## Q&A with Eugene Fama on the Bell Curve in finance, posted without commentary

It would be very enlightening if you would comment on the Nassim Nicholas Taleb (“The Black Swan”) attack on the use of Gaussian (normal bell curve) mathematics as the foundation of finance. As you may know, Taleb is a fan of Mandelbrot, whose mathematics account for fat tails. He argues that the bell curve doesn’t reflect reality. He is also quite critical of academics who teach modern portfolio theory because it is based on the assumption that returns are normally distributed. Doesn’t all this imply that academics should start doing reality-based research?

EFF: Half of my 1964 Ph.D. thesis is tests of market efficiency, and the other half is a detailed examination of the distribution of stock returns. Mandelbrot is right. The distribution is fat-tailed relative to the normal distribution. In other words, extreme returns occur much more often than would be expected if returns were normal.

There was lots of interest in this issue for about ten years. Then academics lost interest. The reason is that most of what we do in terms of portfolio theory and models of risk and expected return works for Mandelbrot’s stable distribution class, as well as for the normal distribution (which is in fact a member of the stable class). For passive investors, none of this matters, beyond being aware that outlier returns are more common than would be expected if return distributions were normal.

For other applications, however, the difference can be critical. Risk management by financial institutions is a good example. For example, portfolio insurance, which was the rage in the early 1980s, bombed in the crash of October 1987, because this was an event that was inconceivable in their normality based return model. The normality assumption is also likely to be a serious problem in various kinds of derivatives, where lots of the price is due to the probability of extreme events. For example, news story accounts suggest that AIG blew up because its risk model for credit default swaps did not properly account for outlier events.

KRF: I agree with Gene, but want to make another point that he is appropriately reluctant to make. Taleb is generally correct about the importance of outliers, but he gets carried away in his criticism of academic research. There are lots of academics who are well aware of this issue and consider it seriously when doing empirical research. Those of us who used Gene’s textbook in our first finance course have been concerned with this fat-tail problem our whole careers. Most of the empirical studies in finance use simple and robust techniques that do not make precise distributional assumptions, and Gene can take much of the credit for this as well, whether through his feedback in seminars, suggestions on written work, comments in referee reports, or the advice he has given his many Ph.D. students over the years.

The possibility of extreme outcomes is certainly important for things like risk management, option pricing, and many complicated “arbitrage” strategies. Investors should also recognize the potential effect of outliers when assessing the distribution of future returns on their portfolios. None of this implies, however, that the existence of outliers undermines modern portfolio theory or asset pricing theory. And the central implications of modern portfolio theory and asset pricing—the benefits of diversification and the trade-off between risk and return—remain valid under any reasonable distribution of returns.

Source

## This is also part of A Mathematician’s Lament, by the way.

From the famous Mathematician’s Lament. Not that I necessarily cosign the whole thing (I definitely don’t!) but I wish more mathematicians were willing to imagine that less math could be part of the solution, someday, even if in today’s world cutting math requirements would introduce dangerous inequities. It’s an idea worth getting used to.

## The Mathematical Modelers’ Hippocratic Oath

The Financial Modelers’ Manifesto was a proposal for more responsibility in risk management and quantitative finance written by financial engineers Emanuel Derman and Paul Wilmott. The manifesto includes a Modelers’ Hippocratic Oath.

The Modelers’ Hippocratic Oath

I will remember that I didn’t make the world, and it doesn’t satisfy my equations.

Though I will use models boldly to estimate value, I will not be overly impressed by mathematics.

I will never sacrifice reality for elegance without explaining why I have done so.

Nor will I give the people who use my model false comfort about its accuracy. Instead, I will make explicit its assumptions and oversights.

I understand that my work may have enormous effects on society and the economy, many of them beyond my comprehension

## Benoit Mandelbrot, Beauty and Finance

Is there anything that shouts “mathematical beauty!” quite like fractals? Fractals are a core pillar of how mathematicians and educators sell math to the public. That makes sense — they really are beautiful in a way that is both visual and intellectually pleasing.

In fact, just a few hours ago I showed this image to my 4th Graders, and I got exactly the “ooohs” and “ahhs” I was hoping for.

