Hurlimann Flashcards
Hürlimann uses expected incremental loss ratios to specify the payment pattern instead of using LDFs calculated directly from the losses. We can calculate a credibility-weighted estimate of the individual and collective reserves that minimizes the MSE of the reserve estimate.
Incremental Expected Loss Ratio
m_k = E[S_ik]/SUM[V_i]
where S_ik are incremental paid claims and V_i is the premium
i is the AY index
k is the development period index
If m_k is summed across development periods, we obtain the total ELR
Expected Value of the Burning Cost
Similar to the a priori estimate U_0
E[U_i(BC)] = V_i * SUM[m_k] = V_i * ELR
Multiplies the premium for AY i by the total ELR to arrive at the expected value of the burning cost/expected loss
Loss Ratio Payout Factor
Represents % of losses paid for each AY
p_i = V_i * SUM[m_k] / E[U_i(BC)] = SUM[m_k] / ELR
where the ELR is the sum of all m_k
Sum the m_k from k = 1 to n-i+1
Loss Ratio Reserve Factor
Represents % of losses yet to be paid for each AY
q_i = 1 - p _i
Individual Total Ultimate Claims
U_i(ind) = C_i,n-i+1 / p_i
Ult(Ind) = paid claims / % paid
Individual Loss Ratio Claims Reserve
R_i(ind) = q_i * U_i(ind) = q_i/p_i * C_i,n-i+1
Ind Reserve = % unpaid/% paid * paid claims
Collective Loss Ratio Claims Reserve
R_i(coll) = q_i * E[U_i(BC)] = q_i * V_i * ELR
Advantage of Collective Loss Ratio Claims Reserve over BF Reserve
different actuaries always come to the same results provided they use the same premiums
Credible Loss Ratio Claims Reserve
R_i(c) = Z_i * R_i(ind) + (1-Z_i) * R_i(coll)
Benktander (GB) Loss Ratio Claims Reserve
Set Z_i = p_i
R_i(GB) = p_i * R_i(ind) + q_i * R_i(coll)
Neuhaus (WN) Loss Ratio Claims Reserve
Set Z_i = p_i * ELR
Iterative Relationships
Using a starting point of U_i(BC)
U_i(m) = (1 - q_i^m) * U_i(ind) + q_i^m * U_i(BC)
R_i(m) = (1 - q_i^m) * R_i(ind) + q_i^m * R_i(coll)
Iteration 1 is the collective method
Iteration 2 is the Benktander method
Iteration inf is the individual method
General Formula for the Optimal Credibility Weights
Assume Var(U_i) = f_i * Var(U_i(BC)) for some constant f_i >= 1
Z_i = p_i / (p_i + t_i) minimizes the MSE and variance of R_i(c)
where t_i = (f_i - 1 + SQRT[(f_i + 1)*(f_i - 1 + 2p_i)]) / 2
Special Case of the Optimal Credibility Weights
The minimum variance optimal credible claims reserve is obtained when f_i = 1
In this case Var(U_i) = Var(U_i(BC)) and t_i = SQRT(p_i)
The optimal Z_i = p_i / (p_i + SQRT(p_i))
Mean Squared Error Formula
MSE(R_i) = Constant * [(Z_i^2 / p_i) + (1 / q_i) + (1 - Z_i^2) / t_i] * q_i^2
Chain-Ladder Method
Similar to the individual loss ratio method with the loss ratio payout factors replaced with the standard CL lag factors
p_i(CL) = 1 / CDF
Cape Cod Method
Credibility mixture of the CL reserves and the Cape Cod reserves
R_i(CC) = q_i(CL) * ELR * V_i
ELR = Losses / SUM(p_i(CL) * V_i)
Optimal Cape Cod Method
Z_i = p_i(CL) / (p_i(CL) + SQRT(p_i(CL)))
BF Method
Credibility mixture of the CL reserves and the BF reserves
R_i(BF) = q_i(CL) * ELR_i * V_i
where ELR_i is some selected initial loss ratio for each AY
Optimal BF Method
Z_i = p_i(CL) / (p_i(CL) + SQRT(p_i(CL)))
Differences between Hurlimann and Benktander methods
- Hurlimann’s method requires a measure of exposure for each accident year (i.e. premiums)
- Hurlimann’s method relies on loss ratios (rather than link ratios) to determine reserves
- Hurlimann’s method is based on a full development triangle, whereas the Benktander method is based on a single accident year
