Some Thoughts on Risk/Return Tradeoff and EMH

So this is partly a response to a question asked by Cullen Roche on Twitter:

@dvegadtime @financequant Q: What's your preferred measure of risk adjusted return? Sharpe, SDR Sharpe or Sortino? Thanks.
— Cullen Roche (@cullenroche) October 19, 2013

But I'm also going to give some initial reservations over the Efficient Markets Hypothesis (EMH). Because in my view, it's not that EMH is false; rather, EMH is just not meaningfully defined in a way that can be tested. I would further suggest that EMH can't be meaningfully defined.

The basic thrust of EMH is that market prices "reflect" (you have to love mirror metaphors) available information. But it's not just that. There's a presumed correct way to reflect information. This is based on the idea that there is a tradeoff between risk and reward.

In general, for EMH to be valid, there must exist (even if unknown to us humans) a function $f(r_1, r_2, . . ., r_n)$ such that if $r_1, r_2, . . ., r_n$ are risk factors then $f$ maps to a unique expected return (or risk premium relative to some "risk-free" rate).

The question I want to ask is this: does such a function exist?

I want to make this perfectly clear. I am not asking whether or not we have such a function (we don't) or whether or not we could create models for such a function. Rather, I'm interested in the question of whether such a thing even exists? Or what reasons do we have for believing such a thing exists?

The reason why EMH requires such a function is that there are no doubt strategies which "outperform the market". EMH supporters often suppose that the market is efficient and the proceed to claim that such strategies must be riskier.

One thing to keep in the back of your mind here is that there is something distinctly normative about this. If I prefer a strategy that is considered "riskier" than another strategy, I'm doing something wrong.

The idea of a tradeoff between risk and return is loosely based upon the idea of risk aversion as indicated by research in psychology. People prefer strategies which are more certain than those that are less certain. To give an example of an informal study (one you can participate in), consider the following two assets:




In any event, I'd like to consider a few questions which, I hope, will leave you skeptical upon the alleged existence of a risk-reward function.

1.  Do all humans have to have identical preferences for various assets? For example, if I prefer Asset 1 over Asset 2, are you permitted to prefer Asset 2 over Asset 1?
   
   I ask this because I think there's a normative element here. While there are general statistical trends on things ("the average person prefers Asset 1 over Asset2"), I think many finance people and economists want to say that people ought to prefer Asset 1 over Asset 2. If they deviate from the "average", then they've done something "wrong".
   
2. If two people prefer Asset 1 over Asset 2, must they do so for the same reasons? Is it possible there are different reasons for preferring one asset over another?
   
3. Is each reason for preferring one asset over another a risk factor? Or can some reasons for preferring one asset over another be something other than a risk factor? For example, consider the following reasons why one might prefer one asset over another:
   
   • Liquidity: one asset is "more liquid" than another.
   • Control: sometimes there is talk of a control premium in which one prefers an asset one has greater control over another asset.
   • Information Cost: One might prefer one asset over another due to difficulty in obtaining information
   Should any of these be considered risk factors or something else altogether?
    

So I think the assumption that there exists a unique function that maps risk to returns is suspect. 

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