Size Matters: The Source of the Small-Cap Premium

size mattersSince at least 1981, when Rolf Banz published The Relationship Between Return and Market Value of Common Stocks, the idea of a small cap premium has been pretty well established.  Thus, over time (though it may take a very long time), we can expect higher average returns for common stocks of smaller companies relative to larger companies.  For example, over the period from 1927-2010, the smallest decile of U.S. stocks outperformed the largest decile by 10.4 percent annually.

The standard explanation for this premium is that small-cap stocks are inherently riskier. The idea is that small-cap stocks are more volatile and more sensitive to overall market movements; they’re also more exposed to systematic default risk and business cycle risk.  But I have a non-standard explanation to offer — one that starts with a physicist named Geoffrey West..

West is both brilliant and amazingly innovative.  He needs to be because his goals are both enormous and elusive.  “I’ve always wanted to find the rules that govern everything,” says the theoretical physicist. “It’s amazing that such rules exist. It’s even more amazing that we can find them.”

The lens through which West undertook that search was his long-term fascination with general scaling phenomena.  His quest led him to range beyond physics to the discovery and analysis of the universal scaling laws that pervade biology and to the development of quantitative models for the design of organisms based on underlying universal principles.  The overarching lesson, West asserts, is that as organisms grow in size they become more efficient. ”That is why nature has evolved large animals,” he said. ”It’s a much better way of utilizing energy…. Small may be beautiful, but it is more efficient to be big.”  This is what we think of as economies of scale.

All mammals grow and develop very quickly early on but their grown curves flatten out over time.  Moreover, mammalian systems work in ways that are remarkably scaled and similar. For example, as they get bigger, mammal pulse rates slow down and lifespans extend such that the number of heartbeats during an average lifetime tends to be roughly the same across species, at around a billion. Smaller mammals simply use their heartbeats up more quickly than larger ones.

Astonishingly, these and a large variety of other phenomena change with body size according to scaling laws which involve a precise mathematical principle called quarter-power scaling. A cat, 100 times more massive than a mouse, lives 100 to the one-quarter power (about three times) longer. This scaling means that lifespan, lifetime number of heartbeats, metabolic rate, and many other measures scale with body mass, but in a way that is sub-linear – less than one-to-one – leading biological organisms to slow down and live longer as they get larger.  As West and his collaborator wrote:

In marked contrast to the amazing diversity and complexity of living organisms is the remarkable simplicity of the scaling behavior of key biological processes over a broad spectrum of phenomena and an immense range of energy and mass.

In biology, the observed scaling is typically a simple power law: Y = Y0Mb, where Y is some observable, Y0 a constant, and M the mass of the organism.1-3 Perhaps of even greater significance, the exponent b almost invariably approximates a simple multiple of 1/4. Among the many fundamental variables that obey such scaling laws – termed “allometric” by Julian Huxley – are metabolic rate, life span, growth rate, heart rate, DNA nucleotide substitution rate, lengths of aortas and genomes, tree height, mass of cerebral grey matter, density of mitochondria, and concentration of RNA.

After these discoveries, West extended his thinking so as to understand quantitatively the structure and dynamics of social organizations.  He began with cities and later examined companies, looking at the relationships between economies of scale, growth, innovation and wealth creation as well as their implications for long-term survivability and sustainability.  Unlike mammals, which scale in a sub-linear way, cities scale super-linearly, for good and for ill. Measures that scale super-linearly increase consistently at a nonlinear rate greater than one-to-one (see below).


The research shows (it is also described in a recent Scientific American article) that with every doubling of city population, each inhabitant is, on average, 15 percent wealthier, 15 percent more productive, 15 percent more innovative, 15 percent more likely to contract infectious disease and 15 percent more likely to be victimized by violent crime regardless of the time period and regardless of location or the particularities of any given city.  Bigger cities have these general statistical advantages and disadvantages because the agglomeration of people and better developed infrastructures invoke efficiencies and speed up the pace at which things happen. Thus if ten million people all live in one city, their economic output will typically be about 15 percent greater than if the same ten million people lived in two cities of half that size.  The bigger city will have more educational institutions, more cultural events, more patents produced, and more innovation generally.

The data also demonstrates that cities’ use of resources follows a similar, though inverted, law. When the size of a city doubles, its material infrastructure — anything from the number of gas stations to the total length of its pipes, roads or electrical wires — does not.  A city of ten million typically needs 15 percent less of the same infrastructure than do two cities of five million each. On average, the bigger the city, the more efficient its use of infrastructure, leading to important savings in materials, energy and emissions.  Again, these patterns of increased productivity and decreased costs (economies of scale) hold true across nations with very different levels of development, technology and wealth.

“If you ask people why they move to the city, they always give the same reasons,” West says. “They’ve come to get a job or follow their friends or to be at the center of a scene. That’s why we pay the high rent. Cities are all about the people, not the infrastructure.”  In sum, “[w]hat the data clearly shows… is that when people come together, they become much more productive.”

Cities provide the architecture that enables our economic potential and unleashes our ingenuity.  It may even explain why residents of cities like San Francisco and New York are so arrogant about where they live and so contemptuous of suburbanites.  Per West, “[w]e broke away from the equations of biology, all of which are sub-linear. Every other creature gets slower as it gets bigger. That’s why the elephant plods along. But in cities, the opposite happens. As cities get bigger, everything starts accelerating. There is no equivalent for this in nature. It would be like finding an elephant that’s proportionally faster than a mouse.”

