Clark's method

David R. Clark published "LDF Curve-Fitting and Stochastic Reserving: A Maximum Likelihood Approach" in the CAS Forum in fall 2003, and the paper is now on the syllabus of Casualty Actuarial Society Exam 7. The idea fits in one curve. Instead of estimating one development factor per column, you assume the share of ultimate reported at age x follows a growth function G(x), loglogistic or Weibull. The loglogistic is x to the ω over x to the ω plus θ to the ω. You fit ω and θ to the triangle's increments by maximum likelihood. Increments follow an overdispersed Poisson: variance equals a scale factor σ² times the mean, which gives process risk and parameter risk on top of the reserve. Two versions. The LDF version estimates one ultimate per accident year, n plus two parameters. The Cape Cod version replaces those ultimates with premium times a common loss ratio and drops to three parameters, which makes it the better choice on a small triangle: ten accident years give only fifty-five increments, which the LDF version already spreads over twelve parameters. Age is measured at the middle of the accident year, six months for an accident year seen at December 31. The gain is twofold, few parameters and a tail supplied by the curve with no separate fit. So is the cost: the loglogistic has a heavy tail. Clark himself calls truncation at a set age an effective way to rely less on the extrapolation; without it, the curve projects development for decades. The curve also imposes a smooth shape the triangle does not always follow, and a change in settlement speed goes unnoticed in two parameters. Not to be confused with Stanard's Cape Cod method, whose common loss ratio logic it borrows, or with a tail factor extrapolated separately.