The Disraeli Room

The Disraeli Room

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Reciprocity: A foundation of financial economics

4th July 2014

Dr Tim Johnson outlines the lessons maths can teach us about ethics and finance

Both UK and US legislators have highlighted a degradation in commercial ethics as a significant cause of financial crises since 2007. This might suggest that mathematics has a limited role in addressing the problems the crises have thrown up. I was the RCUK Academic Fellow in Financial Mathematics between 2006 and 2011 and my experience, scholarship and analysis suggest the opposite; that mathematics can give deep insight into commercial ethics.

In 2009 the science journalist Ehsan Masood was asking me naive, but pertinent, questions about the role of mathematics in the markets. These centred on the nature of the probabilities mathematicians used. Most people see probabilities as measures of relative frequencies, a more sophisticated approach is to think of them as measures of information, financial mathematicians think of probabilities as abstract measures that economists sometimes refer to as ‘fictitious probabilities’. Ehsan’s question was essentially: what was the meaning of these probabilities that are central to mathematics’ Fundamental Theorem of Asset Pricing.

I reviewed the early history of mathematical probability in the sixteenth and seventeenth century and learnt that the field emerged out of commercial ethics. Working in an Aristotelian framework, mathematicians, such as Cardano, Pascal, Fermat, Huygens and Bernoulli, were trying to develop a mathematical representation of fairness, or justice, in exchange. This explicitly moral approach to mathematical probability was a feature of the discipline until around the 1830s.

I had made two observations: that there was a connection between contemporary, abstract probabilities and historical ‘moral’ probabilities. This observation was identified by the UK Research Council’s as one of “One Hundred Big Ideas for the Future” in 2011. The second observation, of wider interest, was that this linkage disappeared in the nineteenth century. Science is built on empiricism but the value of science is in explaining observations, and to explain my findings I had to move outside my field of mathematics and undertake scholarship in the humanities and social sciences.

The coherent explanation I came up with is going to be published in the Journal of Business Ethics. The heart of the explanation is that financial markets are, and should be treated as, centres of ‘communicative action’ with the purpose of achieving a consensus on the ‘just’ price of assets in an uncertain world. The key consequence is that markets should operate on the basis of norms of discourse, such as reciprocity, sincerity and charity. In this framework, mathematics provides the discursive language, rather than being a truth-bearer.

There are some significant implications for policy. My work has focused on how the concept of reciprocity is implicit in contemporary financial mathematics, but not explicit in mainstream economics. This means that most of the theory of financial economics can be retained; it is only the contextual meaning of the theory that needs to change. Specifically, an emphasis on reciprocity in pursuit of social cohesion replaces a focus on profit maximisation in a competitive arena. An immediate consequence is that the concept of ‘intergenerational reciprocity’ as advocated by Lord Stern to support climate change mitigation policy becomes conventional. It also questions the meaning of fiduciary duty in investment. Typically the duty has been interpreted as to maximise profits for the investor but there is significant concern that this approach inhibits the funding of projects that offer long term benefits. In my work I highlight how reciprocity endorses ‘not for profit’ investment strategies, by mutuals, credit unions and community crowdfunding initiatives.

On the basis of reciprocity alone, a lender is entitled to charge a high rate of interest to a borrower who is unlikely to be able to re-pay the loan. Charity (agape), the concern for others, is necessary to prevent these situations and is captured in the proverb of Quaker banking “How much do you seek to borrow? For how long? And how will you repay the loan plus its interest?”. It is not in the best interest of a borrower to induce them into debt bondage.

Sincerity was identified as a key norm by Habermas when he formulated The Theory of Communicative Action. In the context of contemporary markets sincerity allows us to evaluate the legitimacy of some market practices, particularly those associated with the emergence of computer based trading. One practice that has proved difficult is that of ‘order stuffing’, submitting orders to electronic exchange and then cancelling them within a microsecond. This is ‘legitimate’ if markets are seen as competitive arenas where agents are seeking to maximise their profits, but illegitimate if markets are centres of communicative action because the orders are ‘insincere’.

Today, when finance appears to be dominated by the short term profit maximisation of individuals, it seems strange to talk about ethical norms in the context of commerce. To get an idea on the peculiarities of the current state of affairs, consider the fact that Shakespeare personified the virtue Charity in Antonio, The Merchant of Venice. Less prosaic, there is a body of academic research that highlights the role commercial practice has played in stimulating both science and democracy. I believe we should bring these themes to the front when discussing the roles of finance in contemporary society.

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