Brosius' Least Squares Flashcards
When is Least Squared Appropriate?
Fluctuations are caused by random chance rather than changes in the book of business
Otherwise can be adjusted:
* Adjust for inflation
* If book of business is growing (but doesn’t change characteristics), divide each year’s loss by exposure
* Berquist-Sherman type techniques
Advantages / Disadvantage of Lease Squares
Advantage
* Flexible enough to include chain ladder, ELR and B-F method, while allowing for the best fit
* Works well when there are significant random fluctuations year to year
Disadvantage
* Does not work well when fluctuations year to year are systematic (book of business is not stable)
* Sampling error can lead to values of a and b that don’t make sense (slope or y-intercept can be < 0)
Chain Ladder, ELR, Credibility-Weighted Ultimate Loss
All Formula
y = Z * C-L Ult + (1-Z) * ELR Ult
C-L Ult = x * LDF
ELR Ult = avg(y)
LDF = avg(y) / avg(x) or sum(y) / sum(x)
Z = b / LDF where b is the slope
y is ult loss, x is cumulative losses to date
Variance of the Hypothetical Mean, VHM = Var[E(X|Y)]
Formula
VHM = E(X/Y)^2 * sd(Y)^2
= (expected % reported)^2 * (SD of Ult Loss)^2
Variability from loss occurrence process
(X/Y) = expected % reported
Expected Value of Process Variance, EPV = E[Var(X|Y)]
Formula
EPV = std(X/Y)^2 * [E(Y)^2 + std(Y)^2]
= (SD of percent reported)^2 * [(expected ult loss)^2 + (SD of ult loss)^2]
Variability from loss reporting process
(X/Y) = expected % reported
Buhlmann Credibility, Z
Credibility Weighted Expected Ult, L(X)
All formula
Z = VHM / (VHM + EPV)
L(X) = Z * X / E[X/Y] + (1-Z) * E(Y)
(X/Y) = expected % reported
Caseload Effect
E(X|Y=y) & Credibility Weighted Ultimate Loss L(x)
What is it? + Formula
Expected Reported Loss Given Ult Loss = E(X/Y) * y = dy+x0
L(x) = Z * (x-x0)/d +(1-Z) * E(Y)
* Z = VHM / (VHM +EPV)
* E(Y) = ELR Ult
* Do NOT use hypothetical numbers when calculating
d = constant %
