Yes, but where does clustered volatility come from? Only the Santa Fe Institute has what seems to me to be a plausible model. The road to hell is paved with models that assume Gaussian distributions, and ignore heteroskedacity.

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Tony K's comment was a good summary of Warren Buffett's stance on the issue:"I wonder if the problem is our definition of risk. I would contend that volatility may not be the same as risk. Volatility is more a function of time preference of returns. Risk to me, should be default or failure risk."

How many of these academics can predict future results, rather than just finding patterns in the past? Anybody can find statistically significant correlations, given enough data.

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Readers of Tetlock's Expert Political Judgment might suggest the simple statistical methods that compare so favorably to even the best experts in his book, though since he reveals so little about how exactly they were used I wouldn't suggest this with any confidence.

It also seems to me that the domain of "general predictions about the future" is simply not a promising subject for investigation. Just because we want something doesn't mean we can have it.

Finally, but mostly facetiously, how about just asking Warren Buffett or George Soros or any of the small number of other investors who have generated sufficiently high returns with sufficient consistency over a long period that the null hypothesis that they are just lucky can be overturned with extreme predjudice. If the domain is limited to a technical field, how about the scientists from PARC or any other inventors who were right but failed to capitalize on their ideas... There may not be any good institutions for providing legal advice either. One may thus be stuck with using an expert, with only your own judgment and track record data to enable you to recognize one.

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Michael, I don't see that the accuracy of portfolio theory says much about the relative accuracy of prediction markets vs. other institutions. Is there some alternative forecasting institution for which you think we have better theories to predict behavior, and where those theories suggest that that alternative would forecast better than prediction markets?

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Robin: There is no puzzle in explaining an anomaly if there is no general finding that it contradicts. There is no general finding that expected reward rises with risk, so there's no puzzle. We just know that some or all of the postulates behind the logical model that predicted such a relationship are wrong beyond the limits too which the logical relationship could be generalized.

By the way, the failure of such a basic prediction of portfolio theory greatly reduces the credence that I would otherwise place in the potential for financial or prediction markets to usefully predict events.

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I wonder if the problem is our definition of risk. I would contend that volatility may not be the same as risk. Volatility is more a function of time preference of returns. Risk to me, should be default or failure risk.

Time preference of returns can vary over time based on factors like demographics and economic outlook.

The other thing about the generalization that high risk/volatility = high returns. I think that is one of those in-a-perfect-world generalizations. Yes if you want to make money investing in high risk vehicles, you need high returns on the ones that work out. Tha does not mean that the market recognizes the risks inherent in the vehicles. So high risk assets may be overpriced. It is more a function of the efficiency of the market and information around it.

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Ah yes, the risk premium problem. It's too big. Sort of like, we have a law of X, and its usually too small, but occasionally too big. ..Perhaps it's not a law, or tendency, or anything!

The Return on equities over bonds was estimated as 8.5% by Ibbotson in the 90's, now it's assumed to be around 6.0%, but forward looking estimates say 3.5%. Still too much for standard utility functions, which suggest it 'should' be only 0.4% based on a representative agent and standard utility function. So that's a puzzle.

But consider that historically Beta=1.5 stocks have the same return as Beta=0.5 stocks. If risk is what matters, why not just by low beta stocks? So equities have 'too much' a premium, within equities it is nonexistent. Curious.

There is no general tendency of Risk-return positive correlation within a variety of investments

*High beta stocks don't outperform low beta stocks.*Out-of-the money call options on individual securities have significantly negative returns, in contrast to the implications of the a risk model where calls are basically levered stock positions.*Stocks with the greatest opinion dispersions underperformed a portfolio of otherwise similar stocks.*S-Corps and Franchises have returns approximately equal to the equity market, even though these investments are generally large portions of a household’s wealth*Commodities seem to offer a risk premium, but only if one looks at the indices with a heavy energy weighting, as opposed to those favoring agriculture commodities.*Bank capital ratios (higher capital means lower leverage, and thus lower risk) are positively related to bank ROE (Hutchinson and Cox, 2006).*Lotteries with the lowest odds and largest jackpots (higher variance) generate more revenues and are more in demand, than higher odds, lower jackpots.*Levitt and Venkatesh (2000) find that drug dealers engage in highly risky profession with a low average wage.G-rated movies generated lower volatility and higher returns than *R-rated movies, though there was a clear preference towards producing R-rated movies (over 1000 R-rated movies and only 60 G-rated ones).*In sports books there is the favorite-long shot bias, as favorites with low payoffs (low volatility) have higher expected returns than the higher volatility long shots.*B rated and defaulted bonds seem to generate no advantage to BBB rated bonds

On the other side:*Equities seem to return too much*BBB bonds seem to return too much*2 year bonds seem to return too much relative to 3-month T-bill

So it's a mess. I would say equities are best explained by measurement error from several sources:*'peso problems' in historical returns (the survivorship bias of indices (eg, Japan went to zero, US is a winner),*dollar-weighting stock returns (see Dichev)*taxing capital gains and losses asymmetrically* transaction costs

All could easily take a 3.5% equity risk premium to 0%, if not lower. Heck, this isn't a law, it's not a tendency.

Think of it this way. Is it a higher average return strategy to write papers that are less established, that are 'riskier'? Would you tell a young PhD to do work that most people think are not good ideas? A good idea, be it an investment, or research agenda, is not merely risky, but new, true, and important. Risk is one part of it (ie, new), but not the only part.

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Michael and Eric, you are right to note that the celebrated risk-price effect is surprisingly weak. I don't understand though how this squares with there being a puzzle of explaining a "risk-premium puzzle" of higher returns of stocks relative to bonds. Is there really no puzzle to be explained?

Anony, I see nothing at Not Even Wrong about Bostrom.

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If 100 years is a small sample, it's pretty unlikely 200 years is an acceptable sample.

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Sorry, but I don't know where else to post this.

Nick Bostrom, you should go over to Peter Woit's blog "not even wrong", where people have been discussing your recent NYT appearance in a not very nice way. Maybe you should set the record straight.

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how is volatility measured in these studies?

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If at first you don't corroborate your theory on good data...go back 100 years to a much smaller and more suspect data (mainly railroads and banks, no dividend data) that tends to support your data and say "we've got proof!" That kind of empirical strategy seems loaded with bias.

Plus, if, say the CAPM only works over 200 year horizons, as a practical matter for someone with an investment horizon of 30 years, isn't that saying that it isn't important? What is the upside to playing a strategy optimally if you need 7 lifetimes to demonstrate your better strategy?

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This is silly. "Positive and statistically significant p < .05 when the data set is huge" is incredibly far from "linear relationship between market risk and return". Basically, or best financial theories are normative but not predictive.

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