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Bill Miller’s Mutual Fund Streak

With the recent announcement that Bill Miller will be stepping down from running Legg Mason Value Trust fund, a number of people have used this as an opportunity to re-examine his incredible fifteen year streak of beating the S&P 500. Running from 1991 to 2006, this record has never been matched.

However, in recent years, Miller’s performance fared worse than the market’s, with some years losing up to 50%. Many are therefore now saying that Miller’s streak was nothing more than luck. If there are enough mutual fund managers competing to beat the market, surely at least a handful must have performance streaks like Miller’s.

Happily, we needn’t speculate. We can actually subject this sort of statement to rigorous mathematical analysis. I happen to be partial to using math to understand performance streaks, having examined the math behind 1941 Joe DiMaggio’s hitting streak, as well as streaks in mutual funds.

Unlike what others have done, a more subtle approach than multiplying simple probabilities is required. Specifically, we need to recognize that the probability distribution of beating the market can vary from year to year.

As described here (and in more detail here), I worked with Andrew Mauboussin to compare the performance streaks in the real world to ones in a computationally-generated null model. Looking over many decades (1962-2008), our model used the same numbers of funds each year and the fraction of funds that beat the market as the weights for our Bernoulli trials (weighted coin flips), in order to create as realistic a null model as possible, but one where skill would play no part. Running this model 10,000 times, we then checked to see the distribution of long performance streaks.

Unlike the real world, we found nothing that approached Bill Miller’s streak. In fact, streaks of fifteen years only occurred 30 times out of 10,000 runs, far below a reasonable expectation based on luck. In fact, the next longest streaks in the real world were only eleven years long (these occurred twice). Now this doesn’t sound like much of a difference. But remember, these are streaks. To beat the market year after year isn’t a little bit harder, it’s geometrically harder. This can be seen by looking at how often an eleven-year streak occurred in the null model. While still somewhat unlikely, these occurred in nearly a third of our simulations.

The impressiveness of Bill Miller’s streak is magnified by looking at its timing. Our analysis shows that the streak, begun in the early Nineties, occurred during an unlikely period when compared to the Seventies or early Two Thousands. Surprisingly, despite the market’s good overall performance in the Nineties, it was one of the worst decades for active managers. For example, only about one in ten funds beat the market in 1995 and 1997, demonstrating that a long streak during this time is far from inevitable.

While we can never say with certainty that Miller’s streak was due to skill, such a streak is not possible on the strength of luck alone.

Image from Wikimedia Commons | Katrina.Tuliao

America’s Age, Empires, and Mathematics

I had a piece in the Ideas section of the Boston Globe this weekend about understanding the nature of empires and civilizations, seen through the lens of mathematics, entitled How Long Will America Last? An impossible question, answered with math:

With all the chatter about the rise of China, our possible economic collapse, and climate change, it is little wonder that Americans might be growing preoccupied with our nation’s staying power. Is the rise of the United States a fleeting moment in world history, or simply the beginning of many centuries of American ascendancy?

It might seem like a question for pundits to argue over, pessimists against optimists. But there is another way to answer the question as well: with some data.

History is filled with examples of powers much like America?—?nations whose wealth and influence allowed them outsized effects on the world. In the past, they were empires; America doesn’t usually see itself that way, but its wealth and influence put it in this peer group. And once we place it there, we can look at the lifetimes of lots of empires, see how long they’ve lasted, and use this to gain a bit of insight into our American situation.

This kind of approach, using a quantitative approach to understand history, is part of what has recently begun to be called cliodynamics. The field of cliodynamics?—?a term coined by the mathematician, biologist, and social scientist Peter Turchin from the name Clio, the muse of history?—?uses mathematics to understand the shape of history, and has been around for centuries. With a pedigree dating back to such approaches as that of Francis Galton, a relative of Darwin, who used math to understand the extinction of Victorian aristocratic surnames, a cliodynamic approach can be used to understand the ebb and flow of entire civilizations on a grand scale. Now, with the advent of the digitization of vast amounts of data, we can apply a certain precision to history that wasn’t possible before.

So that’s what I set out to do.

It’s based on my journal article The Life-Spans of Empires. The rest of the essay can be found here.

