The Tao of Trading - the case for trading - Ariel Faigon (2008)

Intro   Case   Pivot   Position   Simulation  

The self-evident case for trading

While looking at thousands of charts over time, I was struck many times by the remarkable fact that the overall distance traveled by these charts is so much bigger than the net-distance from start to finish. Just as a river meanders trough rough terrain going through countless wiggles, so do stock charts zig zag seemingly randomly between almost any two points. There's a fractal-like nature to these charts: if one looks at monthly, or weekly, or daily charts there's some surprising similarity between the short term and the long term charts. Wiggles exist in every time frame and the ratio between the overall distance and the net distance seems strikingly similar.

The above chart seems contrived, and it is. So let's look at a real chart. I'd pick some index, a broad index, the quintessential representation of market gyrations and trends. Here's SPY, the S&P500 chart over the recent 100 days of trading.

To make this exercise a bit more illustrative, this time let's add a line trying to smooth-out the wiggles:

This realistic chart, just like the contrived one, has about half of its closing-prices above, and about half below the smoothed line. Obviously -- even without taking into account minute by minute, or second by second price changes -- the overall distance travelled by SPY in the past 100 days is much larger than the net-distance travelled (formalists could use the triangle inequality principle recursively on each small section where the smoothed line crosses the price chart twice to establish some lower bounds on the ratio.)

In fact: it is 17(!) times bigger:

SPY ended 2008-04-10 at a price of 136.02
SPY ended 2008-08-29 at a price of 128.79

Thus, the overall move (for B&H investors) during this period was a net-loss of 5.32%

The simple sum of all daily percentage moves was 90.45% -- 17 times bigger than the overall net movement.

        90.45 / 5.32 = 17.0019

Put it another way, 16 out of 17 "price moves" got cancelled along the way by back and forth movements. Granted, some of these movements were more short-term, some longer-term, but the 17x number vs the buy and hold "net movement" (which was overall negative) is strikingly high.

Moreover, this particular case understates the typical ratio because:

For comparison, I did a similar exercise 4 days later with a more volatile ETF: IYT, the Dow Transportation ETF. Here's IYT's 100-day chart:

The net movement is 1.37% up and the sum of movements is 152.99% a "distance-travelled" ratio of 173.7(!). Note: IYT is a good example but is not one of the best examples. Many leveraged ETFs have 100-day distance ratios in the several hundred range, and a few get close to a ratio of 1000.

If a ratio of 1000 sounds far-fetched to you, pat yourself on the back. Your perception of reality is sound.

Here, one should realize that these ratios greatly overstate cases where the net move happens to be close to zero (small denominator, huge ratio). It is much better to compare the overall distance travelled to the difference between the maximum and the minimum price along the way divided by the average price. This revised and more conservative ratio tends to fall around 3-5 for non-leveraged sector ETFs and go up to 10 for leveraged (2x) ETFs. Now we're back in the reality zone, and the ratios are still very impressive.

Here's a table with these numbers for some ProShares ETFs calculated over the recent 100 day period. 

Ticker  %Net Chg  %(Max-Min) %Total     %Total/%(MaxMin) Ratio
------	--------  ---------- ------     --------------------
DDM:    -22.27    35.53      178.19     5.02
DIG:    -33.20    55.64      308.72     5.55
DOG:     11.14    17.18       89.95     5.24
DUG:     28.72    42.70      312.24     7.31
DXD:     21.10    33.41      179.33     5.37
EEV:     56.09    58.72      247.27     4.21
EFU:     53.22    49.19      177.24     3.60
EFZ:     24.12    25.25       81.86     3.24
EUM:     26.45    31.40      127.39     4.06
EWV:     33.71    42.01      185.86     4.42
FXP:     26.97    50.48      370.94     7.35
LTL:    -22.86    36.12      172.72     4.78
MVV:    -11.48    30.95      187.54     6.06
MYY:      4.07    13.99       93.14     6.66
MZZ:      7.42    28.65      182.67     6.37
PSQ:      4.12    13.71      104.02     7.58
QID:      5.04    25.88      205.17     7.93
QLD:    -11.96    31.72      203.80     6.43
REW:      5.84    29.34      212.33     7.24
ROM:    -15.79    35.08      212.34     6.05
RWM:     -2.66    14.34      113.05     7.88
RXD:     -4.77    23.09      136.70     5.92
RXL:      1.39    22.45      145.11     6.46
SBB:     -1.92    12.93       94.82     7.33
SCC:     -4.13    34.12      238.05     6.98
SDD:     -5.05    27.99      217.79     7.78
SDK:     14.53    31.06      191.67     6.17
SDP:     26.89    31.11      163.46     5.25
SDS:     17.31    31.04      179.42     5.78
SFK:     15.45    26.11      167.42     6.41
SH:       9.10    15.74       86.87     5.52
SIJ:     13.43    32.00      205.37     6.42
SJF:     15.07    36.88      197.83     5.36
SJH:     -8.05    35.66      240.39     6.74
SJL:      6.66    32.93      188.14     5.71
SKF:      3.51    74.86      410.93     5.49
SKK:     -5.05    24.46      217.13     8.88
SMN:     33.12    46.36      299.21     6.45
SRS:     -1.15    39.34      330.74     8.41
SSG:     19.80    43.12      272.28     6.32
SSO:    -19.01    33.58      176.79     5.26
SZK:      3.28    21.25      144.86     6.82
TLL:     10.50    30.61      168.74     5.51
TWM:     -7.19    30.06      220.93     7.35
UCC:     -4.38    37.31      235.04     6.30
UGE:     -6.75    22.05      152.75     6.93
UKF:    -16.95    29.55      165.01     5.58
UKK:     -1.25    25.82      213.89     8.28
UKW:    -15.14    35.55      189.30     5.33
UPW:    -24.20    33.58      158.01     4.71
URE:    -18.88    49.15      331.00     6.73
USD:    -26.32    53.20      277.81     5.22
UVG:    -20.00    39.60      191.69     4.84
UVT:     -1.06    34.04      243.92     7.17
UVU:    -11.20    36.83      188.06     5.11
UWM:     -1.06    30.72      227.02     7.39
UXI:    -17.88    33.86      204.69     6.05
UYG:    -28.98    83.47      410.36     4.92
UYM:    -33.77    56.38      304.64     5.40

