LessonThis is a lesson restated in new words. Any figure in it is an example from the source, not this desk's record.
A Markov chain is a set of states and a rule for the next one. For loans the buckets are current, one month late, two months late, and three or more. A single month can move a loan only along an arrow aging allows. The transition matrix to the twelfth power is the chance of each journey over a year.
Why it matters
Independent draws ignore that constraint. The lecture starts at 100 percent current and draws the next month from historical shares: 81 percent still current, 4 percent one bucket late, 11 percent two buckets late, and 4 percent three or more. The last two are impossible in one month. A chain with zeros on the forbidden arrows never prints that path. The same idea is why a die-roll portfolio, a new independent face every day, is the wrong picture for credit.
The idea in plain words
A random variable is a distribution. The lecture's coin example is 70 percent ones and 30 percent zeros, and a long run of zeros is still possible. The distribution also moves, so a fixed curve calls ordinary draws unlikely. The chain does not freeze that. It only replaces full independence with local dependence.
The diagram has four circles. Current can stay current or slip one bucket. It cannot reach 60-to-89 or 90-plus in one step. The 30-to-59 state can cure, stay, or worsen by one bucket. Each arrow is a month. The transition matrix stores those probabilities, rows from, columns to. A missing arrow is a zero. The lecture's filled example puts 0.99 on current-to-current. The entry from 30-to-59 into 60-to-89 is one cell, called P23 in the indexing they use.
Multi-step odds are the Chapman–Kolmogorov calculation, which here is just a matrix power. Twelve steps is the matrix times itself twelve times. The lecture reads current to 90-plus off that powered matrix and gets about 18.26 percent. Loans that start already deep in arrears have a higher chance of still being there, which is the dependence the independent draw threw away.
A state vector is the share of the book in each bucket. One start is 50 percent current, 20 percent at 30-to-59, 30 percent at 60-to-89, and none at 90-plus. After 12 months the expected 90-plus share is about 7 percent. Start with 30 percent already at 90-plus and the same horizon gives about 27 percent. Stretch the horizon to 10,000 steps and both starts sit near 3.8 percent. Until that mixing, the initial vector matters. The lecture lists the further vocabulary, recurrence, irreducibility, steady states, absorbing default, periodicity, without working each one.
How to use it
States are the easy part: low, medium, and high volatility, or bull, bear, and sideways. The probabilities are the work. Maximum likelihood, under the model's assumptions, is the count of observed i-to-j moves divided by the moves that left i. On synthetic loans the estimated matrix sits close to the matrix that generated them, and more data pulls it closer. Early estimates are noisy. The assumptions are the Markov property, constant transition odds, a unique stationary distribution, enough data, and independent loans. Job losses that hit many borrowers at once break the last one. Odds that drift break time-homogeneity. A borrower who returns from 90-plus and is then riskier than a never-late borrower breaks the memoryless property unless you enlarge the state space.
The properties people like, consistency and a sampling distribution that becomes normal as the sample grows, are conditional on those assumptions. The lecture's stance is that the chain is a first model, reasonable rather than final, and the step before a hidden chain in which the regime itself is not seen.
What usually goes wrong
Drawing next month's mix from the historical pie, as if each loan forgot its bucket, recreates the impossible 90-day jump. Treating 0.99 and the other cells as permanent ignores the lecture's own warning that transition odds move. And stopping at a one-step matrix, when the question was "where is a current loan in a year," skips the power that actually answers it.
Questions people ask
Why is current-to-90-plus equal to zero in one step?
A month cannot age a current loan by 90 days. The arrow is missing, so that cell is zero. Longer horizons can still reach 90-plus by a sequence of legal steps.
What changes between 7 percent and 27 percent?
The horizon is 12 months in both cases. The 7 percent book did not start with 90-plus loans. The 27 percent book started with 30 percent of them.
How are the cells estimated?
By the share of observed departures from a state that landed in each destination. On the synthetic book that share approaches the true matrix as the sample grows, if the assumptions hold.