In a free market economy, production has to keep expanding.  Had cities been dominated by sub-linear scaling (like biology), growth would have been curbed naturally.  Therefore, it’s a very good thing that cities scale in a super-linear way in that it allows for open-ended growth.  But this open-ended growth comes at a price in that it eventually leads to collapse.  Eventually, resources disappear.  After a necessary resource is exhausted, we look for and then exploit an alternative resource, if only to sustain our super-linear growth.  West cites a long list of breakthroughs to illustrate this historical pattern, from the discovery of the steam engine to the invention of the internet. Nothing can keep growing forever.

Each of these major “game changing” innovations essentially resets the clock on growth with new boundary conditions. To get continuous open growth, we need continuous cycles of innovation and for this re-cycling has to take place at an increasing speed.  Indeed, the entire pace of life increases with size. So everything that’s going on in New York today is systematically faster than it is in San Francisco and life in the City by the Bay is faster than it is in my home of San Diego — even the speed of walking (as anyone who has walked in Times Square can attest). This is the essence of cities — not buildings, roads and other things, but people. It seems that these super-linear scaling laws are a manifestation of social networks, the universality of the way human beings interact.

This research and analysis inevitably leads to questions about global sustainability and how we might construct a framework that gives rise to having wealth creation, innovation, this kind of quality and standard of life, wealth production, and so on while avoiding collapse.  Global warming, anyone? This will be a huge challenge going forward.

West subsequently extended the scope of his research some more and discovered that companies scale too. At first glance, cities and companies seem quite similar. They are large combinations of people interacting in a well-defined physical space utilizing infrastructure and human capital.  However, cities almost never die, while companies fail all the time.  The modern corporation has an average lifespan of something like 40-50 years.

West examined growth curves for thousands of companies across a variety of variables including valuation, assets, revenue, profits and employees as a function of time. Almost across the board, the generic behavior is a sub-linear, much more like biology than cities. Using data from all companies publicly traded in the U.S., West saw that sales increased linearly with company size (one-to-one). However, profits and most other measures increased sub-linearly by an exponent of about one-eighth, with fluctuations proportional to the size of the company. Thus profits decrease relative to sales, for example.  Indeed, as companies increase in size from 100 to 1,000,000 employees, their net income, assets and 23 other metrics per person increase only at a 4/5 ratio (see below). Corporate Revenue

Accordingly, by most measures, we should expect companies to grow quickly for a while before slowing and then eventually declining and finally failing.  Like animals and cities they can grow more efficient with size, but unlike cities, their innovation cannot keep pace as their systems gradually decay, requiring ever more costly repair until a crisis of some sort sinks them. Like animals, companies are sub-linear and doomed to die.

We tend to think that business enterprises are dominated by people and the innovations they create.  We hope that they can grow indefinitely.  Instead, the data suggests that as companies grow they become dominated primarily by infrastructure rather than people and by economies of scale rather than innovation.  In other words, like organisms and as opposed to cities, companies grow quickly only for a while, followed by a period of stagnant growth and eventual death.

As cities grow, what West calls their “dimensionality” increases. That is, the space of opportunity, functions and of jobs keeps increasing. In effect, life “opens up.” Thus cities tolerate and support more and extraordinary diversity.  This is in complete contrast to companies generally.  As companies mature, they move from being super-linear to sub-linear scaling because they become dominated by bureaucracy and administration to the detriment of creativity and innovation (see below).  Diversity decreases.  Company mavericks go from being the drivers of the company to becoming legacy assets to getting fired.


WMT sales

Unlike companies, which are managed in a top-down fashion by a team of highly paid executives, cities are unruly places, largely immune to the desires of politicians and planners. As West explains, “Cities can’t be managed, and that’s what keeps them so vibrant. They’re just these insane masses of people, bumping into each other and maybe sharing an idea or two. It’s the freedom of the city that keeps it alive.”  By contrast, very few large company employees think of themselves as free in the workplace.

Traditional living was robust.  It provided subsistence first and only then added “cash crops.”  Modern farming adds risk in that every crop is a “cash crop.”  City living increases the problems.  Indeed, all of the tsunami of problems we’re facing, from global warming and the environment to the questions of financial markets and risk, crime, pollution, disease and so forth, all of them are essentially urban problems.   But these added risks and added problems bring added benefits too, as West has demonstrated. They are the centers of wealth creation, creativity, innovation, and invention. They’re the exciting places. They are these magnets that suck people in. So they are the origin of the problems but they are the origin of the solutions.  And today, over 80 percent of the world’s population lives in cities.

Companies are the infrastructure of economies.  Accordingly, they scale sub-linearly.  The danger, West says, is that the inevitable decline in profit per employee makes large companies increasingly fragile and thus vulnerable to any number of potential disruptions, including market volatility. As they become more and more complex they thus become more and more unstable.  Since huge companies have to support multiple business lines and risks together with a huge infrastructure — overhead costs increase with size — even a minor disturbance can lead to significant or even catastrophic losses.

So perhaps the small-cap premium isn’t (or isn’t just) a function of risk. The inherent inefficiencies of large companies — inefficiencies that grow as companies grow (which may also help to explain why mergers so often fail) — are a drag on productivity and profits.  As West puts it, “Companies are killed by their need to keep on getting bigger.”


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