7.4 Method 3) Direct Development

Directly applying XS LDF to XS loss to date

Different ways to get \(^{XS}LDF^L_t\):

Formula 1: Given unlimited & limited LDFs and \(\chi\)

\[\begin{equation} ^{XS}LDF_t^{L} = \dfrac{^{XS}Ult}{^{XS}Loss_t} = \dfrac{Ult \cdot \chi}{\frac{Ult}{LDF_t} - \frac{Ult\cdot(1-\chi)}{LDF_t^L}} = \dfrac{\chi}{\frac{1}{LDF_t} - \frac{(1-\chi)}{LDF_t^L}} \tag{7.3} \end{equation}\]

• Based on \(\underbrace{Loss_t \cdot LDF_t}_{100\%} = \underbrace{Loss^L_t \cdot LDF^L_t}_{100\% - \chi} + \underbrace{^{XS}Loss_t^L \cdot {^{XS}LDF}^L_t}_{\chi}\)

• Can’t use the actual losses to date for this as it’ll just be equal to the implied method mathematically

Formula 2: Given unlimited & limited LDFs and \(R^L_t\)

\[\begin{equation} ^{XS}LDF^L_t = \dfrac{LDF_t - R^L_t \times LDF^L_t}{1-R^L_t} \tag{7.4} \end{equation}\]

• Based on Question 21 from 2004 (7.1)

• \(R^L_t = \dfrac{Sev^L}{Sev}\), or

• \(R^L_t = \dfrac{Ult^L \div LDF^L}{Ult \div LDF} = \dfrac{\text{Expected Limited Reported}}{\text{Expected Unlimited Reported}}\)

Disadvantages

• LDFs large and volatile, not recommend this method