Are there ways to measure the risk aversion of a representative investor, based on publicly available market data? Public available data could include asset price, volume, and flow data, and may be aggregated according to some criterion (asset class and contract type being the most obvious). Pointers to any existing literature are greatly appreciated.
By definition, the average investor holds the market portfolio. Risk aversion can be measured as the slope (i.e. ratio of expected returns to volatility) on the efficient frontier. Therefore, the risk aversion of the average investor assuming the S&P500 is the proxy for the market portfolio is the expected returns of the S&P 500 divided by the volatility of the portfolio.
Suppose that you have a model of returns, and a representative agent whose form of utility function you have specified right off the bat. This RA can be constructed, under conditions, from a population that is defined to have heterogeneous utility objectives. This is the problem of aggregation, and it's treated in every good asset pricing theory text (e.g. Skiadas' Asset Pricing Theory or Duffie's Dynamic Asset Pricing Theory). @Dirk is unfortunately wrong: virtually all financial models (even B&S!) assume an RA; and the CAPM and market portfolio have nothing to do with the existence or not, in theory or in "reality", of an RA in asset pricing.
If you want to estimate or otherwise assess risk aversion, you have to have such a model explicitly spelled out. Otherwise you will have to estimate a distribution of risk aversion coefficients, that is virtually impossible - unless you have data on the initial positions of every player in the market! You also need a measure of consumption growth - as creative as possible.
Now @Tal Fishman's answer is accurate. Lucas's (1978) "Asset Prices in an Exchange Economy" model, when applied to equity premium data predicts a coefficient of risk aversion for the utility function of the RA that is exorbitantly high - and hence "unreasonable". This is the equity premium puzzle -- there is also a dual puzzle in the relevant literature, called "risk-free puzzle".
Economists have tried to resolve it. Currently, the consensus in the literature seems to accumulate on the solution proposed by Rietz, and put forward by Barro and his student, Ian Martin in a series of very interesting papers. In sum, a model that is consistent with market anticipation of "rare disasters" in consumption growth can produce reasonable (i.e. low to the point of being manageable) coefficients of risk aversion in such an RA asset pricing model.
This is the canonical theory. A Taleb kind of approach, commonly known as "decision theoretic" (cf.@richardh's comment above) can extend standard expected utility constructions in theoretical asset pricing into having different forms of utility objectives, probability assessments of future events, or what have you. Doing empirics with these models is very often impossible.
As a practical solution to all this, one would suggest to estimate the changes in risk aversion of whatever model you predefine, and not its absolute level. But you certainly need an RA model, whose use is put to work every time you compute the beta of a stock or a market portfolio.
There are multiple components to this but in a very general sense one has to look at two things: The "credit" spread and the "cost of insurance" The former tells you how much excess return the representative investor is demanding to be compensated for the risk of counterparty default. The later tells you how much an investor is willing to pay to defease a risk over a given interval. Insurance can come in the form of credit default swaps, option premium or in some cases the contango.
Answering this question is impossible without making many significant assumptions regarding risk preferences, utility functions, a model of returns, etc.
One way to measure the risk aversion of the representative investor is to compare the market's expected return to expected volatility. When applied to the allocation decision between equity and fixed income, this is known as the equity risk premium. The equity premium puzzle, first pointed out by Mehra and Prescott (1985), holds that this difference is too large to be explained by most reasonable models of investor risk preference.
Damodaran details a number of ways in which equity risk premium is commonly measured. For example, combining the implied equity risk premium with implied volatility from the VIX, one can directly obtain a measure of the market's current risk aversion.
Cochrane (2011) discusses the variation among and commonality between risk premia in different asset classes.
Going out on another limb here: No.
First off, there is no representative investor. If we were all the same, the CAPM and the market portfolio would hold. So that part is unobservable.
Second, that is why the sell-side strategists are to happy to create things like misery indices (bad example as it's more macro-based in its sum of inflation plus unemployment), or vix plus credit spreads, or things like AAII sentiment, or speculators/hedgers rations, or put/call ratios, or ... You get the drift.
All that said, I'd love to be proven wrong and see estimates of one such beast.
I second @DirkEddelbuettel here. Measuring aggregate risk aversion implies abstracting away and making an explicitly artificial theoretical construct. There does not 'exist' a representative investor. What we have are investors, traders, with varying risk aversions, utility functions with differing parameters (if not functional forms). The distribution of risk aversion coefficients which @Dimitris says is virtually impossible, is obviously closer to reality, complex yes, but more plausible. Masanao Aoki's work is on lines such as these [ econ.ucla.edu/faculty/regular/Aoki.html ], though not directly at the level of specificity in finance.
Even as far as estimating the utility function of one investor is concerned, that is non-trivial, even as from a more recent text on the field [ http://amzn.com/0521748682 ]. I am not sure how coming up with a summary statistic for a few million investors is any easier.
Also, it might so happen that there are periods when a greater number of investors converge in their utility functions, and at other times, there is greater dispersion. i.e. if there is a distribution of the utility function parameters over the set of investors, the moments are constantly fluctuating in a non-trivial way. Also, the existence of a representative investor, I think, implies that we are concerned only with inter-temporal decision making, and hence investing/trading - whereas, the reality of trading is a mix of inter-temporal decision making coupled with heterogeneity of investor/traders.