# Is it accurate to say that if I was to trade something, my probability of success can't be worse than random?

I'm just trying to visualise the costs of trading. Say I set up an account to trade something (forex, stock, even bitcoin) and I was going to let a random generator determine when I should buy or sell it. If i do this, I would assume I have an equal probability to make a profit or a loss.

Can I also assume that probabilistically speaking, a trader cannot do worse than random? Say, if I had to guess the roll of a dice, my chance of being correct can't be less than 16.667%.

Extending that logic, then for an inexperienced trader, is it right to say then that it's equally difficult to purposely make a loss then it is to purposely make a profit? Because if I can purposely make a loss, I would purposely just do the opposite of what I'm doing to make a profit. So in the dice example, if I can somehow lower my chances of winning below 16.6667%, it means I would simply need to bet on the other 5 numbers to give myself a better than 83% chance of winning.

So then, is the costs of trading from a purely probabilistic point of view simply the transaction costs? No matter what, my chances cannot be worse than random and if my trading system has an edge that is greater than the percentage of the transaction that is transaction cost, then I am probabilistically likely to make a profit?

Note - this question isn't about the RISKS or how to manage it so you don't lose your shirt, I'm simply talking about probability.

• You are mixing the probability of winning or loosing with profit / loss. This is incorrect. In the dice example if you bet 100, and you get 200 for correct guess and 0 for incorrect guess. Then net long term you will lose 400. Markets are more complicated. a winning bet can get you 101 or 10001 or anything and a losing bet can get you 99 or 0 or in derivatives even negative [i.e. you need to pay more]. Jun 22 '17 at 2:51
• I'm voting to close this question as off-topic because its more of research / academic exercise on random probability theory maybe more apt on maths stackexchange. Jun 22 '17 at 2:53
• Example strategy that performs much worse than "chance" (assuming you provide some sort of reasonable definition of chance): every time the market price for a security reaches a record high I'll buy 100 shares, and every time it reaches a record low, I'll sell 100 shares. Jun 22 '17 at 3:25
• I don't think this is a good question for Mathematics; honestly it's probably fine for here, although it's just a very naive question in general.
– Joe
Jun 22 '17 at 4:56
• No, if you always sell low and buy high, you can loose an arbitrarily large amount of money. This unfortunately is what many naive investors do: they buy during market rallies and bubbles, and sell in panics and bear markets. What you may not appreciate is that there are many kinds of random number generators which follow many different distributions. If you found a market that was actually random, AND you knew what distribution it followed you might be able to break even. Actual markets aren't entirely random, and have all sorts of correlations and trends that sit on top of the randomness. Jun 22 '17 at 5:32

The stock market is not a zero-sum game. Some parts are (forex, some option trading), but plain old stock trading is not zero sum.

That is to say, if you were to invest "at random", you would on average make money. That's because the market as a whole makes money - it goes up over time (6-10% annually, averaged over time). That's because you're not just gambling when you buy a stock; you're actually contributing money to a company (directly or indirectly), which it uses to fund activities that (on average) make money. When you buy Caterpillar stock, you're indirectly funding Caterpillar building tractors, which they then sell for a profit, and thus your stock appreciates in value. While not every company makes a profit, and thus not every stock appreciates in true value, the average one does.

To some extent, buying index funds is pretty close to "investing at random". It has a far lower risk quotient, of course, since you're not buying a few stocks at random but instead are buying all stocks in an index; but buying stocks from the S&P 500 at random would on average give the same return as VOO (with way more volatility).

So for one, you definitely could do worse than 50/50; if you simply sold the market short (sold random stocks short), you would lose money over time on average, above and beyond the transaction cost, since the market will go up over time on average.

Secondly, there is the consideration of limited and unlimited gains or losses.

Some trades, specifically some option trades, have limited potential gains, and unlimited potential losses.

Take for example, a simple call option. If you sell a naked call option - meaning you sell a call option but don't own the stock - for \$100, at a strike price of \$20, for 100 shares, you make money as long as the price of that stock is under \$21. You have a potential to make \$100, because that's what you sold it for; if the price is under \$20, it's not exercised, and you just get that \$100, free.

But, on the other hand, if the stock goes up, you could potentially be out any amount of money. If the stock trades at \$24, you're out \$400-100 = \$300, right? (Plus transaction costs.)

But what if it trades at \$60? Or \$100? Or \$10000? You're still out 100 * that amount, so in the latter case, \$1 million. It's not likely to trade at that point, but it could.

