Estimate Market Beta from Aligned Asset and Benchmark Returns

What is an asset's market beta? Suppose you are given one year of returns for a stock. Is that information alone sufficient to calculate its beta, and how would you estimate beta once the necessary data is available? ### Constraints and Clarifications No return observations or benchmark series are supplied in this exercise, so explain the estimator and required inputs rather than inventing a numerical beta. Clarify the return frequency, benchmark, alignment, and whether the model uses raw or excess returns. ```hint Beta compares two return series An asset's own variation does not show how it moves with a market benchmark. Identify the second series and the dates on which the observations must match. ``` ### What a Strong Answer Covers - Beta as a slope or covariance-to-variance ratio relative to a specified benchmark. - The market-return data required in addition to the asset's returns. - Consistent observation dates, frequency, return definitions, and treatment of excess returns. - A regression with an intercept, uncertainty of a one-year estimate, and the requirement of nonzero benchmark variance. - A distinction between beta and the asset's total volatility or expected return. ### Follow-up Questions 1. Can two stocks with the same volatility have different betas? 2. Why might beta change when the benchmark or observation frequency changes? 3. What can make a one-year estimate unstable even when the calculation is implemented correctly?