In one sentence — A moving average mainly passes slow movements and attenuates fast ones. Its response curve says how much of each frequency remains, but not what price will do next.
What question does Appendix IV answer?
An eleven-point moving average replaces each value with the mean of eleven nearby observations. The chart becomes smoother, but “smoother” does not reveal which oscillations have been preserved, removed, or reversed.
In Appendix IV of The Profit Magic of Stock Transaction Timing, Hurst feeds one cosine wave at a time into the filter. Comparing the output's amplitude and phase with the input produces the frequency response of the centred moving average.
From the average to the curve
Let n be an odd number of observations, h=(n−1)/2, and Ω=ωΔt the angular frequency per sample. The centred average assigned to time c is:
y_c = (1/n) Σ x_{c+k}, with k from −h to +h.
Hurst's signed response can be written in two equivalent forms:
H_c(Ω) = (1/n)[1 + 2 Σ cos(kΩ)]
H_c(Ω) = sin(nΩ/2) / [n sin(Ω/2)]
The sum runs from k=1 to h. At zero frequency the response is one, so a constant component passes in full. The absolute value |H_c| is the amplitude ratio; the sign indicates whether output remains in phase or is reversed by 180°.
Reading the response
The first zero occurs at Ω=2π/n, corresponding to period T=nΔt under the book's convention. A component at that period disappears from the output. Further zeros occur at frequency multiples: a moving average therefore has no single clean boundary, but a sequence of lobes.
Immediately beyond the first zero the response turns negative. Hurst summarizes the first minimum as −0.23 and uses it as a graphical rule independent of span. The exact discrete formula shows a small dependence on n: with eleven points the minimum is about −0.223. Thus −0.23 is a useful historical approximation, not an exact universal constant for every length.
Hurst calls these Gibbs error lobes; modern terminology describes them as sidelobes of the rectangular window. High frequencies can reappear in attenuated form, alternately in phase and out of phase. This is creep-through: a smooth line may still contain unwanted residuals.
Zero phase and delay are different questions
An eleven-point centred window uses five observations on the left, the centre point, and five on the right. When the result is plotted at the centre, peaks of positive-response components remain aligned in the historical sample. That is the zero-phase meaning of the centred representation.
The value assigned to c, however, cannot be calculated until c+5 has arrived. The equivalent causal implementation has a group delay of h=(n−1)/2 samples. Zero phase does not mean immediate availability, and the most recent edge lacks half a window.
Hurst calls the lag “half the span” while using span=nΔt. In modern FIR indexing, n points occupy n−1 intervals and the exact causal delay is (n−1)Δt/2. The statements use different span conventions; neither permits filling the edge with future data.
The inverse moving average
The inverse operation subtracts the centred average from the input. Its response is:
H_inv(Ω) = 1 − H_c(Ω).
Slow components are reduced while faster ones pass more strongly, producing high-pass behaviour. When H_c enters its negative lobe, H_inv exceeds one; with eleven points it reaches about 1.223 at the first maximum. This overshoot can enlarge a component, so observed amplitude is not an undistorted measurement.
What the calculation proves—and what it does not
The reproducible check uses a synthetic eleven-point window. It confirms H_c(0)=1, the first zero at 2π/11, the lobe near −0.223, and the identity between a value centred at c and the same causal value becoming available at c+5.
These are mathematical properties of the filter on the same sample. They are not independent validation of a cyclic model, do not measure an operational edge, and are not a forecast error. A predictive use needs a rule fixed in advance and an out-of-sample test.
Hurst justifies practical use of the average through the amplitude-frequency relation proposed for stock prices in Appendix I. That relation belongs to his historical model; it is not a universal law to assume without checking the market, period, and data under study.
Operational limits
Span is not a perfect cutoff. The first zero removes one exact period, while sidelobes allow other frequencies to leak through.
Edges depend on implementation. Dropping half a window, shifting the output, or padding the data creates different results and must be disclosed.
Design error is not forecast error. The gap between a desired and realized response describes the filter; it does not anticipate market direction.
Sources
- J. M. Hurst, The Profit Magic of Stock Transaction Timing, Prentice-Hall, 1970, Appendix IV, pp. 207–211 — derivation, first zero, lobes, inverse, and the implications stated in the book.
- MIT OpenCourseWare, 6.02 Lecture 12: Filters and composition, slide 15 — frequency response of a moving-average filter.
- MIT OpenCourseWare, Digital Signal Processing: linear-phase FIR systems — symmetry, centring, and linear phase in FIR filters.
- SciPy,
freqzandgroup_delay— modern calculation of complex frequency response and group delay.
Connections
DJIA spectrum · Cyclic moving averages · Inverse moving average · Ormsby filters · Hurst appendices