Sham Mathematics in the Investment Industry

The Math in Modern Portfolio Theory Is Unoriginal and Mostly Trivial

Most descriptions of Modern Portfolio Theory include Markowitz’s mean-variance analysis, Sharpe’s Capital Asset Pricing Model, and the Black-Scholes-Merton option pricing model.

Markowitz’s algorithmic solution to the quadratic programming problem, which arose from his need to maximize expected return for a given level of variance, is probably the best piece of mathematics in the history of the industry. But don’t get too excited about it: Despite its elegance, other algorithms that solved the same problem were already available.

Sharpe’s 1964 CAPM paper, for which he was awarded the Nobel Prize, uses no mathematics at all, unless you consider its reliance on a two-dimensional graph for its argument mathematics.

The Black-Scholes formula depended on the solution to the heat equation, a partial differential equation that was solved by Joseph Fourier more than 200 years ago.

None of these would have survived a peer-review process if submitted to a mathematics journal, with the possible exception of Markowitz.

Taleb, in his 2007 book The Black Swan, says that Markowitz and Sharpe “built beautifully Platonic models on a Gaussian base.” He means that they depended on the assumption of a mathematical normal (i.e. Gaussian) distribution. But this is false. Neither Markowitz nor Sharpe depended on an assumption that the distribution of investment returns is normal.

Triana makes the same error. He says, “all the Nobels awarded to financial economics are heavily grounded on the Normal assumption.” But neither Markowitz’s nor Sharpe’s work is.

These errors highlight the fact that much of Taleb’s and Triana’s critiques of financial mathematics are criticisms of the fact that it doesn’t account for “fat tails” — the more-frequent-than-expected outliers that are neglected by the normal distribution, which Taleb dubbed Black Swans. It’s true that this is an important omission in much financial theory, but it doesn’t apply to Markowitz or Sharpe.

The Math in MPT Has No Practical Use

Markowitz’s mean-variance analysis has no meaningful practical application, except for snowjob purposes. It suffers, most importantly, from a fatal garbage-in-garbage-out problem. This is now widely known. For example, it has been highlighted by William Bernstein in articles and books, among many others.

Markowitz’s sole useful observation is little more than a mathematized version of the old saw, “Don’t put all your eggs in one basket.” The reason for this admonition is obvious without mathematics. But when you throw in the mean-variance approach you get “If you distribute your eggs among many baskets, you’ll get the same expected number of hatched chicks as if you put them all in one basket, but you’ll have less uncertainty as to how many hatched chicks you get.” Does this conclusion really need mean-variance analysis?

Sharpe’s CAPM paper did not state any implied practical implication. It was a nice thought experiment. It can be argued that it led eventually to the idea of an index fund. Nevertheless, it had no real immediate practical application. At the time, however, math-nutty theorists and practitioners immediately claimed it meant that you should choose a beta for your portfolio based on your risk preference, then keep that beta constant indefinitely. Sharpe’s theory dictates no such thing. And there is no reason why it should be done.

The Black-Scholes-Merton (BSM) option pricing formula suffers at the outset from a potential garbage-in-garbage-out problem. In order to price an option, you have to input to the formula a sigma, that is, the expected future volatility of the underlying asset. This might be OK if there were a reliable estimate of that number. But even at first use it was discovered that BSM didn’t produce an option price that was even close to the price found in the market. But it got worse. It was also discovered that in order to come close to matching the market prices of a series of options on the same stock with different maturities and different levels of being in-the-money or out-of-the-money, you had to input a different volatility for each option. The result was what is called the “volatility smile.” This thoroughly invalidated any pretense that BSM could be used in practice in real life. Triana’s book includes extensive coverage of this problem.

An Ongoing Problem

In the past, attempts to apply theories like MPT, CAPM and BSM have caused financial crises — and will continue to do so. This claim forms much of the burden of Triana’s book. And he is right. The “portfolio insurance” crash on Black Monday, October 19, 1987, was the result of the belief by creators of portfolio insurance that markets would move continuously, as was assumed by the BSM model. When they moved in jumps on that day, creating a cascade of sales, a crash occurred.¹

Triana also argues, correctly, that taking the measure VaR (Value at Risk) seriously was partly responsible for the 2007–2009 financial crisis. But he also argues, at great length but incorrectly, that the Gaussian copula bears the responsibility for the misrating of CDOs (collateralized debt obligations). It was not the formula itself that was responsible, but the inputs into it, especially the low assumed mortgage default rate. That was too low because the supposedly “evidence-based” ratings agencies relied (as the insane modeling culture requires them to do) on “data,” which of course means past, not future data.

