Markets are Anti-Inductive

I sus­pect there’s a Pons As­ino­rum of prob­a­bil­ity be­tween the bet­tor who thinks that you make money on horse races by bet­ting on the horse you think will win, and the bet­tor who re­al­izes that you can only make money on horse races if you find horses whose odds seem poorly cal­ibrated rel­a­tive to su­pe­rior prob­a­bil­is­tic guesses.

There is, I think, a sec­ond Pons As­ino­rum as­so­ci­ated with more ad­vanced fi­nance, and it is the con­cept that mar­kets are an anti-in­duc­tive en­vi­ron­ment.

Let’s say you see me flip­ping a coin. It is not nec­es­sar­ily a fair coin. It’s a bi­ased coin, and you don’t know the bias. I flip the coin nine times, and the coin comes up “heads” each time. I flip the coin a tenth time. What is the prob­a­bil­ity that it comes up heads?

If you an­swered “ten-elevenths, by Laplace’s Rule of Suc­ces­sion”, you are a fine sci­en­tist in or­di­nary en­vi­ron­ments, but you will lose money in fi­nance.

In fi­nance the cor­rect re­ply is, “Well… if ev­ery­one else also saw the coin com­ing up heads… then by now the odds are prob­a­bly back to fifty-fifty.”

Re­cently on Hacker News I saw a com­menter in­sist­ing that stock prices had nowhere to go but down, be­cause the econ­omy was in such awful shape. If stock prices have nowhere to go but down, and ev­ery­one knows it, then trades won’t clear—re­mem­ber, for ev­ery sel­ler there must be a buyer—un­til prices have gone down far enough that there is once again a pos­si­bil­ity of prices go­ing up.

So you can see the bizarreness of some­one say­ing, “Real es­tate prices have gone up by 10% a year for the last N years, and we’ve never seen a drop.” This treats the mar­ket like it was the mass of an elec­tron or some­thing. Mar­kets are anti-in­duc­tive. If, his­tor­i­cally, real es­tate prices have always gone up, they will keep ris­ing un­til they can go down.

To get an ex­cess re­turn—a re­turn that pays pre­mium in­ter­est over the go­ing rate for that level of risk­i­ness—you need to know some­thing that other mar­ket par­ti­ci­pants don’t, or they will rush in and bid up what­ever you’re buy­ing (or bid down what­ever you’re sel­l­ing) un­til the re­turns match pre­vailing rates.

If the econ­omy is awful and ev­ery­one knows it, no one’s go­ing to buy at a price that doesn’t take into ac­count that knowl­edge.

If there’s an ob­vi­ous pos­si­bil­ity of prices drop­ping fur­ther, then the mar­ket must also be­lieve there’s a prob­a­bil­ity of prices ris­ing to make up for it, or the trades won’t clear.

This el­e­men­tary point has all sorts of caveats I’m not both­er­ing to in­clude here, like the fact that “up” and “down” is rel­a­tive to the risk-free in­ter­est rate and so on. No­body be­lieves the mar­ket is re­ally “effi­cient”, and re­cent events sug­gest it is less effi­cient than pre­vi­ously be­lieved, and I have a cer­tain friend who says it’s even less effi­cient than that… but still, the mar­ket does not leave hun­dred-dol­lar-bills on the table if ev­ery­one be­lieves in them.

There was a time when the Dow sys­tem­at­i­cally tended to drop on Fri­day and rise on Mon­day, and once this was no­ticed and pub­lished, the effect went away.

Past his­tory, e.g. “real es­tate prices have always gone up”, is not pri­vate info.

And the same also goes for more com­pli­cated reg­u­lar­i­ties. Let’s say two stock prices are his­tor­i­cally an­ti­cor­re­lated—the var­i­ance in their re­turns moves in op­po­site di­rec­tions. As soon as ev­ery­one be­lieves this, hedge-fund man­agers will lev­er­age up and buy both stocks. Every­one will do this, mean­ing that both stocks will rise. As the stocks rise, their re­turns get more ex­pen­sive. The hedge-fund man­agers book prof­its, though, be­cause their stocks are ris­ing. Even­tu­ally the stock prices rise to the point they can go down. Once they do, hedge-fund man­agers who got in late will have to liqui­date some of their as­sets to cover mar­gin calls. This means that both stock prices will go down—at the same time, even though they were origi­nally an­ti­cor­re­lated. Other hedge funds may lose money on the same two stocks and also sell or liqui­date, driv­ing the price down fur­ther, etcetera. The cor­rel­a­tive struc­ture be­haves anti-in­duc­tively, be­cause other peo­ple can ob­serve it too.

If mortage de­faults are his­tor­i­cally un­cor­re­lated, so that you can get an ex­cess re­turn on risk by buy­ing lots of mortages and pool­ing them to­gether, then peo­ple will rush in and buy lots of mort­gages un­til (a) rates on mort­gages are bid down (b) in­di­vi­d­ual mort­gage failure rates rise (c) mort­gage failure rates be­come more cor­re­lated, pos­si­bly look­ing un­cor­re­lated in the short-term but hav­ing more fu­ture sce­nar­ios where they all fail at once.

What­ever is be­lieved in, stops be­ing real. The mar­ket is liter­ally anti-in­duc­tive rather than anti-reg­u­lar—it’s the reg­u­lar­ity that enough par­ti­ci­pants in­duce, which there­fore goes away.

This, as I un­der­stand it, is the stan­dard the­ory of “effi­cient mar­kets”, which should per­haps have been called “in­ex­ploitable mar­kets” or “mar­kets that are not easy to ex­ploit be­cause oth­ers are already try­ing to ex­ploit them”. Should I have made a mis­take thereof, let me be cor­rected.

Now it’s not sur­pris­ing, on the one hand, to see this screwed up in ran­dom in­ter­net dis­cus­sions where a gold bug ar­gues from well-known ob­ser­va­tions about the past his­tory of gold. (This is the equiv­a­lent of try­ing to make money at horse-rac­ing by bet­ting on the horse that you think will win—failing to cross the Pons As­ino­rum.)

But it is sur­pris­ing is to hear his­to­ries of the fi­nan­cial crisis in which pres­ti­gious ac­tors ar­gued in crowded au­di­to­ri­ums that, pre­vi­ously, real-es­tate prices had always gone up, or that pre­vi­ously mortage de­faults had been un­cor­re­lated. This is naive in­duc­tive rea­son­ing of the sort that only works on fal­ling ap­ples and ris­ing suns and hu­man be­hav­ior and ev­ery­thing else in the uni­verse ex­cept mar­kets. Shouldn’t ev­ery­one have frowned and said, “But isn’t the mar­ket­place an anti-in­duc­tive en­vi­ron­ment?”

Not that this is stan­dard ter­minol­ogy—but per­haps “effi­cient mar­ket” doesn’t con­vey quite the same warn­ing as “anti-in­duc­tive”. We would ap­pear to need stronger warn­ings.

PS: To clar­ify, the coin ex­am­ple is a hu­morous ex­ag­ger­a­tion of what the world would be like if most phys­i­cal sys­tems be­haved the same way as mar­ket price move­ments, illus­trat­ing the point, “An ex­ploitable pric­ing reg­u­lar­ity that is eas­ily in­ducted de­grades into in­ex­ploitable noise.” Here the coin com­ing up “heads” is analo­gous to get­ting an above-mar­ket re­turn on a pub­li­cly traded as­set.