Secrets Of Success: The answer's mathematical, if you work it out

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The Independent Online

As a one-time alumnus of that notorious number cruncher's laboratory, the Massachusetts Institute of Technology (don't ask why they let me in), I have always had a soft spot for those who inhabit the strange higher planes of maths, physics and economics. It is an unending source of fascination to observe how those who pursue any branch of pure maths have to struggle so hard to reconcile their world, a closed system of rare and genuine beauty, with the awkward facts and frictions of what we like to call the real world.

One of the great triumphs or tragedies of the past 30 years, depending on your point of view, has been the incursion of mathematicians into the field of finance and investment. The two greatest breakthroughs over the past century have both been rooted in mathematics, though many other theories and ideas have proved less successful.

The two big successes, I would say, are Marokowitz's insight into the way that portfolios of securities behave (so radical an idea it was largely ignored for 20 years after its promulgation); and second, the Black-Scholes options pricing model, which appears to capture successfully most of the important elements of option pricing, despite being based on simplifying assumptions that do not, in fact, hold in the real world.

These powerful ideas are discussed with approval in a book by an American professor, John Allen Paulos, A Mathematician Plays the Market, featured on these pages three weeks ago.

Professor Paulos, you will recall, is the successful academic and author of popular maths books who owns up to having behaved like an idiot (and lost a fortune) by repeatedly buying shares in WorldCom all the way down from their giddy heights during the boom until the company's spectacular demise last year.

It is not clear whether Professor Paulos wrote the book to try to explain to himself why he behaved as such an idiot, or whether he hit on the idea of including his misfortunes to guarantee an audience for what is, in the main part, a rather different book, namely a primer on what maths-minded folk should know about the investment markets.

As all Professor Paulos's previous books are also essentially maths primers for the interested layman, something tells me the second is probably nearer the truth, entertaining though the story of his obsessive love affair with WorldCom stock is.

The one I have read before, A Mathematician Reads the Newspaper, is an amusing account of how numerically illiterate most newspapers are, so there is a certain pleasure to be had in reading that mathematical genius need to be no barrier to financial comeuppance. A lot of this new book will cover familiar ground for those who have a good grounding in maths. Yet two small things and one big thing struck me with force after reading it. One is quite how important a difference there is between arithmetical and geometric averages in looking at, say, annual rates of return from different assets.

It is no accident that fund management companies and brokers tend to emphasise the average rate of return from shares over long periods, rather than the geometric, or internal, rate of return.

The former, I knew, is invariably higher than the latter. What I hadn't appreciated until reading the good Professor Paulos is that the geometric rate of return is also the median (or most common) return likely to be achieved by investors, a fact, along with the burden of costs, that helps to explain why fund investors so frequently fail to capture a big chunk of the market return their funds are investing in.

He gives the example of a hypothetical investment that on average doubles in a week half the time, and, on average, loses half its value in a week the rest of the time.

The arithmetic average weekly return you can expect is 25 per cent per week, which would turn £10,000 into £1bn in a year: the geometric return, the one that will happen to you most often, is a more realistic 0 per cent, suggesting your investment will still be worth £10,000 after 52 weeks. Some difference.

The second small thing I liked in the book is the clever demonstration of how often and how successfully random outcomes can appear to be something else.

I believe it is no exaggeration to say that a lot of marketing in the financial services business is based on trying to convince you that an impressive result, which may be random, is due to skill or judgement on someone else's part. It does not follow that skill or judgement do not exist, but it does mean, in Professor Paulos's words, that "the huge element of chance present in the market cannot be denied".

The most striking thing that comes out of the book is that, for all the advances made in understanding markets over the past 70 years, the real nature of the game was actually delineated with devastating elegance by John Maynard Keynes nearly 70 years ago.

As well as being an economist, Keynes was a great mathematician, but one who knew enough about the real world to know the movement of share prices was only partially based on fundamental values.

Just as important for investors was trying to work out how other investors would judge the same piece of information. Markets, in other words, are complex interactive systems in which the attitude and behaviour of other investors is clearly the most important single driving force.

That is why no rules hold forever and why even boffins can and do make spectacular fools of themselves.

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