14.5 Trading the Negative Persistence

The data analysis conducted so far indicates a negative persistence ($ H<0.5$) of the log differences of pairs of voting and non-voting stocks of a company. It should be possible to take advantage of this knowledge. If we found a profitable trading strategy, we would interpret this result as a further indication for the reverting behavior of voting/non-voting log-differences.

The average relationship between voting and non-voting stocks in the sample period may be expressed in the following way,

log(voting) = $ \beta $ log(non-voting) + $ \varepsilon$
where $ \beta $ may be estimated by linear regression. If the log-differences of voting and non-voting stocks are reverting as the R/S analysis indicates, negative differences, $ X_t < 0$, are often followed by positive differences and vice versa. In terms of the Hurst coefficient interpretation, given a negative difference, a positive difference has a higher chance to appear in the future than a negative one and vice versa, implying voting stocks probably to become relatively more expensive than their non-voting counterparts. Thus, we go long the voting and short the non-voting stock. In case of the inverse situation, we carry out the inverse trade (short voting and long non-voting). When initiating a trade we take a cash neutral position. That is, we go long one share of the voting and sell short $ m$ shares of the non-voting stock to obtain a zero cash flow from this action.

But how to know that a `turning point' is reached? What is a signal for the reverse? Naturally, one could think, the longer a negative difference persisted, the more likely the difference is going to be positive. In our simulation, we calculate the maximum and minimum difference of the preceding $ M$ trading days (for example $ M=50,100,150$). If the current difference is more negative than the minimum over the last $ M$ trading days, we proceed from the assumption that a reverse is to come and that the difference is going to be positive, thereby triggering a long voting and short non-voting position. A difference greater than the $ M$ day maximum releases the opposite position.

When we take a new position, we compute the cash flow from closing the old one. Finally, we calculate the total cash flow, i.e.  we sum up all cash flows without taking interests into account. To account for transaction costs, we compute the total net cash flow. For each share bought or sold, we calculate a hypothetical percentage, say $ 0.5\%$, of the share price and subtract the sum of all costs incurred from the total cash flow. In order to compare the total net cash flows of our four pairs of stocks which have different levels of stock prices, we normalize them by taking WMF stocks as a numeraire.

In Table 14.3 we show the total net cash flows and in Table 14.4 the number of trade reverses are given. It is clear that for increasing transaction costs the performance deteriorates, a feature common for all $ 4$ pairs of stocks. Moreover, it is quite obvious that the number of trade reverses decreases with the number of days used to compute the signal. An interesting point to note is that for RWE, which is in the German DAX$ 30$, the total net cash flow is worse in all situations. A possible explanation would be that since the Hurst coefficient is the highest, the log-differences contain less `reversion'. Thus, the strategy designed to exploit the reverting behavior should perform rather poorly. WMF and KSB have a smaller Hurst coefficient than RWE and the strategy performs better than for RWE. Furthermore, the payoff pattern is very similar in all situations. Dyckerhoff with a Hurst coefficient of $ H=0.37$ exhibits a payoff structure that rather resembles the one of WMF/KSB.


Table: Performance of Long Memory Strategies (TotalNetCashFlow in EUR). 29133 XFGLongMemTrade.xpl
Transaction $ M$   WMF Dyckerhoff KSB RWE
Costs     $ H=0.33$ $ H=0.37$ $ H=0.33$ $ H=0.41$
$ 0.00$ $ 50$   $ 133.16$ $ 197.54$ $ 138.68$ $ 39.93$
  $ 100$   $ 104.44$ $ 122.91$ $ 118.85$ $ 20.67$
  $ 150$   $ 71.09$ $ 62.73$ $ 56.78$ $ 8.80$
$ 0.005$ $ 50$   $ 116.92$ $ 176.49$ $ 122.32$ $ 21.50$
  $ 100$   $ 94.87$ $ 111.82$ $ 109.26$ $ 12.16$
  $ 150$   $ 64.78$ $ 57.25$ $ 51.86$ $ 2.90$
$ 0.01$ $ 50$   $ 100.69$ $ 155.43$ $ 105.96$ $ 3.07$
  $ 100$   $ 85.30$ $ 100.73$ $ 99.68$ $ 3.65$
  $ 150$   $ 58.48$ $ 51.77$ $ 49.97$ $ -3.01$



Table 14.4: Number of Reverses of Long Memory Trades
$ M$   WMF Dyckerhoff KSB RWE
$ 50$   $ 120$ $ 141$ $ 132$ $ 145$
$ 100$   $ 68$ $ 69$ $ 69$ $ 59$
$ 150$   $ 47$ $ 35$ $ 41$ $ 42$


Regarding the interpretation of the trading strategy, one has to be aware that neither the cash flows are adjusted for risk nor did we account for interest rate effects although the analysis spread over a period of time of about $ 26$ years.