Modelling Framework For Portfolio Management

Our key proposal is that the underlying idea required to model an investment portfolio is a study of a single random variable in probability theory. This proposal maps the idea that investing is about moving capital through time⁶.  So the math fits investment problems like a clover.

If we treat a dynamic random variable as a fundamental idea for viewing investing, we can break the investment problem into two more components. The components are return modelling and the investor’s objective. 

The challenge of modelling returns

Only and only under the ergodic hypothesis does a single sample path (time series) contain all the information about the randomness of the phenomenon⁷. This raises a question: what is the ergodic hypothesis?

Some people want to be professional actors, but very few succeed. Some people try to learn probability, but very few understand it. What is even rare is to find a professional actor who understands probability.  

In his fine work, Ergodic Theory and Information, the actor and probabilist Patrick Billingsley, notes that you need to study several theorems associated with ergodicity to get what the concept is all about. This is true, so I will use different sources to explain the idea while keeping it simple. 

Ergodicity is the simplest case of a dynamical random variable which describes a situation in which the passage of time does not affect the laws of probability describing the random phenomenon. In connection with section 3 above, the term measure preserving transformations is often used to emphasize that the action of time does not change probabilities. In this sense, ergodicity is also the study of stationary processes. It’s a study of the subclass of stationary processes (continuous stationary process, see Gnedenko) where the focus is on dealing with fluctuations such that sample paths are “smooth” and as close as possible.

Another excellent presentation of ergodicity is given by Michel Loève⁸. Loève highlights that in some situations, we may be interested in the average behaviour of individual sequences (averaging along a sample path), not the behaviour of averages of sequences (averaging across sample paths).

When the focus is on averaging along a path, the term time-average is used to make clear that what is average is “time events”: events from the same sample path, separated by the passage of time. On the other hand, when the attention is on averaging across sample paths at any given point in time, the term ensemble-average is used. The question of interest is: does a time average exist (in a mathematical sense)? 

The ergodic hypothesis asserts that not only does the time average exist, but the time average converges to the ensemble average as time goes to infinity and the ensemble average is the same at all times. Notions such as stationarity, continuity, etc are ingredients which make this hypothesis possible. For instance, the problem of time averaging becomes mathematically tractable if it is assumed that time events come from the same source which is not affected by the passage of time.  That’s the hypothesis, what is a connection to reality?

In finance, we only observe events, not the sources of events. Ayache has even argued that viewing financial events as generated by some underlying mechanism or source is an assumption that can be revisited and dispensed with. His book, An Inverse View of the Market explores getting rid of this notion (although I am not sure I follow his argument). 

The problem would have been somewhat tractable if it was possible to do experiments and test hypotheses in finance. Conducting independent experiments will enable researchers to analyze each trajectory and deduce useful statistics from each path.

It is my understanding that experimentalists in hard science can do what is called single-particle tracking. Ralf Metzler and the co-authors have compiled the outcome of these numerous experiments in their paper. The result is that the time averages obtained from an individual path are irreproducible. It’s an irreproducibility crisis. In addition, the time averages have nothing to do with the ensemble averages. It’s an ergodicity-breaking crisis.

These findings have opened up a search for better models to explain the “anomalous” behaviour of single particles evolving through time.

In finance, the difficulty of the problem increases. There is no way we can set a clock to zero and rerun multiple experiments to learn how a household’s wealth evolves. We can’t reset financial markets and learn about the time evolution of a particular asset class under numerous experiments. Reflecting on this issue, Marco López de Prado observed that the is no way of testing the validity of most empirical findings in finance even if they are true⁹.

Notwithstanding the above points, we can learn from historical data and gain useful information about returns. We can also perform “experiments” on a computer by bootstrapping, resampling data, etc. The key message is about a fundamental epistemological limitation when it comes to making statistical inferences from a single time series. We can only gain probabilistic knowledge, hence is it important to avoid excessive risk-taking based on “evidence” from a single time series. We will see in future discussions that standard quantitative techniques in econometrics and finance can lead us astray in this case.

