Predicting the Time-window for Turns, in all Markets for TVC:DJI by IanSinclaire

Inexplicably, upon publishing this post, the Title Chart becomes distorted beyond recognition thus,
all references are to this Original Chart – below

  • Do markets trend on the medium term (months) and mean-revert on the long run (years)?
  • Does Black’s intuition bear out that prices tend to be off approximately by a factor of 2? (Taking years to equilibrate.)
  • How does Technical Analysis, as a whole, act as a trend following system while Fundamental Analysis matters only once prices get way out of line?
  • Is mean-reversion a sufficient self-correcting mechanism to temper irrational exuberance in financial markets?

We examine these questions in the proceeding;

In his 1986 piece Fisher Black wrote:
“An efficient market is one in which price is within a factor 2 of value, i.e. the price is more than half of value and less than twice value. He went on saying: The factor of 2 is arbitrary, of course. Intuitively, though, it seems reasonable to me, in the light of sources of uncertainty about value and the strength of the forces tending to cause price to return to value. By this definition, I think almost all markets are efficient almost all of the time.”

The myth that “informed traders” step in and arbitrage away any small discrepancies between value and prices never made much sense.

If for no other reason but the wisdom of crowds is too easily distracted by trends and panic.

Humans are pretty much clueless about the “fundamental value” of anything traded in markets, save perhaps a few instruments in terms of some relative value.

Prices regularly evolve pretty much unbridled in response to uninformed supply and demand flows, until the difference with value becomes so strong that some mean-reversion forces prices back to more reasonable levels.

Black imagined, Efficient Market Theory would only make sense on time scales longer than the mean-reversion time (TMR), the order of magnitude of which is set by S√TMR∼d.

For stock indices wit hS∼20%/year, makes TMR = ∼6 years.

The dynamics of prices within Black’s uncertainty band is in fact not random but exhibits trends: in the absence of strong fundamental anchoring forces, investors tend to under-react to news or take cues from past price changes themselves.

In fact, the notorious and unbridled reliance and un-anchored, speculative extrapolation is the mainstay of most investors, as well as Wall Street’s itself, as it is the regular course of everyday “investing” across most asset classes.

In the following a picture emerges (and we test it), whether market returns are positively correlated on time scales TMR and negatively correlated on long time scales ∼TMR, before eventually following the (very) long term fate of fundamental value – in what looks like a biased geometric random walk with a non-stationary drift.

We have looked at a very large set of financial instruments, drawing on data sets from 1800 – 2020 (i.e. 220 years).

We applied the same method to all available data in Stocks, Bonds, FX, Commodity Futures and Spot Prices, the shortest data set going back 1955.

As it turns out that, in particular, mean-reversion forces start cancelling trend following forces after a period of around 2 years, and mean-reversion seems to peak for channel widths on the order of 50-100%, which corresponds to Black’s “factor 2”.

Mean-reversion appears as a mitigating force against trend following that allows markets to become efficient on the very long run, as anticipated previously by many authors.

Regarding the data we used for this study;

Commodity Data sets – Starting date
Natural Gas 1986
Corn 1858
Wheat 1841
Sugar 1784
Live Cattle 1858
Copper 1800

Equity Price data sets – Starting date

USA 1791

Australia 1875

Canada 1914

Germany 1870

Switzerlan 1914

Japan 1914
United Kingdom 1693

From trends to mean-reversion

The relation between past de-trended returns on scale t'< and future de-trended returns on scale t’>. Defining p(t) as the price level of any asset (stock index, bond,commodity, etc.) at time t. The long term trend over some ti scale T is defined as:


For each contract and time t, we associate a point(x,y) where x is the de-trended past return on scale t'< and y the de-trended future return on

scale t>:x:= logp (t)−logp(t−t'<)−mtt'<;y:= logp (t+t’>)−logp(t)−mtm’t’.

Note that the future return is de-trended in a causal way, i.e. no future information is used here (otherwise mean-reversion would be trivial). For convenience, both x and y are normalized such that their variance is unity .

Remarkably, all data,including futures and spot data lead to the same overall conclusions. See in chart; As the function of the past (time) horizon t'< (log scale) for Red & White Bars, the futures daily data and spot monthly data.

To compare the behaviour of the regression slope shown in the chart with a simple model, assume that the de-trended log-price pi(t) evolves as a mean-reverting Ornstein-Uhlenbeck process driven by a positively correlated trending noise m.

It is immediately apparent from the dashed line in the chart that the prediction of such a model with g= 0.22, k−1= 16 years and y’−1= 33 days, chosen to fit the futures data and g= 0.33, k’−1= 8 years and gh’−1= 200 days, chosen to fit the spot data.

In the short term volatility of prices is simply given by S’2k’s’.

Non-linear effects

A closer look at the plot(x,y )however reveals significant departure from a simple linear behaviour. One expects trend effects to weaken as the absolute value of past returns increases, as indeed reported previously. We have therefore attempted a cubic polynomial regression, devised to capture both potential asymmetries between positive and negative returns, and saturation or even inversion effects for large returns.

The conclusion on the change of sign of the slope around yt'<= 2 years is therefore robust. The quadratic term, on the other hand, is positive for short lags but becomes negative at longer lags, for both data sets. The cubic term appears to be negative for all time scales in the case of futures , but this conclusion is less clear-cut for spot data.
The behaviour of the quadratic term is interesting, as it indicates that positive trends are stronger than negative trends on short time scales, while negative trends are stronger than positive trends on long time scales.

A negative cubic term, on the other hand, suggests that large moves (in absolute value) tend to mean-revert, as expected, even on short time scales where trend is dominant for small moves. Taking these non-linearities into account however does not affect much the time scale for which the linear coefficient vanishes, i.e. roughly 2 years


Here we have provided some further evidence that markets trend on the medium term (months) and mean-revert on the long term (several years).

This coincides with Black’s intuition that prices tend to be off by a factor of 2.
It takes roughly 6 years for the price of an asset with 20 % annual volatility to vary by 50 %.

We further postulate the presence of two types of agents in financial markets:
Technical Analysts, who act as trend followers, and Fundamental Analysts, whose effects set in when the price is clearly out of whack. Mean-reversion is a self-correcting mechanism, tempering (albeit only weakly) the exuberance in financial markets.

From a practical point of view, these results suggest that universal trend following strategies should be supplemented by universal price-based “value strategies” that mean-revert on long term returns. As it’s been observed before, trend-following strategies offer a hedge against market draw-downs while value strategies offer a hedge against over-exploited trends.

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