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Market Price Movements
Richard D. Ahrens
There is something distinctly strange about market prices. One moment they will step along smoothly, a moment later they will jump like a frightened cat. Efficient Market advocates tell us, "Unpredictable! Just a normally distributed random walk." Academics echo, "Bernoulli trials, the market has no memory." We will examine these assertions and apply simple, arithmetic tests to verify their accuracy.
Randomness
"Everyone knows" market price movements are random. How do we know this? Because our finance professor said so, and he knew it because his finance professor said so, and so on. If you trace it back to 1900, the man that everyone quotes is Louis Bachelier. People praise his work and say things like "Bachelier is now recognised internationally as the father of financial mathematics..."[1].
There is no doubt that Monsieur Bachelier was a fine mathematician, but his work was little read and widely misunderstood. His doctoral thesis outlines the distribution function for what is now known as the Weiner stochastic process � a remarkable accomplishment. He is widely quoted as saying that market price movements followed that model. It may seem a small point, but if you read his original dissertation, what he actually said was if market prices moved according to that model then certain conclusions about speculation could be drawn.
Bachelier's work was brilliant and his mathematical proofs were well-developed and elegant. But just as Einstein showed that Newton's mechanics had shortcomings, later research by Lorenz, Rossler, Mandelbrot and others have shown that Bachelier was wise to have prefaced his thesis with that "if". As Mandelbrot puts it[2]:
The old financial orthodoxy was founded on two critical assumptions in Bachelier's key model: Price changes are statistically independent, and they are normally distributed. The facts, as I vehemently argued in the 1960s and many economists now acknowledge, show otherwise.
Many people believe that market price movements are normally distributed. In order to test this assertion, I wrote a simple program to record daily changes in stock prices. The program saves each day's change as a percentage. I ran the program for over 7,000 stocks using data from January 1997 to January 2006. This yielded a total of more than 10 million daily changes. Then I had another program count the number of occurrences of each of the daily changes. This resulted in a graph that made the PDF (probability distribution function) of the daily changes visible. This is shown below in Figure 1.
It is clear that this is not the smooth bell-curve of a normal distribution. The center is a spike instead of a mound and the tails go much too far on both the plus side and the minus side. (A graph of monthly changes shows exactly the same pattern.)
In Figure 2 we see a normal distribution (the black line) compared with other Levy Stable Distributions. While the standard distribution is Levy2.0, the distribution of market price changes much more closely resembles the red line in this graph, the distribution referred to as Levy0.5.
Figure 2
The data clearly shows that market price movements are not normally distributed. Even so, it is still possible that daily price movements are statistically independent. If they really are, then the data should bear it out.
One example of a statistically independent series is a sequence of coin flips. The outcome of one coin flip has no influence on the outcome of the next. If you are flipping a "fair coin" � one which is perfectly balanced � then the chance of it landing heads up is 0.5 or 1 out of 2. Even if you have just flipped 5 heads in a row, the chance of the next flip coming up heads is still 1 out of 2.
If the chance of getting heads once is � and the chance of getting heads again is also �, then the chance of getting heads twice in a row is the product of the probabilities or �. This is the basis of what is called the Theory of Runs. A "run" is a sequence of identical outcomes, so the Theory of Runs has to do with things like the probability of flipping 5 heads or tails in a row, or hitting the same number in roulette over and over.
With a fair coin, the chance of getting the same side up as you did last time is always � so the odds of getting three heads or tails in a row is � x � x � = 1/8. Similarly, if the stock market is a random walk, then the chance of it going up or going down three times in a row should also be 1/8.
Here is how runs of up and down moves look for a truly random sequence:
|
Length of Each Run |
Advancing Runs |
Declining Runs |
Advancing + Declining Runs |
Adv+Dec vs Previous Row |
|
1 |
12484 |
12512 |
24996 |
N/A |
|
2 |
6307 |
6344 |
12651 |
50.61% |
|
3 |
3095 |
3160 |
6255 |
49.44% |
|
4 |
1548 |
1540 |
3088 |
49.37% |
|
5 |
790 |
728 |
1518 |
49.16% |
|
6 |
404 |
365 |
769 |
50.66% |
|
7 |
202 |
198 |
400 |
52.02% |
|
8 |
119 |
96 |
215 |
53.75% |
|
9 |
41 |
54 |
95 |
44.19% |
|
10 |
29 |
23 |
52 |
54.74% |
Notice that the numbers in each row decline by just about half each time we add one more to the length of the runs (until we get into rows with relatively few samples) and the average of the values in the last column is 50.44% - very close to the � we expected.
