# What are the main flaws behind Ross Recovery Theorem?

Stephen Ross’ new paper claims that it is possible to separate risk aversions and historical probabilities if the Stochastic Discount Factor is transition independent using Perron-Frobenius Theorem. Carr and Yu have extended the model to a preference free setting with bounded stochastic processes. But a recent paper by Hansen, Borovicka and Scheinkman seems to show that the approach by Ross is misspecified.

Can you explain why Ross’ recovery is misspecified?

This is a loaded question. Ross' recovery theorem has both flaws and insights. The single answer thus far did a great job of addressing the flaws from an economics perspective. No one questions that the math is wrong: it is correct. Here is a mathematical insight from Ross' work. Abstracting from the finance and economics, the purely probabilistic content of Ross' Recovery theorem is the following. Suppose we are given a positive measure that describes the product of transition frequencies of a univariate Markov process and a positive multiplicative functional $m_t = \exp(-\int_0^t g_u du)$, where $g_u$ is some real-valued stochastic process at time $u$. The Ross recovery theorem answers two questions posed together: Question 1) Under what additional probabilistic structure, can we remove the multiplicative functional and thereby obtain transition frequencies, which need not be the same as the transition frequencies multiplying the functional? Question 2) What additional probabilistic structure to Question 1 is needed to uniquely obtain these possibly different transition frequencies?

Ross' answer to both questions posed simultaneously is that the sufficient conditions are that the univariate Markov process is an irreducible time-homogeneous finite-state Markov Chain and that the measure change be transition independent. From a purely probabilistic perspective, Hansen Scheinkman 2009 answer question 1. Not surprisingly, they can do so with weaker sufficient conditions, requiring neither finite states or transition independence. Both papers supply mathematical insights into how to change from a positive measure eg Arrow Debreu security prices to a probability measure eg forward measure or market beliefs or a third probability measure. From a purely probabilistic standpoint, Ross incremental contribution is to answer question 2. To get the stronger conclusion of uniqueness, he makes stronger assumptions. I don't personally consider this a flaw. In fact, identifying the additional structure that leads to the stronger conclusion is usually considered by mathematicians to be a mathematical insight. Some economists argue in contrast that this additional structure is a weakness because 1) it is less general, and rules out some well known successful models, and 2) the particular application Ross suggested ends up being empirically refuted.

So to summarize, I believe that the two incremental assumptions Ross made in answering Question 2 can simultaneously be interpreted as both a mathematical insight and an economics flaw.

• Thank you very much for the answer. It was quite illuminating! But of course, that should be expected by one of the best scholars in his field! – fni Nov 2 '17 at 16:11

Very simply, Ross' framework assumes a great deal to extract the true pricing kernel. Time homogeneity, additively separable state dependent utility, (discrete time Markovian structure - though these have been relaxed.) In particular, there are two schools of criticism, one is that time homogeneity makes little sense in the real market. In fact, the Recovery Theorem would interpret a declining term structure of volatility as a form of mean reversion - if you want to test this calibrate a time homogeneous process to one with time dependent and decreasing volatility.

The other school of criticism is that the utility function assumed by Ross is transition independent, namely if the preferences of the market were correctly aggregated (he assumes the existence of a representative agent) then they would only depend on the final state, the initial state and the path would not enter (except as trivial normalizations.) This is the key to HBS (2014)

Furthermore, Ross' result is a direct consequence of a result by Hansen-Scheinkman which shows that under much more relaxed conditions (non additively separable utility, stochastic volatility) the stochastic discount factor has Ross' structure under a Long Run measure. Which means if Ross' assumptions dont hold (but time homogeneity does) then Ross' procedure actually extracts the long-run measure of HS (upto a multiplicative order).