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I do not have background in economics, so my question may be naive or I might have taken incorrect assumptions somewhere. As far as I understand, efficient market hypothesis applies, saying that the stock prices already contain all the available information, so it is not possible to consistently beat the market without internal information on given stock.

I tried to build a simplistic toy-model of stock prices that have this property and I came with model, where each individual stock is in each time step multiplied by term $e^\varepsilon$, where $\varepsilon$ is randomly selected from uniform distribution centered around zero.

$$ p(t+dt)=p(t)e^{\varepsilon \,dt}\approx p(t)(1+\varepsilon \,dt) $$

Now, since every positive $\varepsilon$ has equally probable negative counter-part and since $e^\varepsilon e^{-\varepsilon}=1$, the mean value of the stock price is zero. Unless I overlooked something, it should not be possible to extract profit from stock prices following such a law, because the information about the future price is not contained in the past and it is random. We can center the distribution around positive number to add growth of the overall economy.

Now I wanted to create similar toy-model, where economic cycles are included. First thing that came to my mind is to multiply the generated prices of stocks from the previous model by a periodic function, for example:

$$ p_\mathrm{cycled}(t)=\left(1+A\sin\left(\frac{2\pi t}{T}\right)\right)p(t) $$

But this is clearly contradicting the efficient market hypothesis, because anyone knowing the period of the function could buy the stock when the prices are low and sell when the prices are high. I thought of possible ways how to make the model consistent with the efficient market hypothesis and I came with these points:

  1. The cycle is not a cycle, but some kind of noise. What makes the strategy of making profit buying low and selling high impossible is, that nobody knows, when the prices will go up and when down.

  2. When buying low, some companies go bankrupt before going up to high period of the cycle. Nobody knows which one though, and it is therefore not possible to make profit on the cycle.

  3. Something else.

The question: Is there simple toy-model of stock prices that manifests the economic cycles and is consistent with the efficient market hypothesis? If there are more mechanisms at play, I would be very interested in seeing them all embedded as a toy model.

I am suspicious about 1., because either the noise is unbounded (as the exponential noise above), or there is memory in the noise. The memory can be used to extract the necessary information to make profit. If the noise is unbounded, then it is not really similar to how net world economy behaves. 2. seems plausible, but I didn't found any toy-model embedding such behavior yet.

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My takeaway: From the answers here and after giving it some thought, I came to conclusion that if the EMH is true, there should not be any cycles with known oscillations in the prices, no autocorrelation in the noise to be exploited and the possible cycle up- and down- should not be bounded (otherwise, everyone would know when is the down-phase, even though they would not be able to predict it).

I came with example toy-model satisfying these requirements with three components:

  1. Normal distribution random walk
  2. Exponential growth of several % per year
  3. Constant small probability of sudden crash with factor 0.75-1

enter image description here

In the model, there are "quasi-cycles" between the crashes and there is no exploitable future information. (I think.)

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  • $\begingroup$ P. S. I am very aware that the toy-models do not predict real stock prices and are oversimplified. I simply want to see efficient market hypothesis + business cycles embedded as a toy model to understand them better and to make sure they are actually consistent. $\endgroup$ – Irigi Nov 22 '19 at 19:52
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If you want some simple model of stock prices consistent with efficient market hypothesis it would be random walk:

$$p_{t+1}= a+ p_t +\epsilon_t$$

You don’t even need to model cycles there explicitly just due to random chance it will exhibit some ‘cyclical-like’ behavior. Although, I know it’s not actual cyclical behavior because it can diverge it’s actually still more correct then using some trig wave function. Economic cycles are neither regular nor symmetric. Empirically most economies experience protracted periods of prolonged economic boom and relatively short but steep recessions.

Also you actually can even add some trending and cyclical behavior by including some linear or non linear trend to random walk and it would still be consistent with efficient market hypothesis. You could even add sinusoidal wave if you would want (although I don’t think you should).

Just to show the power of random walk I made a simulation for you in R. You can see that it basically looks like any stock data yet its just the above mentioned model. It has 'cycles' and all and it is fully consistent with EMH.

enter image description here

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  • $\begingroup$ Thank you for the answer! I understand that the random walk with constant offset is consistent with EMH. You write that we can add cyclical trending. But isn't it a contradiction with EMH, which should be leveled by people buying in minimums of the trending cycle and selling in the maximums? Or are the minimums and maximum simply ups and downs of the random walk? (Pronounced by using heavy-tailed noise, for example.) $\endgroup$ – Irigi Nov 23 '19 at 8:33
  • $\begingroup$ @Irigi EMH does not make assumption about buying behavior just that all available information is taken into account. Thus additional deterministic trend would be consistent. Of course rational agents will want to buy when the price is low and sell when it’s high, but that does not eliminate cycles in stock market in the same way as it does not eliminate cycles in any other market. If most stock prices fall due to some shock that reduces the productive capacity of firms then you will get bust no matter the buying/selling strategy of agents. But, again I don’t think adding trig trend is helpful $\endgroup$ – 1muflon1 Nov 23 '19 at 10:53
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The EMH applies to assets, not just stocks, and its implications are more relevant for investors who own part of the market - not the entire thing. This is important, because it's the difference between looking at a closed system versus an open one, and between populations versus samples. People make money all the time by cycling between stocks, bonds, precious metals, real estate, etc. Flights to "safe havens" occur all the time, rightly and wrongly. Someone buying SNC Lavalin stock three months ago and selling it a few weeks after the Canadian election would have made some fairly predictable money.

The apparent issue you're running into here stems from the erroneous assumption that non-stock assets are "safe". In fact, they have their own risks and cycles, which are (anti-)correlated with the stock market in various ways. The "Risk-free rate" in the Fama-French model isn't constant, and is fundamentally no more predictable than any individual stock price.

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  • $\begingroup$ Are you basically saying that the economic cycles do not even out by actions of people buying low, selling high, because it is not possible to reliably store value between the cycle minimum and maximums? For example if I simply keep fiat currency, my profit from such action would be on average be equal to the loss via inflation? (And similar risks for storing in other assets.) $\endgroup$ – Irigi Nov 22 '19 at 20:22
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    $\begingroup$ This is a really deep topic and there are other issues that contribute to the EMH remaining forever a "hypothesis" rather than a "theory". For our purposes, all I'm saying is that if you revise your model to include all assets - and not just stocks - that is a good first step to dealing with your concern that someone who knows T could exit and re-enter the market. Like perfect competition, the EMH is most usefully understood as a benchmark rather than a description of reality. By having the benchmark, it is easier to notice what reality is doing and find behavior worth trying to explain. $\endgroup$ – heh Nov 22 '19 at 20:45
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An IEEE journal a few decades back proved that within epsilon smaller than the vig brokers would charge you that you cannot predict the stock market well enough to make money.

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  • $\begingroup$ @irigi: Just a heads up that you don't need to model cycles explicitly in order to obtain them in a series. I can't do the concept justice here but see the papers by yule ( 1927 ? ) or slutsky ( 1927 ? ) or frisch. (1933 ). The yule-slutsky effect is another thing that you could google for that will probably give you the exact references. These guys didn't even have computers and were waaaaaaay ahead of their time. truly fascinating stuff to read and, as a physicist, you'll appreciate it. if you can't find the relevant references, let me know and I'll send the exact titles. $\endgroup$ – mark leeds Nov 29 '19 at 13:05

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