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5/25/2005
Understanding the Movements of the Dow

It is interesting to contemplate the occasions when the Dow has not been above its level of many years before. Such periods are of interest of course to those of a doomsday bent because they can immediately proclaim such things as "Ha, ha, in 30 of the last 100 years, you would have lost money if you had bought the Dow 6 years ago, and held until some time in the month. Stuff your buy-and-hold, your Dimson and your Lorie". Of course, such observations are completely consistent with random walks with drifts of 5% a year and it is good to quantify such moves to prove this.

A more important reason to look at such periods is that the Dow has an inexorable tendency to rise. It started the 20th century at 65 and stands at 10500 today. That's a 4.8% a year compounded return. When you add in a 4 1/4% a year dividend return, ( 5% from 1871 to 1945 and 4% from 1946 to 1996) you come up with that 9% a year compounded return that the Triumphal trio has documented in almost all countries.

It stands to reason that if the Dow has such an inexorable tendency to rise, that it's good to buy it when it's in one of its down drafts. In other words, that dollar averaging, putting more of your chips in the Dow when it's down than when it's up might give you an edge. But like all such interesting queries, this must be tested as most plausible things of this nature are a bunch of hot air. And worse yet, because they aren't tested, one can never differentiate the tested from the untested.

In any case the Dow closed 1999 at 10700 and closed 2000 at 11000 versus its current level of some 10500. Thus, at least 5 1/2 years have gone by when an unlucky investor who bought in 1999 and didn't receive dividends would actually have a loss. What happens in such situations, and what variability is attached?

To test it, the Professor, his student Chris Hammond, the master simulator Tom Downing and I looked at all those months in the last 100 years when the closing price was below the year end four years earlier. For example the Dow price at October month end 1907 was 57.7. This was lower than the price at the end of 1902, which was 64. That's four years and 10 months without a profit.

During the period from 1900 to date [May 2005] there were 1,264 months examined. Two hundred seventy, or 22%, of them showed such a loss( the dry years). The average one-month price appreciation the next month was 0.8% -- a standard deviation of 7%. This compares to a return of 0.5% a month with a standard deviation of 5% on all other months. Such a difference is merely a 1-in-10 shot to have arisen if indeed there was no difference between the months. In the usual terms, the difference was not significant. The question arises if the price appreciation over bigger periods was more significant. For example, in the year following the dry months, the average price appreciation was 14%, versus 5% in the bountiful months.

Because of the clustering of dry months, without a gain, it was necessary to do some relatively sophisticated simulation to determine the likelihood of such differences arising by chance. We chose to do it by assuming that the distribution of intervals between dry months was our total sample. We chose a random price from the full 105 years, and then classified it as to whether it was a dry or bountiful year. Then we chose the next 12 months, skipping an appropriate number of months based on the distribution of intervals between consecutive months of dryness in the sample. The results show that the difference is about 1 in 40 to have arisen by chance. Thus, there is some support for the idea that dollar averaging works and the idea that buying is best when the doomsday scenarists are gloating the most. 

5/27/2005
Up the Ladder, Down the Chute, by Chris Hammond and Charles Pennington

Laszlo Birinyi's firm recently prepared a table that partitioned the time from 1962 through the present into periods of rally and decline for the S&P. Among the questions it suggested, one of the most prominent was "Is this behavior consistent with randomness?" Before we could begin to answer this question, we had to first decide how to reconstruct the data using some algorithm. Using monthly data for the S&P since 1953, we settled on the following procedure:

  1. Select a point as being a potential maximum if the price of the S&P at the close of the month is greater than the price at the close of the previous six months as well as the next six months (which means, of course, that one cannot identify such a point until six months after it has occurred). Minima are selected in the same fashion.
  2. Order the set of all maxima and minima. If there are several maxima before a minimum, throw away all but the last one. Do likewise for minima. One is left with a set of points partitioning the time from 1953 to 2005.
  3. The periods from a minimum to a maximum will be called rallies, and the periods from a maximum to a minimum will be called declines.