Now let's look at the probability distribution for the stock market. The next table uses daily price data from the S&P 500 from January, 1950 to the present.
|
Length of Each Run |
Advancing Runs |
Declining Runs |
Advancing + Declining Runs |
Adv+Dec vs Previous Row |
|
1 |
552 |
721 |
1273 |
N/A |
|
2 |
620 |
652 |
1272 |
99.92% |
|
3 |
434 |
375 |
809 |
63.60% |
|
4 |
256 |
256 |
512 |
63.29% |
|
5 |
152 |
150 |
302 |
58.98% |
|
6 |
120 |
109 |
229 |
75.83% |
|
7 |
91 |
46 |
137 |
59.83% |
|
8 |
48 |
26 |
74 |
54.01% |
|
9 |
33 |
17 |
50 |
67.57% |
|
10 |
25 |
9 |
34 |
68.00% |
|
11 |
20 |
5 |
25 |
73.53% |
|
12 |
13 |
2 |
15 |
60.00% |
In order to avoid skewing the analysis too badly, let's ignore that fact that there are just as many runs of 2 as there are runs of 1. This oddity could be due to the fact that we have a relatively low total number of samples � about 14,200. But even with row 2 removed, the average of the values in the last column is 64.46%. And note that this bias does not confine itself to either the advancing runs or the declining runs.
Examination of almost any pair of numbers from the Advancing Runs or the Declining Runs column reveals that there is a significantly greater than even chance of an advance being followed by another advance or a decline being followed by another decline. If these trials were actually independent but biased � like an imbalanced coin or "loaded" dice - then an increase in the advancing runs would come at the expense of a decrease in the chances of declining runs or vice versa. The fact that the probabilities displayed by both the advancing runs and the declining runs is greater than 50% indicates that whether prices are rising or falling, what has happened in the past influences what will happen next.
Conclusion
It is clear that price changes in capital markets are neither normally distributed nor statistically independent. Admittedly, there is a lot of noise in market price movements, but what happens one day does influence what will happen the next. In other words, markets have "memory." So the question is, why do markets have memory and why do they trend?
A lot of people have done sophisticated studies of the markets and come up with all sorts of creative and carefully developed theories about how they work. The interesting thing is that the more scientific they try to become, the more they tend to overlook the single most important fact that underlies everything else: Markets are people.
When you get past the kind of financial instruments being traded; when you factor out money supply and interest rates which do not have risk; when you look beyond econometrics and the contending forces of supply and demand...markets are people. (Even when markets are influenced by automated trading, the computer programs used are reflections of the attitudes and beliefs of the people who created them.)
Therefore, the simple fact that markets are people explains why markets have memory. It also sheds a good deal of light on why they trend � up, down, and horizontally.
Richard Ahrens is a technical analyst, teacher, and developer of automated trading systems. He has been involved in the capital markets since 1987. His research has been complimented in Fortune and Fast Company. He is currently doing risk management work for a start-up hedge fund and he can be reached at [email protected].
Footnotes:
1. http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Bachelier.html
2. The (mis)Behavior of Markets, Benoit MandelBrot, Basic Books, 2004, Page 11
Housing Stocks Not Ripe For Value Pickers� Yet
By D.R. Barton, Jr.
I�ve been reading a lot about the housing market lately, especially from a value perspective. Housing stocks are starting to look like bargains to some analysts. After the recent drops in the housing industry stock prices, many of the largest home builders are selling at prices that are just above book value of assets.
And even for normally clear thinking analysts, this is a bargain that looks too good to pass up.
Don�t be tempted to jump on the housing stock value bandwagon just yet.
There are still some major risk factors in the housing market that point to the fact that this isn�t just a quick �down and back up� correction in the midst of a longer term bull run in housing. Let�s look at a few reasons for caution at this juncture:
� Demand is showing clear signs of drying up. While the number of new housing starts was way up in last week�s report, the number of new houses actually being sold was down again. Worse yet, unsold inventories of houses rose to the point where 8.1 months worth of houses are sitting around � which is a new high.
� Builders are still building. And while home owners are slowing their pace of purchasing, builders are slowing down, even in the face of record inventories.
