Standard deviation as a risk measure and the false reality of modern portfolio theory..

I have meant for some time to pen some further thoughts,held for a long time, on the modern portfolio theory paradigm.  My reply to a recent e-mail provoked a brief encapsulation of these thoughts.  My brief e-mail reply is noted below, and below this a more detailed description of the position.

If markets are efficient, and in a general equilibrium, then historical standard deviation will provide you with an “average” sensitivity of price movements to an average set of new information/events/shocks.  

The actual price movement at a point in time should have a direct relationship to the news event/shock, and the price movement should therefore be predictable for a given shock.  What the average standard deviation does not do is provide you with, in your face as it were, the price movement for all events, both big and small.  This is found in the distribution, and for a given equilibrium, the distribution of returns only represents the dataset for one run, or one string of outcomes, for that equilibrium.   It tells you nothing about the future shocks, just how price reacts to a shock.  

In other words, the standard deviation does not provide the investor, unless they are mathematically inclined, with a full distribution of the price movements of their investment, nor does an historical analysis of price movements provide you with the future profile of price movements.   As stated, it only tells you the qualities of the elastic band, nothing about the forces that will attempt to shape it.

But, once we are out of equilibrium then all the physical qualities (normal distribution) of a risk measurement based on standard deviation go out of the window.    Way out of equilibrium you only need a small shock to cause a large price movement, so the relationship between risk and change is lost.  Standard deviation becomes meaningless.  An analogy, is that an elastic band that is fatigued has different physical properties to a new elastic band: we have a fatigued elastic band and standard deviation only applies to new ones.

I think it worth revisiting some of the dogmas that gave the green light to mankind to go forth and multiply to go forth and leverage him/herself to death, spreading his/her seed risk  amongst those whose collapse would strike at the heart of our universe who could best bear it.   The world was not becoming a safer place at all, but a riskier, far more dangerous and leveraged place.

Modern portfolio theory constructs depend on efficient markets, rationale investors and a general equilibrium both within and between markets and economies, and stable relationships between equilibrium points.

In fact, it is more likely that markets are not efficient, that the economic and market universe is not always at or close to equilibrium, and irrational investors and irrational constructs (regulation, policy, institutions) are significant forces behind market movements and extreme valuations. One could also say that humans are likely to accentuate the negative where this provides immediate reward, irrespective of the long term risks.

These MPT constructs also depend on the assumption that future news and hence future price movements are random and unpredictable and that prices adjust efficiently and quickly to new information.

Under such constructs and under such assumptions, historical price movements, relative price movements and returns hold important information with respect to the risk and return profiles/sensitivities of asset classes and securities’ characteristics of a given economic and market universe (assuming all else being equal) and this data can be used to construct (mix) so called “efficient portfolios” that capture the systematic risk and return characteristics of this universe.

While average risk and return figures represented by mean returns and standard deviations are meant to reflect expected returns and risks, under certain conditions, they only do so for prospective returns under the assumption of the law of large numbers: that is if you were to relive the same time period a large number of times, the average return and risk across all outcomes should more or less equal the expected risk and return, again under certain assumptions; this also means that the current outcome string (the event the client is exposed to) may not lie along the mean return or standard deviation.

Assuming markets are efficient and at equilibrium, there is also a risk that the historical outcome string, which is often used to develop risk/return distributions (and hence mean expected returns and risks), might also be along a line that does not reflect the mean risk/return attributes of the universe and that the past is likewise only one outcome, and not an average of outcomes – note the mean expected return should be the average return from all probable risk/return outcomes and not the average data from one return/risk sequence. In an unstable or non general equilibrium universe, the mean expected risk and return will also shift erratically.

In fact, even if markets were efficient and at equilibrium, it is virtually impossible to work out the long term expected risk/return relationship from the single historical risk/return string – a string in this sense is the long term time series of data.

