It's important to note this is a minority view. That is, while almost everyone thinks beta (or its SDF analogues) are insufficiently rewarded, most, like Asness, Frazzini and Pederson, think that higher risk still begets a higher return, just marginally so. They are wrong, but on one level it doesn't matter: low volatility stocks still dominate the indexes in Sharpe ratios, or any 'risk-reward' metric you can concoct. But it matters a lot theoretically, because if higher risk generates lower returns, the whole 'risk premium' theory that underlies so much economic reasoning is empty, invalidating a lot research.
Bali and Cakici (2006) find no large negative return to high idiosyncratic vol stocks as found by Ang, Hodrick, Xing and Zheng among others. One key point to remember is that idiosyncratic vol is total volatility minus the volatility from systematic factors. Basically, you take an 'excess' return, after subtracting a return from its market beta (standard CAPM), and the returns from 'size' and 'value' proxies.
They find that when you 'equal-weight' quintiles, as opposed to value weight them, the large negative correlation between idiosyncratic volatality and return goes away. Further, when they form quintile portfolios based on market cap, so each portfolio is from largest to smallest, but the first portfolio contains many firms equal to 20% of the market, the top portfolio the fewest stocks, but includes large stocks, the results for value or equal weighted portfolios is not correlated with idiosyncratic volatility. Lastly, they look at large/liquid, lorge/high-price, and liquid/high priced subsamples, and again find no relation.
I really like this paper because most interesting findings are not uncovered via abstruse statistics, but careful slicing of the data. For example, early in equity research, people included all firms, including really small firms no one could really buy much of, and weighting them like the S&P500 stocks. These small firms had really large bid-ask spreads, and so as they traded at their bid of 1.00 to their ask of 1.25, you would record a return of 25%, when instead the closing price merely recorded a last trade at the bid going to the ask. Then when it went down to 1.00 again, the loss was only 20%, resulting in a phantom return of +5% even though the price went from 1.00 to 1.25 to 1.00. Other biases from small firms is they delist frequently, and when they delist the returns were hugely negative, about -50%, but this would be marked as 'N/A', greatly biasing returns. One only has to note the huge premiums documented in 'low price' stocks in the 1980s, and the embarrassing failure of 'low price' stock funds that were popular at the end of the eighties, to note that such findings were artifacts of bias.
Now, size is highly correlated with volatility, so controlling for size and then looking for volatility effects, is like controlling for beauty and looking at how weight affects someone's hot-or-not rating. If 'size' is not really a factor, but mainly captures a return premium via volatility, the null result may simply the result of dilution via this prefiltering. I mean, after you control for size in quintiles, and then via the size factor return, you don't have a lot of volatility dispersion left. As size and value don't have any real basis in my mind, I would try simple 'total volatility' as an alternative, once one has enough size/liquidity.
But forget that, Bali and Cakici's data show the following results, first replicating Ang et al's research, then using the refinement they find rids the data of that result.
Annual Returns to Low to High Idiosyncratic Volatility Quintiles
Now, I just took their monthly returns and multiplied them by 12. In reality, as the average holding period for a stock is around 1 year, you should do a geometric adjustment which lowers the return by the annualized variance divided by two, which makes the high vol quintiles look worse. But forget that. Note that in the 'standard case', there's a massive 10% difference between high and low volatility stocks. This clearly is 'too high'. But their new finding is still 3%, though he states these have t-stats of only 1-ish, insignificant. What Bali and Cakici call 'no robust, significant relation' just means it's not huge, but rather the kind of modest improvement one would expect given that large opportunities in liquid stocks are generally arbitraged out of the market. The relation remains precisely because it is so unsexy.
So, Bali and Cakici's results are consistent with what low-vol proponents like myself and Robeco have been asserting: they generate a couple of basis point higher returns, at 30-40% less volatility/beta. That's what I asserted in my 1994 dissertation, and in my book Finding Alpha, and my SSRN paper. That's what Haugen and Baker (1991), Schwartz 2000, Clarke, de Silva and Thorley (2006), Blitz and van Vliet (2007) all document in their analyses of minimum variance portfolios. Bali and Cakici's returns only highlight this is not a free-lunch to some 10% excess return you might apply within a long-short portfolio.
I have found, in the CRSP data since 1962, that small priced stocks have a pretty high monthly return premium, something I know is the result of bias as opposed to anything real. So, there's still a bias in the CRSP returns, even though they supposedly exorcized those problems in response to Shumway's finding that delisted returns exagerrated the returns of small companies. So, big analyses that look at 'all' stocks are seriously flawed, and their adjustment seems in the right direction. But the result still stands: you can get a couple of percent higher return, at a huge decrease in risk, by focusing on low volatility equities.