No-claims discount simulator
The Markov chain every CS2 paper reaches for: three discount levels, up one after a claim-free year, down after a claim. Set the claim probability and the step-back rule, then watch the portfolio settle into its stationary distribution.
Transition matrix
| From \ To | 0% | 25% | 50% |
|---|---|---|---|
| 0% discount | 0.15 | 0.85 | 0.00 |
| 25% discount | 0.15 | 0.00 | 0.85 |
| 50% discount | 0.15 | 0.00 | 0.85 |
Each row is a current discount level; each entry is the probability of moving to that column's level next year. A claim-free year (probability 0.85) moves you up one level; a claim moves you down two levels, floored at 0%.
Long run
Convergence to the steady state
A brand-new portfolio starts entirely at 0% discount and washes towards the stationary distribution. However it starts, the chain forgets its initial state; that memoryless convergence is the whole point of the Markov property.
The classic exam setup: discount levels 0% / 25% / 50%, claim-free years step up one level. The harsher rule concentrates the steady state at the extremes; try both and watch the middle level drain. Claim probability is assumed the same at every level, which real systems complicate (protected bonuses, claim frequency varying with the discount).
Why examiners love this chain
It is small enough to solve by hand and rich enough to test everything: writing down the transition matrix from a verbal rule, finding the stationary distribution from πP = π, and turning it into money as a long-run average premium. The classic twists are all one slider away here: a harsher step-back rule, a higher claim frequency, or asking what fraction of drivers ever hold maximum discount. If you can rebuild this page's matrix from the rule description alone, the exam question is done.
Make it stick. Markov chains sit next to time series and survival models in CS2; try the time series playground and the survival models playground next. Memori is a flashcard app built by actuarial students, with a ready-made CS2 set in the shop. Join the beta.
Illustrative system and figures. For education only, not financial advice.