Calculate beta from paired returns
For each accepted interval t, define x as the benchmark return and y as the asset return. This page uses simple returns r = P(end) / P(start) − 1, with positive prices and a consistent adjustment convention. Fit y = a + beta × x with an intercept. Use either decimal returns or percentage points for both series; mixing those units makes beta wrong by a factor of 100.
- Require at least two valid pairs and Sxx > 0. Two pairs determine a line but provide no evidence that the relationship is stable.
- Sample covariance and sample variance both divide by N − 1, which cancels. Population versions both divide by N and give the same beta; mixing the conventions does not.
- Raw-return beta and CAPM excess-return beta have different inputs. For the latter, subtract the same interval risk-free return from both series before fitting. Keep the chosen convention in the output label.
| Quantity | Calculation | Meaning |
|---|
| Sxx | Σ(x − mean(x))² | Benchmark variation within the window |
| Sxy | Σ(x − mean(x))(y − mean(y)) | Paired return co-movement |
| Beta | Sxy / Sxx | Asset return slope per unit of benchmark return |
| Intercept a | mean(y) − beta × mean(x) | Fitted asset return when x = 0 |
Match return intervals, not just date labels
A daily close in Tokyo and a daily close in New York occur at different times. Matching their calendar date does not produce simultaneous information. On TradingView, changing the chart timezone changes the displayed clock, not the underlying Pine time calculation. Compare actual timestamps, session selection and daylight-saving rules before treating two bars as paired observations.
- Strict matching can leave little or no data for different exchanges. An alternative is to sample both instruments at a documented common cutoff or use longer intervals, but neither removes stale-price risk automatically.
- For investor-currency beta, convert both price histories to that currency at matching endpoints before calculating returns. Native-currency beta answers a different question.
| Input decision | Explicit policy for this guide |
|---|
| Return endpoints | Require equal start and end timestamps for both instruments. Reject unmatched intervals. |
| Missing bar or trading holiday | Do not invent a zero return or carry forward a benchmark price. Show the rejected pair count. |
| Window | Use the latest N accepted pairs; show their oldest and newest endpoints because calendar coverage can expand. |
| Prices and currency | State price-return or total-return treatment, split/dividend adjustments and quote currency. |
| Closed observations | Update the accepted sample only after both endpoint prices are final for the chosen interval. |
A missing-day example that date joins get wrong
Assume these hypothetical prices share the same closing time whenever both trade. The asset has no Tuesday observation. Its Wednesday return is 104 / 100 − 1 = 4%, covering Monday through Wednesday. The benchmark Wednesday return computed on its own daily series is 204 / 202 − 1 = 0.990099%, covering Tuesday through Wednesday. Joining those returns by Wednesday alone mismatches their starting points.
- Under the strict interval policy, reject the mismatched Wednesday pair. Keep it out of both covariance and variance.
- If the declared policy instead samples consecutive common endpoints, calculate Monday-to-Wednesday for both: asset 4%, benchmark 2%. This is one longer interval, not two daily observations.
- That single common-endpoint pair cannot estimate beta by itself. Filling Tuesday with the asset value 100 would invent a zero return and change the sample.
| Close | Asset | Benchmark |
|---|
| Monday | 100 | 200 |
| Tuesday | Missing | 202 |
| Wednesday | 104 | 204 |
Reproduce a rolling beta series and compare windows
Use the six hypothetical, aligned, completed return pairs below. All returns are percentage points, and every interval has equal duration. The short windows expose the arithmetic; they are not recommended trading settings. For intervals 1 through 4, both means are zero, Sxy = 14 and Sxx = 10, so four-pair beta is 1.40 and the intercept is zero.
- At interval 5, the four-pair window is intervals 2 through 5. Benchmark mean is 0.5 and asset mean is 1.25. Sxy = 5.5 and Sxx = 5, giving beta = 1.10.
- At interval 6, the four-pair window is intervals 3 through 6. Benchmark mean is 1.25 and asset mean is 1. Sxy = -2 and Sxx = 2.75, giving beta = -0.727273.
- The three-pair estimate at interval 5 is 0.50 while the four-pair estimate is 1.10. The same newest observation produces different results because each window retains different history.
