A one-step transition matrix tells you the probability of moving from rating BBB to rating A next year. A risk manager usually wants the probability of moving from BBB to A over a five-year horizon — a genuinely different question, since the bond could pass through several intermediate ratings along the way. This note shows that the -step transition probability is simply the transition matrix raised to the th power, and proves the Chapman-Kolmogorov equation that makes this true — the algebraic engine behind every multi-period Markov-chain calculation in this unit.
Learning objectives. By the end of this note you should be able to:
- compute a two-step transition probability directly from the Markov property and the definition of conditional probability;
- state and prove the Chapman-Kolmogorov equation;
- conclude that is the entry of , the th matrix power;
- compute multistep probabilities for a small chain by hand or by matrix multiplication.
In this lesson
- Warming up: a two-step calculation
- The Chapman-Kolmogorov equation
- Worked example: gambler's ruin over multiple plays
The full lesson, with worked examples and exercise solutions, is for members.
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