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Risk and Return in General: Theory and Evidence

Risk premiums are presumably omnipresent and almost impossible to measure. This paper outlines the origin of the modern theory of risk premiums, the history of its testing, and surveys the current failure of this theory across over 20 different asset classes. It is argued that a relative status oriented utility functions explaisn the absence of a risk premium rather neatly. When agents are concerned about relative wealth, risk taking is then deviating from the consensus or market portfolio. In this environment, all risk becomes like idiosyncratic risk in the standard model, avoidable so unpriced.

Theoretically, globally concave utility functions are necessary and sufficient for agents to be risk averse.  People strictly prefer a certain outcome to an uncertain outcome with the same mean payoff, and so demand payoff premium to be indifferent.  To the degree risk is not diversifiable as in ‘market risk’, someone must hold it, and because it is disliked those who do hold it must be compensated via a risk premium relative to risk-free securities.  Yet it is striking that a first approximation to risk via volatility or beta against the market return generates no positive risk premiums.  Consider that assets such as houses have characteristics that require compensation, such as crime, bad schools, or noise, and these factors and their effects are eminently measurable and consistent with intuition (Black 1999).  Risk, meanwhile, has devolved into the financial equivalence of dark matter, evident solely by its effects.  As asset pricing models have increased in complexity from the simple one-factor CAPM, to “stochastic discount factor(s) … so general, they place almost no restrictions on financial data” (Campbell 2002).  Explaining asset returns via risk is often more calibration than prediction, as when the risk premiums are functions of atheoretically observed risk factors (see Dai and Singleton 2002; Fama and French 1992).

This paper argues a more radical but much simpler solution: There is generally no empirical risk-reward relation, and that the seemingly obvious examples are exceptions to the general rule, explainable as liquidity premiums and measurement error.  I present a model that explains the null risk-return result as an equilibrium when people internalize risky decisions by comparing themselves to others, as opposed to the standard approach to risk based on the absolute volatility of their wealth.  If utility is a status function, specifically the value of wealth relative to one’s peers, only deviations from the consensus are ‘risky’ (i.e., benchmark risk, see Cremers and Petajisto, 2006).  ‘Risk’ can be avoided in this context by everyone holding an identical market portfolio, making it similar to diversifiable risk in a world of absolute returns, avoidable, and so unpriced.  Investors see too little exposure to an asset class is just as bad as too much; one is cowardly and the other reckless, objectively measured by how far these bets take one from the average.

 The main empirical implication of this model is that all assets have the same expected return, and this is a first approximation to expected returns than assuming a positive risk premium exists.  The decline in expected return for the highest volatility assets is rational if these assets are especially subject to overconfidence or have value in signaling; such outside-the-box benefits generates lower-than-average returns in equilibrium.

The value of a model is its ability to explain not merely an outstanding problem such as the well-known empirical failure of beta, but seemingly diverse phenomena.  One nonobvious overidentification is the Easterlin Paradox (Easterlin 1974), where within a country, richer people report higher subjective well-being than poorer people do on average, whereas a comparison between countries or across time reveals only a minor relation between income levels and subjective well-being. Anthropologist Donald Brown (1991) notes a concern for relative status is a ‘human universal’ documented in every known human culture (H.L.Mencken defined wealthy as “any income that is at least one hundred dollars more a year than the income of one's wife's sister's husband.”).  An increasing number of economists have analyzed the association between income and subjective measures of well-being and have concluded that the positive effects of extra income on quality of life are relatively small, whereas relative income effects are significant (e.g., Easterbrook, 2003, Frey and Stutzer 2004; Clark 2003; Blanchflower and Oswald 2004).   Indeed, several researchers have argued that we should be more concerned about inequality than absolute wealth in advanced economies (Bradbury and Katz 2002, Frank 2000, Schwartz 2003).  In a relative utility framework, this makes sense, in contrast to traditional utility functions.

There are other extant puzzles consistent with this approach.  For example, the home bias in portfolio holdings (Coval and Moskowitz 1999) is a rational equilibrium when people are trying to stay even with their peer group, which is mainly their countrymen, not the world's population.  Risk seeking in this context—deviations from the consensus—is shown to necessarily involve expectations of higher absolute returns, and so we should always see higher expected return motivating decisions to deviate from the ‘market’ portfolio, as has been documented by the fact that all brokerage recommendations to buy have higher-than-average stock returns.  If people were standard Markowitzian maximizers a significant fraction, if not half, of attractive stocks would have below average returns but even lower risk; instead, we have none. 

There are several qualitative issues that are better understood through a relative risk utility function.  Shefrin and Statman (1984) argue that if a stock pick has gone down, one regrets the investment, and in hoping that the stock price will rise in the next period and thereby avoid regret, holds the stock, illustrating the disposition effect of individual investors being more hesitant to trade out of losers than winners.  In a relative risk environment, risk taking is not a purely subjective choice along a Security Market Line of fair bets, but rather a choice that can only reflect the belief that one knows better than the market, and so failure reflects on one’s judgment and thus one’s sense of self-worth.  In another vein, Caplan (2007) notes that most voters prefer anti-competitive policies that are seen by economists as irrational because they have counterproductive aggregate productivity implications.  Yet if people care mainly about their status, the negative externalities affect only their absolute position, which is not nearly as important as their relative position, making these beliefs rational.

