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Berquist-Sherman Techniques

Method

  1. Disposal rates
  2. Adjusted Paid claims using interpolation guided by disposal rates

The second step can be done in two ways:

Linear assumption

Assume claims and counts have a linear relationship

Use simple linear interpolation, say we found that the 24-mo adjusted paid count is higher than 24-mo (unadjusted) paid count. Then we must interpolate between 24-mo to 36-mo for the paid claims.

So, the 24-mo Adjusted claim is given by:

\[ \text{24-mo claim} + \dfrac{\text{Selected 24-mo DR} - \text{24-mo DR}}{\text{36-mo DR}- \text{24-mo DR}} \times(\text{36-mo claim} - \text{24-mo claim}) \]
  • Easy to remember if you understand what is happening… we are just adding a portion of the difference \((\text{36-mo claim} - \text{24-mo claim})\)
  • The portion is \(\dfrac{S-A}{B-A}\) where \(A \lt S \lt B\)

Exponential assumption

Assume claims and counts have an exponential relationship

  • Find adjusted paid claim counts = paid counts \(\times\) latest Disposal rate for maturity
  • Find the exponential fit i.e., \(\text{claims} = a\cdot \exp({b\times\text{counts}})\), we are going to find it for the interval "12-mo to 24-mo" (for an arbitrary AY)
\[ b_{\text{24-mo}} = \dfrac{\ln(\text{24-mo claim}) - \ln(\text{12-mo claim})}{\text{24-mo count}- \text{12-mo count}} \]

Thus, \(b_{\text{24-mo}}\) is the fit between 12-mo and 24-mo. Note this convention for it will be needed to decide which parameters to use.

  • For \(a\), just use the first relationship. So, \(a_{\text{24-mo}} =\dfrac{\text{24-mo claims}}{\exp(b_{\text{24-mo}} \times\text{24-mo counts})}\)

Now just apply the relationship to find \(\text{claims}= a \cdot \exp(b \times \text{counts})\) with the following logic (again, using 24-mo claims and counts as an example.)

  1. (GREEN) For 24-mo maturity, if the adjusted count \(\gt\) actual count, the adjusted claims should be \(\gt\) actual claims.
    • Use the 24-mo to 36-mo relationship thus,
    • adjusted claims = \(a_{36-mo}\exp(b_{\text{36-mo}}\times \text{24-mo counts})\)
  2. (RED) For 24-mo maturity, if the adjusted count \(\lt\) actual count, the adjusted claims should be \(\lt\) actual claims. So
    • Use the 12-mo to 24-mo relationship thus,
    • adjusted claims = \(a_{24-mo}\exp(b_{\text{24-mo}}\times \text{24-mo counts})\)

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Case Reserve Adjustment

Method

  1. Find average case reserves (case severity) of the latest period
  2. Use paid severity trend to trend the the latest case severity \(\to\) Adjusted Average Case Severity
  3. Adjusted Cum rept claims = case severity \(\times\) open claim# + paid claims

Notes

  • Detrend everything, even if it it inherently had the same trend as the Year-on-year losses

Misc.

  • Medical malpractice:
    • Cannot use paid loss data to estimate severity trends for CRAd
    • Slow payment of claims for medical malpractice
    • reduces data available by accident years to estimate paid severity at early maturities