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2002issue C101-5

Price regression line versus beta for index tracking

Two equities can share a beta of 1.00 and still post five-year price changes of opposite sign. The historical workflow contrasts that period-gain slope with a price-regression-line on price and index levels, then reads the coefficient of determination before treating the slope as a co-movement forecast.

  • Two equities can share a beta of 1.00 and still post five-year price changes of opposite sign, so beta does not by itself establish that a series follows the market.
  • Beta is the slope fitted to paired period-gains, while the price-regression-line maps a change in the index into an expected price-level change.
  • The same price-on-index fit yields a coefficient of determination that shows how usable the slope is as a co-movement forecast.
  • Sampling by beta near 1.0 or near 0.0 can both leave price paths linearly predictable versus the index, so the two constructions answer different questions.
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Current trackLinear regression112 readings

Beta does not establish market following

Two equities can share a beta of 1.00 and still post five-year price changes of opposite sign. Beta does not by itself establish that a series follows the market.

Beta is constructed from paired period-gains. A period-gain is the percent change from a period's beginning value to its ending value. Each month's percent change in the instrument is plotted against the same month's percent change in the index, and beta is the slope of the fitted line. That slope describes the size and direction of deviations around average gains, not whether prices themselves track the index.

A gain-on-gain slope is not a level path

On twelve monthly gains from February 2001 through early January 2002, that gain-on-gain slope for one gold fund versus the S&P 500 was +0.227. The construction treats that figure as lower gain volatility than the index.

The price-regression-line on levels

The price-regression-line is a different slope. It is the best-fit straight line that relates an instrument's price level to the contemporaneous index level. A change in the index is scaled by this slope to form an expected price change. The slope can be positive, for the same direction as the index, or negative, for the opposite direction.

With only two dated observations, the price-regression-line idea reduces to the ratio of price change to index change. For that gold fund the two-point ratio was negative, implying opposite-direction level moves.

With more than two observations the slope is the ordinary least-squares construction. If X is the index series and Y is the instrument price series, the price-regression-line is the covariance-variance-ratio of those series: the covariance of X and Y divided by the variance of X.

The coefficient of determination of the price fit

The same fit yields a coefficient of determination, the R-squared of the price-on-index regression. It measures how much of the price variation is captured by the straight-line fit and therefore how usable the slope is as a co-movement forecast. In the gold-fund versus S&P 500 example that value was 0.30, which the source treats as a reasonably usable linear indicator of price response to index changes.

A high coefficient of determination with a positive price-regression-line slope is presented as close same-direction tracking. A high coefficient of determination with a negative slope is presented as opposite-direction tracking. A low coefficient of determination is presented as movement that is largely independent of the index.

Price regression of RYURX on the S&P 500

Monthly NAV of RYURX falls as the S&P 500 rises and sits almost on the fitted line (slope −0.00912, R² 0.993). A trader should read that as a tight inverse level-to-level tracker, not as a near-zero beta meaning no link to the market. The twelve close pairs and the fitted Ysl values come from the author’s Excel sidebar table.
Monthly NAV of RYURX falls as the S&P 500 rises and sits almost on the fitted line (slope −0.00912, R² 0.993). A trader should read that as a tight inverse level-to-level tracker, not as a near-zero beta meaning no link to the market. The twelve close pairs and the fitted Ysl values come from the author’s Excel sidebar table.RYURX versus S&P 500 · Monthly closes, February 2001–January 2002 · 2001-02-01T00:00:00.000Z to 2002-01-02T00:00:00.000Z

Workbook fit on 12 monthly closes, February 2001–January 2002: PRL = COVAR(X,Y)/VARP(X) = −0.00912, intercept B = 21.07013, R² = 0.993125. Points are ordered by rising S&P 500 close, not by calendar date.

The two constructions answer different questions

Sampling funds by beta near 1.0 can still produce series whose price paths are linearly predictable versus the index. Funds with beta near 0.0 can also show high price-path predictability. The two constructions answer different questions.

A spreadsheet implementation of the price-on-index line reports, for one inverse-oriented fund, a slope of about -0.00912 and a coefficient of determination of about 0.993. That pair illustrates a tight negative linear mapping from index level to price.

Educational research material, not investment advice. Historical source context does not establish present-day performance.
30 of 43 in the Linear regression track
20031-3 pp.Next on Linear regressionRegression slope with an r-squared trend confidence gateLinear-regression-slope estimates how much a price series is expected to change per sampling interval and encodes whether the fitted trend is positive or negative, but it does not by itself measure how strong that trend is.
All readings on this track · 43 readings
  1. 1990Constructing dollar baselines from rates, inflation, and residuals
  2. 1990Constructing a nominal index value from forward earnings and fitted yield
  3. 1990Constructing a nominal index price from earnings and a fitted yield
  4. 1990Endpoint-pinned price paths are not forecasts
  5. 1990Constructing least-squares polynomial smoothers
  6. 1991Endpoint growth rates versus linear-regression consistency
  7. 1991Out-of-sample checks for linear growth fits
  8. 1991Trend as persistence, not a straight line
  9. 1991Quadratic trend, residual oscillator, and a secondary cycle calendar
  10. 1991Time-origin offset and residual-price divergence on a quadratic least-squares fit
  11. 1991A least-squares trendline from ordered prices
  12. 1992Constructing log-linear growth and reliability screens
  13. 1992Constructing log-linear growth-rate baselines
  14. 1992Next-session high, low, and close from rolling linear regression
  15. 1993Auditing an index price-earnings multiple with short-rate regression
  16. 1994Regression-seeded nested exponential price filter
  17. 1994Constructing the double exponential average from lag cancellation
  18. 1994Evaluating money supply as a linear leading-index baseline
  19. 1995Constructing least-squares trend channels
  20. 1995Linear baseline holdout checks for annual bill-rate forecasts
  21. 1995Projection bands from high and low regression slopes
  22. 1995Evaluating a least-squares end-point moving average on a known test series
  23. 1996Constructing an endpoint moving average from a least-squares line
  24. 1996Scoring equity path consistency with a k-ratio overlay
  25. 1996Evaluating month-end yield gaps for equity regimes
  26. 1996Constructing session-indexed standard error bands
  27. 1998Evaluating linear regression baselines for index valuation
  28. 1998R-squared as a two-state trend filter from a price-time fit
  29. 2000Second-order moving-average lag correction
  30. 2002Price regression line versus beta for index tracking
  31. 2003Regression slope with an r-squared trend confidence gate
  32. 2003Constructing finite-volume-element divergence with slope comparison
  33. 2004Building a daily score from regression, retracement, and volume
  34. 2004Constructing least-squares trendlines from ordered prices
  35. 2007Rectangle breakout targets beyond height
  36. 2007Confirming a price trend with regression slope and r-squared
  37. 2008A linear-regression angle assembled as one trend filter
  38. 2010A two-state swing machine from four running extremes
  39. 2016Score oil-complex tightness before divergence or regression
  40. 2017Nikkei-yen intermarket divergence as a regime case study
  41. 2017Constructing Calmar ratio and linear regression baselines
  42. 2019Pair-trade layer construction versus average-spread management
  43. 2020A convolution slope built from nested linear regression
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