If the standard asset pricing model is correct, we estimate expected returns from risk premiums and their factor loadings (eg, beta*equity return premium), and we can infer the risk loadings and risk premiums from an expected return. Given we don’t observe expected returns, but assume that with large samples, average returns estimate expected returns, this is why the small cap risk factor is supposed to exist, because it tautologically explains the higher return to small cap stocks over large cap stocks. In this way there should be a risk-return duality from expected return to risk.

There has been a long tradition in Western thought of trying to develop a set of beliefs that are objectively true and rational. This usually involved some basic assumptions and then logic, as in dialectical materialism, or Russell and Whitehead’s Principia Mathematica. It is very comforting to think that important beliefs are can be rationally justified, in contrast to beliefs based on faith or aesthetic preferences. When finance created asset pricing theory in the 50’s and 60’s, this kind of rational reductionism was dominant, and it seemed obvious that applying the logic would render previously intractable, qualitative problems into unambiguous solutions within linear programming or dynamic programming. Expected returns in this framework have a rational, objective grounding, a la Logical Positivism.

ProShares has revolutionized trading with their plethora of levered funds that provide 2 times the daily return of some target index, such as the SSO ETF which replicates a strategy that generates 200% return of the S&P500 index over a single day. These ETFs are very popular, and obviously have the precise doubling or tripling of the covariance as the targeted index. Yet, due to trading costs and compounding, the returns over longer periods are significantly lower than 2 times the return on the index. Of the 24 ETFs that promise a 200% daily return of the index, where the index has risen since the end of 2009 through August 2011, the average Ultra ETF has returned about 10% less than 2 times the index. If returns are a function of covariances, investors are being shortchanged here, and the ETF should trade at a discount to net asset value, but they don’t.

Imagine a world where expected returns are solely a function of covariances, as standard theory implies. Then if you create assets with specific covariances, the market should give them specific expected returns. Everything should be consistent. People should expect risk and return to be positively correlated.

Instead, Sharpe and Amromin find that people expect volatility and returns to be inversely correlated: when they are bullish they expect low volatility, and when they are bearish they expect high volatility. This is counter to standard theory, where basically expected equity returns should be a linear function of market variance. Given the inverse correlation between returns and volatilities, where increases in stock prices correspond to decreases in implied volatilities and vice versa, this makes sense. People are clearly assuming a specific return, and then deriving the volatility consistent with that scenario.

For buyers this means they expect volatility to decrease as the price rises, as it does in practice (and is implicit in the volatility skew in equity options). It is not possible to expect a really large expected return without assuming a large amount of risk, merely because it necessarily follows that if you expect, say, a 20% return on an asset, it must be capable of generating a fluctuation that high, which unfortunately also implies it could fall by 20%. The conditional volatility on an asset where a relatively large return is significantly probable must be larger than for an asset where a relatively large return is less common. Expected volatility is a function of expected return, and expected returns are a function of expected volatility (higher volatility assets have higher conceivable returns), but they are collectively wrong, which is why average returns are decreasing as a function of volatility.

Investors understand that stocks prices fluctuate randomly, but just as ‘no conqueror believes in chance,’ no active investor believes in chance either when they take risk. People are thinking about the collapse of the wave function, as opposed to the wave, because it’s more common for investors to think about outcomes as opposed to probability distributions. They understand they can be wrong, and experience much anxiety about their investments, but that’s different than thinking that expected returns are random, and very few investors base their expected return on risk factors and their loadings, as opposed to some specific outcome.

There is a duality, but it involves two sets of expected returns. If people invest based on the delusion that their expected returns are single outcomes, they are constantly evaluating value. Many investors are making this work, investing by applying some discounted cash flow logic. They are sufficiently right that a stock price is a decent estimate of its future profitability, but sufficiently wrong so that the stocks with the greatest expected variances, which necessarily include those stocks with the greatest expected returns among its investors, tend to be most ‘over-bought’, and thus have the lowest returns. Expected returns, derived from a model that statistically estimates expected returns, as opposed to canvassing investor opinion, are negatively related to risk. Low volatility investing is truly outside-the-box, an expectation that comes from noting the emergent pattern caused by investor beliefs, which contain a systematic bias.

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