1995issue C041-7

Market z-score residuals for style pair construction

A historical workflow restated a stock and the S&P 500 as z-scores, subtracted the index, and rebuilt a residual-price-path. Linear-regression then compared original prices, residual prices, and the market index for high book-to-price AHM and low book-to-price KO.

Closing prices of a stock and of the S&P 500 were converted to z-scores so both series sat on one standard-deviation scale, treated as ranging from -4 to +4.
A residual-price-path was formed by subtracting the S&P 500 z-score and then restoring the stock mean and standard deviation.
AHM stayed close to its residual path while KO diverged, and the residual-versus-original contrast was offered as an indirect reading of systematic-risk versus unsystematic-risk.
Editorial reading: size a book-to-price-pair from residual correlation-analysis after index-subtraction, not from the high versus low book-to-price labels themselves.

Entries in this reading3 entries

Correlation analysis3 readings Linear regression7 readings Current trackZ-score normalization11 readings

Editorial framing: a two-name sleeve

Editorial framing: teach a two-name sleeve as an index-residual construction problem. Put each holding and the market on one z-score scale, peel off the common factor with index-subtraction, rebuild the residual-price-path, and size the pair from residual correlation-analysis instead of from book-to-price style labels.

The archive itself recorded the z-score restatement, the rebuilt residual-price-path, and the linear-regression comparisons. Those steps are historical workflow, not a present-day recommendation.

Putting stock and index on one z-score scale

A z-score is a closing observation restated as distance from that series mean, in units of that series standard deviation. Closing prices of a stock and of the S&P 500 were converted to z-scores by subtracting each series mean and dividing by that series standard deviation, placing both on a shared standard-deviation scale.

On that z-score scale, values were treated as ranging from -4 to +4, which was used to justify subtracting an index score from a stock score.

Rebuilding the residual-price-path

A residual-price-path is a rebuilt stock series formed by subtracting the market z-score and then restoring the stock mean and standard deviation. Index-subtraction removes a shared market-factor layer so remaining variability can be compared across names.

The worked book-to-price-pair

The worked sample used closes from 2 January 1985 through 30 September 1993 for high book-to-price AHM, low book-to-price KO, and the S&P 500. A book-to-price-pair is a high versus low book-to-price pairing used as a value-style name and a growth-style name.

Original AHM and residual AHM paths stayed similar, which the comparison treated as little S&P 500 influence on that high book-to-price name. Original KO and residual KO paths diverged sharply, which the comparison treated as substantial S&P 500 influence on that low book-to-price name.

Linear-regression readings

Linear-regression is a fitted line relating an original series, a residual series, or the market index so slope, intercept, and correlation can be read together. Linear-regression of original AHM on residual AHM produced a correlation of 0.614. Regression of the S&P 500 on original AHM produced 0.247 and on residual AHM produced 0.614.

Linear-regression of original KO on residual KO produced a correlation of 0.183. Regression of the S&P 500 on original KO produced 0.933 and on residual KO produced 0.183. Correlation-analysis is a check of how tightly original prices, residual prices, and the market index still move together after the factor is removed.

Systematic-risk and unsystematic-risk

The residual-versus-original contrast was offered as an indirect reading of systematic-risk versus unsystematic-risk that diversification can reduce. Systematic-risk is the market-linked share of variability that a common index factor is meant to capture. Unsystematic-risk is the name-specific share of variability that diversification is meant to reduce.

KO close versus residual path after removing S&P 500

Coca-Cola close (circles) rose from about $5 to the mid-$40s from 1985 through 1993, while the residual-price path rebuilt after subtracting the S&P 500 z-score stayed in a $10–$30 band and did not share the same climb. A trader should read that as market factor dominating the growth-stock sleeve; residual correlation, not the book-to-price label, is what remains to size the pair. Values are read off Figure 5, not from a printed table.KO · daily close · 1985-01-02T00:00:00.000Z to 1993-09-30T00:00:00.000Z

Daily closes 2 Jan 1985–30 Sep 1993. Residual series is (((SS KO − SS S&P) × SD KO) + mean KO). Digitised from the plotted curves; about 45 equally spaced samples, y rounded to 0.5 because the raster cannot support finer precision. Source regressions: KO vs residual r=0.183; S&P vs KO r=0.933.

Educational research material, not investment advice. Historical source context does not establish present-day performance.

2 of 11 in the Z-score normalization track

19951-11 pp.Next on Z-score normalizationConstructing scaled z-score normalization for model inputsForecast models that work best on a limited numeric range often first map unbounded ordered observations, such as prices, through a bounded preprocess.

All readings on this track · 11 readings

By fieldPortfolio construction18 readings By year199510 readings

All 11 readings tagged Z-score normalization

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