If you were to trade "at random", you'd probably run into one of those types of situations. That's because there are lots of potential trades out there that nobody expects anyone to take - but that doesn't mean that people wouldn't be happy to take your money if you offered it to them.

That's the reason your 16.66 vs 83.33 argument is faulty: you're absolutely right that if there were a consistently losing line, that the consistently winning line would exist, but that requires someone that is willing to take the losing line. Trades require two actors, one on each side; if you're willing to be the patsy, there's always someone happy to take advantage of you, but you might not get a patsy.

• It is a misconception that when I buy Caterpillar stock the money will actually go into Caterpillar's coffers and will be used to make more tractors, unless you are buying from Caterpillar itself, and it is not the case in the vast majority of blind market transactions. money.stackexchange.com/a/67489/27572 Jun 22 '17 at 16:40
• @Mindwin It's not at all a misconception of the above answer. indirectly is the important word. Increasing the market value of a company does effectively give them more potential capital - it increases their ability to borrow, it increases the price at which new share offerings could be made, etc. I am not saying at all that you hand them cash. This question for example.
– Joe
Jun 22 '17 at 18:13

One key piece missing from your theory is the bid/ask spread. If you buy a stock for \$10, you usually can't immediately turn around and sell it for \$10. You can only sell it for whatever someone is willing to pay for it. So virtually any random investment (stocks, bonds, forex, whatever) immediately loses a small amount of value, and over the long run you will almost certainly lose money if you buy/sell at random.

• I'd assume that by "transaction cost" OP includes bid/ask spread and fees. Jun 22 '17 at 14:41
• no, the bid/ask issue is totally different from transaction costs, and they have nothing to do with each other. Jun 22 '17 at 19:46

I'm just trying to visualize the costs of trading. Say I set up an account to trade something (forex, stock, even bitcoin) and I was going to let a random generator determine when I should buy or sell it. If I do this, I would assume I have an equal probability to make a profit or a loss.

Your question is what a mathematician would call an "ill-posed problem." It makes it a challenge to answer. The short answer is "no." We will have to consider three broad cases for types of assets and two time intervals.

Let us start with a very short time interval. The bid-ask spread covers the anticipated cost to the market maker of holding an asset bought in the market equal to the opportunity costs over the half-life of the holding period. A consequence of this is that you are nearly guaranteed to lose money if your time interval between trades is less than the half-life of the actual portfolio of the market maker. To use a dice analogy, imagine having to pay a fee per roll before you can gamble. You can win, but it will be biased toward losing.

Now let us go to the extreme opposite time period, which is that you will buy now and sell one minute before you die. For stocks, you would have received the dividends plus any stocks you sold from mergers. Conversely, you would have had to pay the dividends on your short sales and received a gain on every short stock that went bankrupt. Because you have to pay interest on short sales and dividends passed, you will lose money on a net basis to the market maker. Maybe you are seeing a pattern here. The phrase "market maker" will come up a lot.

Now let us look at currencies. In the long run, if the current fiat money policy regime holds, you will lose a lot of money. Deflation is not a big deal under a commodity money regime, but it is a problem under fiat money, so central banks avoid it. So your long currency holdings will depreciate. Your short would appreciate, except you have to pay interest on them at a rate greater than the rate of inflation to the market maker.

Finally, for commodities, no one will allow perpetual holding of short positions in commodities because people want them delivered. Because insider knowledge is presumed under the commodities trading laws, a random investor would be at a giant disadvantage similar to what a chess player who played randomly would face against a grand master chess player. There is a very strong information asymmetry in commodity contracts. There are people who actually do know how much cotton there is in the world, how much is planted in the ground, and what the demand will be and that knowledge is not shared with the world at large. You would be fleeced.

Can I also assume that probabilistically speaking, a trader cannot do worst than random? Say, if I had to guess the roll of a dice, my chance of being correct can't be less than 16.667%.

A physicist, a con man, a magician and a statistician would tell you that dice rolls and coin tosses are not random. While we teach "fair" coins and "fair" dice in introductory college classes to simplify many complex ideas, they also do not exist. If you want to see a funny version of the dice roll game, watch the 1962 Japanese movie Zatoichi. It is an action movie, but it begins with a dice game.

Consider adopting a Bayesian perspective on probability as it would be a healthier perspective based on how you are thinking about this problem. A "frequency" approach always assumes the null model is true, which is what you are doing. Had you tried this will real money, your model would have been falsified, but you still wouldn't know the true model.

