The current approach to investment management has no sound basis and doesn’t work. There is a better way.

I have written many articles pointing out that the current approach to investment management doesn’t actually work. The theory is all single-point-in-time. This is absurd for a process of investment, saving, and spending that takes place over the long term and aims at goals that are often far in the future. To extend the single-point-in-time theory to a continuing process over time, applications of the theory simply assume that the single-point-in-time solution applies to every moment or interval in time in the future.

That this approach is inapplicable has been made obvious by the need for “target-date funds” with their “glide paths,” which do not apply the same solution at every point in time. Yet the supposedly brilliant concept has nothing to offer that would provide an appropriate glide path for a particular set of future needs. Hence, glide paths are derived by rudimentary rule of thumb with no sound theoretical basis and little or no relationship to the single-point-in-time theory.

The current theory, which is taught in schools, and tested in CFA institute exams for certification purposes, has as its bedrock the mean-variance optimization model. In a truly appalling 2010 book, The Endowment Model of Investing: Return, Risk, and Diversification, the authors admitted that the model had to be “tortured” – by which they meant rigged – to produce acceptable outputs. They then went ahead and “applied” the model throughout the entirety of their 352-page book, quite openly manipulating the model to get the results on which their conclusions depend. It should be no wonder that the endowment model has failed so badly.

As I have shown before, most of the articles on investment management in financial journals are deeply flawed. They rest on performative mathematical exercises which are then interpreted by the authors to support their own preconceived non-mathematical conclusions, even when, which is very often the case, they do not actually support them.

**There is a better way**

A 1964 book edited by MIT professor Paul Cootner, The Random Character of Stock Market Prices, with contributed chapters by numerous finance researchers, found that “[s]ome investigators now conclude that stock price changes are best approximated by classical Brownian motion.”

It is still the case that no evolutionary model – i.e. a model that evolves over time – better fits the pattern of securities prices than what is commonly known as the random walk model, or in continuous time, geometric Brownian motion.

The random walk model is frequently used in Monte Carlo simulations. Such simulations explore the array of results that are likely to accrue, with their probabilities, given an investment strategy and a pattern of saving and spending.

Less explored is the reverse: what investment strategy would produce a desired probabilistic array of results, given a pattern of saving and spending? Asset-liability models intended to help determine pension fund contributions do try to answer this question, but by a simple trial and error process, not analytically.

The only recent example of such an analytical exploration of which I am aware is Martin Tarlie’s paper, “A Case Study in Multiperiod Portfolio Optimization: A Classic Problem Revisited,” which is the basis for GMO’s asset management platform Nebo. In that paper, Tarlie derives an optimal glide path given a series of desired withdrawals from a portfolio over time.

Tarlie’s paper should show the way for a whole series of papers exploring the same question. This could be a field filled not only with rich mathematics, but with important practical applications as well. That would be better than the useless theories and sham practices the field indulges in now.

**Possible tweaks to the theory**

The random walk model is often accused of being unrealistic because it does not, for example, model “fat tails.” In other words, because it assumes that the underlying probability distribution of one-period rates of return is a normal distribution, it does not model the non-normal, fat-tailed distribution that is observed in practice.^{1}

But there is another feature of the series of securities returns, even more obvious from the data, that is not modeled by the standard geometric Brownian motion model. That is the fact that the standard deviation of returns is not constant, it varies over time. Furthermore, it varies not in a totally unpredictable way – it exhibits some persistence. That is, periods of high standard deviation – high “volatility” – tend to be followed by periods of high standard deviation, and vice versa. This is not a feature of the standard geometric Brownian motion model. In that model, the standard deviation of returns – the volatility – is assumed to be constant.

Volatility can instead be modeled as stochastic and mean-reverting by adding stochastic volatility to the standard Brownian motion model. According to Nassim Taleb, this will also serve to fatten the tails of the probability distribution of return. It is not difficult mathematically – at least algorithmically – to build stochastic volatility into the Brownian motion model. But it is difficult (and less representative of observed sequences of returns and their volatilities) to build independent, identically distributed (from one period to the next) fat-tailed distributions into the model.^{2} Adding stochastic volatility to the model would, however, make mathematical analysis more difficult. However, that would simply mean a greater challenge for the mathematician.

