The mean-variance model for portfolio optimization minimizes portfolio risk (covariance matrix), which is the second statistical moment of multivariate asset returns, and sometimes simultaneously maximizes portfolio return, which is the first statistical moment of asset returns.

Is it possible to do asset allocation without any consideration to the moments of asset returns whatsoever? If so, what else is there to use? Do any theories and techniques exist that optimize investment portfolios without estimation of asset return moments? Do non-moment based portfolios perform well?

(please not looking for the well-known heuristic methods either (equal weight, market cap weight, etc)


3 Answers 3


The answer is sort of. I am going to provide you a history of the mathematics so that you will understand why this discussion is challenging to have in economics. Also, you are probably going to have to ground yourself in math you haven’t looked at before.

In 1867, a young man by the name of Jules Regnault became the world’s first quant, publishing a book about it. It was not rigorous and was built around the practice. The next step would by Ysidro Edgeworth at a meeting of the Royal Society. I am working from memory, but I think it was in 1888, but it might be as early as 1882. He links Gauss’s “law of errors,” to the trading of banknotes. He also anticipates game theory in the same discussion. At that point, you are not at the discussion of moments, but you are very close.

Just a little before, and just a little after, two important mathematical debates happen that will impact us at the end of this posting. The first is the rise of the marginalists. That allows the use of calculus in economics. The second is Georg Cantor’s number theory that permits a preference-based grounding for utility theory.

Our next stop is with Bachelier and his doctoral thesis in 1900. A work of brilliance, it is ignored. Because it was unknown, Einstein and Kolmogorov had to reinvent it. He was not studying equity securities but rentes. That will come to matter at the end of this discussion.

We are going to jump over Frank Ramsey, Frank Knight, Bruno de Finetti, Ronald Fisher, Egon Pearson, Jerzy Neyman, and Abraham Wald. That will turn out to be bad judgment, but based on your question, you will need to walk back into their work to get out of the moments conundrum.

We are going to leap forward to 1940 to Hiyoshi Ito and Ruslan Stratonovich. From them, we will jump to Richard Bellman and Harry Markowitz. Ito and Stratonovich, independently, invent stochastic calculus. It is a method built on the assumption that the parameters are known. Bellman and Markowitz pick up slightly separate discussions of this, and two linked paths are created.

Markowitz’s work is not rigorous. You should pick it up. It would now be considered shocking. However, working from the base of knowledge, it shouldn’t be. He has to explain what a mean or a variance is in the article. He also doesn’t know the solution yet. Markowitz’s method is built around estimation. Ito’s approach assumes that you do not need to make an estimation. That fact ends up being important.

What makes Markowitz’s approach interesting is that it does not depend upon utility. It would imply that the utility optimizing arrangement is also statistically optimal. That is a novel thought. The problem would be that it isn’t yet clear what has to be true for this idea to work.

In 1953, John von Neuman wrote a short warning note that the underlying mathematics necessary for modern portfolio theory was yet to be solved and that the proofs in economics may not be actual proofs. Whether it was because nobody read the note, or because they didn’t perceive the weight of it, economists plowed onward.

In 1958, John White, a mathematician working in another area unrelated to finance, proved that models such as the Capital Asset Pricing Model or Black-Scholes have no empirical solution if the parameters are unknown. Nobody noticed his proof. Because this proof undergirds other essential areas, this proof can be treated as valid. For example, if White’s proof is wrong, then you have to toss the unit root tests. We know that they work.

In 1963, Benoit Mandelbrot wrote an article that roughly said, if this is your model, then this data cannot be your data, and this is your data. He then argued that the data couldn’t be drawn from a distribution with a first moment. Eugene Fama takes up and later drops this line of work. I believe he dropped it in error. I almost made the same error, but I had an accidental advantage. I solved the problem using a Bayesian solution first. The discrepancy between the Frequentist and the Bayesian outcome made me understand where economists were tripping up on the math. I tripped in the same places but had a second frame of reference to work with. Fama did not.

Then is a string of articles over the late sixties and early seventies, the classical versions of the CAPM and Black-Scholes were developed. At the same time, Fama and MacBeth did an extensive empirical study falsifying the CAPM. As Black-Scholes can be derived from the CAPM, logically, it was falsified as well. Each of these models is built around parameters that are assumed to be from distributions with a mean, variance, and covariance. Under the strongest forms, the only logical solution is to use some variant on least-squares estimation. The only problem is that it does not work.

The next generation arises out of what is hoped to be a solution to the observation that the relationships seem to be non-stationary. That leads to things such as ARIMA and ARCH/GARCH. It is here that you can begin to get an inkling of the moments problem as it is related to another issue, statistical sufficiency.