The name associated most strongly with fractals is Benoit Mandelbrot’s. Mathematician, visionary, Jewish WW2 refugee, early adopter of computational visualizations, really a very cool person, and discoverer of the Mandelbrot Set. He seems likely to join that canon of Great Mathematicians. (There is such a canon, right?)

Plus, there is an awesome song about him:

And it was only since I’ve started reading more about finance that I realized there was a part of the story that wasn’t being told. Because while there is no denying the beauty of fractals, Mandelbrot’s study of fractals is caught up in the “wild randomness” that he sees in financial markets.

Just to be clear, I’m not saying that this is some sort of big secret. Mandelbrot wrote and spoke frequently about this, and I came across this nearly immediately while studying finance. Because within finance, Mandelbrot emerges as a critic of the sort of quantitative financial engineering that I’m trying my best to understand.

(Shoutout, Nassim Nicholas Taleb.)

The sort of financial models I read about in textbooks all represent the movement of a stock as a kind of random walk, weighted by certain probabilities. So IBM may be more likely to go up or go down, but essentially its price at a given time has to do with how the stock travels through this field of uncertainty — and it is moving randomly. Maybe its price is more likely to rise than fall, but still it’s governed by the same laws as coin flips.

In other words, the value of IBM follows a normal distribution, i.e. the conventional bell curve.

That’s how most quants (apparently) think about finance, but Mandelbrot thought this was a big mistake. He thought the markets were governed by wild randomness, a propensity to run towards the extremes. Here is a piece where Mandelbrot (co-written with Taleb) describes the difference between his model and the conventional one:

These two models correspond to two mutually exclusive types of randomness: mild or Gaussian on the one hand, and wild, fractal or “scalable power laws” on the other. Measurements that exhibit mild randomness are suitable for treatment by the bell curve or Gaussian models, whereas those that are susceptible to wild randomness can only be expressed accurately using a fractal scale.

I’m not quite ready to try to describe what he means by “fractal scale.”

Does it mean anything that the financial side of Mandelbrot’s work is less often shared with students and the public? Maybe not. It certainly doesn’t exhibit the same gut-punch beauty as his images, and finance is probably not the quickest way to gain an appreciation for fractals.

And yet…so often, mathematicians and educators are eager to emphasize the beauty of mathematics. And I agree — math can be beautiful! But it doesn’t seem quite right to say that mathematicians simply chase beauty. Mandelbrot wouldn’t have studied fractals if he wasn’t trying to model financial data, and the story of how math at all levels is entangled with finance is deeply undertold.

## How much real-world complexity can we tolerate in math class?

I saw this, and I have a reaction:

I think of this as a matter of the complexity that we are exposing students to. Part of what makes these math modeling experiences engaging is that they are relatively simple. Not to say of course that these problems are easy for students. They aren’t. But when they are engaging it’s because the context is new, but the structure is discernible to students. That’s what engagement is: that feeling of novelty, along with the feeling of I-can-do-this.

And the reason why students can do this is typically because we’ve chosen a dataset that matches either a proportional, linear, exponential or quadratic function, or a pair of these functions. (True: it’s hard to choose the appropriate variables, and if you choose the wrong variables you won’t have very much fun. For that reason teachers usually make the smart choice to help students choose useful variables before they have time to tackle some task.)

Anyway I don’t want the point of this to be that the modeling experiences we create in math class are easy or pointless. That’s not what I’m trying to say. What I am trying to say is that part of what makes them engaging is that they are relatively simple modeling problems. And what makes them relatively simple is what makes them mathematical modeling problems as opposed to scientific ones. What I mean is, students don’t know the context in depth, we are telling them that they know enough to make predictions — they are relatively protected from the complexity.

What makes scientific modeling a different kind of engaging is that it’s in a way more serious contextual work. You learn to see new variables and new factors, and then you’re trying to coordinate them into newer, more powerful models. But there is a sense in which this is more serious work than mathematical modeling, which (by dint of being part of math class) engages less with the context.

