Estimate Market Beta from Aligned Asset and Benchmark Returns

How is an asset's market beta defined? If you are handed a single year of returns for one stock, does that data by itself let you compute its beta, and assuming the needed inputs are on hand, what procedure would you use to estimate it?

Constraints and Clarifications

This exercise provides neither return observations nor a benchmark series, so describe the estimator and the inputs it requires instead of fabricating a numeric beta. State the return frequency, the benchmark, how the series are aligned, and whether the model works with raw returns or excess returns.

Hint — Beta compares two return series: An asset's own fluctuation says nothing about how it co-moves with a market benchmark. Pin down the second series and the dates on which the observations must line up.

What a Strong Answer Covers

• Beta framed as a slope, or as the ratio of covariance to variance, against a stated benchmark.
• The market-return series that must accompany the asset's returns.
• Matching observation dates, a consistent frequency, agreed return definitions, and how excess returns are handled.
• A regression including an intercept, the uncertainty attached to a one-year estimate, and the need for the benchmark variance to be nonzero.
• A clear separation between beta and either the asset's total volatility or its expected return.

Follow-up Questions

1. Is it possible for two stocks with identical volatility to have different betas?
2. What causes beta to shift when the benchmark or the observation frequency is changed?
3. Even with a correct implementation, what makes a one-year estimate unstable?

Overview: State what market beta means, name the benchmark data required on top of a stock's returns, and cover regression estimation together with its uncertainty.