Looking at charts and numbers like the above, and assuming no leverage or overweighting, I find a reason to believe that an investor can afford to have a pretty large miss ratio and still beat Buy and Hold. If one trades against the shorter-term movements and especially against the sudden, larger movements which are statistically speaking, out of the norm. One needs to be wrong more often than not in order to under-perform Buy and Hold. Yes, every buy/sell is essentially a coin flip, but probability, in this case, is on your side. Flip often, and you should do well over the long run.

Here's another way to look at it: after a daily move up of X%, where X% is, statistically, an unusually-large one-day movement, the B&H investor just sits there and does nothing. In contrast, the short-term contrarian investor thinks "thank you mr. market!", and realizes the profit by selling (or partially selling). The short-term contrarian does so knowing that with good odds of 1:3 to 1:10, this movement is likely to be cancelled in whole or in part soon.

Here I want to emphasize one very important point: by trading this way one is not trying to predict the future. All the statistical decisions are based on past movements. I think this is a truly wonderful and powerful feature of the "being active" system. You don't need to know or try to guess the future at all!

Based on this data, the reversion to the mean force is in the contrarian trader's favor, assuming he knows where the "mean" is. The potential is there, the challenge, of course, is finding the critical points when to buy/sell and how much.

Here I must stop and admit that I "cheated" a bit by putting the cart before the horse. The smoothed line above was drawn (using R) by an algorithm called LOESS (or LOWESS), locally weighted scatterplot smoothing which looks at neighboring points on both sides of the smoothed point in order to calculate the smoothed point. Obviously, in real-life we cannot peek into the future like LOESS does. We can only rely on points to the left (in the past) and the current time (present) to calculate the smoothed line. It follows that we cannot do as well as with LOESS. Still, we can get close enough to make the effort worthwhile. We'll get there soon.

The point to take from this section is that it is self evident that by buying below the "average price" and selling above it we can do a very large number of "trading trips" back-and-forth resulting in an overall performance that is greatly superior to buy and hold (where there's only one buy from start to finish).

Optimally, one could do 3 to 10 times more. Realistically, if you could do even 1.5 times more, this is golden. This is not even taking into account, the fact that this strategy is symmetric (market neutral) whereas buy and hold is profitable only during bull markets. In fact, based on observation of successful short-term contrarian traders (notably James Simons' Renaissance Medallion) they tend to perform much better during bearish and nervous periods. Score another point for being active.

If the demonstrated active trading performance is much larger, than B&H after deducting what we pay in extra commissions and taxes, then we've proven the case for being active traders. The good news about commissions and taxes are that they are both easy to calculate, since there are no elements of uncertainty in them.

The harder part is to solve the two subproblems of optimal trading. Let's describe them:

The Optimal Trading Problem

For simplicity, we break the optimal trading problem into two separate probabilistic sub-problems:
  1. Finding the optimal "pivot point" for an oscillating asset. The pivot can also be called the "mid-point" or "equilibrium-point"

    The pivot-point moves with time so it is a function of time:

    pivot = f(t)

    The pivot point metric is the "mean" price of an asset to which we expect it to return in the (not yet defined) short term. i.e. the asset is expected to drop in price if it is above the pivot-point, and increase in price if it is below the pivot-point.

    Remember, nothing is guaranteed, in pure theory, an asset may keep going and going in one direction and "never" return to its calculated pivot point. Fortunately, this is pure theory. In practice, assets return to their pivot points in pretty short time-frames much more often than not whenever the deviation from the mean is large enough for the time-frame.

  2. Finding the optimal position size in time, given a "pivot-point".

    The position-size is a real number with a range:

    position = (-infinity, +infinity)

    1.0 means we're fully invested (100%), i.e. no cash. 0.0 means we're not invested at all, i.e. we sit with 100% cash, and -1.0 means we're fully short (with all our balance).

    Numbers greater than 1.0 or smaller than -1.0 mean that we're leveraged: we use more than our capital (borrowing cash) to establish short or long positions.

    Practically, given standard brokerage limitations in a good-standing account, we can assume our position will be in the range of (-2.0 .. +2.0). I.e. our broker allows us to take up to 100% margin (buy or short assets up to 2x of our own total balance).

    This limited amount of funds places a constraint on how much risk we can take. Limiting risk is good since it can save our ass in periods of multiple misses. But it also places a limit on what we can do and achieve.

Let's call problem #1: "The optimal pivot problem"
Let's call problem #2: "The optimal position size problem"

I hope with the above I established a reasonable case for being active. Admittedly, this is reasonably convincing hand-waving and not a proof.

We're now ready to move to the first subproblem.

  • The Optimal Pivot sub-problem

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    Disclaimer: this should not be considered as investment advice. It is merely describing my own thoughts and actions.

    Feedback is welcome.

    -- ariel