Let’s end a section with wise words from the poet Paul Valéry: 

“The subject is immense, requiring every order of knowledge and endless information. Besides, when such a complex whole is in question, the difficulty of reconstructing the past,  even the recent past, is altogether comparable to that of constructing the future, even the near future, or rather they are the same difficulty. The prophet is on the same boat as the historian, let us leave them there”~From The Crisis of the Mind.

Investor’s objectives 

There are two ideas which we will unpack to get to the bottom of the investor’s objective. These ideas are often used in finance and economics literature as a micro foundation for investor’s objectives.

The first idea is about the interaction between the individual and the collective. About 32 years ago, the clear-eyed economist Alan Kirman declared the notion of the representative agent in economic theory a bankrupt idea (he used the word defunct). His research paper culminated in a book, Complex Economics: Individual and Collective Rationality published in 2010, which opens with an apt comment: “The economic crisis is also a crisis of economic theory”.

Put simply by Kirman, the notion of a representative agent is the idea that the aggregate behaviour of market participants can be deduced from the behaviour of the average or representative individual. As discussed above, this is simply an ergodic hypothesis: a dynamic non-change, in the midst of change.

We can do better by defining a single random variable over time that undergoes a repeated transformation(s). That’s what we did in section 3 above and that is all we need. No further assumptions (equilibrium, partial general equilibrium, rational expectation, Martingale, independence, Makovian, ergodicity, converges, continuity, etc) are required. We agree with Kriman that the economy should be viewed as a complex adaptive system moving from state to state while reorganising itself. As a result of these dynamics, individuals constantly adapt to emerging economies to capture new opportunities. In the context of an investment portfolio, the blend of capital allocated to different assets and the strategies which inform the allocations might be reviewed and adjusted from time to time.

The second idea is about the role of human behaviour in the financial decision-making process. Research in this area evolved through three stages. The first wave is characterised by a focus on the mathematical mapping of individuals’ preferences through utility functions and a proposal of different utilities to describe human preferences.

The second wave is more behavioural-oriented and less mathematical. Chapter 6 of the World Bank report: Mind, Society and Behaviour summarizes behavioural issues that households face very well. I also consider the less-mentioned work by Gigerenzer on the role of heuristics in decision-making under uncertainty as part of this stage.

The third wave seeks to establish household finance as another subfield of finance at the same level as other subfields such as corporate finance and public finance. This aim is expressed succinctly in John Campbell’s 2006 Presidential Address to the American Finance Association. The goal is to identify and address problems limiting individuals’ ability to achieve their financial goals. It’s a broad goal which includes and extends the previous research to cover some less-studied aspects of household finance. 

While pursuing this goal, Francisco Gomes and the co-authors observed that research in finance focuses a lot more on financial markets and households are typically viewed through the price formation process in the markets as noise traders. Moreover, households are modelled as representative agents which we discussed above. Households are key nodes in the economic and financial network, so we need ideas and theories which appreciate that. 

My goal in this discussion is simple and narrow. The goal is to connect household finance with a dynamic random variable. The connection becomes clear once we view household finances as similar to corporate and public finances. The second step is to generalise problems studied in corporate and public finances to household finance. 

In particular, we are interested in asset and liability management problems. The question is: is the entity in a position to settle its obligations as and when they become due? This question led Harald Cramér to spend 50 years of his life in probability and the outcome of this intellectual pursuit resulted in Risk Theory: a quantitative framework used to model insurance claims¹⁰. He made it clear in his recollection, half a century with probability theory, that this problem is about a dynamic random variable. 

In other words, household financial decisions are influenced by many factors (age, education, level of wealth, lifestyle preference, risk tolerance, behavioural mistakes, regulatory landscape, etc) and these decisions result in an accumulation of liabilities and perhaps assets in a household’s balance sheet. That process happens sequentially through time. Therefore, the time evolution of assets and liabilities in a household’s balance sheet can be modelled using the approach mentioned in section 3 above.