Brosius

Question 1

Define Least Squares Method

Answer 1

A credibility weighted expectation of link ratio method (a=0) and budgeted loss method (b=0)

Question 2

Assumptions using Least Squares Method

Answer 2

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

Question 3

Formula for b and a in Least Square method

Answer 3

Question 4

Problem with Least Squares Method

Answer 4

can result in weird numbers (a<0 or b<0)

Question 5

Solutions to resolve weird results from Least Squares Method

Answer 5

(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

Question 5

For Least Squares method, what are the problems if a<0 or b<0? what are some of the possible corrections?

Answer 5

(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

Question 6

Pros of Least Square Method

Answer 6

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

Question 7

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 -

Answer 7

Budgeted Loss Method - decrease loss reserve estimate
BF Method - loss reserve is unchanged
Link Ratio Method - increase loss reserve estimate

Question 8

What’s the purpose of Least-Squares Method

Answer 8

Least-Squares method fits a regression line through the data to estimate developed losses (or loss ratios, as below)

Question 9

Estimates for developed losses for Link Ratio, Budgeted Loss, and Least Squares

Answer 9

Question 10

In what special cases is Least Squares Method equal to Link Ratio or budgeted loss method

Answer 10

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

Question 11

How is the Least Squared Method more flexible than BF method

Answer 11

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

Question 12

what situations will result in problems for estimating parameters for the Least Squares Method

Answer 12

• 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)

Question 13

When is the Least Squares method appropriate to use? When is it inappropriate?

Answer 13

• 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

Question 14

What adjustments should be made to data prior to using Least Squares Development

Answer 14

• 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

Question 15

Bayesian Credibility in a changing system

Answer 15

X/Y is the % emerged during current period

Question 16

What is a caseload effect?

Answer 16

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

Question 17

Calculate Least Square Method Loss with credibility weights

Answer 17

where c is from Link ratio method and d is selected CDF

Question 18

Brosius Bayesian credibility purpose

Answer 18

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

Question 19

Formula for Brosius-Bayesian credibility

Answer 19

Z = VHM/(VHM+EVPV)