Beta depends on its window and benchmark

A relative slope rather than total risk

For aligned asset and benchmark returns, the simple regression beta with an intercept equals their sample covariance divided by the benchmark's sample variance. Both moments use the same paired rows and the same divisor, so that divisor cancels in their ratio. Beta describes linear sensitivity to this particular benchmark, not the asset's total volatility or its maximum possible loss. Equivalently, it is correlation multiplied by the ratio of asset volatility to benchmark volatility when both are nonzero. An asset can have substantial standalone variation and a small beta because much of its variation is unrelated to the selected benchmark.

The benchmark defines the comparison

Changing from a domestic equity benchmark to a sector or global benchmark changes the explanatory return series, not merely the label on an unchanged statistic. Currency, dividend treatment and hedging conventions also alter that series. Damodaran's discussion of risk-parameter estimation identifies benchmark choice as a material source of variation in reported betas. A regression on a narrow index answers a narrower exposure question than a regression on a broad index. A numerically lower slope under the second benchmark does not establish lower absolute risk. The measurement remains conditional on what moves in the denominator and how it covaries with the asset.

Raw returns and excess returns differ

A raw-return market regression and a regression of asset excess returns on market excess returns are different specifications. Kenneth French's factor documentation explicitly defines the market factor as a market return less a risk-free return. Subtracting a constant from both series leaves centered covariance and variance unchanged, but subtracting a time-varying risk-free series generally does not. The intercept's interpretation also changes. Raw returns, excess returns and benchmark-relative active returns are not interchangeable input names. A reproducible beta identifies which transformation was made, its frequency and the matching risk-free observation, without inferring CAPM validity from a successful slope calculation.

Window length and return interval are separate

The lookback window determines which history enters estimation; the return interval determines the units of each observation within it. A longer window supplies more observations at a fixed interval but can mix materially different exposures. Shortening the interval increases row count without necessarily creating proportionately more independent information. Hawawini's published abstract documents that beta can change with the return interval. Non-synchronous closes and stale prices can move measured covariation across adjacent observations. There is no universal square-root-of-time conversion for beta. Re-estimating it from weekly returns is a different calculation from multiplying a daily beta by an annualization factor.

A controlled window and benchmark comparison

In this original synthetic hypothetical, benchmark A repeats the four decimal returns -0.02, -0.01, 0.01 and 0.02 twice. Asset returns are 1.5 times A in the first block and 0.5 times A in the second. Every block has zero mean. The table gives beta of 1.5 for the early window, 0.5 for the recent window and 1.0 for all eight observations. Benchmark B is defined as twice A, leaving the asset unchanged; the full-window beta becomes 0.5. B is an algebraic comparison series, not a real index or an investment recommendation. All estimates include an intercept.

What the pooled number conceals

The full-window beta happens to equal the average of the two block betas because these constructed blocks have equal lengths, equal predictor variation and equal means. That is not a general pooling identity. Differences in block means and benchmark variation can change the combined slope substantially. In the example, reporting only the pooled value conceals the deliberately changing relationship. Reporting only the recent value suppresses its much shorter estimation history. Neither estimate forecasts the next block by arithmetic alone. Standard errors, structural-change analysis and any shrinkage adjustment are additional methods, not consequences of displaying the raw covariance-to-variance ratio.

The boundaries of a reproducible estimate

A beta estimate is interpretable when its window endpoints, observation count, benchmark identity, currency, return convention and missing-pair rule are available together. Dropping different rows for covariance and variance destroys the stated slope identity. A constant benchmark makes beta undefined; very small benchmark variation can make the estimate highly sensitive to small data changes. Multiplying both series by the same unit conversion leaves beta unchanged, whereas changing only one series changes its scale. These are mathematical checks, not evidence that beta is stable. A historical exposure estimate and a forward risk assessment remain separate claims even when their displayed values coincide.

Primary references

These sources support methods and conventions. The example values are original constructions, not observations extracted from these documents.

• Aswath Damodaran: Estimating Risk Parameters
• Gabriel Hawawini: Why Beta Shifts as the Return Interval Changes, public abstract
• Kenneth R. French: Description of Fama/French Factors

Related investigations