While a relative-status utility function is not standard in finance, there is considerable piecemeal precedence in the academic literature.  Popular utility models with context-dependent reference points include the hedonic treadmill of Brickman and Campbell (1971), the habit formation model of Constantinides (1990), and the ‘keeping up with the Joneses’ utility function of Abel (1990) and Gali (1994).  Economists have shown an incorporation of a concern for social context alters results of traditional growth models (Cole et al 1992), public goods (Frank 1997), and repeated interactions (Sobel 2005).  DeMarzo, Kaniel, and Kremer (2004) show that positional goods can cause investment in risky assets even if they have a zero risk premium.  Some of these models are discussed below.  Academics seem to like the idea in piecemeal applications applied to assets; I simply think these applications are too convoluted and ambiguous. 

Why should people care so much about status?  As humans are quintessentially social animals, much of the human brain is dedicated to processing social information, including in-group status.  As per capita economic growth prior to 1750 was essentially zero, anything hard-wired by evolution was efficient in the zero-sum world that existed for most of human history.  Biologists Insel and Fernald (2004) argue that because information about social status is essential for reproduction and survival, specialized neural mechanisms have evolved to process social information, making status orientation hard-wired into our brains as a consequence of evolutionary selection (mates are the ultimate status good).  Pesendorfer (1995) finds a status-oriented utility function can be an efficient signaling device, because status goods signal ability.  Rayo and Becker (2007) model a scenario in which a concern for status is evolutionarily efficient because it allows one to evaluate the value of decisions better in a stochastic environment.  Thus unlike the context dependent utility functions mentioned above, the relative wealth approach is more than merely intuitive; it is consistent with deeper laws of evolutionarily stable strategies (see Smith 1982).  

Though the bulk of a shock in income comes from the relative social position it conveys (Easterlin 1995; Oswald 1997; Frank 2004), people are cognizant of both their relative and absolute wealth in making decisions, and surely weight these objectives differently in different contexts.  Different formulations of relative wealth — ratios as opposed to differences, relative to the median as opposed to the arithmetic mean, etc. — yield similar but slightly different results than those presented here.  Heterogeneity in wealth and risk aversion creates myriad complications outside of the scope of this paper.  While reality is much richer than that presented here, the models presented are sufficient to capture the essence of what drives the traditional risk-return relation, and show that a relative comparison eliminates this implication. Empirically,  as an approximation, the relative risk model dominates the absolute risk model.

Although most financial researchers consider the search for risk productive work in progress, consider physical beauty, a characteristic similar to risk in that people intuitively rank it. Beauty is based on subjective preferences, intuitively more subjective than risk. Yet empirically, beauty is much easier to define than risk. Most everyone can agree that certain individuals such as that Jessica Alba and George Clooney are physically attractive. Indeed, one can mathematize certain metrics of subjective beauty that are consistent across cultures such as waist-to-hip ratios, facial symmetry, and other correlates with virility. The key is that one can mathematize subjective preferences and correlate them to reality when they exist.

In contrast, if Forbes had to put a list of 10 risky companies on its cover, it would not be obvious at all, because volatility and beta are actually inversely correlated with returns when you control for size. What does one choose? Value companies? Value companies are not risky in any obvious way, relative to growth companies, which, at high P/E multiples and higher volatility fit the more intuitive description of risky. Distressed companies? In fact, distressed companies have poor returns, so in the risk-return framework such deathbed companies have some strange low-risk characteristic. Some assets, like short term debt or AAA bonds, have very low returns and are intuitively not risky: low volatility, correlation with the market, the business cycle, yet these characteristics do not extrapolate along the yield curve or within corporate ratings. In ranking things from highest to lowest, why should identifying risk be infinitely more difficult than identifying beauty? One thing finance never mentioned in the early advanced textbooks on asset pricing, because they never expected this problem, is that risk as a practical matter is insanely subtle.

While utility functions are too abstruse for most practitioners, benchmarking is something they are very familiar with, and basically embodies the same idea.  Bill Sharpe consults for pension funds evaluating asset managers and states his first objective in is that ‘I want a product to be defined relative to a benchmark’ (Tanous 1997). Fund Manager Kenneth Fisher‘s book Only Three Questions that Count, in the index next to Risk, it has ’see  Benchmarking.’ When asked about the nature of risk in small stocks, Eugene Fama noted that in the 1980‘s, “small stocks were in a depression”, and Merton Miller noted the underperformance of the Dimensional Fund Advisors small-cap portfolio against the S&P500 for 6 years in a row was evidence of its risk (Tanous 1997). But smaller stocks actually had comparable total returns, and higher returns relative to the risk-free rate, in the 1980‘s compared to the 1970‘s.  It was only relative to their benchmark (the S&P500, large cap stocks) that they had ‘poor‘ returns’ highlighting that even Fama and Miller’s practical intuition on risk is purely relative, and these are champions of the standard model.  It seems reasonable to presume that for these investment professionals and academics, risk, intuitively, is a return relative to a benchmark.  If all investors act as if they are benchmarking to aggregate indices, risk will not be priced in equilibrium.

In section 1, I outline the creation of the risk premium over the past 100 years, why the stochastic discount factor appears to imply a positive risk premium.  In section 2 I outline the history of the empirical testing of this theory, from the beginnings of the CAPM through the APT theory.  In section 3, I outline the relative status (aka benchmarking or envy ) utility, and why it implies a zero risk premium.  Lastly, in section 4 I outline the empirical failure of this theory.  Versions of this are available at SSRN here. Section 5 concludes.

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