# Are there optimal portfolio theories than instead of the expected value they were based on the Mode of distributions

Are there optimal portfolio theories than instead of the expected value they were based on the Mode of distributions?

During my engineer student days I saw the Markowitz theory for portfolio selection and there is something that always bothers me, and it is that are based on maximizing the expected value... let me explain why:

If the price of assets follows a skewed LogNormal distribution (as is suggested by the Black-Scholes model), the expected value is going to be always above the Mode, meaning that the most probable value I will find if I look the asset at a random time (the Mode), will be different from the expected value.

Since when making a portfolio I bought assets where I don't really know when I am going to sell them, I should be trying to achieve the higher prices for the most possible times, so, if I decide to sell at a random time, their observed values are the higher probable ones.

But if I have the average value always, other will be asking the same amount of prize since is the expected value for that asset, but the most probable value I should see is going to be below this value so I will be in a bad situation.

So, instead of this strategy, I should be trying to be over that value to sell it with a higher margin.

Or thinking it on this other way... If I maximized the expected value, and I go to sell at some random time, since for LogNormal distributions the expected value is above the Mode, I am going to be below the market value of the asset the most of the time, so I will be losing money.

This is why my intuition says that a portfolio optimization strategy should be maximizing the Mode of the distribution instead of the expected value (constrained to other figures like minimizing the overall variance, as example, but for now I don't want to close the question to other possible figures).

This is why I want to know If there exists porfolio strategies that maximized the Mode instead of the Expected Value of the assets. Please any reference are welcome, since I didn't find any on Google (maybe because I am not using the right terms gived I am not a researcher on this area, neither a native English speaker).

Summarizing, in my opinion: the "most expected value" in a skewed probability distribution is the Mode and not the "Mean value" (it is just that in symmetrical distributions they match to be the same value, and because of of the wide use of the Normal distribution on physics, due it is the maximum entropy distribution for finite mean and power, the idea is stack in everybody minds), so I want to know about alternatives where the Mode instead of the Mean Value is used to characterize the variables (also, if I am right for some distributions where the mean value is undefined, the Mode does indeed exist anyway).

Beforehand, thanks you very much.

Added later

Don't know if this is something easy to do or not, but maybe fastly comparing portfolios, one made with classic Markowitz, an the other choosing portfolios with a modified efficient frontier made with Modes instead of Expected values... It will behave better or not?

2nd Added Later

Due the few numbers of comments and answers, I would like to ask also... Is my intuition wrong?... or this is an example of being this an understudied assumption?

• Just to clarify (didn't read your whole post), the Black Scholes model implies that returns are Normal, and prices are LogNormal. May 15 at 1:54
• @rubikscube09 thanks you very much for your comment. I will correct it now. Hope later you could fully read it. May 15 at 6:03
• @Joako When do you maximize expected value? Are you referring to maximising the Sharpe ratio? Or expected value of utility? May 16 at 23:30
• @PontusHultkrantz Actually I am asking it in the wider form possible, if the efficient frontier is matched to the Sharpe ratio or other utility curves is not main issue, but instead, why in first place the expected values were used to made the efficient frontier where distributions are skewed... but I am not aiming to the Markowitz portfolio theory on specific, but to figure out if there exist newest portfolio optimization strategies based on the Mode of asset's prices' distributions instead of their expected values -as another example, Martingales are studied through their expected value also May 16 at 23:49
• @Joako:If the distribution is skewed, and the investor cares about it, rather than including the mode, skewness can be included in the risk-adjusted measure, often by including relevant moments (if they exist). Expected value is useful bcs most investors want long term growth. Describing the Mode is much more vodoo, and estimating it isn't trivial, and there might also be no unique one. For discrete distributions, why would the most expected single value be of interest? An investor plays Russian roulette many times, the mode is he survives, so he doesn't really care about the other outcomes? May 18 at 23:34

## 1 Answer

Yes, I am proposing a new branch of stochastic calculus. It drops Ito's assumption that the parameters are known. There is a Bayesian branch and the conjecture of a Frequentist branch. It is possible that the Frequentist branch is only valid in certain conditional cases. If true, it would send a wrecking ball through some vector regression models.

In order to get a point from a distribution, whether a Bayesian predictive distribution or a similar construction on the Frequentist side, we would normally impose a utility function. For the expected value, we would impose quadratic loss. However, what I did was propose an indirect utility function in lieu of Abraham Wald's utility or loss function.

For a truncated distribution, the center of location is often the mode, which is found by minimizing the all-or-nothing loss function over a distribution, with some caveats of course. The distribution couldn't have mass at a single point and zero everywhere else.

If one views the equilibrium as being the point where there is no subjective error, then it is also the center of location. When bankruptcy truncates most symmetric distributions, $$\mu$$ remains at the mode. Of course, many heavy-tailed distributions lack a mean. The indirect utility function allows for a piecewise construction of a utility function. It is, effectively, the system's utility. It does not require that any person hold that function. Indeed, it is simple to think of heterogeneous utility functions that would generate a system function that is unlike it.

It does not always use the mode, of course, it depends on the distribution involved. It becomes a theory, not so much based on the mean, median or mode, but of the equilibrium price, which happens quite often to sit at a standard type of center of location.

You could also develop a portfolio allocation set of rules on the mode simply by constructing a model where your utility function is the negative of the all-or-nothing function. You would then maximize the negative of the all or nothing function, which is the mode. Your solution would be driven by $$\mathcal{U}(\theta,\hat{\theta})=0\text{ when }\theta-\epsilon<\hat{\theta}<\theta+\epsilon,\text{ else }-c,c>0.$$

You can find the paper at: Harris, David E., A Generalization of Stochastic Calculus--A Conjecture (November 29, 2018). Available at SSRN: https://ssrn.com/abstract=3197451

I will present it at WEAI on July 3rd, 2022, so you can also find it in the proceedings once the conference is over.

Do note that this is not a theory of the mode, but it allows you to build a math for the theory around the mode. I do this elsewhere but the paper is not online. I replace the options models and because many distributions are truncated, it revolves around the mode. There is also a modal regression possible as well.

It is rather important to realize that when distributions lack a mean value, both the median and the mode have poor properties compared to distributions that do have a population mean. For two variables to be related via the mode in a Bayesian construction, the strongest statement you could make for $$y=\beta{x}+\alpha+\varepsilon$$ is that the most common relationship between $$y$$ and $$x$$ is $$y=\beta{x}+\alpha+\varepsilon$$.

Nothing mathematically prohibits them from having other, less common, relationships. That is a very weak statement. That takes you very far down the road with Leonard Jimmie Savage's subjectivist statistics.

• Thanks you very much for taking the time for this complete answer. Is much more than what I were expecting, so I will take the time to read the paper with detail, it looks really interesting, but is probably a bit advanced to my current knowledge (so it will help to improve it). Good luck with your research. May 15 at 6:11