Another tweak that has been proposed is to add mean-reversion to the returns themselves rather than to the volatilities. This is because there have been some articles in finance journals finding that returns have momentum in the short term but mean-revert in the longer term. I believe the evidence for these phenomena is weak and tends to change from time to time, and therefore does not merit being added to the model.

**Alternative: the brute force method**

With today’s ultrafast computers, it is conceivable that one could run every possible investment strategy through a Monte Carlo simulation. Those strategies could, at least in principle, include not only every possible sequence of annual stock-bond mixes, but additional investments such as single premium fixed or deferred annuities or other instruments. Then, the results of all of these simulations could be compared to see which one best optimizes some aggregate figure of merit; for example, Tarlie used as the figure of merit the expected shortfall (i.e. the deficit or negative assets) at some future date, after all or almost all of the planned outlays have occurred.

However fast are modern computers, what I have just stated is not really feasible in its entirety. The number of sequences of 40 years of either 100 percent stocks or 100 percent bonds is two raised to the 40th power, or more than a trillion. The number of 40-year sequences in which stock and bond percentages vary by increments of 10 percent is well beyond the capacity of a supercomputer.

But it might be possible in a reduced form – for example, only certain patterns of stock-bond allocation over time could be allowed – and then only because of the phenomenal power and speed of modern computers. Better, however, is to find mathematical shortcuts that replace or at least supplement the brute force method. That is the challenge I am posing. It could and should lead to mathematical developments that are truly elegant and yield actual practical results, as opposed to the purely performative and largely useless mathematics that fills journals of finance now.

**Mathematical theory vis-à-vis applications**

When I got my degree in pure mathematics, pure mathematics had nothing to do with finance. In fact, it had little or nothing to do with anything in the real world. That was intentional. The belief in the pure mathematics department was that we were pursuing an art form. Any application to the real world would taint it.

That was why I was so surprised when at that time, I saw a fellow graduate student carrying a book with the title Waves in Shallow Water. I asked him what he was doing with that book. Presumably it would be a book about applications of mathematics in the real world.

No, that’s not what it was. It was a book about a set of partial differential equations that was posited to be used to model waves in shallow water. The objective of the work was to determine whether a mathematical solution to those equations existed and whether it was unique. It wasn’t important what the solution was – in fact the investigations might not even determine what the solution actually was, only whether it existed and was unique. That was all that mattered to the pure mathematician.

This might seem like a silly thing to pursue for a whole PhD dissertation. Of course, a solution exists, in the real world – there are in fact waves in shallow water; it’s observable. And it is unique; you can’t have at the same time two different sets of waves in shallow water resulting from the same set of forces impacting on them, like two alternative universes.

This is an example of a piece of pure mathematics motivated by – but not aiming to solve – a specific physical problem. All that the mathematician cares about is doing the exercise to determine whether a solution to the set of equations exists, and whether that solution is unique. It’s a math exercise, not an exercise to solve a physical problem.

Since that time much pure mathematics has become motivated not by physics, but by finance. The purpose is mostly the same: not to solve a problem for practical application, but to solve an intellectual problem without regard to its possible application in the real world. Of course, people outside the pure mathematical field might believe or assume it is intended to solve a problem in the real world – and this is a boon to pure mathematicians because it helps them obtain funding, even though it is a misconception of what they are doing.

Unfortunately, there is very little overlap between academic finance and pure mathematics. Finance would benefit from more peer-reviewing by actual mathematicians. It might weed out much of the error and illogical conclusions.

When, in the few cases I’ve seen of real mathematicians working along with finance professionals (or of real mathematicians writing in math journals and then somehow writing differently – coming to different conclusions – in finance journals), the mathematical work has no practical implications, but it is then interpreted by the finance professionals intent on selling a financial product as having practical implications that it does not have.

I say this to own up to the fact that some of the investigations I am implying should be carried out in the field of finance may already be carried out in the field of pure mathematics, motivated by financial problems in the same way that waves in shallow water motivated my fellow grad student. The purpose would not necessarily be to get a result that can be applied in practice, but to perform a satisfying intellectual investigation. Or they might already be carried out in the field of mathematics but not related to finance at all – possibly not knowingly – and yet could be applied to finance.

To the extent that this may be true, there should be better collaboration between the mathematicians working on theoretical problems and the finance professionals – with the mathematicians, one hopes, keeping the finance professionals honest.

*Economist and mathematician Michael Edesess is an 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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