NBER WORKING PAPER SERIESTHE RECOVERY THEOREMStephen A. RossWorking Paper 17323http://www.nber.org/papers/w17323NATIONAL BUREAU OF ECONOMIC RESEARCH1050 Massachusetts AvenueCambridge, MA 02138August 2011The author is a Managing Partner of Ross Farrar, LLC, a Registered Investment Adviser that managesportfolios of derivative securities including options. The data used in the paper were obtained undera non-disclosure agreement with a financial firm. The views expressed herein are those of the authorand do not necessarily reflect the views of the National Bureau of Economic Research.© 2011 by Stephen A. Ross. All rights reserved. Short sections of text, not to exceed two paragraphs,may be quoted without explicit permission provided that full credit, including © notice, is given tothe source.The Recovery TheoremStephen A. RossNBER Working Paper No. 17323August 2011JEL No. E1,G0,G11,G12,G17ABSTRACTWe can only estimate the distribution of stock returns but we observe the distribution of risk neutralstate prices. Risk neutral state prices are the product of risk aversion – the pricing kernel – and thenatural probability distribution. The Recovery Theorem enables us to separate these and to determinethe market’s forecast of returns and the market’s risk aversion from state prices alone. Among otherthings, this allows us to determine the pricing kernel, the market risk premium, the probability of acatastrophe, and to construct model free tests of the efficient market hypothesis.Stephen A. RossMIT Sloan School of Management100 Main Street, E62-616Cambridge, MA 02142and [email protected] Because financial markets price securities with payoffs extending out in time the hope that they can be used to forecast the future has long fascinated both scholars and practitioners. Nowhere has this been more apparent than with the studies of the term structure of interest rates with its enormous literature devoted to examining the predictive content of forward rates. But with the exception of foreign exchange and some futures markets, a similar line of research has not developed in other markets and, most notably, not in the equity markets. While we have a rich market in equity options and a well-developed theory of how to use their prices to extract the martingale risk neutral probabilities (see Cox and Ross (1976a, 1976b)), there has been a theoretical hurdle to using them to make forecasts or, for that matter, to speak to issues in the natural world. Risk neutral returns are natural returns that have been ‘risk adjusted’. In the risk neutral measure the expected return on all assets is the risk free rate because the risk neutral measure is the natural measure with the risk premium subtracted out. The risk premium is a function both of risk and of the market’s risk aversion, and to use risk neutral prices to inform about real or natural probabilities we have to know the risk adjustment so we can add it back in. In models with a representative agent this is equivalent to knowing that agent’s utility function and that is not directly observable. Instead, we infer it from fitting or ‘calibrating’ market models. Furthermore, efforts to empirically measure the aversion to risk have led to more controversy than consensus. For example, measurements of the coefficient of aggregate risk aversion range from 2 or 3 to 500 depending on the model. The data are less helpful than we would like because we have a lengthy history in which U.S. stock returns seemed to have consistently outperformed fixed income returns – the equity premium puzzle (Prescott and Mehra [1985])– and that has given rise to a host of suspect proscriptions for the unwary investor. These conundrums have led some to propose that finance has its equivalent to the dark matter cosmologists posit to explain the behavior of their models for the universe when observables don’t seem to be sufficient. Our dark matter is the very low probability of a catastrophic event and the impact that changes in that perceived probability can have on asset prices (see, e.g., Barro [2006] and Weitzmann [2007]). Apparently, though, such events are not all that remote and five sigma events seem to occur with a frequency that belies their supposed low probability. When we extract the risk neutral probabilities of such events from the prices of options on the S&P 500, we find the risk neutral probability of, for example, a 25% drop in a month, to be higher than the probability calculated from observed stock returns. But, since the risk neutral probabilities are the natural probabilities adjusted for the risk premium, either the market forecasts a higher probability of a stock decline than occurred historically or the market requires a very high risk premium to insure against a decline. Without knowing which, it is impossible to separate the two out and find the market’s forecast of the event probability.3Finding the market’s forecast for returns is important for other reasons as well. The natural expected return of a strategy depends on the risk premium for that strategy and, consequently, it has long been argued that any tests of efficient market hypotheses are simultaneously, tests of a particular asset pricing model and of the efficient market hypothesis (Fama [1970]). But if we knew the kernel we could estimate how variable the risk premium is (see Ross [2005]), and a bound on the variability of the kernel would limit how predictable a model for returns could be and still not violate efficient markets. In other words, it would provide a model free test of the efficient markets hypothesis. A related issue is the inability to find the current market forecast of the expected return on equities. Unable to read this off of prices as we do with forward rates, we are left to using historical returns and resorting to opinion polls of economists and investors - asking them to reveal their estimated risk premiums. It certainly doesn’t seem that we can derive the risk premium directly from option prices because by pricing one asset – the derivative – in terms of another, the underlying, the elusive risk premium doesn’t appear in the resulting formula. But, in fact, all is not quite so hopeless. While quite different, our results are in the spirit of Dybvig and Rogers [1997], who showed that if stock returns follow a recombining tree (or diffusion) then from observing an agent’s