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In fact, if a filter is chosen that is tuned to a slightly higher frequency than the filter with the most resonance, there will be a slight lead in the filter output, which will make it slightly predictive One problem with band-pass filters constructed using Butterworth high- and low-pass components is the dramatically large phase shifts that occur as the periodicity of the signal moves away from the center of the pass-band Such phase shifts can completely throw off any attempt at trade timing based on the output of the titter.

Year 1 Retention rate Retained customers Average order Orders per year Total revenue Cost rate Variable costs Acquisition cost ($1,400) Retention costs ($90/yr) Total costs Gross pro t Discount rate Net present value of pro t Cumulative NPV of pro t Lifetime value 70% 45,000 $2,500 8 $900,000,000 75% $675,000,000 $ 63,000,000 $ 4,050,000 $742,050,000 $157,950,000 1.01 $155,779,959 $155,779,959 $3,462 Year 2 75% 31,500 $2,700 9 $765,450,000 71% $543,469,500 0 $ 2,835,000 $546,304,500 $219,145,500 1.09 $200,894,344 $356,674,303 $7,926 Year 3 80% 23,625 $2,900 10 $685,125,000 69% $472,736,250 0 $ 2,126,250 $474,862,500 $210,262,500 1.17 $179,159,636 $535,833,940 $11,907

Butterworth filters are not necessarily the optimal filters to use when applying a filter bank methodology to the study of market cycles. The drawbacks to using Butterworth filters include the fact that they do not necessarily have the response speed desired when trading markets and making quick, timely decisions. Problems with measuring the instantaneous amplitude of a particular cycle are another consideration. And as noted earlier, the phase response of Butterworth filters is less than ideal. Wavelets, on the other hand, provide an elegant alternative. The theory of filter banks has recently become much more sophisticated with the introduction of wavelet theory. On a practical level, wavelet theory enables the construction of fairly elegant digital filters that have a number of desirable properties. The filters used in the tests below are loosely based upon the Moreler wavelet. The Morelet wavelet behaves very much like a localized Fourier

percentage of the selling price of company products that goes for manufacturing, distribution, and other costs. The balance is pro t or marketing costs. The acquisition cost is computed by adding together all the expenses involved in acquiring new customers during a given year advertising on TV, radio, print, and direct mail; sales force salaries, commissions, and bonuses and dividing that total by the number of customers actually acquired during that year. The retention costs are those costs spent to keep customers happy and buying. This could include an annual retreat, customer communications, or customer services. The discount rate is used to compute the net present value of the pro ts received. Money that will arrive in 1 year or 2 years is obviously not as valuable today as money that arrives today. The discount rate is a number that puts all these amounts on the same basis so that they can be added together. As discussed in earlier chapters, the formula for the discount rate is D (1 i * rf )waiting time

transform It captures information about cyclic activity at a specific time, with as rapid a decrease as possible in the influence (on the result) of data points that have increasing distance from the time being examined Unlike Butterworth filters, Morelet wavelets are maximally localized in time for a given level of selectivity or sharpness This is a very desirable feature when only the most current information should influence estimates of potentially tradable cycles The filters constructed for the tests below also have the benefit of a much better phase response, highly important when attempting to obtain accurate market timing in the context of vary ing cycles The advanced filters under discussion can easily be used in banks, employing a methodology similar to the one used in our May 1997 study The kind of wavelet-based filters used in the tests below are designed to behave like quadrature mirrorfilters; ie.

, there are two outputs for each filter: an in-phase output and an in-quadrature output The in-phase output coincides precisely with any signal in the market with a frequency that lies at the center of the pass-band of the filter The in-quadrature output is precisely 90 degrees out-ofphase, having zero crossings when the in-phase output is at a peak or trough, and having peaks and troughs when the in-phase output is crossing zero In a mathematical sense, the outputs can be said to be orthogonal Using these filters, the instantaneous amplitude of the cyclic activity (at the frequency the filter is tuned to) can be computed by simply taking the square of the in-phase output, adding it to the square of the in-quadrature output, and taking the square root of the sum.