No-claims discount simulator

Each line is the share of the portfolio at that discount level, year by year, from the starting mix you pick above; the dashed lines are the stationary distribution. The thing to see: switch the start and the lines begin somewhere completely different, yet land on the SAME dashed lines — the chain forgets where it started, which is the Markov property, and it is why the steady state can be found from πP = π without knowing any history. The landing is fast, and that is a finding, not a flaw: under "down two" every claim goes straight to 0%, so after year one the 0% share already equals p exactly, and (with both step-up timers at one year) the whole mix is EXACTLY stationary from year two — the third dot, from any start: this chain's memory is precisely two years deep. "Down one" only approaches its dashed lines gradually, and raising the step-up timers below stretches the settling out further — more states to work through means a longer memory.

The chart above averages the whole portfolio, so this card follows one actual driver instead: gold dots are claim years, each one a knock down the ladder followed by the climb back at one level per claim-free year. Run it a few times: the path never settles, but the share of years spent at each level (the bars) hovers around the same steady-state percentages as the portfolio chart. That is the other reading of the stationary distribution: it is both the long-run mix of the portfolio at one moment and the long-run mix of one driver's years.

The classic exam setup: discount levels 0% / 25% / 50%. A claim-free year steps up one level (staying put once at 50%). On a claim, "down one" means 50% → 25% and 25% → 0%; "down two" sends 50% straight to 0%. From 25% a claim hits the 0% floor under either rule, so with three levels the toggle changes only the 50% row of the matrix. Under "down two" every claim lands at 0% wherever it starts, which is why the 0% share of the steady state then equals the claim probability exactly (with both timers at one year). The step-up timers make promotion slower than demotion, as real NCD systems do; push them up and watch the steady state slide toward the lower discounts and the average premium rise. Claim probability is assumed the same at every level, which real systems complicate (protected bonuses, claim frequency varying with the discount).