Appendix I — The DJIA spectrum, 44 years (Hurst)

Source: J. M. Hurst, The Profit Magic of Stock Transaction Timing, Prentice-Hall, 1970 — Appendix I, The Not-to-Be-Expected "Order" of Spectral Relationships in Stock Price Data (pp. 188–200, Figs. A I-1 → A I-8).

Prerequisites

Fourier analysis and the Ormsby filters — the tools these results were produced with.

44 years in one spectrum

The high-resolution harmonic analysis was run on the DJIA's weekly closes from 29 April 1921 to 25 June 1965: 2229 points, 0.568 radians/year resolution. Three symptoms of order leap out of the plot:

1. The coarse structure: amplitudes gather into broad segments, with minima at ≈0.95 · 1.65 · 2.8 · 4.75 · 7.0 · 9.8 rad/year — and the central peaks of those lobes correspond to the periodicities observable by eye in the chapters' charts (4.5 years, 3 years, 18 months, 1 year…). Persisting for at least 44 years: any significant drift would have averaged peaks and valleys away.
2. The fine structure: between the valleys, regular peaks ≈0.8 rad/year apart — and while the coarse lobes widen as frequency rises, the fine spacing stays equal everywhere, right up to frequencies resolvable only with trade-by-trade data. The first hint of a line spectrum.
3. The upper bound: the dotted curve joining the peaks has an equation — and here the appendix plays its ace.

The a = k/ω law

The spectrum's upper bound follows aᵢ = k/ωᵢ. If each component is Cᵢ = aᵢ·sin(ωᵢt + φᵢ), differentiating: Ċᵢ = aᵢωᵢ·cos(ωᵢt + φᵢ) = k·cos(ωᵢt + φᵢ) — the maximum time rate of change of every spectral element is identical to that of every other line in the spectrum, and this delicate balance holds for decades.

"Each modulated component of the model has exactly the same maximum impact on price motion as that of any other, no matter what the disparity in amplitude may be."

It is the deep justification of the book's most-used technique: counting how many components push up and how many down (the state table) without weighting them — because at their maximum push, they all weigh the same. "The very fact that this technique works constitutes a test of the general hypothesis" — and Fig. IX-4 shows how thoroughly it passes.

Comb filters and the spectral lines

To validate and extend Fourier, the appendix uses combs of overlapping band-pass filters — every frequency viewed by at least four filters. If spectral energy were not concentrated in discrete lines, the filters should respond more or less everywhere. Instead the outputs cluster in narrow bands ("The Incredible Frequency-Separation Effect!", Fig. A I-4): filters straddling the gaps produce outputs outside their own passbands — discarded as meaningless — and the clusters coincide with Fourier's fine structure.

The robustness test is systematic: random passbands, varied spacing, different overlaps, non-equal spacing — same groupings, same regions. And ranking the means of the least-squares lines by increasing frequency, the slope yields the minimum line spacing: 0.3676 radians/year — about half the Fourier estimate, with traces visible between the fine-structure peaks.

The line spectral model

Regular lines every 0.3676 rad/year, organized as in radio engineering: a central group of lines behaving like a carrier, with fine-structure lines as amplitude-modulation sidebands — some themselves modulated, generating the composite's frequency modulation. "Exactly the observed nature of the periodic fluctuation of price with time": the magnitude-duration fluctuation seen from the frequency side. The apparent lower limit (≈18 years) is imposed only by resolution; at the high end, no limit found to the order of the spectral signature.

Summary card

Links

• Fourier analysis — the starting tool
• The Ormsby filters — the comb's teeth
• Appendix II — the extension to individual issues
• Cyclic X motivation — what this is the signature of
• The appendices — index
• Hurst tradition — chapter index