I’m not trying to hate on mathematical modeling here, but I am trying to articulate something that I’m moving towards. I really do think kids deserve a chance to connect with applied math problems, “real world” math. At the same time, I’ve been frustrated with what passes for “real world” in math because it doesn’t take the context seriously. I have a great respect for complexity and people who study it. I’d hate for students to get the message that math can just march in and math all over the place and solve everything. You have to have some humility and learn about the world before diving in with an equation!

One question I’m asking myself these days is whether there is a way to take the contexts more seriously in math class. Is there a way I could bring more applied math into my classes without getting lost in the complexity, or ignoring it entirely?

## Trying to write arbitrage puzzles

I’ve been trying to write little arbitrage puzzles. I’m hoping to make them accessible to kids. Here’s one I’m working on now.

It might rain on April 15th. Then again, it might not.

Your first friend agrees to make a bet. If it rains, you get \$10. If it doesn’t, you have to pay your friend \$5.

You have a second friend who wants the opposite bet. If it rains, you pay your friend \$5. If it doesn’t, they’ll pay you \$10.

What do you do, and why?

My take on a solution in the comments.

## Axiomatization of “Story”

Axiom 1: At least one story exists.

Axiom 2: There exists an “empty story,” i.e. a story where nothing happens.

Axiom 3: Two stories are the same if and only if they contain the same events in the same order.

Axiom 4: If $X$ is a story and $Y$ is a story, “$X$ then $Y$” is also a story.

Axiom 5: For every story $X$, there exists a story $Y$ that contains story $X$. In this case we say that $Y$ is a telling of $X$.

Axiom 6: A story exists that contains the empty story, the story containing the empty story, the story containing that story, the one containing that, etc. forever. This is called the neverending story.

Since there is a story where nothing happens, there is also a story that is a telling of that empty story. As a result, the story where nothing happens and then someone tells a story about nothing happening is also a story. This can keep on going indefinitely, populating the entire universe of stories with retellings about nothing.

## Geometry and theology

Boiled the elements down into the axioms
Mistook a fax for remedial tediums
It seems Ezekiel speaks to some
My mind was hazy and numb
And left hand gripped a clump of palladium
Saw the beast with the wings and the talons
The simple answer but it felt out of balance
Bad news like a blue screen of death
Besides the point, but which hue seems best?

I’ll keep conducting these autistic symphonies
These sentences have sentenced me
Like I didn’t have the sense to sense the mere
The presence grows weird
Doesn’t make sense but I don’t fear, not a damn thing
Like I had a motorcycle but my name is Zed
(I’ll be missed), said another clumsy alchemist
Like he just learned predicate calculus
The existential quantifier, a backwards EX
My rap career is a cataloging of defects
Copy edited by Ryan Seacrest
Like he must be new to this
Milo doesn’t exude hubris, chuuch
Like he must be new to this
Milo doesn’t exude hubris

## Is this math?

Look at those lines, that perspective, the symmetry…there is a clear sense of geometry in this picture, and if geometry is math (it is) then there’s a case to be made that this picture is mathematical. Hence the artist is a mathematician. So the picture is math.

Except that’s not quite right, is it? It’s not fair to this piece of art to see it as essentially mathematical. That’s not what the human person who created it was going for. Math doesn’t get to make a claim on any use of parallel lines. Back off, math! Art gets to be art.

I’ve been thinking a lot lately about what counts as math, and what it means to expand what counts as mathematical. Certainly, if you think math is just what’s covered at school, you’re wrong. But then we make these expansions — kids are doing math when they’re playing, they’re doing math when they’re drawing, when they’re braiding hair, when they’re building…

Maybe the issue is in thinking of mathematics as a field or a single subject, as opposed to a mode of thinking that is used by artists, philosophers, lawyers, Wall Street, cashiers, kids, hair-braiders, engravers, everyone. Mathematics isn’t a distinctive activity that you do, it’s a certain way of doing other things.

But then what is that certain way of doing things? Probably not a clear set of criteria, but a kind of family relation: if you’re using numbers, if you’re paying attention to repetition, using the properties of shapes to create something new — you’re not necessarily doing mathematics, but you’re doing whatever it is you are doing in a mathematical way.

But math doesn’t get to claim art.