Even Taleb Got Seduced Into It

Taleb is viciously critical of mathematical modeling in finance, and I usually agree with him on this score and on many others. But I was surprised to find that he had fallen for sham mathematics produced by someone he seemed to admire, Ole Peters of the London Mathematical Laboratory and Santa Fe Institute. Taleb’s seduction may have been due to the fact that much of Peters’s observations center on fat tails. However, those are not actually fat tails of the non-normal distribution type, but of the highly skewed lognormal type — a distribution that is functionally equivalent to the normal.²

I interviewed Peters and wrote an article about him in 2016. In my interview, I asked him what practical implications his work has for investing. He replied that fortunately, he doesn’t have to worry about that. But in that article, I speculated that “Even so, it would not surprise me to hear that next week some investment manager has claimed to have a market-beating strategy based on the work of Ole Peters.”

It turned out I was right. The eventual claims would be bought into by Peters himself, as well as the investment consulting firms and managers that engaged his services. Because of Taleb’s later enthusiastic sponsoring of Peters, I read virtually all of Peters’s papers, in which he claims two or three practical applications of his work to investments. On close scrutiny, I found that none of those practical applications were valid.

Dripping Roasts

A friend of mine, a law professor, calls anything that makes lawyers salivate at the prospect of huge fees a “dripping roast.” For example, in 1980 the U.S. Congress created CERCLA, an act also known as Superfund, to create a trust fund “for cleaning up abandoned or uncontrolled hazardous waste sites.” The fund eventually accumulated $1.6 billion. The idea was to assess “Potentially Responsible Parties” (companies that contributed to the pollution at the site) to pay compensation for damages. The parties were assigned “joint and several liability,” which meant that any single polluter could be held responsible for the entire cleanup cost. If one contributor to the pollution could not be found or went bankrupt or could not pay, the liability shifted to others.

In 1992 the RAND Corporation produced a report on the Superfund program. It found that up to 88% of the fund went to lawyers.

In the investment business, the dripping roasts are gigantic funds like pension funds, endowment funds, and foundation funds. Participants in the management of these funds are fund administrators, consultants hired by the administrators, investment managers recommended by the consultants, and fund trustees. All of them get paid out of the corpus of the fund, and paid well. Who pays? In the case of public pension funds in the United States, it’s the taxpayers. In 2020, $68 billion was paid to money managers alone, an average cost of $500 a year to the average taxpayer. Of course, almost no individual taxpayer is aware of paying this cost. It is a dripping roast to the consultants, money managers, administrators and trustees of the fund, who can all take a delicious bite.

You could throw into the dripping roast category high net worth individuals who accumulated many millions of dollars plying the trade in which they have expertise. For example, I once met a man who had made $800 million by buying humongous spools of sheet metal, which are bought mostly by automobile manufacturers, and dividing them into smaller but still large pieces and selling them to buyers who need only relatively small amounts. Such a man will assume that in investing the money he made, he should seek out someone who is equally expert in the money management field as he is in his field. Of course, he can’t tell that those money managers actually have no practical expertise at all. But he knows they are reputed to be experts in the sophisticated mathematics of finance.

When there are such dripping roasts to be chewed on, who could resist?

But Some Could

In a passage that I felt spoke directly to me, Triana says:

The problem with all this is that it is expected that many peddlers of the quantitative snake oil would be perfectly aware of the unworldliness of some of the solutions. Of their meaninglessness.

By continuing to sell their wares, they would be incurring an intellectual fraud. For people endowed with prestigious doctorates from the world’s leading universities, and who might have previously dreamed of academic glory and yearned for the pure discovery of knowledge, the moral dilemma may prove taxing. They would be corrupting their beloved and sacred scientific methodologies in the pursuit of material comfort. They would be (knowingly) contributing to a lie that clouds understanding and that may put the world at large in undue danger. They would have contributed to transforming advanced mathematics and statistics into misleading sales pitches in search of a quick buck.

There’s a solution to this dilemma: Get out of the field. That’s what I did, long ago. Do something different, something better. Never mind the money. It caused some significant financial pain, but I would never change that decision. I wish more people would make it. We would all be better off.

Endnotes

¹ When I pointed out in 1986 to Mark Rubinstein of Leland O’Brien Rubinstein, who were the creators of portfolio insurance and for whom I was a consultant on the mathematics, that it wasn’t really insurance, he replied “It is in normal markets.” (Not meaning normally distributed, but moving smoothly and continuously.) Of course, insurance is not meant for normal, ordinary, day-to-day conditions but for abnormal occurrences. Otherwise, it is not useful.

² The lognormal distribution is actually nothing but a mirror image of the normal distribution of rates of return — but when rates of return are stated as continuously compounded rather than periodically compounded.

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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