The Road Not Taken

The road mistakenly taken

In a recent online conversation, I was asked whether I believed in the validity of at least some of the tenets of modern portfolio theory (MPT). I said that it depends on what you include in MPT.

So, I decided to look up what is generally regarded as MPT.

At all of the five principal websites that emerge on a search for a definition of modern portfolio theory, Investopedia, Wikipedia, Britannica, risk.net, and the Corporate Finance Institute, MPT is defined as, to quote the Investopedia entry, “a practical method for selecting investments in order to maximize their overall returns within an acceptable level of risk,” or to quote the Corporate Finance Institute entry, “An investment theory that allows investors to assemble a portfolio of assets that maximizes expected return for a given level of risk.” The Wikipedia entry simply says, “Modern portfolio theory (MPT), or mean-variance analysis, is a mathematical framework for assembling a portfolio of assets such that the expected return is maximized for a given level of risk.”

In every case, risk is defined as the standard deviation of returns (though the Investopedia and Britannica sites give lip service to the unimportant criticism of standard deviation that it equally weights the risk of loss and the “risk” of gain).

But, so defined, the only benefit that MPT has provided is the reorientation of securities regulation to locate investment risk in a portfolio as a whole rather than every individual investment in the portfolio. It could be argued – and is often argued – that one of the main revelations of MPT was the benefit of diversification, but that would have been obvious to anyone familiar with the old saw, “Don’t put all your eggs in one basket.”

The mean-variance optimization that many people – and some of those web sites that define MPT – apparently think is the “Gee Whiz” brilliant mathematical methodology of the theory doesn’t work at all in practice. And now, an analysis has found that it is not even true that long-term investment returns increase with greater “risk” – i.e. with greater standard deviation of returns – thus falsifying a bedrock assumption of the model.

In short, MPT has been a 70-year red herring. And it still is.

Its inadequacies spawned the weaknesses of subsequent developments. Because MPT says nothing about the portfolio allocation one should hold at various future times but only at a point in time, it was assumed – evidently because one’s “risk” level had presumably been chosen once and for all – that whatever MPT said should be your point-in-time portfolio should be your portfolio at all times after that; hence, rebalancing.

But then it was realized that as an individual investor ages, she might want – rather than rebalancing her portfolio to the original mix – to alter her allocation between stocks and bonds to make it less risky over time. That led to arbitrarily defined “glide paths” of portfolio allocations tending toward less stocks as one ages. This glide path and target-date fund development has no roots in theory, and certainly not in MPT theory. This is indicative of how little MPT has helped.

And most recently a concern has focused on “sequence of returns risk,” a concern that could only arise because the accepted definition of risk was so limited, and the accepted theory had no temporal (over-time) component.

The investment world arbitrarily pasted these onto MPT, continuing for decades to pretend it was using it.

The road not taken

In 1964, a book was published that provided the real breakthrough in investment theory. Titled “The Random Character of Stock Market Prices,” it contained a compendium of articles by various authors, including the book’s editor, MIT professor Paul Cootner.

Says Cootner’s introduction, “…most of the papers test one simple question: Is there any evidence that any historical data about the price of a stock will enable us to improve our forecasts of the future profit from holding this stock?”

The conclusion is: “…except possibly for a trend which is related to the [long-term] rate of return, future changes in stock prices could just as well be determined by a flip of a coin as by any elaborate analysis of past data. It is from this concept of stock prices having independent increments that the idea of stock prices describing a ‘random walk’ arises.”

The evidence for this hypothesis in Cootner’s book is overwhelming. I had the opportunity on two separate occasions to research the largest databases of stock and bond portfolio returns available, and I found the same thing. Anyone who interrogates a large database of returns data will find it too.

Although the early mathematical formulations of this random walk theory were incomplete, a complete and internally consistent model is easy to formulate. Indulge me while I explain – or skip the next paragraph if you prefer.

While ordinary investment returns (called “holding-period returns” or HPRs by Zvi Bodie et al’s standard textbook “Investments”) compound over time, their logarithmic counterpart, continuously compounded returns (CCRs), add over time. Since returns are infinitely divisible (they can be calculated in principle from any instant in time to any other instant in time), any continously compounded return over a time period is the sum of an arbitrary number of returns over its sub-periods. Therefore, by the central limit theorem, if returns (CCRs) are independent and have a finite standard deviation then they must be normally distributed. (If they do not have a finite standard deviation, they may have a “stable,” or “fat-tailed” distribution.)

The result of the derivation in the previous paragraph is that stock prices – and therefore stock portfolio values – follow a “geometric Brownian motion,” with an upward drift over time due to the tendency toward secular upward movement of stock prices in aggregate.

This is a powerful model. And it has definitely been used. But it has not been used to full advantage. It has been underused for some purposes for which it could be extremely useful, completely replacing MPT, and it has been overused for other purposes for which it is not useful.

The useful application of the Brownian motion model

Application of the random walk model (the discrete counterpart of Brownian motion – i.e. Brownian motion that is computer programmable) can solve all the problems we have been pretending that MPT solves. In particular, it can solve the asset allocation problem far better than MPT would if it even worked.

The inputs to the Monte Carlo simulation are very few, far fewer than the required inputs for mean-variance analysis, and the results are much less sensitive to the inputs.

For any investment strategy over time – call it a glide path or whatever – it is possible to determine its resulting probability distribution of final, or intermediate, wealth at any point in time using the model that implements random walk theory, Monte Carlo simulation.

