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)

Question 13

Identify which of Rcoll and R_GB is preferable reserve from a statistical point of view

Answer 13

Rcoll is preferable because it minimizes the MSE of the reserve

Question 14

Explain why Neuhaus method may not be appropriate when premium volume is shrinking over time

Answer 14

Premium shrinking may indicate a change in mix of business.
Since Neuhaus method assumes a constant ELR for all accident years, a change in mix of business may violate the constant ELR assumption.