Bad Math about Infertility in the WSJ

From the Wall Street Journal:

Infertility, defined as the inability to conceive after one year of unprotected sex, affects one in six couples of childbearing age in the U.S. In 40% of cases, the problem is with the man; in 40% it’s with the woman, and in 20%, something is amiss with both, say Zev Rosenwaks and Marc Goldstein, fertility experts at New York Presbyterian/Weill Cornell Medical College and co-authors of the 2010 book, “A Baby at Last!”

Probability is often confusing, but after running the numbers, I’m fairly certain this math is wrong. If we assume that infertility affects one in six couples, then 5/6 pairings are fertile. Assuming that people choose who they try to have children with independent of fertility, at least initially, and assuming the infertility can be equally due to either the man or the woman, the results follow quite clearly: only about 5% of cases of infertility are due to both members of the couple being infertile, while the other 95% of the time it’s either due to the man or woman only.

Anyone know how the WSJ numbers were calculated?

How my calculations were done: if 5/6 pairings are fertile, this means that the square root of 5/6, or about 91%, of the population is fertile. Of those pairings then that result in infertility (that remaining 1/6), the ones that are due to both the man and woman being infertile are only (1-0.91)^2/(1/6), or about 5%.

Gaussian Genealogy: Math Masters Trace Their Intellectual Lineage

For my first piece in Wired (June issue, page 56, if you’re playing at home), I explored the Mathematics Genealogy Project, which examines the academic lineage of mathematicians and is well-known to all mathematically-inclined academics. For example, play with my “family” tree for awhile, and you’ll find a lot of academic inbreeding. For Wired, I created a heavily-annotated ancestry for Gerald Sussman, a professor at MIT, who has quite the pedigree (Leibniz, Gauss, Euler, and many more). Check out the piece here.

Cultural Ontogeny Recapitulates Phylogeny

In evolutionary biology, there is a now-discredited idea that “ontogeny recapitulates phylogeny.” In other words, the development of an organism follows its evolutionary history. Human embryos look like they have gills because people evolved from fish, we have tails in utero because of the same origins, and so forth.

In a recent paper in PLoS ONE, Alex Mesoudi, a professor at the University of London, discusses this briefly, but in the realm of culture. Mesoudi’s paper, entitled Variable Cultural Acquisition Costs Constrain Cumulative Cultural Evolution, explores how to model the exponential increase in cultural complexity, whether scientific knowledge, technological innovation, or other cultural products. Mesoudi argues that in order to create any new innovation that builds on previous knowledge, an individual must first learn and master all the innovations that came before it. In other words, cultural ontogeny recapitulates phylogeny.

And Mesoudi demonstrates this in an elegant way, by looking at the age at which British students first learn various mathematical concepts, as compared to the year these concepts were actually discovered. Here is the resulting figure:

As can be seen, there is a clear, albeit nonlinear, relationship between these quantities (original data here). More complex concepts–those learned later in life–are in fact those that were discovered more recently. Specifically, since the function is actually a logarithmic curve, this means that newer concepts are being discovered more quickly, and learned more rapidly.

It’s unlikely that this works for all topics–if a field’s college courses don’t require prerequisites, this relationship is highly unlikely to hold–but it’s fascinating to see the regularity of this shape.

Mesoudi A (2011). Variable cultural acquisition costs constrain cumulative cultural evolution. PloS one, 6 (3) PMID: 21479170

Applied Math at the Movies (including supplement)

This morning I had an article entitled The Mysterious Equilibrium of Zombies in the Boston Globe Ideas section about applied math in movies. I mentioned a number of movies, math and articles. For those who are interested in more details, here are some references, film clips and stills:

Casino Royale and Fractals

Harry Potter and The Millennium Bridge

The Dark Knight and Game Theory

Zombie Epidemiology

Six Degrees of Separation

Balance Theory

Braess’s Paradox

Braess’s Paradox, named after Dietrich Braess, is when you add roads or capacity for cars, and thereby worsen traffic (or alternatively, you lower traffic costs by removing roads). Formally, this simply means that the current traffic equilibrium state is not the optimal one. Dietrich Braess, on his website, notes that this concept has applications to computer networks in addition to traffic networks.

Gibrat’s Law

Gibrat’s Law states that the proportionate growth of a city (or corporation or other social entity) is independent of its size. Here’s an example from Economy Professor:

If a company with sales of $10m doubles in size over a period of time, it is likely the same will happen for a company beginning with sales of only $1m.

This kind of growth can yield stable distributions, such as power laws. An article called Gibrat’s Law for (All) Cities, has more about this law as applied to cities, as might be expected by the title.