� Home builders� book values may not be all they�re cracked up to be. Buried on page C16 of today�s Wall Street Journal is a very interesting article about the current plight the home builders. If they stop development of existing or contracted projects, they have surety bonds with local governments that were written to make sure the builders complete promised infrastructure projects (like adding sewer & water, roads, etc.). If a real slowdown comes and projects have to be halted, the value of these surety bonds could take major chunks out of home builders� book value � knocking out as much as 50%.
Indeed, the picture is looking not-too-attractive for all but the most ardent bargain hunters.
What to do?
A great place to start would be to take Van�s excellent advice for value investors: don�t pull the trigger until you see clear signs that the market is heading back up. Too many people only look at one or two items to make investment or trading decisions (in this case, it might be price to book value). When the fundamentals look good, they leap in with both feet.
In our book, Safe Strategies for Financial Fre-edom, we list one of the characteristics of safe and successful value investing as �waiting for a move in your direction.� This is great advice to keep us from trying to catch a falling knife. Realize that this extra little bit of price movement is just a small price to pay for confirmation that a reversal of trend has actually happened.
Great Trading!
D. R.
About D. R. Barton: A passion for the systematic approach to the markets and lifelong love of teaching and learning have propelled D.R. Barton, Jr. to the top of the investment and trading arena where he is one of the most widely read and followed traders and analysts in the world.
He is a regularly featured guest analyst on both Report on Business TV, and WTOP News Radio in Washington, D. C., and has been a guest analyst on Bloomberg Radio. His articles have appeared on SmartMoney.com and Financial Advisor magazine.
It is Tax Time Again...
We often get emails and calls from our clients concerning their taxes. As all of you know, Van is not an accountant or tax guy and we can�t really answer all of your questions. The best advice that we can give you is to refer you to one company that has specifically been in the business of helping traders with their taxes for close to ten years - Traders Accounting.
As an active trader, if you are having problems understanding what forms to use and how to file them and if you are interested in finding someone who will get your taxes done correctly and keep you out of trouble with the IRS, then please take the time to visit them and see what they have to offer.
www.tradersaccounting.com/2006tax
Life of No Regrets
by Melita Hunt
We�ve all heard the saying �I have no regrets� and then the common phrase that usually follows is �I wouldn�t change a thing if I had to do it all again.� Well I just can�t quite agree. I am currently in Australia and now one of the key things that I do every time I visit home is to question my mother about an array of various things from current events in her life to past experiences. I especially focus on things that I didn�t really know or understand (or they didn�t tell me) when I was growing up. Sometimes it�s simply to find out what I don�t know, and often it�s to gain a new perspective or her opinion on something that has probably shaped my beliefs.
It�s surprising how many things weren�t talked about because it wasn�t deemed as the correct thing to do, or someone may be embarrassed, shunned by society or some other excuse (you all know what I�m talking about).
A lot of this stems from the fact that ten years ago my father passed away and I have learned so much about him since that time. It�s as though everyone wanted to tell me stories after the fact, but wouldn�t dare talk about it when he was alive. And now there are many questions left unanswered; things that I would love to talk to him about. Do I regret that I can�t do that? Well, I thought I did, but when I looked up the word �regret� in the dictionary it has a very strong meanings such as, feeling sorrow, loss, longing or disappointment. But that�s not exactly how I feel. It�s more of a sense of missing out or pity that we didn�t talk honestly about his life. This took me one step further to the �wouldn�t change a thing� comment and that�s the part that I can�t agree with. I don�t have any regrets as such because I don�t dwell on these types of things or feel sorrow, disappointment etc., about them but I realized that there are many things that I would change if I could do it all again.
Yes I know the argument that everything happens to shape us, make us who we are, but I would be a fool not to want to change a few things! There are certain investments I would certainly not make again, people I would listen to and others that I would be sure to avoid. I would make wiser vocational choices and put into practice the array of other lessons that have been learned. And I would talk to my dad civilly rather than fight, rebel or need to be right in every encounter that we had.
What�s the point? Simply that we shouldn�t waste time on regret but we should be wise about the things that we didn�t �quite do� or are currently �not doing.� We can�t change what is or what was, but we can gain two very valuable lessons from the past:
- We can change our perspective or opinion about anything at anytime.
- We can learn from past mistakes and be sure not to repeat them.
So I encourage you to draw a line in the sand and let go of any regrets. Ask more questions, listen a lot, have fun with your family, learn about one another, re-connect with someone you miss. And be sure to tell all of your secrets regardless of the consequences, it is much more fun and freeing to be who you really are, warts and all. The people who care about you love you anyway!
You can contact Melita at [email protected]
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