shift)
— other methods such as Berquist-Sherman may be better
What adjustments should be made to data prior to using Least Squares Development
- If using incurred losses, correct data for inflation to put losses on a constant-dollar basis
- If there is significant growth in the book, divide losses by an exposure basis to correct the distortion
Bayesian Credibility in a changing system
X/Y is the % emerged during current period
What is a caseload effect?
when the caseload is low, a claim is more likely to be reported in a timely fashion than when caseloads are high
thus we expect the development ratio to be not a constant but a decreasing function of y
Calculate Least Square Method Loss with credibility weights
where c is from Link ratio method and d is selected CDF
Brosius Bayesian credibility purpose
if the book of business changes significantly, we can’t use the regular Least Squares. If we make a few assumptions about expected ult losses and percent reported then we can still use the data to calculate a Bayesian credibility of ult loss
Formula for Brosius-Bayesian credibility
Z = VHM/(VHM+EVPV)
