Sahasrabuddhe: Development in Layers

Study Strategy¶

The source material is daunting, and unnecessarily rigorous. I feel start basic and once you have grasped the mechanism of adjustments, then go for the source material (optional)
Solve questions and work your way back to theory to stitch everything together!
First learn how to calculate the trend indices, and practice.
Claim Size model will require you to refer back to material.
- Calculate LEV depending on distribution, formula given on the exam
- \(L(d,p)\) for a layer is just difference

Big Picture¶

We have ultimate claim size mean → adjust it for trends then you have it for every cell → use these to find limited expected values in a layer and relativities between layers \(a\) and \(b\).
\(a\) = Basic limit; \(b\) = Limits of our given table
Use relativities to calculate cumulative triangles at current year cost levels and basic limits → Denominator = current @ original limits, Numerator = latest @ basic limits
Calculate LDFs from this Adjusted Cumulative triangle

Checklist¶

My Notes¶

The point

Adjust LDF for one claim layer to produce LDFs appropriate for another layer.

Useful for high deductible and XOL reinsurance policies (with thinness in excess/ceded layer)

Notation¶

Exposure period (AY) \(i\)
Development period \(j\)
\(C^L_{i,j}\) → Cumulative claims in \(L\)
\(C_{i,\infty}^L\) → Ultimate claims in \(L\)
\(f_{n,j}^L\) → Age-to-age (LDF)
\(F^L_{n,j}\) → Age-to-ult (CDF)
\(L(d,p)\) → claims layer (\(0 \leq d < p \leq \infty\))
- truncated below \(d\) → deductible
- censored above \(p\) → policy limit
- \(d=0, p=\infty\) → ground-up unlimited basis

Trend¶

Trends apply to and are calculated from ground-up unlimited claims layer

Complication about selection of trend factors

Trend typically acts on incremental values
Analysis uses cumulative data
Then exactly what trend factors to apply to this data?

Steps¶

Calculate AY trend (row-wise)
Calculate CY trend (diagonally)
Total trend = AY \(\times\) CY (for \(T_{ij}\) values)

Claim Size Model¶

Calculate LEVs from the distribution → Every distribution has a limited version, in Exam P, we had seen this as \(E(X \wedge L)\) and is provided in the tables for each distribution. Each cell has a mean value which can be used to determine it. We will typically be given the exponential distribution.¹
\(LEV(L(d,p)) = LEV(p) - LEV(d)\) for a given distribution \(\Phi\)

How \(\theta\) impacts adjusted LDFs

If claim size parameters (\(\theta\)'s) are larger
- adjustment of LDFs below limit is impactful ← Greater difference between adjusted and unadjusted factors
- the impact from a larger trend would be the same.
If \(\theta\)'s are smaller (less claims are being censored at policy limit)
- limited LEVs are closer to unlimited LEVs
- Adjustment of LDFs below limit is less impactful (← difference between adjusted and unadjusted factors is minimal)
- but above the limit is more impactful (← fewer losses in excess layer)

Adjusting LDFs¶

Why adjust LDFs?

Traditional Chain Ladder to calculate LDFs is appropriate if our loss data
- is on a ground-up (\(d=0\))
- unlimited basis (\(p = \infty\))
- and trend only acts in AY direction
If either of these aren't true \(\implies\) We need to adjust LDFs

Use 500k as Basic limit unless told otherwise

Basic limit = data is sufficiently credible for estimating claims development patterns
To adjust LDFs
1. Adjust Original loss triangle to latest period's cost level @ basic limit
2. Use \(\uparrow\) to calculate LDFs → CDFs
3. Adjust CDFs for any layer

Let's look at how to do the last step. For layer \(X\)

\[ F_{ij}^X = F_{n,j}^B \times \dfrac{\text{Ult in Layer}/\text{Ult Latest Basic}}{\text{Dev(j) in Layer}/\text{Dev(j) Latest Basic}} \]

Its best to create an "in Layer" table (using the relation \(LEV(L(d,p)) = LEV(p) - LEV(d)\))
And apply this adjustment to CDFs \(= \dfrac{\text{Ult in Layer}}{\text{Dev(j) in Layer}} / \dfrac{\text{Ult Latest Basic}}{\text{Dev(j) Latest Basic}}\)
- So, to remember, you are calculating \(\dfrac{\text{CDF for Layer}}{\text{CDF for Basic @Latest}}\) and multiplying it to selected CDFs from Basic @Latest

Claims Layer Ratios¶

The Goal

Find LDFs for

There is no claim size model for maturities other than ultimate.
But don't try to develop one. Just estimate the ratios

\[ \text{Claims Layer Ratio} = R_{j}(B,X) = \dfrac{LEV(B)}{LEV(X)} \]

Adjustment factors for for LDFs
- Basic \(=\dfrac{\text{LEV(Basic, Ult)}}{\text{LEV(Limited, Ult)}}/\text{Claims Layer Ratio}\)
- Basic → Limit (X) \(=\left(1-\dfrac{\text{LEV(Basic, Ult)}}{\text{LEV(Limited, Ult)}}\right)/(1-\text{Claims Layer Ratio})\)

Properties of Claims Layer Ratio

For \(R_{j}(B,X) \lt 1\) (i.e. ignoring negative dev for which it would be \(\geq 1\))

\(R_{a} \gt R_{b}\) (\(a < b\)) for earlier maturities → less development in excess layer earlier on. (e.g. \(R_{12} \gt R_{24}\))
\(R_{a} \geq R_{\infty}\) → Pre-mature (reported) ratios will be greater than ultimate \(\impliedby\) more development in higher layers than lower layers²
If LDFs calculated from unlimited data → \(\max{R} = R_{\infty} \times F_{j}\) #doubt

Distribution needs to meet requirements:
1. Parameters can be adjusted for inflation
2. LEV and unlimited means can be calculated with reasonable effort ↩
Lower layers don't develop once they hit the top of the layer ↩