Crypto beta is the slope of a strategy's returns against a chosen benchmark; alpha is the fitted intercept. For an excess-return regression, subtract the same period's cash reference return from both series first. Synchronize the return windows, currency and units before fitting. These estimates describe the selected sample and benchmark, rather than proving trading skill or predicting the next market move.
A strategy can beat its benchmark because it carried more exposure during a rising market. Regression separates an estimated linear sensitivity from an intercept, subject to a model and its limitations. It answers a different question from simply subtracting two ending account returns.
Define the two series before fitting
Let p be the strategy's simple return for one observation interval, m the benchmark's return and f the cash reference return for that same interval. Define strategy excess return y = p − f and benchmark excess return x = m − f. Fit:
y = a + b × x + e
Here a is estimated alpha, b is estimated beta and e is the residual for an observation. Include an intercept in the fit; forcing the line through zero prevents it from estimating alpha. BYU's Silver Fund calculation guide provides a primary example of estimating beta as the slope and alpha as the intercept of matched excess returns. Its equity trading-day annualization convention should not be imported automatically into a crypto dataset.
A BTC benchmark produces BTC-relative estimates. It is not automatically the complete market portfolio assumed by textbook CAPM. Call the result benchmark-relative regression alpha, and state the proxy. Choose that benchmark before examining which one makes the intercept most attractive; the strategy benchmark comparison guide covers selecting an appropriate alternative.
Use an actual, documented reference series appropriate to the reporting currency and interval. A quoted annual yield requires a stated conversion convention before it becomes a daily return. A zero cash return is an assumption that must be disclosed. A stablecoin balance or advertised lending yield is not automatically a risk-free asset.
A measurement contract for crypto data
Build paired account and benchmark returns, rather than regressing token prices or individual trade profits. A trade lasting three hours cannot be paired with a whole-day benchmark return without changing the question.
| Item to freeze | Required record |
|---|---|
| Strategy denominator | Whole allocated equity, including idle balances and marked open positions |
| Cash flows | Treatment of deposits and withdrawals before calculating returns |
| Benchmark | Instrument or index, return construction and any constituent rules |
| Clock | Exact interval boundaries, such as consecutive 00:00 UTC marks |
| Units | Same reporting currency; simple returns on both sides; both decimals or both percentages |
| Costs and reference | Strategy costs included; benchmark cost convention; matched cash return |
| Sample | Start/end dates, paired observations, missing-data rule and estimation window |
The time-weighted versus money-weighted return guide explains why deposits cannot simply become portfolio profit. For a daily regression, prepare a consistent daily return series after handling those external flows.
Do not fill an unavailable benchmark return with zero. Investigate the gap and disclose exclusions. If a missing mark creates a two-day return, it must not be labeled or paired as a one-day return. Likewise, a USD account and a BTC-denominated benchmark do not become comparable merely because both columns display percentages.
Six invented days: alpha differs from the gap
This is a hypothetical arithmetic example, not market data, a KLINEO result or evidence of an attainable return. Assume six synchronized daily observations, simple returns in the same currency, no external flows and a stipulated cash return of 0.01% each day. Strategy returns are after all modeled costs. The benchmark is an idealized reference return series.
The table already subtracts that 0.01% from each raw return. Thus a benchmark excess return of −2.00% corresponds to a raw return of −1.99%. Residuals are measured in percentage points.
| Day | Benchmark excess x | Strategy excess y | Fitted 0.20% + 1.50 × x | Residual |
|---|---|---|---|---|
| 1 | −2.00% | −2.30% | −2.80% | +0.50 pp |
| 2 | −1.00% | −1.80% | −1.30% | −0.50 pp |
| 3 | 0.00% | 0.20% | 0.20% | 0.00 pp |
| 4 | 1.00% | 1.70% | 1.70% | 0.00 pp |
| 5 | 2.00% | 2.70% | 3.20% | −0.50 pp |
| 6 | 3.00% | 5.20% | 4.70% | +0.50 pp |
Ordinary least squares chooses the line that minimizes squared residuals. NIST's straight-line least-squares derivation gives the slope as the centered cross-product sum divided by the benchmark's centered squared sum, followed by the intercept from the means.
Using percentage-point values, the benchmark mean is 0.50 and strategy mean is 0.95. The centered cross-product sum is 26.25; the benchmark centered squared sum is 17.50. Therefore beta is 26.25 / 17.50 = 1.50, and alpha is 0.95 − 1.50 × 0.50 = 0.20, or 0.20% per daily observation.
The mean raw-return gap is 0.45 percentage points per day because the common cash subtraction cancels. That gap contains both the intercept and the extra sensitivity to the benchmark: 0.20 + (1.50 − 1) × 0.50 = 0.45. It is neither alpha alone nor the difference between compounded six-day returns.
Scaling both input columns to decimals preserves beta at 1.50 and expresses alpha as 0.002. Scaling only one column changes the apparent slope by a factor of 100. Save units with the output so that “beta 150” does not survive a formatting mistake.
Read the coefficients without turning them into promises
A beta of 1.50 describes the fitted change in strategy excess return associated with a one-percentage-point change in benchmark excess return: 1.50 percentage points in this sample. The residual can still dominate a particular day. Beta is not an order-sizing instruction, a causal effect or the account's leverage setting.
Beta also differs from correlation. Correlation standardizes both series; beta retains their relative variation. For portfolio co-movement and common-loss scenarios, use the separate correlation-risk worksheet.
The intercept is the fitted strategy excess return at zero benchmark excess return. It can reflect omitted drivers, changing exposure, implementation choices or sampling noise. If you instead regress raw returns in this constant-cash example, the intercept becomes 0.195%, because 0.20% + (1 − 1.50) × 0.01% = 0.195%. Comparing that raw intercept directly with zero changes the interpretation. Aswath Damodaran's NYU chapter on risk measurement explains the raw-versus-excess intercept distinction.
Keep alpha in its estimation interval. Multiplying a daily intercept by an observation count is an arithmetic annualization convention, not a compounded account return or CAGR. Compounding a constant daily alpha as though it were an investable standalone strategy introduces another assumption.
What can make a neat regression misleading?
If benchmark excess returns never vary, the slope denominator is zero: beta is undefined, and the intercept and slope cannot be separately identified. Near-zero variation can also make estimates unstable. Report that failure instead of substituting beta zero.
Six invented days cannot support a claim of statistically reliable alpha. With real data, report the sample size, coefficient uncertainty, residual diagnostics and window sensitivity. Returns or residuals that depend on earlier observations require special care; the return autocorrelation guide addresses that diagnostic. The statsmodels robust covariance documentation distinguishes heteroskedasticity-robust and heteroskedasticity-and-autocorrelation-robust inference. The latter requires choices such as a lag count; it does not repair an unsuitable benchmark or unstable relationship.
A spot/perpetual comparison may leave funding and collateral effects inside the residual. A strategy that changes leverage halfway through the window may have an average slope that describes neither half well. Inspect predeclared subwindows and retain the disagreement. Avoid choosing a new window only because it restores positive alpha.
An account-level regression also cannot show the adverse path inside a trade. Pair it with trade MAE and MFE analysis when reviewing exit rules. A modest fitted beta can coexist with a severe intratrade loss or liquidation event.
Method and limits
This AI-assisted educational article uses the linked primary sources and an original, locally recalculated fictional dataset. It estimates no current asset beta, tests no trading strategy and establishes no KLINEO analytics feature. A useful report preserves its benchmark, cash reference, units, timestamps and uncertainty beside the coefficients so another reviewer can reproduce what they mean.