who owns S&P 500 index funds and gold bars, the index would combine the S&P 500 and the price of gold. However, the resulting beta would no longer be a market-beta in the typical meaning of the term. The choice of whether to subtract the risk-free rate (from both own returns and market rates of return) before estimating market-betas is similarly inconsequential. When this is done, usually one selects an interest rate equivalent to the time interval (i.e., a one-day or one-month Treasury interest rate.) Empirical estimation It is important to distinguish between a true market-beta that defines the true expected relationship between the rate of return on assets and the market, and a realized market-beta that is based on historical rates of returns and represents just one specific history out of the set of possible stock return realizations. The true market-beta could be viewed as the average outcome if infinitely many draws could be observed---but because observing more than one draw is never strictly the case, the true market-beta can never be observed even in retrospect. Only the realized market-beta can be observed. However, on average, the best forecast of the realized market-beta is also the best forecast of the true market-beta. Estimators of market-beta have to wrestle with two important problems: The underlying market betas are known to move over time. Investors are interested in the best forecast of the true prevailing market-beta most indicative of the most likely future market-beta realization (which will be the realized risk contribution to their portfolios) and not in the historical market-beta. Despite these problems, a historical beta estimator remains an obvious benchmark predictor. It is obtained as the slope of the fitted line from the linear least-squares estimator. The OLS regression can be estimated on 1–5 years worth of daily, weekly or monthly stock returns. The choice depends on the trade off between accuracy of beta measurement (longer periodic measurement times and more years give more accurate results) and historic firm beta changes over time (for example, due to changing sales products or clients). Improved estimators Other beta estimators reflect the tendency of betas (like rates of return) for regression toward the mean, induced not only by measurement error but also by underlying changes in the true beta and/or historical randomness. (Intuitively, one would not suggest a company with high return [e.g., a drug discovery] last year also to have as