This investment strategy could be stated as an annual mix of stocks and bonds and any other investment one might contemplate, including annuities, TIPS or bond ladders, and virtually anything else. The statement of the strategy could extend over 50 to 80 years or more.

Then, a method is needed to compare these probability distributions to determine which are more desirable than others. For this, a good method is to prescribe a desired pattern of cash contributions and withdrawals over time, and then to define risk as Martin Tarlie of GMO defined it in creating the asset management platform called Nebo: the risk of “not having what you need, when you need it”; in other words, the risk of not being able to make the withdrawals that you believe you will need.

This risk can be quantified in a manner similar to Tarlie’s: the expected shortfall in funds after all the withdrawals have been made. Minimizing that expected shortfall will, if the minimum is small enough, make it very unlikely that funds will be insufficient.

This risk quantity can be minimized in principle by trying every conceivable investment strategy over time. But that would be very computation-intensive, though doable with today’s fast computers.

Better is to find a mathematical shortcut. Tarlie does this by applying the calculus of variations to find the solution. But other methodologies, or combinations of methods or approximate methods, might be applied depending on what assets are included in the strategy portfolio and at what times.

Finding and refining those methodologies could – and should – be a main focus of articles in financial journals, which would, like they are now, be replete with mathematics, but in contrast to what those articles contain now they would have truly practical application.

There are, of course, other important practical applications of Monte Carlo simulation. For example, I used it to estimate the real value of tax loss harvesting, as opposed to the very poorly estimated and wildly exaggerated values that had been touted by some robo-advisors. It has been used for years for asset-liability modeling for pension funds, to estimate how much the fund sponsors will have to contribute in the future. But it has not been applied as widely as it could and should be.

The not-so-useful applications of the Brownian motion model

Since the Black-Scholes model, which used the Brownian motion model to derive its formula for option pricing, the model suddenly became all the rage in the academic finance world. It is taught in courses in “quantitative finance,” though the instruction in it, as far as I can tell, typically does not have much depth, focusing mainly on one formula, the Ito formula. Most people who have gotten degrees in quantitative finance do not, in my experience, have much of an understanding of probability theory in general on which the Brownian motion model is grounded, or for that matter of mathematics.

But using the Brownian motion formula as window dressing in an academic article is almost de rigueur. I have seen many articles that begin their mathematical treatment by stating the Brownian motion model, then proceed to use it to derive something inconsequential, from which they will often imply a real-world conclusion that is not in fact implied by the mathematics. (Paul Romer called this “mathiness”: “slippage between statements in natural versus formal language.”)

Of course, the pressure to publish is incessant, and success in publication depends largely on producing an article that is in keeping with the common wisdom or looks like it is. Sometimes I wonder what would happen if Albert Einstein sent the four astonishing breakthrough articles he produced in 1905 to a journal now.

Bob’s suggestions

As I mentioned above, Bob Huebscher suggested that the beneficial contributions of the finance industry might include index funds, options, annuities, and TIPS ladders. I agree on index funds, annuities, and TIPS ladders but not options.

Index funds, of course, are a great boon to investors, not so much because they are “optimally” diversified but because their fees are low. The only variable that has ever reliably correlated with an investment vehicle’s future net investment returns is its fees. It is believed that index funds are a derivative of MPT but that is not true. Random walk theory (which in its full expression includes the fact that variability of return is reduced by diversification), which incorporates the assumption that you can’t predict securities price changes and therefore security selection is futile, together with the fact that fees correlate negatively with net return, suffice to imply that a broadly diversified, low-cost index fund is the best bet.

Annuities – and by this I mean single-premium immediate annuities (SPIAs) or deferred-income annuities (DIAs), not variable annuities – are also a great boon to individual investors, because they provide near-certainty of income for the investor’s lifetime (especially if they are inflation-protected). They are a product of the insurance sub-industry of the financial industry, not the securities sub-industry. They are certainly instruments to be considered for inclusion in an individual’s investment portfolio, and their inclusion or non-inclusion can be modeled by the Brownian motion-related methodology described above.

TIPS ladders are of course valuable for the same reason, they provide certainty of income for a fixed period. Recently it was found that, due to heightened interest rates, a 30-year TIPS ladder could satisfy the fabled “4% rule” of systematic withdrawals. With certainty like that, an investor might forego all or much of a risky investment portfolio.

But as to options. Options have existed for centuries, but it was only in the early 1970s that the Chicago Board Options Exchange and other exchanges were established to enable more liquid trading in them, and that the Black-Scholes formula was derived to assign to them an estimated value. But apart from providing another outlet for addictive traders’ gambling urge, and a lucrative money-making opportunity for those who sell supposedly highly successful trading strategies and courses on how to implement them, I don’t see that any of that provides a benefit to serious investors.

People in the securities industry are generally highly regarded – though perhaps less since the 2007-2009 financial crisis – because they make and handle money and are believed to know a lot about arcane technical matters. It would be better if they actually did.

Economist and mathematician Michael Edesess is adjunct professor and visiting faculty at the Hong Kong University of Science and Technology. In 2007, he authored a book about the investment services industry titled The Big Investment Lie, published by Berrett-Koehler. His new book, The Three Simple Rules of Investing, co-authored with Kwok L. Tsui, Carol Fabbri and George Peacock, was published by Berrett-Koehler in June 2014.

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