Mack 1994 & Venter Factors Flashcards
What are the three implicit assumptions under Chain Ladder
- Expected losses in the next development period are proportional to the losses-to-date (linearity)
- Losses are independent between accident years
- Variance of losses in the next development period is proportional to losses-to-date with proportionality constant alpha-sq that varies by age
Three separate formulas to find the best fk.
Plots to test for implicit assumptions
- Linearity - plot Ci,k+1 aginst Cik
- Variance - plot weighted residuals agiant Cik
Mack Testing for correlations between subsequent development (should be uncorrelated)
- calculate ATA
- create a table of rik and sik (rik the rank of colk, sik the rank of rik-1 removing the most recent
- let T = see below
- I-k is the number of ranks we’re testing. Denom = (I-k)*[(I-k)-1]
Whats the importance to perform Calendar year influence test for AY correlation?
In general, AY uncorrelated is violated when we have calendar year influences that affect multiple AYs includes but not limited to
- Major changes in claims handling practices
- Changes in payment processes
- Unexpectedly high or low inflation
- Significant changes due to court decision
CY influence test general process
- calculate ATA
- convert ATA into ranks only
- convert rank into S, L, *. [S = below-avg ATA, L = above-avg ATA]
- create table of Sj and Lj for each CY j except j=1 (bc that only has 1 item, nothing to compare with) aka diagonal
- Zj = min(Sj, Lj), Mack assumes Z ~ Norm(E[Z], Var[Z])
- n= Sj+Lj. m = rounddown((n-1)/2, 0)
- See E[Zj], Var[Zj], E[Z], Var[Z] below
Mack Chain Ladder assumption #1 elaborate?
Expected losses in the next development period is proportional to losses-to-date
- The chain-ladder method uses the same LDF for each accident year
- use most recent loss-to-date to project losses, ignoring losses as of earlier development periods
Mack Chain Ladder assumption #2 elaborate?
Mack Chain Ladder assumption #3 elaborate?
Lognormal Confidence Interval formula
formula for alpha^2
Venter-Correlation Test procedure
- calculate correlation between 2 incremental columns
- calculate T using below, where n is the # of pairs used and df = n-2 (why???)
Venter’s testable implications of assumptions
- significance of factor f(d) (aka. incremental %): abs value of f twice the std rule
- Superiority of factor assumption to alternative emergence pattern
- Linearity of model: Look at residuals asa function of c(w,d)
- Stability of model: look at residuals as a function of time
- no correlation among columns
- no high or low diagonals
Superiority of factor assumption: alternative emergence pattern #2: Parametrized BF steps
- E(q(w,d)|data to w+d-1] = f(d)h(w) where f(d) is the incr lag factor, h(w) is the ult loss amount for an AY
- for a complete triangle with m accident years, 2m-2 params
- initiate f(d), can use incremental % emerged
- calculate h(w) using squared weights h(w) = sum(f(d) * IncLoss_w,d) / sum(f(d)^2)
- calculate first iterated f(d)_1 using h(w) as weights, f(d)_1 = sum(h(w) * IncLoss_w,d) / sum(h(w)^2)
- E[IncLoss_w,d] = f(d) * h(w)
what’s the purpose testing significantly high or low diagonals
Mack’s high-low diagonal test counts the number of high and low factors on each diagonal, and test whether or not that number is likely to occur due to chance
