Hurlimann Flashcards
2 Differences between Hurlimann and Benktander methods
- Hurlimann’s method is based on a full development triangle, whereas the Benktander method is based on a single origin period
- Hurlimann’s method requires a measure of exposure for each origin period (i.e. premiums)
Main Result of Hurlimann method
Provides an optimal credibility weight for combining the chain-ladder (individual) loss ratio reserve with the B-F (collective) loss ratio reserve
Sik
paid claims from origin period i as of k years of development
Total Ultimate Claims from origin period i
k=1nΣSik
Cumulative Paid Claims (Cik)
Cik= j=1kΣSij
Ri (i-th period Reserve)
Ri = k=n-i+2nΣSik
where i = 2, … , n
R (Total Claims Reserve)
R = i=1nΣRi
Expected Loss Ratio (mk)
The incremental amount of expected paid claims per unit of premium in each development period (i.e. an incremental loss ratio)
mk = E [i=1n-k+1ΣSik] / i=1n-k+1ΣVi
where k= 1, … , n
Expected value of the burning cost of the total ultimate claims
This quantity is simlar to the prior estimate Uo from Mack
E[UiBC] = Vi · k=1nΣmk
By summing up the incremental loss ratios, we obtain an overall expected loss ratio. When we multiply the overall expected LR by the premium, we obtain an expected loss for each origin period
Loss Ratio Payout Factor (pi)
represents the percent of losses emerged to date for each origin period
pi = k=1n-i+1Σmk / k=1nΣmk
Individual Total Ultimate Claims (Uiind)
Similar to Chain- Ladder estimate
Uiind = Ci,n-i+1 / pi
Individual Loss Ratio Claims Reserve (Riind)
Riind = Uiind - Ci,n-1+1
=qi * Uiind
=(qi/pi) * Ci,n-1+1
Collective Loss Ratio Claims Reserve (Ricoll)
Ricoll = qi * UiBC
Collective Total Ultimate Claims (Uicoll)
Similar to BF estimate from Mack
Uicoll = Ricoll + Ci,n-i+1
Advantage of Collective Loss Ratio Claims reserve over BF reserve
difference actuaries always come to the same results provided they use the same premiums
Credible Loss Ratio Claims Reserve (Ric)
mixture of the individual and collective loss ratio reserves
Ric = Zi * Riind + (1-Zi) * Ricoll
where Zi is the credibility weight given to the individual loss ratio reserve
Benktander Loss Ratio Claims Reserve (RiGB)
obtained by setting Zi = ZGB = pi
RiGB = pi * Riind + qi * Ricoll
Neuhaus Loss Ratio Claims Reserve (RiWN)
obtained by setting Zi = ZiWN = pi * k=1nΣmk
RiWN = ZiWN * Riind + (1- ZiWN) * Ricoll
Zi*
(optimal credibility weights which minimize MSE)
Zi* = pi / (pi + ti)
Estimation of ti
Special case when fi = 1
ti* = sqrt(pi)
MSE(Ric)
(MSE for the credible loss ratio reserve)
MSE(Ric) = E[ai2(Ui)] * [(Zi2/pi) + (1/qi) + ((1-Zi)2/ti)
fkCL
(chain ladder age to age factors)
i=1n-kΣCi,k+1 / i=1n-kΣCik
FkCL
(Chain ladder Age to Ultimate factors)
j=kn-1ΠfjCL
piCL
(Chain Ladder Lag Factors)
1/Fn-i+1CL
qiCL
(Chain Ladder Reserve Factors)
1-piCL
RiCL
(qiCL/piCL) * Ci,n-i+1
Cape Cod Method
Benktander-type credibility mixture with the following components:
Riind = (qiCL/piCL) * Ci,n-i+1
Ricoll = qiCL * LR * Vi
Zi = piCL
LR = i=1nΣCi,n-i+1 / i=1nΣpiCL * Vi
Optimal Cape Cod method
Identical to the Cape Cod method, but with the following credibility weights:
Zi = piCL/ (piCL + sqrt(piCL))
BF Method
same as CC method, but LRi is some selected inital loss ratio for each origin period