10.4 Bayesian Model for the Bornhuetter-Ferguson Method

Use the ODP model from (10.4)

For the BF there is expert opinion about the level of each row, so we’ll specify the prior distribution for it below:

\[\begin{equation} x_i \mid \alpha_i, \beta_i \sim \Gamma(\alpha_i, \beta_i) \:\: ; \:\: x_i \perp\!\!\!\!\perp x_j \:\: i \neq j \tag{10.9} \end{equation}\]

Easiest to consider the mean and variance of the gamma distribution to parameterize

\[\begin{equation} \mathrm{E}[x_i] = \dfrac{\alpha_i}{\beta_i} = M_i\\ \mathrm{Var}(x_i) = \dfrac{\alpha_i}{\beta_i^2} = \dfrac{M_i}{\beta_i}\\ \tag{10.10} \end{equation}\]

  • For a given choice of \(M_i\), the variance can be altered by changing the value of \(\beta_i\)

  • Larger \(\beta_i\) means we are more sure about the value \(M_i\) and vice versa

Now consider the effect of using these prior distributions on the model for the data, after some lengthy proof:

\[\begin{equation} \mathrm{E}[c_{ij}] = Z_{ij} \times \underbrace{(\lambda_j -1)D_{ij-1}}_{\text{Chainladder}} + (1 - Z_{ij}) \times \underbrace{M_i \: y_i}_{\text{BF}} \tag{10.11} \end{equation}\]

Remark.

  • Result is a blend of Chainladder and BF (like Benktander)

  • \(y_j = \dfrac{\lambda_j -1}{\lambda_j \lambda_{j+1} \cdots \lambda_n}\)

Credibility factor \(Z_{ij}\)

\[\begin{equation} Z_{ij} = \dfrac{p_{j-1}}{\beta_i \varphi + p_{j-1}} \tag{10.12} \end{equation}\]

Remark.

  • \(p_{j-1} = \sum_{k=1}^{j-1} y_k\) (Mack notations), percentage paid at the prior age

    • Large \(p_{j-1}\) means losses are more mature so more weight to the Chainladder method

  • We can influence the balance of the weighting (\(Z_{ij}\)) through \(\beta_i\)

    • Larger \(\beta_i\) the more weight goes to the BF method

    • Prior with large variance will give results close to Chainladder

      • Lowers the prediction error but not by much

    • Prior with small variance will give results close to BF

      • Will lower the prediction error due to low variance in the prior

  • \(\varphi\) the process variance on \(c_{ij}\) also have impact on the credibililty

    • If there’s more variability in the \(c_{ij}\) then we trust the BF method more

10.4.1 Calculation Example

Given

  • Incremental paid triangle

  • ODP mean and variance for every AYs

  • Dispersion factor \(\varphi\)

Procedure

  1. Covert triangle to cumulative and select LDFs

    (Use all year vol. wtd. average)

  2. Calculate Cumulative LDFs and \(p_j\) and \(y_j\)

  3. Calculate Chainladder ultimate and use \(y_i\) to calculate the incremental means for each of the future cells

  4. Calculate a-priori mean based on the given ODP mean for each AY \(\times\) \(y_j\) for the incremental means for each of the future cells

  5. Calculate \(\beta_i\) for each AYs using \(\beta_i = \dfrac{\mathrm{E}[x_i]}{\mathrm{Var}(x_i)}\)

    • Note that the \(\beta_i\)’s are unit dependent so need to be consistent

  6. Calculate \(Z_{ij}\) with (10.12) for each future cells

  7. Finally credibility weight the Chainladder and BF estimates