In financial markets, it has been suggested by Nassim Taleb, things are very different indeed. One might even be given to thinking that thousands of companies have raised huge amounts of money to finance vast numbers of projects and every time there was a philosophically challenged lender on the other end of the trade. Popular finance suggests that your fund manager placed bets on the red hot favorite: the survival of a company, and exposed you to less likely but potentially disastrous outcomes. She sought higher yields but, as with taking the 16/1-on favourites at the trots, this came at great price: the occasional massive setback more than countering all the gains. Moral hazard exacerbates the problem, we are warned, because fund managers share the upside with investors but not the downside.

Despite the persuasiveness of Mr Taleb there is no compelling reason to think that the polarity of the longshot bias - should it exist in a broad sense at all - is flipped as we move from the racetrack to finance. Those choosing to adopt an allegedly philosophical lens known as Black Swan theory should take a harder look at bond markets, because if that theory is correct corporate bonds buyers should be amongst the worst offenders. The facts that get in the way include a recent paper by Giesecke, Longstaff, Schaefer and Strebulaev surveying 150 years of corporate bond issuance and 150 years of corporate defaults. It may be surprising to Black Swan theory fans that over the long term buy and hold investors were compensated for

*twice*the risk they were taking, not undercompensated.

What makes this even more remarkable is the fact that their data includes not only the Great Depression but a period that was even worse for fixed income investors. In percentage terms, the railroad crisis of 1873-1875 hit the bond markets with the same force the Black Death hit Europe.

This leaves the reverse longshot theory on shaky ground, to put it mildly, and individuals would be crazy to piss away their retirement funds on pseudo-science of this caliber. As I noted in this post, even if there were a widespread reverse longshot bias in finance it would

*still*not justify a bar-bell portfolio.

**A Subtle Bias?**

I am prompted to mention the longshot bias just now for a different reason. While reading a seemingly unrelated, purely statistical paper the other day it occurred to me that the longshot bias is more subtle than I (an erstwhile assistant to a professional gambler in Sydney) previously appreciated. The longshot bias has been dissected many times over the years, most recently by Snowberg and Wolfers, but what I find interesting is a mathematical angle that may have gone overlooked (at least in my somewhat cursory reading of the longshot literature). I therefore draw tge readers attention to two facts:

(a) [From the economics literature] In the absence of a track take, parimutuel markets effect convex combinations of the subjective probabilities of participants.

(b) [From the statistical literature] No convex combination of calibrated forecasts is calibrated.

The first observation is quite well known, I believe, because betting proportional to one's best estimate of true probabilities is optimal under surprisingly weak assumptions. And there are other markets, I dare presume, which effect linear pooling of opinions. But the second observation might be less well known. The terminology 'calibrated' must be read as in this paper, by the way, where the linear combination of probabilistic forecasts is contemplated and improvements suggested. Those improvements attempt to overcome the problem of underconfident forecasts: those that assign lower probabilities to odds on favourites and higher (than true) probabilities to longshots. The point: the linear combination of forecasts creates this type of bias even if the individual bettors are themselves subject to no longshot bias at all.

The authors do not contemplate markets where convex combinations of probabilistic forecasts are implicitly combined, but it seems a few simple theorems would follow easily and it might be able to prove underconfidence in a broad number of markets. In particular, it is already apparent from the two obervations listed above that at least one stylized marketplace comprising only individuals whose forecasts are free of statistical bias can nevertheless conspire to create underconfident forecasts in the risk-neutral measure. That is, a longshot bias can exist in the absence of any individual behavioural biases whatsoever.

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