The results of performing this procedure to the S&P are shown below. For each period, we take note of its duration in months, and the annualized percent change in price over the period. This provides a reasonable approximation to the Birinyi table. However, it is also a little finer, giving us a larger data set.

In order to address the question posed, we found the percent change over each month, and we stored it in a list. We then simulated the S&P time series by starting at the same initial value and moving to the next month's value by randomly selecting one of the percent changes in our list and using that as the current month's percent change. We sample without replacement, i.e., we use all of the same values but in a random order. We perform our algorithm on each of the simulated time series and keep track of the data. Twenty trials were performed.

We find a significant distinction between the actual S&P data and the simulations. The average duration of a rally in the S&P is 22 months, and there were 17 rallies. For each simulation, there was an average rally duration. Taking the average of these yields 15 months, with 25 rallies on average. The standard deviation is 3.4 months. The actual value for the average rally duration is about 4 standard deviations away from the mean, a significant finding. This indicates that actual rallies tend to last longer than in the simulations.

This could be the result of correlation in returns between successive months, meaning that when you are rising, you tend to continue doing so, making for a longer run. When you remove these correlations, you get choppier time series. However, there are some misgivings regarding this approach. One objection is that "If its not predictive, then it is consistent with randomness," and our study has no predictive value. These are questions that ought to be addressed and which will require significant thought.

Table 1: Rallies and Declines in the S&P 500 Since 1953, Monthly Data

                           Duration        Annualized Return
Stage     Start   End      Rally  Decline  Rally Decline
Rally     Aug-53  Jul-56      35            0.29
Decline   Jul-56  Feb-57               7           -0.20
Rally     Feb-57  Jul-57       5            0.28
Decline   Jul-57  Dec-57               5           -0.35
Rally     Dec-57  Jul-59      19            0.30
Decline   Jul-59  Oct-60              15           -0.10
Rally     Oct-60  Dec-61      14            0.29
Decline   Dec-61  Jun-62               6           -0.41
Rally     Jun-62  Jan-66      43            0.16
Decline   Jan-66  Sep-66               8           -0.25
Rally     Sep-66  Sep-67      12            0.26
Decline   Sep-67  Feb-68               5           -0.17
Rally     Feb-68  Nov-68       9            0.29
Decline   Nov-68  Jun-70              19           -0.22
Rally     Jun-70  Apr-71      10            0.54
Decline   Apr-71  Nov-71               7           -0.16
Rally     Nov-71  Dec-72      13            0.23
Decline   Dec-72  Sep-74              21           -0.30
Rally     Sep-74  Jun-75       9            0.71
Decline   Jun-75  Feb-78              32           -0.03
Rally     Feb-78  Aug-78       6            0.41
Decline   Aug-78  Oct-78               2           -0.43
Rally     Oct-78  Nov-80      25            0.22
Decline   Nov-80  Jul-82              20           -0.15
Rally     Jul-82  Jun-83      11            0.63
Decline   Jun-83  May-84              11           -0.11
Rally     May-84  Aug-87      39            0.27
Decline   Aug-87  Nov-87               3           -0.76
Rally     Nov-87  May-90      30            0.20
Decline   May-90  Oct-90               5           -0.34
Rally     Oct-90  Dec-91      14            0.31
Decline   Dec-91  Jun-94              30           -0.02
Rally     Jun-94  Aug-00      74            0.22
Decline   Aug-00  Sep-01              13           -0.29
Rally     Sep-01  Feb-04      29            0.04

Average                     22.1    12.3    0.31   -0.25

Table 2: Results of Simulating the S&P Time Series

           Rallies                              Declines
           Number Duration  StDev   Annualized  Number Duration  StDev   Annualized
                                    Return                               Return
Average     24.55    14.94  11.80   0.33         24.55    10.40   8.74  -0.23
StDev        2.52     1.68   3.40   0.05          2.40     2.04   3.11   0.03
StErr        0.56     0.38   0.76   0.01          0.54     0.46   0.70   0.01
Actual         17    22.06  17.47   0.31            16    12.29   8.94  -0.25
T-score     -2.99     4.23   1.67  -0.41         -3.57     0.93   0.06  -0.64

 

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