8.4 Other Practical Uses

If we don’t have the severity distribution by age (as required by the denominator of equation (8.4)), we can work with the severity at ultimate and estimate \(R_j(X,B)\), for lower layers \(X < B\)

\[\begin{equation} F_{ij}^X = F_{nj}^B \times \frac{LEV(X;\Phi_{i\infty})\div LEV(B;\Phi_{ {\color{red}i} \infty})}{R_j{(X,B)}} \tag{8.6} \end{equation}\]

  • Note the \(i\) in the second part of the numerator (I guess it’s to be consistent with formula below)

  • See how this change pictorially (Compare to fig. 8.1)

Ratio of limited losses at layer \(X\) to layer \(B\) at age \(j\)

\[\begin{equation} R_{j}(X,B) = \dfrac{LEV(X;\Phi_{ij})}{LEV(B;\Phi_{ij})} \tag{8.7} \end{equation}\]

  • The ratio is calculated on the diagonal

  • This needs to be estimated

Equation (8.7) assume:

\[\dfrac{LEV(B; \Phi_{n \infty})}{LEV(B; \Phi_{nj})} \approx \dfrac{LEV(B; \Phi_{i \infty})}{LEV(B; \Phi_{ij})}\]

  • Assumes the ratio of losses at different cost layers is immaterial

  • The expected LDF in different AYs (row \(n\) and \(i\)) are similar when losses are capped at \(B\)

  • This is a reasonable assumption with low inflation

Estimate \(R_j(X,B)\) (8.7)

\(R_j(X,B)\) is bound by the following:

\[U_i \leq R_j(X,B) \leq U_i \cdot F_{ij}^{\infty} \leq 1\]

Where:

\[U_i = R_{i\infty}(X,B) = \dfrac{LEV(X;\Phi_{i\infty})}{LEV(B;\Phi_{i \infty})}\]

Remark.

  • For a given \(j\), \(R_{ij}(X,B)\) decreases as we move across the age \(i\) since more losses pierce through the layer

  • We expect \(U_i\) to \(\downarrow\) for more recent AYs due to loss trend; Larger % of losses pierce the lower layer

    Larger losses are capped in the numerator but not as much in the denominator

  • \(U_i\) increases as we move to more mature years (??)

  • \(U_i\) is the ratio at ultimate of limited means; same as \(R_{i \infty}\)

Empirical example from text:

  • Select a decay from 1.000 and approaches \(U_i\) as maturity increase

  • Overlay with empirical \(R_j(X,B)\) along the diagonal

  • Overlay with the \(U_i\) ultimate ratios as well to serve as a floor

Estimate doesn’t work when:

  • There is expected negative development

  • XS layer develops more quickly than a working layer

Upper bound:

  • Get upper bound for \(R_j\) by using unlimited losses for \(B\) and the loss development factors from unlimited losses \(B\infty\)