232 THE JOURNAL OF BUSINESS NOTES TO TABLE Q In applying the filter technique, the data P(? the closing price of security j for the ermine whether the first position taken day on which transaction t for filter will be long or short. With an a per cent filter, an initial position is taken as soon as Ii?= the total dollar profit on transaction there is an up-move or a down-move(which t of filter i when applied to security er comes first) where the total price j. The profits ar tal change is equal to or greater than a per cent dividends, which are positive for he position is assumed to be taken on the long transactions and negative fc first day for which the price change equals or n(p= the duration in terms of total trad eeds the a per cent limit. Any positions ng days of transaction t for filter i open at the end of the sampling period are when applied to security j. disregarded. Thus only completed transac- N(= the total number of trading days tions are included in the calculations during which positio The closing price on the day a position is under filter i when al opened defines a reference price: a peak in ty j. Thus the case of a long transaction and a trough in the case of a short transaction. On each N() subsequent day it ssary to check whether the position should be closed, i. e, whether the current price is a per cent below where r(s is the total number of the reference (peak)price in a long position transactions initiated by filter i for security j price if the open position is short. If the cur- ri= the rate of return with daily com- rent position is not to be closed, it is then necessary to check whether the reference when applied to security j. It is price must be changed. In a long position this will be necessary when the current price P{2[1+2]y=P{+r exceeds the reference price so that a new peak has been attained, whereas in a short rp"=the over-all rate of return with daily. position a new trough will be defined when the current price is below the reference when applied to security j. It price. Of course, when the reference price changes all subsequent testing uses the new On ex-dividend days the reference price r()=I[1+r2]p/NY value as ba s adjusted by adding back the amount of the dividend. Such an adjustment is neces- Rp"= the nominal annual rate of return sary in order to insure that the filter will not for filter i when applied to com- iggered simply because the stock's pri pany ]. It is computed as typically falls on an ex-dividend date. In R4=260r4 addition, if a split occurs when a position is open, the price of the security subsequent to R( are the returns shown for the the split is adjusted upward by the appropri- filter technique() in Table 1 ate factor until the position is closed. R (=the nominal annual rate of return With this background discussion we shall from buy-and-hold during the time ow consider the rate-of-return calcula period for which filter i had open summarized in Table 1. The following are positions in security] e basic variables in the calculations