Prior to reading the book, allow me to present a simple counter-argument to the premise of Flash Boys.[1]

I will do so in the manner of the ancient Greeks, who sought to prove axioms using principled methods of proof, rather than obtaining empirical evidence, thus freeing them from the need to actually do work. All else equal, I would prefer not to do work, so if I can sometimes prove something without actually having to go do things, I’ll do that.

Right then. Allow me to prove the following theorem:

A*ny unlevered investor who purchases publicly traded securities (I’ll use stocks here, because I buy stocks) based on fundamental valuation (i.e., the informed estimation of the present value of expected future cash flows, hereinafter referred to as “V”) cannot be made unilaterally worse off with respect to competition from any other market participant*[2]. This includes actions by other market participants possessing a speed advantage (uh oh!).

Let us proceed!

Let the market price of a security hereinafter be referred to as “P”.

Axiom (1): there are only three possible outcomes while trying execute orders to buy stock, (a) P, goes down, (b) P stays flat, (c) P goes up.

Axiom (2): a fundamental investor will only purchase shares when P <= V, i.e., when shares can be bought at a discount to estimates of intrinsic value.[3]

Assume the following initial condition: P < V, meaning the investor begins to attempt to accumulate a position when shares are currently trading for less than the investor’s estimate of intrinsic value.

Now, let us consider the actions of the investor in (2) during the possible events described by (1).

*Primo*, P goes down. If P continues to decrease due to the actions of others, the investor will simply continue accumulating shares, as the condition in (2) is still met (and now offers a higher expected return, too!).[4]

*Secondo*, P stays flat. The investor will continue accumulating shares, given that (2) is still satisfied. The expected return remains constant, and is above the investor’s desired expected return.

*Terzo*[5], P goes up. (A wild high frequency trader appears!) During accumulation, the share price is pushed up due to the purchase actions of traders who have detected the presence of a buyer (“front-running”). If P increases to a level such that P > V, the investor stops purchasing shares, since (2) is no longer satisfied. For simplicity, assume this occurs early on in the trade, where the cost of the orders filled is negligible relative to the investor’s total capital.

Now let us consider the economic result in each outcome. In (a) and (b), the investor is made better off by the amount of (P – V) multiplied by the number of shares purchased[6], since, presumably, cash was exchanged for securities whose value exceeds the amount of cash expended. In (c), where our pesky high frequency trader has interfered, the investor is neither better nor worse off. The investor’s portfolio after the attempted trade is identical to the portfolio prior to the trade. Because no orders were executed, the composition of cash and securities is the same. I don’t know how many other ways to say “nothing has changed”. Additionally, if we do not make a prior assumption, and a non-negligible percentage of the desired trade could be filled prior to the run-up in price, then the outcome in this case is analogous to the outcome of (a) or (b).

Therefore, the investor described herein is indifferent to the presence of high frequency traders who may seek to “front-run” in the market, so long as limit orders are used to maintain price discipline. Regardless of outcome, the investor is either better off or unaffected.

Q.E.D. Scooby snacks for everyone!

For further reading pleasure (or masochistic pain, whatever), I recommend you read this paper by Cliff Asness, because he’s a lot smarter than I am. Specifically section 8.

Feel free to continue reading on for some fun corollary arguments.

Corollary (i): now that Lewis’ villainous high frequency trader has accumulated all these shares in the hope that our hero, the fundamental investor, is going to allow him to unload at a profit by buying all these inflated shares off of him, he’s pretty screwed since this isn’t going to happen. Whoops! Hopefully there’s another high frequency trader to assume the role of greater fool.[7]

Corollary (ii): Our axiom also leads us to the conclusion that, for Lewis’ argument to remain credible, there must exist some subset of investors who (1) do not trade securities based on fundamental valuation, and (2) do not possess a speed advantage, or (3) employ significant amounts of leverage, such that they blow up every time the prices of their holdings move around more than a few percentage points. (I bet you can see where this is going! And my guess is it’d make for a much more interesting book!)

Corollary (iii): Eric Schneiderman should consider doing other stuff. Also the SEC. Though admittedly, it’s quite possible there’s some HFT somewhere doing bad things. Just like there are real investors who do bad things. I mean, well, in any any sufficiently large group of agents anywhere, there is probably at least one bad apple, right? That’s just the Law of Large Bad Things.

[1] Hey, I’m in the middle of other books! I’ll get around to it one day, I swear!

[2] Or said another way: if you invest based on fundamentals, no one can screw you over. Note that this ignores possible reflexivity between the actions of market participants and the value of the businesses traded. Like, supposedly you could short a stock, then blow up the company’s headquarters and factories. There are probably lots of other possibilities. I encourage to you exercise creativity while trying to invalidate this axiom.

[3] If this isn’t intuitive, consider the opposite. Obviously, you aren’t going to buy shares if the price is above what you think they’re worth. I mean, if you are, then I can’t help you, and other people probably deserve to take your money.

[4] Obviously, if you just bought a dollar for 75 cents, then someone offers you another dollar for 50 cents, you buy that one too. In reality most people don’t. But that’s because we have these things called behavioural biases, which is academic speak for “people are silly”. Somehow when we buy a dollar, and then someone offers us another dollar for less, we get upset. Who knew.

[5] That means “third”. Ten points (points are redeemable in exchange for free coffee chats!) if you can guess who I stole this choice of diction from. Also, I know the Greeks didn’t use Italian. But I just did, okay?

[6] More formally, P would be the volume-weighted average price of the investor’s purchases in this case. Ignore the formal stuff though, I’m still right.

[7] Otherwise, what’s the HFT going to do, buy and hold? Hahahahahahahaha.