# Does entropy pooling apply to distributions with time-varying drift?

I have a returns process that is drawn from a normal distribution with a nonlinear time-varying drift, so I was wondering if the entropy pooling method still applies or if I need an invariant ?

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So long as it is possible to simulate the distribution of log returns to the investor's horizon and convert them to prices that serve as an input into the portfolio optimization, it is possible to apply the general Entropy Pooling algorithm to either the log returns or the prices. Whatever care needs to be taken when applying the EP algorithm in its traditional use, should be applied here as well (e.g., the numerical procedure does not work well when you take extreme views and have not simulated the extreme parts of the distribution).

It can also be applied to the slightly more complicated case of multiple period optimization (e.g., if you want to account for the fact that in the short-run you might be in a bad regime but in the long-run it will go to the steady state). I have not seen it in the literature, but I have set up EP problems that account for this by treating the distribution at every horizon as one distribution and then applying the EP algorithm. Whatever time dependence that results from the simulation should be accounted for in this fashion.

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In this case it's possible to simulate the distribution of log returns if I fix my investment horizon, then my drift becomes just a constant at the projected horizon. I actually have the exact case that you describe, where I have an unsustainable regime that will change at some critical point (incorporated in the model) and then reverts back to a steady state. I am curious as how you would set up a dynamic framework, I'm not sure I understand this part correctly: "treating the distribution at every horizon as one distribution", it's not the same distribution at every horizon right ? – Bytesize Nov 27 '12 at 11:29
You normally would simulate to $\widetilde{X}_{t+k}$ and apply EP to that. I'm saying simulate $\widetilde{X}_{t+1},\ldots,\widetilde{X}_{t+k}$ and concatenate them into one matrix $\widetilde{Y}\equiv\left[\begin{array}{ccc} \widetilde{X}_{t+1} & \cdots & \widetilde{X}_{t+k}\end{array}\right]$ and treat that as though it is one distribution. – John Nov 27 '12 at 14:34

I can't get access to the full version to Meucci's original paper on Entropy Pooling (EP), Fully Flexible Views: Theory and Practice, and I hence had a look again at the abstract:

We propose a unified methodology to input non-linear views from any number of users in fully general non-normal markets, and perform, among others, stress-testing, scenario analysis, and ranking allocation. We walk the reader through the theory and we detail an extremely efficient algorithm to easily implement this methodology under fully general assumptions. As it turns out, no repricing is ever necessary, hence the methodology can be readily applied to books with complex derivatives. We also present an analytical solution, useful for benchmarking, which per se generalizes notable previous results. Code illustrating this methodology in practice is available through author's homepage.

This confirmed my initial thought that one of the great advantage of EP is that the approach is very general and can adapt to various models.

So I think the answer is yes, I believe you can use EP even with nonlinear time-varying drift.

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I also think it's possible, however it didn't show clearly in his paper that his model parameters (normal or fat-tailed) could be time-varying. I can send you the paper if you want to have a look at it. – Bytesize Nov 27 '12 at 11:24

Meucci's original paper doesn't state any limitations on prior distribution to which Entropy Pooling (EP) is applied. However, I see two possible issues.

The first problem is that it currently seems to be no place for incorporating time changing parameters or views on a prior distribution in the EP. Therefore, some additional work is required to apply it in your case (although it doesn't look not very complicated).

Assuming that your goal is to include complex views in order to use the posterior distribution in portfolio optimization, another potential problem is that you most probably want to have a market invariant (as per my understanding it's still the cornerstone of the (strategic) asset allocation theory)

The invariants are market variables that can be modeled as the realization of a set of independent and identically distributed random variables at least over the investment horizon. For example, equity invariants are compounded returns, ﬁxed-income invariants are changes in yield to maturity (for a detailed treatment, see the book "Risk and Asset Allocation" by Meucci, chapter 3).

Nonlinear time-varying drift definitely can violate the assumption of identical distribution, therefore making the asset allocation under such distribution meaningless (unless you constantly rebalance your portfolio to make it optimal under the updated drift).

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I had the same concern that we need a market invariant for asset allocation, but from what I see in the Meucci paper "The Prayer", invariance is needed to project the invariant distribution to the investment horizon. In my case however, I already have the projection and the pricing step at the investment horizon with my time-varying drift. Provided I fix my investment horizon, my drift becomes known. I don't think the quest for invariance should be something to always strive for, since there is large evidence that bubble regimes, for example, typically break the invariance. – Bytesize Nov 27 '12 at 11:20
@Bytesize Yes, I agree with you. If your drift is known in your investment horizon, then you are fine. But you still need to adapt the method to account for the time dimension. – Alexey Kalmykov Nov 27 '12 at 11:28