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Loss Development using Credibility - Brosius

Study Strategy

  • Learn to solve by example for Least Squares (=linest), Link ratio method (\(\bar{y}/\bar{x}\)) and budgeted method (\(\bar{y}\)).
  • Next, learn to see the credibility weighting perspective (\(\dfrac{b}{LDF}\))
  • First get acquainted with different Buhlmann credibility terms. Work out the derivation and be convinced about the VHM and EVPV formulae.
  • Go through Hugh White's question. Understand what he asked for and how Least Squares' approach works for each situation.

My Notes

Least Squares approach

\[ y = a + bx \]
  • where \(y\) denotes ultimate values of \(x\). Least Squares is a generalization of
    • Chain Ladder: \(a=0\)
    • Expected loss: \(b =0\)
    • BF: \(b=1\)
  • Appropriateness of least squares?
    • Can we assume the same \(Y\) and \(X\) distribution pair for each year?
    • We can, only if
      1. No systematic shifts in business. All fluctuations are random
      2. Adjust suspected distortions
        • Inflation-adjustment
        • Book expands then use Pure premium (divide losses by exposure)
        • B-S adjustment where-ever appropriate
    • When talking against least squares…
      • There are systematic shifts
      • LS assumes variance in losses is due to random fluctuations only
    • Sampling error may lead to \(a\) and \(b\) values which don't make sense.
      • If \(a\lt0\), use Link Ratio method
      • If \(b\lt 0\), use Budgeted Loss Ratio method
  • Pros
    • Flexible (best fit of CL, ELR, BF)
    • If random year to year fluctuations
  • Cons
    • If Systematic fluctuations (unstable book)
    • Sampling error can give unreasonable \(a\) and \(b\) values (say, less than zero). Use CL (\(a=0\)) and ELR (\(b=0\)) in such cases.

Credibility adjustment

When "small or adjustable year-on-year changes" assumption doesn't hold.

  • LS = Credibility weighting of CL and ELR.
    • \(LDF = \dfrac{\bar{y}}{\bar{x}}\) (weighted average LDF)
    • \(Z = \dfrac{b}{LDF}\)
    • \(d:\) % reported/paid (essentially \(d = \dfrac{1}{LDF}\))
      • \(d = E(\frac{x}{y})\)

  • If the assumption of "small year-to-year changes or that it can be corrected for" doesn't hold!

    • Use Buhlmann credibility
    • Keyword: change! (Before and after)

  • Buhlmann credibility

    • Ultimate loss elements
      • \(E[Y]\)
      • \(\sigma(Y)\)
    • Percent reported elements
      • \(E[X/Y]\)
      • \(\sigma[X/Y]\)
    • Latest reported: \(x\)
    • \(VHM = Var[E(X|Y)] = E(X/Y)^{2} \sigma^{2}(Y)\)
      • Think: We pull out \(E(X/Y)\) as a constant \(c\) so it becomes \(c^{2}\)
    • \(EVPV = E[Var(X|Y)] = \sigma^{2}(X/Y) E[Y^{2}]\)
      • Think: We pull out \(Y\) as \(c\) so it becomes \(c^{2}\) and take its expectation.

Hugh White's Question

Situation: Reported losses > expectations
Decision Thinking Justification Method
Reduce IBNR "Rate of reporting must have increased, the total claims won't change." Accelerating rate of reporting Budgeted Loss
Leave IBNR "Perhaps due to a random fluctuation like a large loss" Random fluctuation Bornhuetter-Ferguson
Increase IBNR "Maybe I believed the expected losses too much, I should probably follow chain ladder more." Lack of confidence in a priori Link Ratio
  • Brosius says…
    • Least squares formula: \(L(x) = a + bx\)
    • \((\bar{Y} - b \bar{X}) + bx = (x-\bar{X})b +\bar{Y}\)
    • \(b = \dfrac{Cov(X,Y)}{Var(X)}\)
    • So, \(L(x) = (x-E(X)) \dfrac{Cov(X,Y)}{Var(X)} + E(Y)\)
  • So, our \(Cov(X,Y)\) and \(Var(X)\) take the decision
    • \(x = E(X)\)
      • As per expectation, our initial estimate of \(E(Y)\)
    • \(x \neq E(X)\)
      • \(Cov(X,Y) = Var(X)\) → Increase only by \(x-E(X)\) → Leave IBNR (2)
      • \(Cov(X,Y) \lt Var(X)\) → Increase by less than \(x - E(X)\) → Reduce IBNR (1)
      • \(Cov(X,Y) \gt Var(X)\) → Increase by more than \(x-E(X)\) → expect increase to continue in future → Increase IBNR (3)
Method Situation
Budgeted Loss \(Cov < V\)
Bornhuetter-Ferguson \(Cov = V\)
Link Ratio \(Cov > V\)

Brosius: Least squares works in any situation.

Muffs

  • Fall 2000 Exam 6 - Q41: When you are given 48 months to ult tail factor but one of the months have 60 month maturity, then know that 60 month doesn't mean ultimate. Thus, you will have to use 48 month loss and multiply it by the given tail factor.
  • Remember, we always do operations with ULTIMATE values. If you are not given ultimate values, first start looking for them.
  • When answering "why inappropriate?"
    • Think of what exactly would go wrong. Like for e.g. → "The calculated reserves would be two low".
    • And also why would it? → "Because it assumes constant loss reserves"
    • Also mention other assumptions → "(assuming %reported remains constant)"
  • When \(b<0\), Least Squares method is invalid

    - Use the Budgeted Loss method instead.