10.3 Incorporating Expert Opinion about the Development Factors

Intervene in the estimation of Chainladder factors

• Intervention in a development factor in a particular row

• How many years of data to use in the estimation

Procedure

Step 1. Use ODNB from (10.6) for our incremental claim distribution

Step 2. Choose prior distribution

• Prior distributions (e.g. gamma, log-normal, etc) are chosen so that the numerical procedures in software (i.e.g winBUGS) work as well as possible

• We just choose if we want a strong or vague prior

  • Vauge priors (i.e. large variances)

    • Closer to Bayesian Chainladder

    • Prediction error will be similar to CL or slightly larger

  • Strong priors (i.e. small variances)

    • We think prior means are appropriate

    • Prediction error will decrease

Examples below for a \(n \times n\) triangle

10.3.1 Reproduce the Chainladder

This is the form if we just want to have just a regular Chainladder

\[\begin{array}. \lambda_{i,j} = \lambda_j & \text{for } &i = 1,2,...,n-+1 ;\\ & &j = 2,3,...,n\\ \end{array}\]

Vague prior distributions for

\[\lambda_j \:\: ; \:\:(j=2,3,...,n)\]

• Use vague prior so \(\lambda_j\) is based on data

10.3.2 Intervention in a development factor in particular rows

Example expert opinion: 2^nd development factor (\(\lambda_3\), from col 2 to 3) should be 1.5 for the recent 3 years (i.e. row 8,9,10) while the others being the same

\[\begin{array}. \lambda_{i,j} = \lambda_j & \text{for } &i = 1,2,...,n-+1 ;\\ & &j = 2,4,5,...,n\\ \lambda_{i,3} = \lambda_3 & \text{for } &i = 1,2,..,7 \\ \lambda_{8,3} = \lambda_{9,3} = \lambda_{10,3} \\ \end{array}\]

Prior distribution mean and variance is chosen to reflect the expert opinion

• \(\lambda_{8,3}\) has prior distribution with mean 1.5 and variance \(W\)

  • \(W\) is selected to reflect the strength of the prior information

• \(\lambda_j\) have prior distributions with large variances

10.3.3 Intervention in using L-years average

Example expert opinion: Use 5 years weighted average for LDF selection

We divide the data into 2 parts using the prior distributions:

\[\begin{equation} \begin{array}. \lambda_{i,j} = \lambda_j & \text{for } &i = n-j-3, n-j-2, n-j-1,\\ & &n-j, n-j+1\\ \lambda_{i,j} = \lambda_j^* & \text{for } &i = 1,2,...,n-j-4 \\ \end{array} \tag{10.8} \end{equation}\]

Both \(\lambda_j\) and \(\lambda_j^*\) have prior distributions with large variance so they are estimated from the data

The first part of the (10.8) is to adjust for later development years where there are less than 5 rows

• For those columns there is just one development parameters \(\lambda_j\)