Actuarial Notes

Bayesian Bornhuetter-Ferguson Model

Bayesian Bornhuetter-Ferguson Model is Verrall's stochastic version of the Bornhuetter-Ferguson method. The row parameters x_i of the over-dispersed Poisson model, each accident year's expected ultimate loss, are given gamma prior distributions that carry an expert's view of the ultimate. Each future incremental loss then has a predictive mean that is a credibility weighting of the chain ladder and the Bornhuetter-Ferguson estimates. The strength of the prior moves the result along a spectrum from the chain ladder to BF.

x_i \sim \mathrm{Gamma}(\alpha_i, \beta_i)

E[x_i] = M_i = \frac{\alpha_i}{\beta_i}

\mathrm{Var}(x_i) = \frac{M_i}{\beta_i}

\begin{aligned}
E[C_{ij}] ={}& Z_{ij}\,(\lambda_j - 1)\,D_{i,j-1} \\
&+ (1 - Z_{ij})\,(\lambda_j - 1)\,\frac{M_i}{\lambda_j\lambda_{j+1}\cdots\lambda_n}
\end{aligned}

Z_{ij} = \frac{\sum_{k=1}^{j-1} y_k}{\beta_i\,\phi + \sum_{k=1}^{j-1} y_k}

  • C_{ij} is the incremental and D_{ij} the cumulative loss of accident year i at development year j, and \lambda_j are the chain ladder factors. M_i is the expert's ultimate, for example from the premium calculation. y_k is the share of ultimate emerging in period k, so \sum_{k<j} y_k = 1/(\lambda_j\cdots\lambda_n) is the share emerged to date. \phi is the over-dispersion parameter, which Verrall plugs in from the ODP model rather than giving it a prior.
  • The prior sets the credibility. For a given M_i, a larger \beta_i is a smaller prior variance and more certainty about M_i. That pushes Z_{ij} toward 0, which is the Bornhuetter-Ferguson method. A small \beta_i, a vague prior, pushes Z_{ij} toward 1, the chain ladder. Everything between is a complete spectrum of models. In Verrall's Taylor and Ashe illustration, a prior standard deviation of 1{,}000 on means of 5.5–6 million reproduced BF, and 1{,}000{,}000 gave reserves between the two.
  • Credibility grows with maturity. Z_{ij} rises with the share emerged, as BF's own weight 1/\text{CDF} does, but the prior's variance and \phi now decide how fast. The gamma prior is conjugate to the ODP likelihood. The posterior mean of x_i is Z \cdot D_{i,j-1}/\sum_{k<j} y_k + (1 - Z) M_i, and every future cell is y_j times it. So the same Z blends the chain ladder and BF reserves for the whole accident year.
  • Implementation. The column parameters get improper priors and are estimated first, so they are the chain ladder's. The model is then rewritten as an over-dispersed negative binomial recursive down the rows, with mean (\gamma_i - 1)\sum_{m<i} C_{m,j}. Verrall's equation (5.4), as corrected in the errata, converts the prior values of x_i into the \gamma_i. MCMC simulation in winBUGS gives the full predictive distribution, whose standard deviation is the prediction error.
  • What it adds to deterministic BF is a prediction error and a predictive distribution, with the uncertainty in the a priori built in instead of ignored. Verrall notes the method can also be adapted so that the expert specifies the reserve rather than the ultimate.

Turning an Expert's View into a Prior At 12 months an accident year has cumulative paid losses of 1{,}500 ($000s). The factors are \lambda_2 = 2.00, \lambda_3 = 1.25 and \lambda_4 = 1.10, with development complete at 48 months. The underwriter expects an ultimate of 6{,}000 with a standard deviation of 1{,}500, and \phi = 50.

Find the gamma prior, the credibility weight, the mean of the next incremental payment, and the blended reserve for the year.

Answer Prior.

\begin{align*}
\beta &= \frac{M}{\mathrm{Var}(x)} \\
&= \frac{6{,}000}{1{,}500^2} \\
&= 0.002667 \\
\alpha &= M\beta \\
&= 16
\end{align*}

Credibility. The share emerged is 1/(2.00 \times 1.25 \times 1.10) = 1/2.75 = 0.3636, and \beta\phi = 0.1333.

\begin{align*}
Z &= \frac{0.3636}{0.1333 + 0.3636} \\
&= 0.7317
\end{align*}

Next incremental. The chain ladder mean is (2.00 - 1)(1{,}500) = 1{,}500 and the BF mean is (2.00 - 1)(6{,}000/2.75) = 2{,}181.8.

\begin{align*}
E[C_{i2}] &= 0.7317(1{,}500) + 0.2683(2{,}181.8) \\
&= 1{,}097.6 + 585.4 \\
&= 1{,}682.9
\end{align*}

Reserve. The chain ladder gives 1{,}500 \times 1.75 = 2{,}625.0 and BF gives 6{,}000 \times (1 - 0.3636) = 3{,}818.2.

\begin{align*}
\hat R &= 0.7317(2{,}625.0) + 0.2683(3{,}818.2) \\
&= 2{,}945.1
\end{align*}

The weight on the data is already 73\% at 12 months, well above BF's own 1/2.75 = 36\%, because a 1{,}500 standard deviation on 6{,}000 is a weak prior. By 36 months, with 1/1.10 = 90.9\% emerged, Z would be 0.9091/(0.1333 + 0.9091) = 0.872.

Verrall's Taylor and Ashe Illustration, Accident Year 10 Verrall's accident year 10 has one observed payment of 344{,}014. The chain ladder factors multiply to 14.4466, and its chain ladder reserve is 4{,}625{,}811. The prior mean ultimate is 6{,}000{,}000, and his program's plug-in scale parameter is 52.8615 in thousands, so \phi = 52{,}861.5.

Estimate the Bayesian reserve with prior standard deviations of 1{,}000 and of 1{,}000{,}000.

Answer BF reserve, which the precise prior should reproduce:

\begin{align*}
R_{BF} &= 6{,}000{,}000\left(1 - \frac{1}{14.4466}\right) \\
&= 5{,}584{,}677
\end{align*}

The share emerged is 1/14.4466 = 0.06922.

Standard deviation 1{,}000: \beta = 6{,}000{,}000/1{,}000^2 = 6, so \beta\phi = 317{,}169 and Z \approx 0.0000002. The reserve is the BF 5{,}584{,}677. Verrall's Table 9 shows a simulated Bayesian mean of 5{,}587{,}000.

Standard deviation 1{,}000{,}000: \beta = 6 \times 10^{-6}, so \beta\phi = 0.3172.

\begin{align*}
Z &= \frac{0.06922}{0.3172 + 0.06922} \\
&= 0.1791 \\
\hat R &= 0.1791(4{,}625{,}811) + 0.8209(5{,}584{,}677) \\
&= 5{,}412{,}899
\end{align*}

Table 10 shows 5{,}390{,}000, between the chain ladder and BF as expected. The gap of under 0.5\% reflects simulation error and the way the program enters the prior, on the outstanding part of the ultimate. With a single payment to go on, even a prior as loose as \pm 1 million on 6 million keeps about 82\% of the weight on the expert, and Table 10's prediction error for the year falls to 20\%, from the chain ladder's 43\%.