It is this reality (assuming efficient markets and general equilibrium) which the Monte Carlo simulation of the distribution of potential outcomes demonstrates. But a Monte Carlo simulation can only represent the expected distribution of outcomes if the mean return, correlations and standard deviations of assets and asset classes are representative of the mean returns and risks.

As stated, the data set from which historical sensitivity and correlation analysis is drawn is itself only one run through a universe: in other words, while the sensitivities would be correct (if markets are efficient and in general equilibrium), the probability distribution, average standard deviation and mean return may not be. In truth, we may never know the true mean and standard deviation of a universe in equilibrium, even if the universe is in equilibrium and markets are efficient.

If all prices are at equilibrium, and backed up by equilibrium relationships, then price movements tell you how sensitive a given security and asset class are to new information/shocks that drive the movement of relationships to a new equilibrium point. This assumes of course that the characteristics of an asset class are stable relative to equilibrium relationships within the universe; there is of course a limit to the relevance of historical price movements to current relationships; the relevance of the past to the present may degrade over time and the relevance of the future to the present, likewise. Even in equilibrium, therefore, relationships change and so will the risk/return relationships (non stationarity), meaning that an historical risk/return distribution may no longer have relevance and the precision by which current relationships are modelled over time likewise become blurred.

Assuming a) a certain sense of stability with respect to macro and micro equilibrium relationships (stationarity) and b) that the data sample captures the full probability distribution of outcomes, historical data could be used to define the risk/return distribution of assets and asset classes and additionally entire portfolios. These distribution profiles could then be used to project forward the probability distribution of portfolio outcomes, including the impact of taxes and liabilities on portfolios over time through the use of Monte Carlo simulations. If a and b are not met, then mean returns, average standard deviations and correlations are likely to be skewed by the current path of the equilibrium outcome.

If a given economic and market universe is in equilibrium, and if you have a long enough sample period, and the data in the sample period shows stable relationships, you may be able to define, with reasonable confidence, the probability profile of shocks to the system. The distribution profile is defined by equilibrium relationships, its stability and the type of shocks that impact the universe over time, over a long enough time. If you are starting from a position of equilibrium and future “shocks” are random, you can neither position a portfolio to outperform nor can you provide a relevant deterministic projection of future risk. What you can do is ascertain the impact on a given portfolio of the range of probable outcomes and select the portfolio which best manages the risks given those probabilities and your own risk aversion.

If the universe is a stable one, this should show up in the data. If it is an unstable one, this should show up in the larger number of extreme price reactions, or fat tails as is the common nomenclature. A stable universe at equilibrium will be more likely to absorb shocks than an unstable equilibrium, impacting the risk distribution profile.

Of course, out of equilibrium, an economic and market universe is more exposed to shocks and hence price movements are likely to be more extreme. If the economic universe is indeed out of equilibrium, then historical risk, returns and relative price movements cannot be used to construct efficient portfolios – on the contrary, you risk constructing the opposite, a higher risk, lower return portfolio.

There are no fat tails in a universe that is out of equilibrium, since the fat tail is the probability of a significant random event impacting equilibrium relationships in unstable equilibria, whereas out of equilibrium, a fat tail event is the impact of any event on a significant structural imbalance and has a much higher probability than a typical fat tail. At extreme out of equilibrium positions, you need only a minor shock to create a significant price movement, which is the opposite of the shock needed to produce an outlier at equilibrium.

If one believes that the economic universe we inhabit moves in and out of equilibrium, and at times moves to extremes, then modern portfolio theory constructs cannot be used to construct efficient portfolios and Monte Carlo simulation cannot be used in modelling the potential distribution of outcomes.

If one believes that we are at equilibrium, that markets are efficient, but that the fundamentals governing risk, returns and relative price movements are either non stationary or the historic data does not provide a large enough sample to get a close enough fix on the mean returns and standard deviations and correlations, then MVOs and Monte Carlos can be used with adjustment and proviso.

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