Yes, you can do much worse than 1/6th of the time. Even if you are trying to be "fair," you have not accounted for the variance.

Extending that logic, then for an inexperienced trader, is it right to say then that it's equally difficult to purposely make a loss then it is to purposely make a profit? Because if I can purposely make a loss, I would purposely just do the opposite of what I'm doing to make a profit. So in the dice example, if I can somehow lower my chances of winning below 16.6667%, it means I would simply need to bet on the other 5 numbers to give myself a better than 83% chance of winning.

If the game were "fair," but for things like forex the rules of the game are purposefully changed by the market maker to maximize long-run profitability. Under US law, forex is not regulated by anything other than common law. As a result, the market maker can state any price, including prices far from the market, with the intent to make a system used by actors losing systems, such as to trigger margin calls. The prices quoted by forex dealers in the US move loosely with the global rates, but vary enough that only the dealer should make money systematically. A fixed strategy would promote loss.

You are assuming that only you know the odds and they would let you profit from your 83.33 percentage chance of winning.

So then, is the costs of trading from a purely probabilistic point of view simply the transaction costs? No matter what, my chances cannot be worse than random and if my trading system has an edge that is greater than the percentage of the transaction that is transaction cost, then I am probabilistically likely to make a profit?

No, the cost of trading is the opportunity cost of the money. The transaction costs are explicit costs, but you have ignored the implicit costs of foregone interest and foregone happiness using the money for other things.

You will want to be careful here in understanding probability because the distribution of returns for all of these assets lack a first moment and so there cannot be a "mean return." A modal return would be an intellectually more consistent perspective, implying you should use an "all-or-nothing" cost function to evaluate your methodology.

If i do this, I would assume I have an equal probability to make a profit or a loss.

The "random walk"/EMH theory that you are assuming is debatable. Among many arguments against EMH, one of the more relevant ones is that there are actually winning trading strategies (e.g. momentum models in trending markets) which invalidates EMH.

Can I also assume that probabilistically speaking, a trader cannot do worst than random? Say, if I had to guess the roll of a dice, my chance of being correct can't be less than 16.667%.

It's only true if the market is truly an independent stochastic process. As mentioned above, there are empirical evidences suggesting that it's not.

is it right to say then that it's equally difficult to purposely make a loss then it is to purposely make a profit?

The ability to profit is more than just being able to make a right call on which direction the market will be going. Even beginners can have a >50% chance of getting on the right side of the trades. It's the position management that kills most of the PnL.

It seems to be that your main point is this:

No matter what, my chances cannot be worse than random and if my trading system has an edge that is greater than the percentage of the transaction that is transaction cost, then I am probabilistically likely to make a profit?

In general, yes, that is true, but...

Consider this very bad strategy: Buy one share of stock and sell it one minute later, and repeat this every minute of the day. Obviously you would bleed your account dry with fees. However, even this horrible strategy still meets your criteria because: if this bad strategy had an edge beyond the transaction fees you would likely still make a profit. In other words, your conclusion reduces to an uninteresting statement:

If there were no transactions fees, then if your trading system has an edge then you will likely make a profit.

Sorry to be the bearer of bad news, but IMHO, that statement, and others made in the question are just obvious things stated in convoluted ways.

I don't want to discourage you from thinking about these things though. I personally really enjoy these type of thought experiments. I just feel you missed the mark on this one...

The previous answers make valid points regarding the risks, and why you can't reasonably compare trading for profit/loss to a roll of the die. This answer looks at the math instead.

I have an equal probability to make a profit or a loss.

Can I also assume that probabilistically speaking, a trader cannot do worst than random?

Is "yes". But only because the question is flawed. Consequently it's throwing people in all directions with their answers. But quite simply, in a truly random environment the worst case scenario, no matter how improbable, is that you lose over and over again until you have nothing left.

This can happen in sequential rolls of the dice AND in trading securities/bonds/whatever. You could guess wrong for every roll of the die AND all of your stock picks could become worthless. Both outcomes result in \$0 (assuming you do not gamble with credit). Tell me, which \$0 is "worse"?

Given the infinite number of plays that "random" implies, the chance of losing your entire bankroll exists in both scenarios, and that is enough by itself to make neither option "worse" than the other.

Of course, the opposite is also true. You could only pick winners, with an unlimited upside potential, but again that could happen with either dice rolls or stock picks. It's just highly improbable.

my chances cannot be worse than random and if my trading system has an edge that is greater than the percentage of the transaction that is transaction cost, then I am probabilistically likely to make a profit?

