6.1 Expected Loss Emergence

Definition 6.1

1. \(x =\) average age of AY

   (e.g. 6mo for the most recent instead of 12mo)

2. \(G(x \mid \omega, \theta)\) = % paid to date growth function

   • 2 forms of the growth function below

   • The curves move smoothly from 0 to 1

Loglogistic:

\[\begin{equation} G(x \mid \omega, \theta) = \dfrac{x^{\omega}}{x^{\omega} + \theta^{\omega}} \tag{6.1} \end{equation}\]

Weibull:

\[\begin{equation} G(x \mid \omega, \theta) = 1- \mathrm{exp}\left\{ { - \left( \dfrac{x}{ \theta } \right)^{\omega}} \right \} \tag{6.2} \end{equation}\]

Remark.

% Emergence in period \(x\) to \(x+12\) = \(G(x+12) - G(x)\)

• Equivalent to the \(f(d)\) in Venter Table 5.3 for the alternative pattern \(f(d)h(w)\)

  Given \(d = x / 12 - 1\)

Advantages (over Venter):

• Uses only 2 column parameters: \(\theta\) for mean; \(\omega\) for s.d.

• Can use data @ different age

  • Not the same maturity as prior years

  • Only last few CYs

• Output is a smooth curve \(\Rightarrow\) Can interpolate between ages and extrapolate a tail

Motivation for using a curve is to recognize that adjacent LDFs are related

Doesn’t work when there is expected negative loss development

6.1.1 Expected Ultimate Loss Methods

Method 1: Cape Cod

\[Premium_{AY} \times ELR\]

Remark.

• A single row parameter \(h\) for the entire triangle

• \(h\) here is the \(ELR\)

• Similar parameters in Venter (5.3)

• This method is preferred:

  • Only need 3 parameters

  • Includes extra information, exposure base

Estimate Future Emergence:

\[[G(y \mid \omega, \theta) - G(x \mid \omega, \theta)] \times [Premium_{AY} \times ELR]\]

Method 2: LDF

\[ULT_{AY}\]

Remark.

• \(h(w)\) for each row (i.e. parameter for each AY)

• \(h(w)\) here represents the ultimate loss for each \(w\)

• Similar parameters in Venter (5.3)

Estimate Future Emergence:

\[[G(y \mid \omega, \theta) - G(x \mid \omega, \theta)] \times ULT_{AY}\]