The equations for each of the described methods:
Link Ratio: L(x) = cx [works with fixed reporting case]
Budgeted Loss: L(x) = k [works with fixed prior case]
Bornhuetter-Ferguson: L(x) = a + x [works when claims are produced by a
Poisson-Binomial mixed process]
The Least Squares Method: L(x)=a + bx [works with each of these scenarios
plus more including the Negative Binomial-Binomial mixed process]
b=[E(xy) - E(x)E(y)]/(E(x^2)-E(x)^2) {if less than 0, substitute link
ratio method}
a=E(y) - bE(x) {if less than 0, substitute budgeted loss method}
* assumes years with common Y and X, may need to adjust losses for
inflation or exposures
The three Development Formulae
Development Formula 1:
L(x) = (x-E(X))[(COV(X,Y))/(VAR(X)] + E(Y)
so if COV(X,Y)<VAR(X) a large reported amount should decrease the
reserve
COV(X,Y)>VAR(X) a large reported amount should increase the reserve
COV(X,Y)=VAR(X) a large reported amount shouldn't change the reserve
Development Formula 2 & 3:
Suppose there is a real d<>0 and xo such that E(x|Y=y)=dy +xo for all y
L(x)=Z((x-xo)/(d)) + (1-Z)E(Y)
where Z=(VHM)/(VHM+EVPV)
This is development formula 3. Development formula 2 is the same where
xo=0.
I am also going to memorize the method used in developing the losses in the
final example, in case there are any applied problems related to this
paper.
Am I missing any major points of the paper? It would be nice to have a
discussion of this papers and the others to try to learn what everyone
thinks are the most important points. Brian
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