If you have worked with time series, then someone somewhere taught you that ordinary least squares was an unbiased estimator, but was not the minimum variance unbiased estimator. That is because it suffers from information loss. Bayesian methods cannot lead to information loss. Bayesian information uses all available information about parameters possible. The Bayesian likelihood function is minimally sufficient. Frequentist slope coefficient estimates are not sufficient for the parameters. As a result, Frequentist methods are lossy methods. However, as White showed, the information loss is total for models like the CAPM or Black-Scholes.

‘Tis a strange field that creates estimates using estimators known not to work. ‘Tis stranger still to have decades of data showing it does not work, yet be unwilling to change the textbooks.

That brings us to your question, now that we have stepped away from the finance textbooks.

Allocation questions depend on predictions of future values. I am going to posit two unrealistically simple asset classes so that the question can be illustrated compactly. The first will be a simple binary lottery. The second will be an equity security that cannot pay dividends over the relevant period or go bankrupt.

There are two predictive choices available, a Frequentist and a Bayesian. It is critically important to understand that the choice made here is dangerous to make incorrectly.

Let us assume that your concern is purely academic. You want an unbiased prediction of other people’s future asset allocations. The Frequentist estimator, when it exists, will ignore moments. That is not obvious. It is important to remember, for the normal distribution, the first moment is $\mu$ and it is not $\hat{\mu}$. It is important to remember that $$\hat{\mu}=\bar{x}=\sum_{i=1}^N\frac{x_i}{N}$$ is independent of $\mu$. That is why sufficiency is important.

To see why, imagine the only way to make a good decision is to know $\mu$, and you do not know it. You are totally dependent on knowledge that you cannot acquire. You need a decision tool that contains all the information available about $\mu$ but does not depend on knowing the true value of $\mu$. That has been the hope and illusion behind portfolio theory. The hope was that by creating estimators, decisions could ignore the requirement in Ito calculus that you know the parameters.

So, asset allocations predictions for non-applied purposes where an unbiased, sufficient predictor exists should use a Frequentist variation on decision theory. The complications come into being in three cases.

First, where an unbiased estimator does not exist, the justification for a Frequentist prediction method becomes rather thin. Second, when a sufficient prediction statistic does not exist, the loss created from using bad predictors can be substantial. The third is where a minimax utility function isn’t needed. Rephrased, is a guarantee of an $\alpha$ percent chance against false positives and the ability to control the level of false negatives relevant?

Now let us assume that your concern is applied and that you have real money to allocate. Then the only choice is to use Bayesian decision theory. Frequentist methods violate de Finetti’s coherence principle and violate the Dutch Book Theorem.

The Dutch Book Theorem arises from a weakened version of the no-arbitrage assumption. It is possible to rig the market against an asset allocator that is using Frequentist statistics. If the entire field is using Frequentist estimators, then it is possible to collect free money from the system. I wrote an article on it at https://www.datasciencecentral.com/profiles/blogs/tool-induced-arbitrage-opportunities-also-how-to-cut-cakes.

On the Bayesian side, the moments vanish as well, as they must as the Bayesian prediction is always sufficient.

That brings up a different host of problems. The Bayesian posterior predictive distribution minimizes the Kullback-Leibler Divergence. That is to say, it is impossible to create a prediction that is closer to nature on a systematic basis. That will not help you, though. The predictive distribution is precisely that, a distribution of asset allocation predictions. You need point statistics, not an infinite number of choices.

Bayesian methods separate inferences from decisions. You need to decide on how much to allocate. That solution comes from imposing a utility function onto the posterior predictive distribution. You need to determine the type of losses that you would take from overestimating or underestimating a parameter.

I wrote an article on one possible solution to this case. It can be found attached to the blog post at https://www.datasciencecentral.com/profiles/blogs/a-generalized-stochastic-calculus.

Now that brings us to the binary case and the equity case. In the binary case, the moments are well defined but unnecessary to know. Imagine that you were going to be a bookie setting odds on a set of $n$ binary events. You have seen $\alpha$ successes and $\beta$ failures in the past. You will be setting payouts on a set of potential future counts of future successful outcomes, $K=\{k_1,\dots{k_n}\}$. The predicted probability that $K=k_i$ is $$\Pr(k_i|n,\alpha,\beta)=\frac{n!}{k!(n-k)!}\frac{(k+\alpha-1)!(n-k+\beta-1)!}{(n+\alpha+\beta-1)!}\frac{(\alpha+\beta-1)!}{(\alpha-1)!(\beta-1)!}.$$ Since you are the bookie, you control the “vig” and as such can convert this over into a Kelly bet.

The physical phenomenon is now separated from the gambling phenomenon. The moments of the binary event no longer match the moments of the raw gamble because the vig separates them.

Now let us move onto equity securities. In the simplest case, returns can be defined as $$r_t=\frac{p_{t+1}q_{t+1}}{p_tq_t}-1.$$ As returns are a function of data, returns are a statistic and is not data.

The distribution of returns depends, in part, on the ratio of prices. Under relatively mild assumptions, the ratio distribution involved cannot have a first moment, so moments have to absent from the solution. See this video for a short discussion: https://youtu.be/R3fcVUBgIZw.

