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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 = (xy-x*y)/(x² - (x)²) and a = y - bx
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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)
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Loss reporting distributions
- X = rpt nb clm, Y = ult nb clm
- Q(x) = E(Y|X = x)
- R(x) = E(Y - X|X = x)
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Poisson-Binomial 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
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LS: Poisson-Binomial 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
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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
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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)
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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
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