Classifying Bull and Bear markets with a two-state Markov chain

Daily returns are famously hard to predict. But the condition the market is in — a multi-year climb or a sustained drawdown — tends to persist. That persistence is exactly what a Markov chain models: a system that jumps between a small number of states, where tomorrow's state depends only on today's, not on the full path that got us here.

This study turns ten years of S&P 500 data, 2,244 trading-day observations from 2015 through 2024, into a two-state chain: Bull or Bear. The classification signal is deliberately simple — the 20-day moving average of daily returns — so the interesting work lives in the estimation, not in feature engineering.

Daily returns were computed from S&P 500 closing prices, smoothed with a 20-day moving average, and thresholded into two states. From the resulting regime sequence, the one-step transition matrix was estimated by maximum likelihood: count every Bull-to-Bull, Bull-to-Bear, Bear-to-Bull, and Bear-to-Bear transition, then normalize by the number of departures from each state.

With the transition matrix in hand, NumPy and Pandas carried the rest of the analysis: the steady-state distribution (the eigenvector of the matrix with eigenvalue 1), first-passage probabilities (the chance of hitting one state from the other for the first time), and multistep transition matrices Pⁿ that answer where the market is likely to be n days out from any starting state.

Bull-to-Bear is the complement of Bull-to-Bull; steady-state shares sum to one.

Three modeling choices drive everything downstream, so they are worth stating explicitly. First, the Markov property: tomorrow's regime is treated as depending only on today's, not on the full path of recent days. That is a simplification — markets have memory — but it is the simplification that makes a transition matrix the right tool, and the estimated matrix can be checked against the data for plausibility.

Second, the regime definition: the sign of the 20-day moving average of daily returns. A shorter window reacts faster but flickers; a longer window (50 or 100 days) is smoother but lags regime changes. Twenty days was chosen as a defensible middle: responsive enough to catch the COVID-era swing, smooth enough to avoid daily noise. Third, two states rather than three (adding a flat/Sideways regime) — two states keep the matrix readable and the first-passage results interpretable, which mattered more here than extra granularity.

Alternatives considered and set aside: hidden Markov models with more states, which add parameters and estimation risk without data to justify them at this sample size; and price-level signals, which react more slowly than return-based moving averages. The chosen design optimizes for auditability: every number in the results section can be re-derived from the transition counts.

The dominant result is persistence. Once the market is classified as Bull, it stays Bull with probability 93.6% on the next day — the diagonal of the transition matrix is heavily loaded, which is the mathematical signature of a system that does not flip states casually. Bear regimes, while less frequent in the long run, are exactly where that persistence cuts both ways: entering a drawdown means expecting a run of Bear days, not a quick bounce.

The steady-state distribution puts numbers behind a fact every investor learns the hard way: over 2015 to 2024, the market spent roughly 68.3% of its time in the Bull regime and 31.7% in Bear. The first-passage and multistep results round out the picture — given enough days, the chain converges to that 68/32 split no matter which state it starts in, and the multistep matrices show the convergence path for any horizon n.

The finished analysis is a Python pipeline in Pandas and NumPy that takes raw price data and returns the full Markov toolkit: the estimated transition matrix, steady-state distribution, first-passage probabilities, and multistep matrices. Every headline number — 93.6% Bull persistence, 68.3% long-run Bull share — falls out of that one matrix.

For an operations-minded view of markets, that compression is the point: two states, four transition probabilities, and a clear statement of how long each regime is likely to last. It is the same discipline as a time study on a factory floor — observe the system, count the transitions, and let the steady state speak.

The chain is estimated from a single decade (2015–2024) that includes the COVID crash and recovery — an unusually violent episode that affects the transition counts. A different decade would produce a different matrix, and the steady state should be read as descriptive of this sample, not as a law of markets.

The regimes are defined by construction (the 20-day moving-average rule), so the model characterizes persistence of that definition; it does not discover regimes independently. And nothing here is a trading strategy: the analysis quantifies how long regimes tend to last, not what any asset will do next.

Constraining the model to two states forced clarity about what a “regime” actually is, and the 20-day moving-average threshold turned out to be a defensible, reproducible definition rather than a fitted one. The more valuable skill was the estimation work behind it: building the transition counts correctly, recovering the steady state from the matrix itself, and using first-passage and multistep results to check that the model behaved sensibly at every horizon.

It also reinforced a habit from the rest of my work: report the quantity that answers the actual question. Here, that means regime persistence and long-run occupancy, not a noisy day-ahead prediction no model can honestly claim.