# Why is random trading minus transaction costs not zero expected value?

Somebody was telling me that if you buy randomly and assuming no transaction costs in todays market place, you wont make money 50% of the time and lose 50% of the time because of adverse selection. Im not quite sure I follow, does anyone know what this could be about?

-
Would you consider the bid ask spread be considered a transaction cost? If not, a very simple counterexample could be provided. But I assume that is not what you are asking about? –  AdAbsurdum Nov 17 '12 at 15:13
Why don't you ask that person who told you this? –  chrisaycock Nov 17 '12 at 16:44
@chrisaycock, good point, but I'm glad that Palace Chan asks the question here too because I'm also interested in the answer. –  Hugues Fontenelle Dec 3 '12 at 14:02

Assuming the following:

2. That you're interested in passive trading since you mentioned adverse selection.
3. As a random passive trader you simply place a bid to buy and an offer to sell at price $P$ and $P + Spread$, respectively.
4. You do no active order management and are therefore a naive trader with respect to order execution. You simply place your bid and offer and wait.

Given your question this would appear to be the only realistic scenario that makes sense. It would not make much sense to consider the case where you can, on demand buy at the bid and sell at the ask since it is impossible.

You can conceptually convince yourself that you will lose money given a random placement of passive, non-managed limit orders if you assume there are both informed and uninformed traders in the market.

First, define "adverse selection" as an uninformed participant trading with a more informed participant. An informed participant trades because he believes, rightly or not, that he has information relevant to the future price. As an uninformed trader you have no information that might impact your belief about the future price.

Second, consider that as a naive liquidity adder randomly placing bids and asks you will often only trade when it is worst to do so. Your bid at price $P$ will be hit with when the market is about to go down and your ask at price $P + Spread$ will be lifted when the market is about to go up.

Given this you'll typically start in a hole when comparing to the midpoint, having bought at the bid immediately prior to a tick down, or sold at the ask immediate prior to a tick up. This is why passive traders place a premium on queue position. The less orders in front of them the better, and the more orders behind them the better.

To illustrate, on average your starting unrealized $PnL$ for your bid order will be the future midpoint minus your current bid:

$PnL_{start} = P_{mid_{t+1}} - P_{bid}$

If you believe the above that the future midpoint, $P_{mid_{t+1}}$ will typically be below your current bid due to adverse selection, then you can see that you will always have a negative starting $PnL$ for the majority of your trades.

-

I would actually venture to say it will be very close to 50% over many transactions. Of course given you do not consider any transaction costs, meaning you buy and sell at the mid-point of cash equity traded prices, without slippage, or any other execution related expense. Also, my claim only holds if you consider cash equities, which are NOT a zero-sum "game". This assumption leads to the logical conclusion that even if there are "informed" (a.k.a. illegally acting on insider information) market participants in the market this does not necessarily stack the odds against a randomly picking trader. I am aware and have browsed through the papers which brought about "adverse selection bias" however, I find their cited empirical evidence very weak and do not believe that it necessarily means the rest of the uninformed traders trade at worse than 50% odds. The story can be quite different in options and futures markets. But regarding cash equities I would be very careful in supporting adverse selection. In that I would first like to make sure which asset class we are exactly talking about.

-
@down voter, care to comment on the down vote? –  Matt Wolf Nov 19 '12 at 1:56

The first formal model to explain this was Kyle (1985).

Oversimplifying, imagine there is a group uninformed traders and an informed trader active in the market for a given security. Uninformed traders cannot make correct directional predictions. The informed trader knows --- with some uncertainty --- what the price will be at the end of a certain time horizon (e.g. he has inside information about earnings). The informed trader wants to profit maximally from his knowledge and achieves that by slowly trading in the direction he predicted. His profit will be equal to the aggregated returns of the uninformed traders.

Bottomline: As an uninformed trade, you trade randomly. If you are lucky, you counterparty is just as uninformed and if there are no transaction costs you stand a 50% of making a profitable trade. If you are not so lucky, you are ''adversely selected'' and your counterparty is an informed trader and your transaction will very likely a losing one. All in all, your returns from trading randomly will be less that the benchmark lucky case.

Several details need to be hammered out to make this intuition consistent, the most important ones are (1) that uniformed traders are noise traders and (2) how the interaction between these two groups and revisions to the price are mediated by a market maker. This intuition extends to toy models of the limit order book (for example, Rosu (2010)).

-