Hurlimann

Question 1

Calculate ELR using the incremental loss ratio approach (collective loss ratio)

Answer 1

mk = sum of Inc Loss_i,k / sum of Premium_i
i is the AY
k is the dev period

ELR = sum of mk

Question 2

Calculate the loss ratio payout factor

Answer 2

pi = sum of mk / ELR

Represents the % of losses paid to date for each origin period.

Question 3

Calculate the loss ratio reserve factor

Answer 3

qi = 1 - pi

Question 4

Calculate the individual loss ratio claims reserve estimate

Answer 4

Uind_i = Loss_i / pi

Rind_i = qi*Loss_i / pi

Question 5

Calculate the collective loss ratio reserve estimate

Answer 5

Rcoll = qiPremELR

Question 6

Calculate the collective total ultimate claims

Answer 6

Ucoll = Rcoll + Loss

Question 7

State an advantage of the collective loss ratio claims reserve over the standard BF reserve.

Answer 7

Different actuaries always come to the same results provided they use the same premiums because expected loss ratio is based on actual claims data rather than a judgmental selection.

Question 8

Calculate credibility-weighted loss ratio claims reserve

Answer 8

Rc_i = ZiRind_i + (1-Zi)Rcoll_i

Zi is the credibility weight

Question 9

How can we obtain Benktander as a credibility-weighted loss ratio claims reserve?

Answer 9

Set Zi = pi

R_GB = piRind + qiRcoll

Question 10

How can we obtain the Neuhaus as a credibility-weighted loss ratio claim reserve?

Answer 10

Set Zi = pi*ELR

R_GB = piELRRind + (1-piELR)Rcoll

Question 11

Calculate the optimal credibility weighs which minimize the MSE of R and the Var(R),

Answer 11

Zi = pi / (pi+pi^0.5)

If we assume V(U_i) = V(UBC_i) thus fi = V(U_i)/V(UBC_i) = 1

Otherwise:
Zi = pi / (pi+ti)
ti = (fi - 1+(fi+1)(fi-1+2pi))^0.5) / 2

Question 12

Calculate MSE for the credible loss ratio claims reserve

Answer 12

mse(Rc_i) = E(alpha^2(U_i))(Zi^2/pi + 1/qi + (1-Zi)^2/ti)qi^2

If Zi=0 we obtain mse(Roll)
If Zi=1 we obtain mse(Rind)