6.4 LDF Method (Method 2)

Can use either Loglogistic (6.1) or Weibull (6.2) for \(G(x)\)

Estimate \(\theta\), \(\omega\), and \(ULT_{AY}\) with MLE

  • Recall \(ULT_{AY}\) estimate from prior section

  • Then estimate \(\mu\) for the whole triangle

\[\mu = ULT_{AY} \times [G(y) - G(x)]\]

Calculate \(\sigma^2\) for residual or reserve process variance

\[\sigma^2 = \dfrac{1}{n-p}\sum\limits_{i \in \Delta} \underbrace{\dfrac{(c_i - \mu_i)^2}{\mu_i}}_{\chi^2 \text{ term}}\]

  • n = all data points (not just the predicted like in Venter)

  • p = 2 + 10 AYs

  • This is for incremental losses

6.4.1 Residual Review

Normalized residuals:

\[r_i = \dfrac{c_i - \mu_i}{\sigma \sqrt{\mu_i}}\]

  • Divide by the square root of the variance \(\sigma^2 \mu\) for ODP

  • Same as \(\dfrac{\sqrt{\chi^2 \text{ term}}}{\sigma}\)

Plot residuals (should be randomly scattered around 0):

  1. age \(x\) vs \(r_{i,x}\)

  2. expected loss \(\mu_i\) vs \(r_{i,x}\)

    Test of the constant \(\frac{Variance}{Mean}\) assumption

    • If fail, can try alternative variance assumptions (e.g. \(\mathrm{Var} \propto \mu^2\))

  3. AY, CY, etc vs \(r_{i,x}\)

6.4.2 Reserve Estimate

Untruncated

  1. Get \(G(x)\): % paid(reported) to date

  2. Ultimate = Paid to date (reported to date) \(\div\) \(G(x)\)

Truncated @ age \(x_t\)

  • To cut of the tail at some point and stop the development

  • Remember we’re \(x\) is in mid year (so year times 12 and minus 6 months)

  • Use \(G'(x) = \dfrac{G(x)}{G(x_t)}\) instead just like above

Process Variance:

\[\begin{equation} \sigma^2 \sum_i \mu_i = \sigma^2 \times \text{Unpaid} \tag{6.8} \end{equation}\]

Parameter variance is huge and computational intensive as it requires inverting a big matrix

  • We expect the Cape Cod to have parameter variance (as we only estimate 3 parameters)

Total variance can be calculated by summing the process and parameter variance