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Standard Risk Models Create a Snipe Hunt, Better to Ignore the Risk Premium as in Derivatives Pricing

Assume you were the designer of a species of conscious agents: God, the program developer of self-motivated avatars in a computer world, or anthropomorphize natural selection. The objective you face is to give these agents a utility function such that they are motivated to create buildings, art, and of course children by themselves based on some instinct. So, as a designer you can add a mechanism so that people feel hungry if famished, and lustful when in the presence of mating opportunities, so they survive over generations. Yet, each of these desires has a clear governor that has a 'high' and 'low' setting, when you feel full or empty: you don't want people eating or having sex so much they ignore everything else, such as getting ready for eating and having sex tomorrow. Now consider the governor that signals the will to want more 'stuff'. The person one creates must have a specific function if utility is increasing at a decreasing rate in 'stuff', as there are an infinite number of functions, and if it is off just a little bit, it can lead to disaster as the economy grows or shrinks. That is, say people developed a utility function x^½, which is satisfies our basic intuition that self interest is both increasing in wealth, but at a decreasing rate. Back in 1800 it worked pretty well. But now we are 10 times wealthier, so we should have much lower risk aversion if our utility were not of that very specific formulation, , such that risk means roughly the same thing in Babylonian times as it does now. If risk aversion today is correct, then back then you were afraid to look at you shadow if you people's utility is like x½, and interest rates would not have been 15% or so as we know from Medieval times, but rather, 100%. Our DNA needs a knife-edged functional conception of happiness that seems patently absurd. One could always invoke the anthropic principle and leave it at that (i.e., if it were not so, we wouldn't have enough wealth to be discussing this on the internet).

In contrast, it seems more likely our little governor simply said: be above average to your peers. As humans always lived in societies with others, benchmarks are not lacking, so this is a very feasible goal. Then, things take care of themselves. People are constantly doing more each generation because even if you are on top you have to run fast just to stay in the same place. Desire, striving, want, has no abstruse functional knife-edge, but a more reasonable feedback mechanism that does not lead to, or start with, an absurdity.

The idea that risk is priced is highly intuitive, and implied by globally concave utility functions, which are also highly intuitive as a general rule.  Yet the empirical support for a priced risk premium is absent across a large spectrum of investable assets (this excludes catastrophic insurance premium which are outside this analysis).  The empirical anomalies to the high risk-high return hypothesis are not exceptions to a general tendency: There is no general tendency within a variety of investments, such as equities, individual equity options, most of the yield curve, high yield corporate and bankrupt bonds, mutual funds, commodities, small business owners, movies, lottery tickets, and horse races.  Indeed, it seems many very high risk assets contain some extra attributeIt is possible the equity premium is actually zero after suitable adjustments for transaction costs and selection biases, and the other two prominent examples of a positive risk-return tradeoff — the short end of the yield curve, the spread on investment grade bonds — do not generalize when extended in the ‘risk’ spectrum for these asset classes.   

As Mark Rubinstein says about the CAPM and its extensions:

More empirical effort may have been put into testing the CAPM equation than any other result in finance. The results are quite mixed and in many ways discouraging. . . . At bottom . . . the central message of the CAPM is this: Ceteris paribus, the prices of securities should be higher (or lower) to the extent their payoffs are slanted toward states in which aggregate consumption or aggregate wealth is low (or high). . . . The true pricing equation may not take the exact form of the CAPM, but the enduring belief of many financial economists is that, whatever form it takes, it will at least embody this principle (Rubinstein (2006)).

The current consensus is that evidence for any robust risk factor is weak at best, yet anticipates a refinement subtle enough to escape notice for 40 years, yet still strong enough to explain these data, implying investors currently recognize and price this undiscovered metric. 

Diether et al (2002), in their paper on analyst uncertainty and returns, note higher estimate dispersion is positively related to Beta, volatility, earnings variability, yet, because the returns go the wrong way (lower return for higher volatility), they note “our results clearly reject the notion that dispersion in forecasts can be viewed as a proxy of risk”.  Thus, in spite of being correlated with all things intuitively risky, like beta, volatility, and size, but uncorrelated with value or momentum, the correlation with returns suggests to these researchers that analyst uncertainty cannot be correlated with risk, because the one thing they know about risk, is that it is positively correlated with returns.

The belief that risk, properly measured, must be positively related to returns is very deep among academics.  Risk is supposedly not only important and everywhere, but subtle, requiring that investors implicitly have skills similar in sophistication and imprecision to what is needed to distinguish between a good and great wine.  As an alternative, the irrelevance of risk to return is implied by a status-conscious investor benchmarking himself against others, and holds in both a utility and arbitrage argument.  Risk is simply allocating an ‘unusual’ amount of wealth to any asset that would generate a significant deviation from the market portfolio. 

The implications of this approach are profound.  We should expect never to see a robust metric of something positively correlated with our wealth’s volatility that is positively related to average returns.  One should treat expected returns the way derivative pricing does, as some trivial correspondence to a labor curve.  As most government pensions in the US currently have significant equity risk premiums built in to their expectations, we should expect many will experience defaults and repudiations as these deviate more and more from their targeted, expected nominal values. 

The standard assumption that relevant volatility is absolute wealth may be a good normative theory, but a relative wealth orientation generates a more accurate positive theory, and its assumption is generally considered more accurate by those doing research on the essence of subjective well-being.  That is, happiness seems unaffected by large increases in wealth across developed countries, and over time in these countries.  A relative status orientation can be modeled as a rational objective function when signaling, and uncertainty, are important (see Rayo and Becker 2007, or Pesendorfer 1995).  The relative status utility function generates a more accurate description of the world. 

Consider the following risk and return chart taken from Campbell Harvey’s website


Figure 5.1

A Common Insinuation

 

The implication is that if you take more risk, as reflected by the standard deviation of annualized returns, you will be rewarded with a higher return on average, which is the quid pro quo.  Many investment textbooks show the returns and standard deviations of Treasury bills, bonds, equities, and small cap stocks, as if the mean-volatility correlation in these asset classes generalizes the law at work (see Brealey, Meyers, and Allen 2003, or Malkiel 2003).  As no serious researcher suggests that mere volatility is correlated with expected returns any more, this conflation of risk and standard deviation seems a convenient error, one that suggests to the uninitiated that they understand returns mathematically when in fact there is no simple correlation that extrapolates.


I submit the following is much more appropriate.

Figure 5.2

Risk and Return in Practice

One can generate a risk premium moving from 3 month to 3 year Treasuries, and then to Baa bonds.  After that, it gets perverse.  The highest returning stocks have the lowest volatility, and from thereon it is all downhill, higher ‘risk’ in any intuitive sense is associated with lower returns. 

A relative-status oriented utility function generates a factor model consistent with the familiar CAPM and APT models, except the risk premiums are zero.  Beta is still descriptive of relative volatility, and generates normative predictions for volatility minimization.  However, there is no robust cross-sectional return to any b, no upward sloping security market line.  The portfolio optimization algorithm for an investor with typical preferences is trivial and mimics practice: allocate assets to the standard categories of conventional wisdom, because this minimizes relative wealth volatility and maximizes returns.  The idea that nondiversifiable risk is not priced because it is unnecessary is augmented, so that diversifiable risk is also not priced because it too is unnecessary in the sense of avoiding deviations from the average portfolio.  As the saying goes, one has to take risk to generate high returns, but there is no greater expected return merely for taking risk of any sort.

 

 

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Eric Falkenstein 9/10