Nope. This is where it all falls apart. Just because your chances of losing it all are similarly improbable, does not make you more likely to win with one method or the other. Regression to the mean, when given infinite, truly random outcomes, makes it impossible to "have an edge".

Also, "probabilistically" isn't a word, but "probably" is.

In theory, in a perfect world, what you state is almost true. Apart from transaction fees, if you assume that the market is perfectly efficient (ie: public information is immediately reflected in a perfect reflection of future share value, in all share prices when the information becomes available), then in theory any transaction you would choose to take is opposed by a reasonable person who is not taking advantage of you, just moving their position around. This would make any and all transactions completely reasonable from a cost-benefit perspective.

ie: if the future value of all dividends to be paid by Apple [ie: the value of holding a share in Apple] exactly matches Apple's share price of \$1,000, then buying a share for \$1,000 is an even trade. Selling a share for \$1,000 is also an even trade.

Now in a perfectly efficient market, which we have assumed, then there is no edge to valuing a company using your own methods. If you take Apple's financial statements / press releases / reported information, and if you apply modern financial theory to evaluate the future dividends from Apple, you should get the same \$1,000 share price that the market has already arrived at.

So in this example, why wouldn't you just throw darts at a printout of the S&P 500 and invest in whatever it lands on? Because, even if the 'perfectly efficient market' agrees on the true value of something, different investments have different characteristics.

As an example, consider a simple comparison of corporate bonds:

Corporations make bond offerings to the public, allowing individual investors to effectively lend money to the corporation, for a future benefit. For simplicity, assume a bond with a 'face value' (the amount to be repaid to the investor on maturity) of \$1,000 has these 3 defining characteristics: (1) The price [What the investor pays to acquire it]; (2) Interest payments [how much, if any, the corporation will pay to the investor before maturity, and when those payments will be made]; and (3) a bond rating [which is a third party assessment of how risky the bond is, based on the 'health' of the corporation].

Now if the bond rating agency is perfect in its risk assessment, and if the price of all bond's is fair, then why does it matter who you loan your money to? It matters because different people want different things out of their investments. If you are waiting to make a down payment on a house next year, then you don't want risk - you want to be certain that you will get your cash back, even if it means lower returns. So, even though a high-risk bond may be perfectly priced, it should only be bought by someone willing to bear that risk. If you are retired, and you need your bonds to pay you interest regularly as your sole source of income, then of course a zero-coupon bond [one that pays no interest] is not helpful to you. If you are young, and have a long time to invest, then you may want risk, because you have time to overcome losses and you want to get the most return possible.

In addition, taxes are not universal between all investors. Some people benefit from things that would be tax-heavy to their neighbors. For example in Canada, there is a 'dividend tax credit' which reduces the taxes owing on dividends received by a corporation. This credit exists to prevent 'double-taxation', because otherwise the corporation would pay its ~30% of tax, and then a wealthy investor would pay another ~45% of tax. Due to the mechanics of how the credit is calculated, however, someone who makes less money, gets an even lower tax bill than they normally would. This means that someone making under the top tax bracket in Canada, has a tax benefit by receiving dividends. This means that while 2 stocks may be both fairly priced, if one pays dividends and the other doesn't [ie: if the other company instead reinvests more heavily in future projects, creating even more value for shareholders down the road], then someone in the bottom tax brackets may want the dividend paying stock more than the other.

In conclusion: Picking investments yourself does require some knowledge to prevent yourself from making a 'bad buy'; this is because the market is not perfectly efficient. As well, specific market mechanics make some trades more costly than they should be in theory; consider for example transaction fees and tax mechanics. Finally, even if you assume that all of the above is irrelevant as a theoretical idea, different investors still have different needs. Just because \$1,000,000 is the 'fair' price for a factory in your home town, doesn't mean you might as well convert your retirement savings to buy it as your sole asset.

Don't compare investing with a roll of the dice, compare it with blackjack and the decision to stand or hit, or put more money on the table (double down or increase bet size) , based on an assessment of the state of the table and history. A naive strategy of say "always hitting to 16" isn't as awful as randomly hitting and standing (which, from time to to time will draw to 21 fair and square) , but there's a basic strategy that gets close to 50% and by increasing or decreasing bet based on counting face cards can get into positive expectations.

Randomly buying and selling stock is randomly hitting. Buying a market index fund is like always hitting to 16. Determining an asset allocation strategy and periodically rebalancing is basic strategy. Adjusting allocations based on business cycle and economic indicators is turning skill into advantage.