7.6 Method 5) Development Method

Method overview:

  1. Select trend to adjust historical layers

  2. Develop ground up claim count

  3. Calculate severity \(^{XS}LDF^L_t\) based on \(R^L_t\)

Data Required

  • Ground up WC claims level data

    (Allows individual losses to ber capped at different amounts)

  • Includes indemnity, medical and ALAE together

  • Can split the data by account, injury, and state as a future step

7.6.1 Severity Trend

Need to account for loss trend when capping losses

  • Apply different limits to the historical data by AYs

    (or else there will be more and more losses piercing the limiting layer \(\therefore\) Distorting the LDFs)

Adjusting for loss trend

  1. Select a trend by fitting an exponential curve to the average severity of unlimited losses by AY (e.g. \(y = ce^{bx}\))

    • Optional: Use different trend for different years

    • Optional: Adjust for large losses

  2. Cap the historical losses at lower amount to compensate for the loss trend by detrending the historical layer

7.6.2 Claim Count Development

Split claim count development from severity development

For claim count development we focusing on ground up claims

Advantages:

  • Most claims are reported not too deep in the tail even if the full severity in not yet known

  • If you only count claims once they pierce the deductible layer you have to deal with uncertain claim count development deep into the tail

  • If your claim counts depend on the limits, every time you update the severity trend assumptions the claim counts will change

7.6.3 Severity Development

Fit LDFs for each layer to the power curve

  • Power curve needs a cutoff

  • Using the same cut off for all layers might introduce a bias as lower layers will cease development earlier

  • Different layer LDFs need to respect the relationships lay out below (Proposition 7.1, 7.2 and 7.3)

7.6.3.1 Relativities

Definition 7.1 Relativity

\[R_t^L = \dfrac{\text{Limited Sev @ }t}{\text{Unlimited Sev @ t}} = \dfrac{S^L_t}{S_t}\]

\[R^L = \dfrac{\text{Limited Sev @ Ult}}{\text{Unlimited Sev @ Ult}} = \dfrac{S^L}{S}\]

Remark. Relativity (for different limits) over time

  • Relativity starts close to 1.00, and drops over time before reaching the ultimate relativity

  • The losses limited at higher limits start with a very high relativity (close to 1.00), and take longer to come down

  • Here they define \(LDF_t = \dfrac{S}{S_t}\) (So severity LDFs, not sure what’s the implication)

Proposition 7.1 \[LDF_t^L = LDF_t \dfrac{R^L}{R_t^L}\]

Proof.

\(LDF_t^L = \dfrac{\text{Ultimate Loss}}{\text{Losses @ t}} = \dfrac{C \cdot S^L}{C_t \cdot S^L_t} = \dfrac{C \cdot S \cdot R^L}{C_t \cdot S_t \cdot R^L_t} = \underbrace{\dfrac{C \cdot S}{C_t \cdot S_t }}_{LDF_t} \times \dfrac{R^L}{R^L_t} = LDF_t \dfrac{R^L}{R_t^L}\)

Remark. \(C\) is claim count and \(S\) is severity

Proposition 7.2 \[^{XS}LDF_t^L = LDF_t \dfrac{(1-R^L)}{(1-R_t^L)}\]

Remark. Analogous relationship for XS LDFs

Proposition 7.3 \[LDF_t = R^L_t \cdot LDF^L_t + (1 - R^L_t) \cdot {^{XS}LDF}^L_t\]

Remark.

  • This follows from Proposition 7.2 and 7.3

  • This formula doesn’t require \(R^L\)

  • Formula only work once claim reporting is finished

  • Relativity based on average from historical losses

  • All the above can be adjusted to work with incremental LDFs if we don’t use the ultimate relativities

7.6.3.2 Incremental LDFs and Change in Relativities

Proposition 7.4 \[\dfrac{_{inc}LDF^L_t}{_{inc}LDF_t} = \Delta R^L_t = \dfrac{R^L_{t+1}}{R^L_t}\]

Remark.

  • Difference of limited and unlimited incremental LDFs is driven by the change in relativity

  • With \(R_t\) and \(R_{t+1}\) we can get the limited or unlimited LDF given one or the other

Proposition 7.5 \[\dfrac{_{inc}^{XS}LDF^L_t}{_{inc}LDF_t} = \Delta (1 - R^L_t) = \dfrac{1 - R^L_{t+1}}{1 - R^L_t}\]