Hurlimann

Question 1

Key Notation Hurlimann

Answer 1

p_i is the proportion of tot ultimate claims from origin period i expected to be paid in development period n-i+1 (loss ratio payout factor or loss ratio lag-factor)

q_i is the proportion of total ultimate claims from origin period i which remain unpaid in development period n-i+1 (loss ratio reserve factor)

U_i^BC=U_i⁽⁰⁾ is the burning cost of total ultimate claims (a priori)

U_i^coll=U_i⁽¹⁾ is the collective total ultimate claims (BF)

U_i^Ind=U_i^(infinity) is the individual total ultimate claims (CL)

R_i^WN is the Neuhaus loss ratio claims reserve

m_k= expected loss ratio in development period k

n= number of origin periods

V_i=premium belonging to origin period i

S_ik paid claims from origion period i as of k years of development. (incremental)

C_ik are cumulative paid claims from origin period i as of k years of development

Question 2

Total ultimate claims from origin period i

Answer 2

Σ_k=1-n S_ik

Question 3

Cumulative paid claims

Answer 3

C_ik = Σ_j=1-k (S_ij)

Question 4

i-th period claims reserve

Answer 4

R_i=Σ_{k=n-i+2 to n} (S_ik)

Question 5

Total claims reserve

Answer 5

R=Σ_{i=2 to n} (R_i)

Question 6

Expected Loss Ratio

Question 7

Expected Value of the Burning Cost

Answer 7

• Similar to prior estimate U₀
• E[U_i^BC]=V_i*Σ_{k=1 to n}(m_k)
• Summing up all the incremental loss ratios (m_k) gives an overall expected loss ratio
• Multiplying by V_i gives an expected loss for each origin period

Question 8

Loss Ratio Payout Factor

Answer 8

• Represents the percent of losses merged to date for each origin period
• p_i=Σ_{k=1 to n-i+1}(m_k) / Σ_{k=1 to n}(m_k)

Question 9

Individual Total Ultimate Claims

Answer 9

• Obtained by grossing up the latest cumulative paid claims for an origin period
• Considered “individual” since it depends on the individual latest claims experience of an origin period
• SImilar to estimate of the Chain-ladder estimate
• U_i^ind = C_i,n-i+1 / p_i

Question 10

Collective Loss Ratio Claims Reserve

Answer 10

• Obtained by using the burning cost of total ultimate claims
• Considered “collective” since it depends on the portfolio claims experience of all origin periods
• R_i^coll = q_i * U_i^BC

Question 11

Individual Loss Ratio Claims Reserve

Answer 11

• R_i^ind = U_i^ind - C_i,n-i+1
• R_i^ind = U_i^ind * q_i
• R_i^ind = q_i / p_i * C_i,n-i+1

Question 12

Collective Total Ultimate Claims

Answer 12

• This estimate is similar to the BF method
• U_i^coll = R_i^coll + C_{i, n-i+1}
• An advantage of the collective loss ratio reserve over the BF reserve is that different actuaries always come to the same results provided they use the same premiums

Question 13

Credible Loss Ratio Claims Reserve

Answer 13

• Individual ignores burning cost and collective ignores paid claims
• Mixture of individual and collective loss ratio reserves
• R_i^C = Z_i * R_i^ind + (1-Z_i) * R_i^coll

Question 14

Benktander Loss Ratio Claims Reserve

Answer 14

• Obtained by setting Z_i = Z_i^GB = p_i
• R_i^GB = p_i * R_i^ind + q_i * R_i^coll
  *

Question 15

Optimal Credibility Weights

Answer 15

• Z_i^* = (p_i)/(p_i + t_i)
• t_i^* = sqroot(p_i)
• Unless told otherwise, assume t_i is as above
• t_i^*= sqroot(pi) <= 1, Z_i^*<= .5
• Estimate appeals as it yields smallest credibility weights for the individual loss reserves, which places more emphasis on the collective loss reserves

Question 16

Means Squared Error for Credible Loss Ratio Reserve

Answer 16

• mse(R_i^C) = E[α_i²(U_i)] * [Z_i²/p_i +1/q_i + (1-Z_i)²/t_i] * q_i²
• Mean squared error for collective is Z_i=0
• Mean squared error for individual is Z_i=1