As above, the parameters fall out of any Bayesian prediction, and the Bayesian prediction doesn’t care that there are no moments to the underlying. The vig is reversed here as the market maker is taking a spread. You have to account for liquidity costs formally, or you will get the wrong results. You will mistake the returns from the company as your returns.

Allocation now depends on a predictive distribution and a utility function. As a side note, since no first moment exists, you cannot minimize squares and get a meaningful answer.

Your point allocation would then be the allocation that maximizes expected utility over the posterior prediction. I would point out that value investing is an interesting special case of the above. Indeed, it is a stochastically dominant strategy (though not uniquely).

One additional side note, log utility maps to the same solution as the Kelly criterion, although it does allow for constraints as well. You won't have moments with equities and you will maximize asymptotic returns.

  • $\begingroup$ so your nudging the possible answer to Bayesian stats because frequentist stats, which mean-variance Markowitz is based on, estimates unacquireable unknowns (moments). could you expand more on bayesian and ito approaches to portfolio theory? could non-moment based portfolios in fact exist outside of the two worlds of frequentism and bayesianism, in some third math world? $\endgroup$
    – develarist
    Commented May 25, 2020 at 19:44
  • $\begingroup$ @develarist There could be another interpretation of probability. The three main interpretations Frequentist, Likelihoodist and Bayesian are certainly not the only ones. Still, a discussion is dependent on axioms. You should pick up the textbook by Parmigiani and Decision Theory to get you started. You will need to do the calculus yourself as there isn't a prebuilt solution. Moreso, there isn't an analytic solution to these problems so you will be building numerical approximations. $\endgroup$ Commented May 27, 2020 at 17:20

Remember that asset returns are there because of the expected utility theory. More precisely, as long as you can assume a "reasonable" expected utility function to be approximated by a quadratic function, you will always end up with some mean-variance tradeoff in wealth allocation.

You want to base portfolio allocations on different quantities? Hypothesize a different utility function for your investor. You can select any other "thing" that your investors may like/dislike (for example the age of the components of the Board) and you will find optimal allocations based on that.

Let me be clear: I'm not making fun of your question. Your question is a perfectly valid one but asset returns distributional properties comes from the very deep set of assumptions. If you want to get rid of them, you have to introduce different aspects in the utility function of the investor. By the way this would not lead to a different theory: it will still be utility maximization. If you talk about "performance" of an allocation strategy, you have to define what performance mean and this is going to depend on the chosen utility function.

I'm currently not aware of allocation strategies not relying on returns distributional properties because at the core there's always a quadratic (therefore mean-variance sensitive) or higher order (therefore mean-variance and higher moments sensitive) utility function.


Some allocation approaches that are not based on moments -

  1. Fixed weight strategies (e.g. 60/40 or equal weight)
  2. Allocation proportional to market capitalisation (often called passive investing or indexing) or proportional to some other measure of size, like book value or sales (often called fundamental indexing)
  3. Building long-short portfolio using terciles, quartiles, quintiles etc.
  4. Long-short portfolios using ranking.
  5. Any number of technical trading rules eg RSI (for direction) and ATR (for trade size)

Point 3 is the approach used to build Fama-French portfolios. The idea is to rank stocks according to some quantity of interest (eg book to price ratio, sales to enterprise value, earnings yield etc) and form portfolios consisting of the top, middle and bottom third of stocks on this ranking (or quartiles, quintiles, deciles etc). Typically the portfolios are rebalanced every month, quarter or year. Within terciles, stocks could be equal weighted, cap weighted or some other weighting scheme. A long-short portfolio consists of a portfolio which is long the top third and short the bottom third, so that it is approximately market neutral.

Point 4 ranks stocks according to some quantity of interest in the same way, so that the ‘best’ stock has rank N and the ‘worst’ stock has rank 1. Positions in the portfolio are proportional to the cross-sectionally de-meaned rank of the stocks, usually normalised so that the total dollar amount of the long and short sides of the portfolio are equal to some target (eg 1x your capital). This approach is used in eg “Carry” by Koijen eg al.

  • $\begingroup$ i meant also none of these heuristic methods $\endgroup$
    – develarist
    Commented May 25, 2020 at 0:16
  • $\begingroup$ Short term you could look at market microstructure but in my view that doesn't belong to portfolio building. I can't think of anything else. $\endgroup$
    – Bob Jansen
    Commented May 25, 2020 at 6:22
  • 1
    $\begingroup$ Before I go any further, are there any other methods you haven’t specified, but want to rule out? $\endgroup$ Commented May 25, 2020 at 6:32
  • $\begingroup$ Can you explain what 3 and 4 on that list entail? $\endgroup$
    – Oscar
    Commented May 25, 2020 at 11:49
  • 1
    $\begingroup$ I edited to add some additional explanation. $\endgroup$ Commented May 25, 2020 at 12:32

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