Formula for Least Squares Method (raw formula, Excel)
Excel: LINEST(known y’s, known x’s) => returns b, a
Least Squares as a credibility weighting of the Link Ratio and Budgeted
Loss methods
link ratio: CL; budgeted loss: expected loss
Estimates of developed losses for each method in Brosius
In what special cases is the Least Squares Method equivalent to the link
ratio or budgeted loss methods?
- If developed losses (y) are totally uncorrelated with undeveloped losses (x),
=> then b = 0 and L(x) = a
o This is the budgeted loss method - If the regression line fits through the origin,
=> a = 0 and L(x) = bx
o This is the link ratio method
How is the Least Squares method more flexible than the BF method?
BF Method
Ultimate losses are estimated as expected unobserved loss plus actual observed loss, L(x) = a + x → b = 1
LS Method
Allows b to vary according to the data, L(x) = a + bx
→ b isn’t constrained to 1
→ LS method allows for negative development
What situations result in problems for estimating parameters for the Least
Squares method?
- Significant changes to the nature of the loss experience in the book of business
- Normal sampling error will lead to variance in a and b estimates
For the 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) will be negative for small values of x
o Substitute the link ratio method instead
b < 0
* Estimate of y decreases as x increases
o Substitute the budgeted loss method instead
When is the least squares method appropriate to use? When is it
inappropriate?
- Appropriate if we have a series of years of data where we can assume stable distributions for Y and X
o We assume fluctuations are driven by random chance - Inappropriate if year-to-year changes are due to systematic changes in the book of business (e.g. mix shift).
o 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
What is the caseload effect?
Credibility development formula allowing for the caseload effect
Formula for the best linear approximation to Q(x), the Bayesian estimate
3 advantages of the best linear approximation as a replacement for the Bayesian estimate
- simpler to compute
- easier to understand and explain
- less dependent upon the underlying distribution
Using the best linear approximation to Q(x), how does the relationship
between Cov(X,Y ) and Var(X ) impact the response to loss reserves for a
large reported loss
A large reported loss (increasing x) changes the loss reserves according to
the three different answers to Hugh White’s question. For x > E[X ]:
* Cov(X,Y ) < Var(X ) - loss reserve decreases
* Cov(X,Y ) = Var(X ) - loss reserve unaffected (ultimate loss increases
. by the increase to x)
* Cov(X,Y ) > Var(X ) - loss reserve increases
Based on Hugh White’s Question, if reported losses are higher than
expected, what are three different responses to estimated loss reserves and
the reasoning behind each response?
If x is greater than E[X ], possible responses to the loss reserve estimate are:
- Reduce the reserve by a corresponding amount
o You believe loss reporting is accelerating, this is the Budgeted Loss method (fixed prior case) - Leave the reserve at the same percent of expected losses
o You believe there was a random fluctuation (e.g. large loss), this is the BF method - Increase the reserve proportionally to the increase in actual reported loss
o You’re not very confident in E[Y ], this is the link ratio method
