# Why is Portfolio Theory not using the distribution of portfolio returns

Portfolio Theory uses things like Expected Value, Risk, Confidence.

I wonder why it's not using the Probability Distribution of a Portfolio?

Isn't it more representative? And things like Expected Value, Risk, Confidence are just a simplified metrics derived form the Probability Distribution?

Or do I miss something?

portfolio construction currently relies on the probability distribution of asset returns because the asset means $$\boldsymbol{\mu}$$ and asset volatilities $$\boldsymbol{\sigma}$$ are estimated from historical time series (data vectors) as $$\hat{\boldsymbol{\mu}}$$ and $$\hat{\boldsymbol{\sigma}}$$ and used as inputs in the Markowitz mean-variance model.

The probability distribution of a portfolio, on the other hand, is not necessarily an input for portfolio construction, so you can’t think of it in the ex ante sense for practical usage since it’s an ex post object.

Nevertheless, the distribution of special portfolios considered to be optimal in portfolio theory have been derived: the global minimum variance (GMV) portfolio, which has the lowest portfolio risk, and the tangency portfolio, which has the highest reward-to-risk (Sharpe) ratio. Analytical solutions for these can be found here. What you'll see is that the reason why their distributions are not used as inputs for portfolio construction directly is that their mean and variance estimates use for inputs the asset return moments themselves, meaning that the probability distribution of asset returns is the top-level input for portfolio construction, not that of the portfolio itself.

It is not uncommon to speak of the statistical distribution of same random vector in terms of its first two moments, the mean $$\mu$$ and its volatility $$\sigma$$, but, since an individual portfolio is commonly solved as a point estimate, with only one single $$\mu$$ and $$\sigma$$, you would seldom hear about the distribution of a portfolio being discussed since these are scalars. Whereas, if you don’t resort to taking the expected value of the weighted portfolio returns (a vector), $$\boldsymbol{\omega^\top \mu}$$, or their standard deviation, you obviously have the datapoints necessary to draw a probability density function.

Risk and probability are not mutually exclusive. Check out page 80 at the source:

https://www.math.ust.hk/~maykwok/courses/ma362/07F/markowitz_JF.pdf

Many people ignore the fact that Harry Markowitz actually defines variance as a forward variance. Without knowing your probability distribution, you can't define your variance! They are interwoven together. In short: MPT does use a probability distribution of a portfolio, just like you expect.

Let me begin with a disclosure; I am a staunch opponent of Modern Portfolio Theory. I believe there is a deep and profound error in its mathematics, which is why it does not work empirically. I also think that I have identified the error. That disclosure aside, let me defend its use of point estimators, confidence intervals, and risk.

Let me provide an observation that is important in finance but not necessarily important elsewhere. Finance, of necessity, is primarily concerned with fixed points because prices are fixed points. If I decide that the fair value of IBM is \$53 per share, that implies that I should ignore it at any price above that and consider purchasing it at any price below that if I am considering buying shares.

I can only make one of three choices. I can buy; I can sell; I can do nothing.

No matter what else I may do, I am trapped in that trinary position. That can be split into two binaries. It can either be, do something or do nothing; and, buy or sell contingent on deciding to do something.

Because decisions are based on prices, at least the buy or sell has to come from a fixed point. The do something or do nothing decision must be based on another variable other than price. Yet, it is still a binary choice. That second dimension must also be built around a fixed point.

An expected value is one example of a way to construct a fixed point. It is not the only way. In a trinary choice, a distribution isn't adequate to create choices.

Risk is always a function. You may make a choice based on expected risk, but risk is always a function. If it is a bivariate function, then it could also contain the points that make your trinary choice. While risk depends on the distribution involved, the distribution is not adequate alone to make risk judgments.

Finally, confidence isn't heavily used in finance, but it should be understood. It is important to remember that in a confidence interval, it is not the interval that you have confidence in but in how the interval is calculated.

With a 95% percent confidence intervale, it is not true that there is a 95% chance the parameter is inside the interval. There is either a 100% chance it is inside the intervale because it is, or there is a zero percent chance it is inside the interval because it is not. What a 95% interval states are that if calculations were performed on infinitely many different samples, then at least 95% of the intervals will cover the true point. It says nothing about the specified interval that you are using. The confidence is not in the one decision you may make, but on an infinite number of future decisions.

With respect to portfolio theory, if and only if the model is correct, then you could set a cutoff percentage, usually called $$\alpha$$ in statistics, which would determine the maximum percentage of time that you would be made a fool of on infinite repetition of the portfolio decisions.

One aspect of confidence intervals that is really useful is that they do not directly depend on the distribution of data in question and do not depend at all on the true value of an underlying parameter. Since that implies that the true distribution could lie anywhere, then for a correctly specified model that implies that a confidence interval is independent of the actual distribution.

That is very freeing because you can see the historical data, but no one can ever see the actual distribution since it requires an infinite amount of data.

You could think of historical data as a vector statistic or matrix statistic rather than the scalar statistics that you are thinking about. You could then use the entire set of data to form inferences or make decisions, but you would still reduce them down to that trinary choice. That trinary is still a fixed point scalar statistic; it is just one with three states. You could perform that over the entire set of possible asset choices, but then you end up with a vector of trinary choices.

While I believe Modern Portfolio Theory holds the same mathematical validity as $$2+2=77$$ has validity, Markowitz did something insanely valuable. He took the unidimensional decision-making that preceded his work and pointed out that it was at least a two-dimensional problem, not one. His successors in that line of thinking then reduced it down to a vector of scalars called allocations. The solution they created is invalid, but focusing it down to fixed points from distributions was valuable.