Hurlimann

Question 1

2 Differences between Hurlimann and Benktander methods

Answer 1

1. Hurlimann’s method is based on a full development triangle, whereas the Benktander method is based on a single origin period
2. Hurlimann’s method requires a measure of exposure for each origin period (i.e. premiums)

Question 2

Main Result of Hurlimann method

Answer 2

Provides an optimal credibility weight for combining the chain-ladder (individual) loss ratio reserve with the B-F (collective) loss ratio reserve

Question 3

S_ik

Answer 3

paid claims from origin period i as of k years of development

Question 4

Total Ultimate Claims from origin period i

Answer 4

_k=1ⁿΣS_ik

Question 5

Cumulative Paid Claims (C_ik)

Answer 5

C_ik= _j=1^kΣS_ij

Question 6

R_i (i-th period Reserve)

Answer 6

R_i = _k=n-i+2ⁿΣS_ik

where i = 2, … , n

Question 7

R (Total Claims Reserve)

Answer 7

R = _i=1ⁿΣR_i

Question 8

Expected Loss Ratio (m_k)

Answer 8

The incremental amount of expected paid claims per unit of premium in each development period (i.e. an incremental loss ratio)

m_k = E [_i=1^n-k+1ΣS_ik] / _i=1^n-k+1ΣV_i

where k= 1, … , n

Question 9

Expected value of the burning cost of the total ultimate claims

Answer 9

This quantity is simlar to the prior estimate U_o from Mack

E[U_i^BC] = V_i · _k=1ⁿΣm_k

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

Question 10

Loss Ratio Payout Factor (p_i)

Answer 10

represents the percent of losses emerged to date for each origin period

p_i = _k=1^n-i+1Σm_k / _k=1ⁿΣm_k

Question 11

Individual Total Ultimate Claims (U_i^ind)

Answer 11

Similar to Chain- Ladder estimate

U_i^ind = C_i,n-i+1 / p_i

Question 12

Individual Loss Ratio Claims Reserve (R_i^ind)

Answer 12

R_i^ind = U_i^ind - C_i,n-1+1

=q_i * U_i^ind

=(q_i/p_i) * C_i,n-1+1

Question 13

Collective Loss Ratio Claims Reserve (R_i^coll)

Answer 13

R_i^coll = q_i * U_i^BC

Question 14

Collective Total Ultimate Claims (U_i^coll)

Answer 14

Similar to BF estimate from Mack

U_i^coll = R_i^coll + C_i,n-i+1

Question 15

Advantage of Collective Loss Ratio Claims reserve over BF reserve

Answer 15

difference actuaries always come to the same results provided they use the same premiums

Question 16

Credible Loss Ratio Claims Reserve (R_i^c)

Answer 16

mixture of the individual and collective loss ratio reserves

R_i^c = Z_i * R_i^ind + (1-Z_i) * R_i^coll

where Z_i is the credibility weight given to the individual loss ratio reserve

Question 17

Benktander Loss Ratio Claims Reserve (R_i^GB)

Answer 17

obtained by setting Z_i = Z_GB = p_i

R_i^GB = p_i * R_i^ind + q_i * R_i^coll

Question 18

Neuhaus Loss Ratio Claims Reserve (R_i^WN)

Answer 18

obtained by setting Z_i = Z_i^WN = p_i * _k=1ⁿΣm_k

R_i^WN = Z_i^WN * R_i^ind + (1- Z_i^WN) * R_i^coll

Question 19

Z_i^*

(optimal credibility weights which minimize MSE)

Answer 19

Z_i^* = p_i / (p_i + t_i)

Question 20

Estimation of t_i

Answer 20

Special case when f_i = 1

t_i^* = sqrt(p_i)

Question 21

MSE(R_i^c)

(MSE for the credible loss ratio reserve)

Answer 21

MSE(R_i^c) = E[a_i²(U_i)] * [(Z_i²/p_i) + (1/q_i) + ((1-Z_i)²/t_i)

Question 22

f_k^CL

(chain ladder age to age factors)

Answer 22

_i=1^n-kΣC_i,k+1 / _i=1^n-kΣC_ik

Question 23

F_k^CL

^{(Chain ladder Age to Ultimate factors)}

Answer 23

_j=k^n-1Πf_j^CL

Question 24

p_i^CL

(Chain Ladder Lag Factors)

Answer 24

1/F_n-i+1^CL

Question 25

q_i^CL

(Chain Ladder Reserve Factors)

Answer 25

1-p_i^CL

Question 26

R_i^CL

Answer 26

(q_i^CL/p_i^CL) * C_i,n-i+1

Question 27

Cape Cod Method

Answer 27

Benktander-type credibility mixture with the following components:

R_i^ind = (q_i^CL/p_i^CL) * Ci_,n-i+1

R_i^coll = q_i^CL * LR * V_i

Z_i = p_i^CL

LR = _i=1ⁿΣC_i,n-i+1 / _i=1ⁿΣp_i^CL * V_i

Question 28

Optimal Cape Cod method

Answer 28

Identical to the Cape Cod method, but with the following credibility weights:

Z_i = p_i^CL/ (p_i^CL​ + sqrt(p_i^CL​))

Question 29

BF Method

Answer 29

same as CC method, but LR_i is some selected inital loss ratio for each origin period