Actuarial Notes

Credible Loss Ratio Reserve

Credible loss ratio reserve R_i^c is Hürlimann's credibility mixture of two reserves for origin period i, both built from a triangle's incremental loss ratios. The individual loss ratio reserve grosses up the period's own paid claims, like the chain ladder. The collective loss ratio reserve applies the whole portfolio's burning cost, like Bornhuetter-Ferguson with an a priori taken from the data.

R_i^{c} = Z_i\,R_i^{ind} + (1 - Z_i)\,R_i^{coll}

R_i^{ind} = \frac{q_i}{p_i}\,C_{i,n-i+1}

R_i^{coll} = q_i\,V_i\sum_{k=1}^{n} m_k

  • V_i is the premium of origin period i and C_{i,n-i+1} its latest cumulative paid claims.
  • m_k is the incremental loss ratio of development period k: the paid claims S_{ik} of that period, summed over the origin periods that have reached it, divided by those periods' premium. \sum_k m_k is the expected loss ratio.
  • p_i = \sum_{k \le n-i+1} m_k \big/ \sum_k m_k is the loss ratio payout factor, the expected share of ultimate paid by now, and q_i = 1 - p_i is the reserve factor. The method works from loss ratios where the standard methods use link ratios.
  • Choosing Z_i. Benktander's weight is Z_i = p_i, the Benktander loss ratio reserve. Neuhaus's is Z_i = p_i \sum_k m_k. Hürlimann's optimal weight minimizes both the mean squared error and the variance of R_i^c when \mathrm{Var}[U_i] = \mathrm{Var}[U_i^{BC}]:

Z_i^{*} = \frac{p_i}{p_i + \sqrt{p_i}}

  • Its properties. Z_i^* \le \tfrac{1}{2}, with equality only when p_i = 1, and it falls as origin periods get younger. It is below Benktander's p_i whenever p_i > 0.382, and Hürlimann notes that the Benktander and Neuhaus methods usually give the individual reserve more weight. In his examples, all three credible reserves are close to one another and well ahead of either extreme on mean squared error.
  • Mean squared error. Hürlimann uses Mack's (2000) conditional payout model, \mathrm{Var}[C_{i,n-i+1} \mid U_i] = p_i q_i\,\alpha_i^2(U_i). With t_i the credibility parameter (\sqrt{p_i} above):

\mathrm{mse}(R_i^c) = E[\alpha_i^2(U_i)]\left(\frac{Z_i^2}{p_i} + \frac{1}{q_i} + \frac{(1 - Z_i)^2}{t_i}\right)q_i^2

  • Z_i = 1 gives \mathrm{mse}(R_i^{ind}) = E[\alpha_i^2]\,q_i/p_i and Z_i = 0 gives \mathrm{mse}(R_i^{coll}) = E[\alpha_i^2]\,q_i(1 + q_i/t_i). The minimum is at Z_i = p_i/(p_i + t_i).
  • Relations to the standard methods.
    • The collective reserve is Mack's (1997) loss ratio method. Its a priori is the portfolio's burning cost, so actuaries using the same premiums always agree, unlike a judgmental BF prior.
    • Iterating U \to C_{i,n-i+1} + q_i U from the burning cost V_i\sum_k m_k gives first the collective reserve, then the Benktander reserve, and converges to the individual reserve.
    • With chain-ladder lag factors p_i^{CL} = 1/\text{CDF} in place of p_i, the individual reserve becomes the chain ladder reserve. Mixing it with a Cape Cod or BF collective reserve at Z_i = p_i^{CL}/(p_i^{CL} + \sqrt{p_i^{CL}}) gives Hürlimann's optimal Cape Cod and optimal Bornhuetter-Ferguson methods.

Collective, Individual and Credible Reserves From One Triangle Incremental paid claims and premium ($000s), with no development after 36 months:

AY 0–12 12–24 24–36 Premium
1 800 600 200 2{,}000
2 1{,}100 750 2{,}500
3 1{,}100 3{,}000

Compute the individual, collective, Benktander, Neuhaus and optimal credible loss ratio reserves.

Answer Incremental loss ratios:

\begin{align*}
m_1 &= \frac{800 + 1{,}100 + 1{,}100}{2{,}000 + 2{,}500 + 3{,}000} \\
&= 0.40 \\
m_2 &= \frac{600 + 750}{2{,}000 + 2{,}500} \\
&= 0.30 \\
m_3 &= \frac{200}{2{,}000} \\
&= 0.10
\end{align*}

So \sum m_k = 0.80, p_2 = 0.70/0.80 = 0.875 and p_3 = 0.40/0.80 = 0.50.

AY 2: C = 1{,}850 and q = 0.125:

\begin{align*}
R^{ind} &= \frac{0.125}{0.875}(1{,}850) \\
&= 264.29 \\
R^{coll} &= 0.125 \times 2{,}500 \times 0.80 \\
&= 250.00
\end{align*}

AY 3: C = 1{,}100 and q = 0.50:

\begin{align*}
R^{ind} &= \frac{0.50}{0.50}(1{,}100) \\
&= 1{,}100 \\
R^{coll} &= 0.50 \times 3{,}000 \times 0.80 \\
&= 1{,}200
\end{align*}

Weights and credible reserves:

AY 2 Z AY 2 R^c AY 3 Z AY 3 R^c
Benktander, p 0.875 262.50 0.500 1{,}150.00
Neuhaus, p \times 0.80 0.700 260.00 0.400 1{,}160.00
Optimal, p/(p + \sqrt{p}) 0.483 256.90 0.414 1{,}158.58

For example, the optimal AY 2 weight is 0.875/(0.875 + 0.9354) = 0.4833, and 0.4833(264.29) + 0.5167(250) = 256.90.

The totals are 1{,}364.29 (individual), 1{,}450.00 (collective), 1{,}412.50 (Benktander), 1{,}420.00 (Neuhaus) and 1{,}415.48 (optimal). The three credible reserves lie between the extremes, as Hürlimann found. For AY 2, the optimal weight (0.48) is far below Benktander's (0.875), so it leans more on the collective reserve for the mature year.

Which Reserve Has the Smallest Mean Squared Error? For AY 3 above (p = q = 0.5), take E[\alpha^2(U)] = 20{,}000 and t = \sqrt{p}. Compare the mean squared errors of the individual, collective, Benktander, Neuhaus and optimal reserves.

Answer Here t = 0.7071, and \mathrm{mse} = 20{,}000\,(Z^2/0.5 + 2 + (1-Z)^2/0.7071)(0.25):

Reserve Z MSE Root MSE
Individual 1 20{,}000 141.4
Collective 0 17{,}071 130.7
Benktander 0.500 14{,}268 119.4
Neuhaus 0.400 14{,}146 118.9
Optimal 0.414 14{,}142 118.9

The optimal row, for example:

\begin{align*}
\mathrm{mse} &= 20{,}000\left(\frac{0.1716}{0.5} + 2 + \frac{0.3431}{0.7071}\right)(0.25) \\
&= 20{,}000 \times 0.7071 \\
&= 14{,}142
\end{align*}

Blending cuts the error by about 17\% against the better extreme. Among the blends the differences are tiny: Benktander's MSE is only 0.9\% above the optimum and Neuhaus's 0.02\%. That is why Hürlimann recommends all three simple credible methods when the optimal weights cannot be estimated more precisely.