1.1 Method Assumptions

Table 1.1: Formula for a given method
Method Formula Restrictions
Least Squares \(y = a + bx\) No Restriction
Chainladder \(y = bx\) a = 0
BF \(y = a + x\) b = 1
ELR \(y = a\) b = 0

Proposition 1.1 (Least Squares Formula)

\[b = \dfrac{\overline{xy} - \bar{x}\bar{y}}{\overline{x^2}-\bar{x}^2} = \dfrac{\mathrm{Cov}(X,Y)}{\mathrm{Var}(X)}\]

\[a = \bar{y} - b\bar{x}\]

Notes:

  • In the example here we are looking at the ATA development

  • This is effectively a credibility weighting system (giving more or less weight to the observed x as appropriate)

Above calculation can be done with the table features on TI-30XS

  • Note that TI-30XS use the formula y = ax + b, different from us

1.1.1 Caveat with Traditional Method

  • Chainladder:

    Difficult to select LDF when they vary greatly from year to year

  • BF:

    Doesn’t work well with negative development

  • ELR:

    Ignores actual experience

  • Difficulty with parameter estimation when loss patterns are changing

    • When the nature of loss experience is changing the use of unadjusted data can lead to errors

  • Variability within a stable book still have sampling error

    \(\Rightarrow\) b that doesn’t reflect the true underlying characteristics

1.1.2 Least Squares Pros & Cons

Pros

  • Good when distn is the same across multiple years

    • As we are assuming a common \(Y\) and \(X\) over the years

  • Good when there is little data and fluctuations in the year to year losses

  • Good when the randomness of the data is primarily driven by process variance

Cons

  • Bad when systemic shift year to year e.g. inflation, legal environment (Tort reform)

    • \(\therefore\) best to adjust for inflation and putting all the years on a constant dollar basis before using the LS method

    • Also should adjust for exposure

      • As the expected value increase \(\propto\) exposure and the covariances increase \(\propto\) squared exposure

1.1.3 Practical Considerations for the Least Square Method

Normalize the losses by dividing with premium since LS assume constant distribution and also adjust for inflation

Calculates ATU, so needs the tail factor first and also start from the oldest period

Recursively go backwards using the previous estimates

Notes on parameters:

  • \(a\) = Projected ultimate if no losses are reported

  • AY matures \(\Rightarrow a \: \downarrow\); \(Z \uparrow\); \(c \downarrow\)

  • If a < 0 \(\Rightarrow\) Use Chainladder to make a = 0

  • If b < 0 \(\Rightarrow\) Use ELR

  • If b = 1 \(\Rightarrow\) Same as BF