
Link ratio, budgeted loss, least square
 Link ratio: L(x) = cx
 where c = y/x
 Budgeted loss: L(x) = k
 Least square: L(x) = a + bx
 where b = (xyx*y)/(x²  (x)²) and a = y  bx

Hugh White's Questions
 If rpt loss is greater than expected, do you
 Reduce bulk reserve by corresponding amt (BL)
 Leave bulk reserve a same % of exp loss (BF)
 Increase bulk in proportion (LR)

Loss reporting distributions
 X = rpt nb clm, Y = ult nb clm
 Q(x) = E(YX = x)
 R(x) = E(Y  XX = x)

PoissonBinomial distribution
 Poisson(μ), Binomial(r,δ)
 Q(x) = x + μ(1  δ)
 R(x) = μ(1  δ)
 BF is optimal in this case
 Note: no answer optimal for Neg Bin

LS: PoissonBinomial Case
 Poisson(μ), Binomial(r,δ)
 Q(x) = x + μ(1  δ)
 R(x) = μ(1  δ)
 BF is optimal in this case
 Note: no optimal case for Neg Bin

When is least square method appropriate
 If yr to yr chg are due largely to systematic shifts in the book of business, other methods may be more appropriate
 If rdm chance is the primary cause of flucuations, the least square should be considered

Lest square cred dvpt formula
 L(x) = Z(x/d) + (1  Z)E(Y)
 Z = VHM / (VHM + EVPV)
 VHM = Var(dY) = d²V(Y)
 EVPV = E(Var(X/Y)*Y²)

Least square dvpt conclusions
 When rdm yr to yr fluctuations are severe, least square tends to produce more reasonable estimates of ultimate than link ratio
 Does not require a great deal of additional data
 Works best when used w. understanding of its limitations
 When significant exposure chgs, can go astray unless make necessary chgs
 Subject to sampling errors due to parameters estimation
 Can be helpful in developing losses for small states or for lines subject to serious fluctuations