- For a real comparison, hold benchmark, return frequency and sample policy fixed while inspecting, for example, 20, 60 and 120 accepted pairs. Longer windows retain more old information; shorter windows react more to individual observations. None is universally correct.
| Interval | Benchmark return | Asset return | 3-pair beta | 4-pair beta |
|---|
| 1 | -2% | -3% | Not ready | Not ready |
| 2 | -1% | -1% | Not ready | Not ready |
| 3 | +1% | +1% | 1.285714 | Not ready |
| 4 | +2% | +3% | 1.285714 | 1.400000 |
| 5 | 0% | +2% | 0.500000 | 1.100000 |
| 6 | +2% | -2% | -0.750000 | -0.727273 |
Separate beta from correlation and price regression
When both return standard deviations are positive, beta = correlation × asset return standard deviation / benchmark return standard deviation. For intervals 1 through 4 above, correlation is 0.989949 and the volatility ratio is √2, giving beta 1.40. High correlation alone therefore does not tell you the size of the fitted return response.
- Multiply every asset price by 10 while leaving benchmark prices unchanged: a price-regression slope multiplies by 10, but asset returns and return beta stay unchanged. This is a simple implementation sanity check.
- Beta of 1.40 describes a 1.40 percentage-point difference in fitted asset return per one-percentage-point difference in benchmark return within this sample. The intercept and residual still matter; it is not a promise of a 1.40% gain tomorrow.
- If asset returns are constant while benchmark returns vary, covariance is zero and beta is zero. Correlation is undefined because asset return standard deviation is zero.
| Calculation | What it measures | Failure if called beta |
|---|
| Asset returns on benchmark returns, with intercept | Relative return sensitivity | Valid beta definition for the stated inputs |
| Asset price on benchmark price | Price-unit slope | Depends on quote scale and can reflect common trends |
| Correlation alone | Standardized linear association from -1 to +1 | Omits relative return volatility |
| Benchmark returns on asset returns | Reverse regression slope | Generally not the reciprocal of asset beta |
Inspect a TradingView implementation before using alerts
Choose the benchmark symbol explicitly and inspect the script source or its documented methodology. Pine requests map another symbol into the chart context. Keeping gaps as missing avoids forward-filled values, but does not prove that the source intervals have equal endpoints. Request the source timestamps as well as prices and enforce the pairing rule before adding observations to the rolling sample.
- Calculate each return in its source context, retain its previous and current close timestamps, and accept the pair only when both intervals match. Reject missing prices and nonpositive price denominators under this guide's policy.
- Store each accepted pair once. After N valid pairs, compute both means and the centered sums over that same set. Remove the oldest pair when a new accepted pair arrives; do not separately skip missing values in the two series.
- Display unavailable during warmup or when benchmark variance is zero. Very small positive variance can produce an unstable large beta; expose that variance rather than clipping beta into a plausible-looking range.
- A confirmed chart bar does not automatically confirm a requested higher-timeframe bar. Use completed source observations and avoid future data. An alert requires a new accepted sample, a valid beta and the declared close confirmation; a stale carried display should not trigger it.
- Check splits, dividend treatment, feed revisions and outliers when beta jumps. Beta summarizes benchmark exposure and does not measure all risk, liquidity or expected profit.
Turn the calculation contract into a Pineify request
Use the Pine Script AI Coding Agent to draft an indicator from the endpoint-matching policy, rolling sample definition and arithmetic fixture above. Ask for the benchmark input, beta plot, valid pair count, sample dates and benchmark variance. Then request syntax checking and compare completed TradingView observations with an independent calculation before adding an exposure alert.
- Specify raw simple returns or excess returns, price adjustments, currency, session and timeframe. Define whether unmatched intervals are rejected or resampled at common endpoints.
- For a threshold alert, specify the threshold, crossing direction, warmup rule and confirmation timing. A beta crossing is a change in a historical estimate, not a complete trading strategy.
- The stock beta calculator linked below provides a separate company lookup and calculation workflow. Its benchmark, historical coverage and return alignment must match before its number can serve as a comparison.
