8.2 Part 1) Setup Base Layer Triangle

Setup a consistent base layer triangle for LDFs selection

  • We only need to go up to calculating LEV if we already have the base LDFs \(F^B_{nj}\)

8.2.1 Setup the trend triangle

Total Trend = AY Trend \(\times\) CY Trend

  • No adjustment needed if only AY trend is present

    Since AY trend increases losses proportionally down each row

  • CY trend should be on incremental losses

These are ground up trend, which is consistent if taken from external sources

Trend to the last row of the triangle

  • Don’t stop just at the diagonal

  • We typically starts trending from the top left corner (but it doesn’t have to)

    i.e. Have the 1.000 at the top left corner

  • Get the AY and CY trend triangle separately and the multiply them together

Caveat:

  • The paper applies the trend to cumulative loss which is problematic

  • Cumulative trend depends on the incremental trend and the emergence pattern

Output: Trend triangle

8.2.2 Calculate unlimited mean

We need the unlimited mean for each cell in triangle (Avg sev paid to date)

  1. Based on mean for the latest AY (last row \(n\))
\[\begin{equation} \begin{array}{c} C_{nj} \sim \Phi_{nj} = Exp(\theta_j) \\ \mathrm{E}[C_{nj}] = \theta_j \\ \tag{8.1} \end{array} \end{equation}\]

  1. Detrend the mean back up from the bottom row to fill the whole square

    • Practically, we usually just need to calculate this for 4 cells for the LDF conversion formula (8.4) (if we have the LDF already)

Output: Unlimited mean loss table adjusted for trend

8.2.3 Calculate LEV

We need the LEV for each cell in:

  • For calculating \(B\) triangle: Triangle @ \(L\) and last row @ \(B\)

  • For converting LDFs: Diagonal and ultimate column @ \(L\) and last row @ \(B\)

\[\begin{equation} LEV(X; \Phi \sim Exp(\theta)) = \theta \: \left[ 1 - \operatorname{exp}\left\{-\left(\dfrac{x}{\theta}\right)\right\} \right] \tag{8.2} \end{equation}\]

Remark.

  • Use the table of means (\(\theta\)’s) calculated from Step 2

  • Similarly, we only need 4 points for the calculates in part 2 if we don’t need the LDFs

  • This gives us everything for equation (8.4) if we already have the Base level LDF

Output: \(LEV\) for each cell in triangle for layer \(L\) and the last row for layer \(B\)

8.2.4 Calculate the Base Layer Triangle and LDF

Convert triangle of actual losses by dividing it’s LEV at layer \(L\) then times it’s LEV at layer \(B\)

\[\begin{equation} C_{ij}' = C_{ij}^L \times \underbrace{\dfrac{LEV(B;\Phi_{nj})}{LEV(L;\Phi_{ij})}}_{\text{ILF w/ on-level to }nj} \tag{8.3} \end{equation}\]

Remark.

  • See definition 8.2 on the ILF component

  • This takes losses from layer \(L\) at trend level \(ij\) to layer \(B\) at trend level \(nj\)

Once we have the triangle we can select the base layer LDFs \(F^B_{nj}\)