Focusing on Fuzzy Finance

Financial professionals are inundated with information. With input from continuous trading systems, specialised publications, press agency dispatches and real-time data vendors, "information overload" would appear to be inevitable. However, the real problem is not the information itself but what to do with it. For any given market, four situations can apply: imprecision ("The CAC40 will trend upwards "); uncertainty ("I reckon the Buba will trim the discount rate by 50 basis points"); contradiction (simultaneous forecasts from different sources, one bullish, the other bearish); total consensus, with precision and certainty (generally known as "wisdom in hindsight"!) Moreover, the first three situations can coincide, making the decision-making process particularly complicated. But, as we realise every day, not impossible. Moreover, there is a tool that can be used systematically to process information from a wide variety of sources. Known as "fuzzy calculus", it is studied in detail in issue no. 23 of Quants Focusing on fuzzy finance , written by Laurent Bellity, a fund manager with CCF Gestion, and Richard Dalaud, a researcher at the DRI. Fuzzy technology A statement such as "The market is underpriced" actually contains information that can be extracted and used. This realisation led to the development of a rigorous discipline known as fuzzy set theory, which has applications, inter alia, in the field of possibility theory. A logical extension of classical sets, fuzzy sets are those in which the membership function is not limited to the two values 0 and 1 but can take on any value between 0 and 1. The statement: "Based on P/E, the market is underpriced" can be represented by a fuzzy set, whose grade of membership is 0 for a P/E of 30, 1 for a P/E of 4 and, for example, _ for a P/E of 10. The technological applications of fuzzy systems are both numerous and spectacular. For example, the underground railway (subway) network of the Japanese city of Sendai has an automatic system based on fuzzy logic. Camcorders have fuzzy settings for stabilisation and focusing. Fuzzy systems can also be found in washing machines, rice cookers, photocopiers and a host of everyday items. At the industrial level, we find products that adapt gradually to the degree of hardness in water, to particle size and to contrast medium. Fuzzy logic, which was very much in vogue in Japan five years ago, continues to gain ground and win over new aficionados. Choosing a portfolio in the presence of imprecise expectations Returning to the world of finance, let us take the example of a portfolio manager investing in the equity markets of France, Japan and the USA. He can make imprecise forecasts about the future level of the three leading indices: "The CAC40 will finish the year on a rise of around 15%, the Dow Jones will advance by about 10% while the Nikkei will make the strongest gains of all, moving ahead by some 18%". With fuzzy calculus, it is possible to take account of the imprecision of "around" and "about", using a fuzzified version of a Markowitz optimisation process to determine the best risk/return profile. In such an approach the risk stems from the elements that are uncertain (volatility) and imprecise (informational quality). It can safely be assumed that an investor who is averse to uncertainty will also shun imprecision. For a given degree of aversion to uncertainty, we will consider two cases. First, we rely on market volatility to measure the level of imprecision. Second, we consider that the information is more precise in our own market than in the US market and that, because of the distances involved, Japan is the least well known of the three. In the first case, the relative weightings of France, the USA and Japan are 35%, 20% and 45%, regardless of the degree of aversion to imprecision we have used in our assumptions. Fuzzy calculus not only allows us to take these expectations into account but also produces stable results. In the second case, the asset allocation depends on aversion to imprecision. The most averse manager invests on the basis of 50% CAC, 40% Dow Jones and 10% Nikkei, while the least averse opts for 35%, 20% and 45%. This time, fuzzy calculus lets us handle the differing quality of the expectations. Naturally enough, aversion to imprecision is reflected in a preference for the domestic market, which is considered to be the best-known. Matching cash flows due on fuzzy dates A similar problem faces the corporate treasurer who has to disburse set amounts Ð cash on delivery, payment for a project start-up, etc. Ð at dates that are imprecise. He wants to play the yield curve, investing his available funds on the longest possible horizon while remaining able to meet a demand for liquidity. With fuzzy calculus, he can match the cash flows to his needs by using imprecise information on payout dates. In short, fuzzy techniques make it possible to give due consideration to information that is qualitative or unformalised. We frequently find ourselves in situations where we know more than we think. But did you know that something so fuzzy could be handled with such precision?