In an effort to move away from the assumption of normality in Economics, Professor Jean-Louis Arcand has written a paper titled “Dynamic Mean Preserving Spreads” together with Max-Olivier Hongler from EPFL and Daniele Rinaldo, Graduate Institute PhD Student in Development Economics. They find that there is only one functional form that satisfies the famous Rothschild and Stiglitz definition of risk. More with Professor Arcand.

You wrote this paper with one of your PhD students, Daniele Rinaldo, and a somewhat unusual co-author for an economics paper, Max-Olivier Hongler, who is a retired physics professor from EPFL. How did this interdisciplinary collaboration start?

The person who brought Max-Olivier and me together was actually Urs Luterbacher, who is an emeritus professor in our Department of International Relations/Political Science. I was friends with him and he was separately friends with Max-Olivier, and he connected us in the context of a project that we didn’t end up pursuing any further, but we anyway decided to keep working together on other things. I’ve learned enormously from this collaboration, especially when it comes to advanced stochastic differential equations. I’ve always wanted to learn about them, but they are just not part of the standard economics curriculum, so this sort of shows that you can teach an old dog new tricks. Finally, Daniele just came naturally on board with his background and interest in mathematics, and he came up with one of the examples we present in the paper.

What was your motivation for writing this paper?

A lot of what we do in economics comes originally from statistical mechanics and physics in general. An example is early work by Solow on economic growth, or Samuelson’s famous Foundations of Economic Analysis book. Economics and physics used to have a lot more in common than they do today; the two fields have gone their separate ways over the past half-century, not only in terms of content but also in terms of scientific methodology. Economics has remained stuck in something of a rut in terms of the methodologies that we use. For example, we tend to assume normality for everything. By modelling the world using Gaussian (normal) distributions we get a good first approximation and we can say a lot of things. The justification for assuming normality is often that we get something that we can write down in an elegant, closed form. However, the world is not a normal place, fat tails characterise many stochastic processes, and so not everything can be explained in terms of a standard bell curve. Finance is probably the one area in economics that has remained the closest to physics, and one of the most famous equations in finance is the Black-Scholes option pricing formula. Both Black and Scholes had backgrounds in physics. The formula uses plain vanilla Brownian motion, so it assumes normality, and the result is a very simple, closed-form solution for option pricing. However, stocks returns are known to have heavy tails, which means that extreme realisations of values are much more likely than would be predicted by a normal distribution. What we are doing in this paper is to move away from this assumption of normality and still get simple solutions that can be used in applied modelling.

The paper deals with “dynamic” mean-preserving spreads and is an extension of the famous 1970 paper on mean-preserving spreads by Rothschild and Stiglitz. What are your main findings?

The basic definition of risk in economics, which was established by Rothschild and Stiglitz (1970), is the mean-preserving spread. This should not be conflated with variance: an increase in risk increases variance but the converse is not necessarily true. The two integral conditions used by Rothschild and Stiglitz to define risk are not specific to a normal world. There are many results using mean-preserving spreads for static problems which do not depend on normality, but up until now no one had looked explicitly at dynamic problems. What we establish in this paper is that there is, somewhat surprisingly, only one specific functional form for what is known as a Brownian bridge diffusion process which satisfies the dynamic counterpart of the Rothschild and Stiglitz definition of an increase in risk. Our second contribution is to show that this functional form is not normal. Rather, it is constituted by the mixture of two normal distributions. This is very important because even though the functional form itself is not normal, we can still do the regular calculations for things such as option pricing, entry and exit decisions by competitive firms, or investment under uncertainty, precisely because of the underlying normal distributions. People in mathematical statistics had actually already noticed this parametric form, which is called super-diffusive ballistic noise, in the 1980s, but we are the first to make the link between the fundamental economic concept of risk of Rothschild and Stiglitz and this functional form. It is very surprising that there is only one functional form that characterises dynamic increases in risk, as Rothschild and Stiglitz’s integral conditions are just conditions on the derivatives of the densities, and are completely general.

In the second part of our paper we present three applications – Black-Scholes option pricing, entry and exit decisions by firms (based on Dixit 1989) and investment under uncertainty (Abel 1983; Abel and Eberly 1994) – where our result can be applied, and we show that this leads to significantly different formulae. Our main contribution is therefore a tool that allows one to easily calculate non-normal versions of many existing results.

How will you use this result in your future research?

The paper is a relatively complicated one, and it is now under review at the Journal of Mathematical Economics. Once we publish it somewhere, we have several other papers – which we are either currently writing or which are already finished – that apply our basic result. Some of my PhD students are also starting to use this result in their research, applying it to areas such as queuing or contract theory. What we’ve seen so far is that going beyond normality does indeed help one to better understand the world.

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Read “Dynamic Mean Preserving Spreads” >

Full citation of the paper:
Arcand, Jean-Louis, Max-Oliver Hongler, and Daniele Rinaldo. “Dynamic Mean-Preserving Spreads.” May 2017.

Works cited

• Abel, Andrew B. 1983. “Optimal Investment under Uncertainty.” American Economic Review 73 (1): 228–33. http://www.jstor.org/stable/1803942.
• Abel, Andrew B., and Janice C. Eberly. 1994. “A Unified Model of Investment under Uncertainty.” American Economic Review 84 (5): 1369–84. http://www.jstor.org/stable/2117777.
• Dixit, Avinash. 1989. “Entry and Exit Decisions under Uncertainty.” Journal of Political Economy 97 (3): 620–38. doi:10.1086/261619.
• Rothschild, Michael, and Joseph E. Stiglitz. 1970. “Increasing Risk: I. A Definition.” Journal of Economic Theory 2 (3): 225–43. doi:10.1016/0022-0531(70)90038-4.

Interview by Nadia Myohl, Master student in International Economics.