The paper notes several papers that show that under certain circumstances, some investors can prefer particularrisky assets over-and-above that dictated by standard risk pricing models. He references Abel (1990), DeMarzo, Kaniel and Kremer (2004), and Roussanov (2009), and others. These papers generally focus on why people underdiversify, or hold too much of particular assets relative to standard theory. In a way, these are similar to papers that argue that asset skew (positive, negative, co-skewness--depending on author), could explain why some risky assets have lower returns than expected via the standard approach. Basically, add something really desirable as an extra to an asset (it increases your status), and in some situations its price is higher than otherwise.
Why I think these papers are inferior to my argument is there are too qualified when applied to the basic point: there are no risk measures that consistently, positively (let alone linearly) related to average returns--and average returns, in the data, is all we got. They attempt to explain a special case. Proposing a model that you can then selectively apply to assets and eras where the model appears invalid is lame given that the welter of data that does not even generate a positive sign for the risk-return is considered 'too short' to say anything (because expected returns are not actual returns). So, if the main theory is considered by many to be untestable because we haven't had enough time (like the Soviet Union's 70-years of dought), how about an ad hoc theory applicable towards some assets?
Here's the bottom line, which apparently is more difficult to understand than the theory, but ultimately much more convincing once you see it. The a negative risk-return relation holds for volatility (cross-sectionally and over time, total and idiosyncratic), beta, options, private investments, leverage, currencies, country returns,yield curves, financial distress, sportsbooks, lotteries, IPOs, junk bonds, and analyst uncertainty. It's pretty absent in country returns, commodities, movies, and private investments. It does work nicely in the the AAA-BBB spread or the short end of the yield curve. Now, if you were an alien, what's your generalization, that in general risk is related to return, not related, or the opposite? The Academy teaches that there's a linear relation, positively sloped!
I'm the only one who writes down several cases to show if people are status oriented, there is no risk-premium in general. End of story. There's no reason to subtilize this idea within a larger model so that it has only a selective implication. Embedding it within a larger model makes it easier for the academy to digest, in that like Millikan electron measurements, new is just slightly better than the old, but not so much as to say anyone in the past was 'wrong', we are just extending their approach. As in kids games, Everyone Wins! Yet, the result is a rather weak assertion, almost by definition non-falsifiable (some assets may show no risk premium--ORLY?), and does not advance the 'science' of finance because it allows professors to continue teaching the manifestly unhelpful risk-return story withing Modern Portfolio Theory (ie, that expected returns are increasing in risk, and only risk, properly defined).
Paul Erdös, the mathematician who collaborated with more people than anyone else on the planet, reckoned that up in heaven God had a book that contained all the best proofs. If Erdös was really impressed by a proof, he declared it to be "from The Book". G.H. Hardy explained such proofs are inevitable, succinct, and unexpected. Examples are Euclid's proof there are an infinite number of primes, or Cantor’s theorem of the non-enumerability of the continuum.
My proofs are in my SSRN paper Risk and Return in General: theory and evidence, but this simple example should suffice. In the table below, asset X is usually considered riskier, with a 30 point range in payoffs versus a 10 point range for Y. Yet on a relative basis, each asset generates identical risk. In State 2, X is a +5 out performer; in State 1, X is a -5 underperformer, and vice versa for asset Y. In relative return space, the higher absolute volatility asset is not riskier. The positions, from a relative standpoint, are mirror images. Buying the market, going with the consensus, generates zero risk.
Relative Payoffs Symmetric
Everything really flows from this simple insight. Implicitly the utility and arbitrage equilibria derive from the fact that when relative wealth is the objective, risk is symmetric, as the complement to any portfolio subset will necessarily have identical — though opposite signed — relative return. Thus arbitrage exits if there is a risk premium in a relative status world. That's it.
In contrast, job-applicant Krasny and the authors he references have incredibly hedged proofs that hold in certain cases to certain degrees (depending on the nature of the parameters), and follow from the third proposition of the fifth theorem. Gali's (1994) paper in this vien actually has a parameter that allows such increased risk to increase, or decrease, its expected return depending on whether the 'externality' from holding that asset is positive or negative. This is explains nothing and everything.
It reminds me of David Hakes's anecdote, that one of his papers was continually rejected, but then his coauthor suggested that the problem with the paper might be that we had made the argument too easy to follow, and thus referees and editors were not sufficiently impressed. He said that he could make the paper more impressive by
generalizing the model. While making the same point as the original paper, the new paper would be more mathematically elegant, and it would become absolutely impenetrable to most readers.After which, it was immediately accepted for publication.
I guess I'm rather impolitic on this issue, and so my straightforward approach not only prevents me from getting published, but also getting referenced. I don't find any gain from playing this game, which I find disingenuous, pretentious, and wasteful. Further, there is no value to the old paradigm, it doesn't help fresh-faced MBAs become better at ascertaining true value, merely better at becoming eloquent confabulators to whatever result you arrived at through totally orthogonal means. These piecemeal status approaches bury the lede within a welter of potential implications and parameter values. Any model with more parameters or a more complex functional form can explain more--including having some higher 'risk' assets with lower expected returns--I'm arguing change the argument in the utility function (relative vs. absolute wealth) and you get a novel result: no risk premium in general. My above table lays out the status-risk-return nexus without hiding it behind specious generality. If it's in Erdös' Book, it's my above 'proof' and not any of these qualified papers that don't even directly state this explains most things, not some things, in asset pricing. Take the jump guys! You have nothing to lose but your irrelevance.
Funny that your theory is about the implications of relative status but you aren't mentioning why people aren't willing to go stray far from consensus: relative status.
People have a lot to lose from taking the jump, relative to their current status.
Now, I *can* see why you would be annoyed with them for not taking the jump, however. :)
I follow your math, but I still think that "X" is a "riskier" investment. If "X" represents yearly returns, I think most people would believe that "X" is more risky, despite your math.
Has there ever been a Thomas Kuhn-style paradigm shift in finance, or is MPT its first real paradigm?
MM: I agree that your interpretation is intuitive, yet you have to first remember these are hypothetical models, so they ignore the obvious idiosyncratic risk that is relevant when forced to choose here. Instead of X and Y, think, savings bond or housing. Many people think having a large percentage of their wealth in housing is prudent. But, housing has much more risk than savings accounts, and over the long run, no risk premium.
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