Mr Dye is head of fund management at PDFM, one of the country's biggest institutional investors with around pounds 50bn to look after. He has attracted attention recently because he keeps a relatively large proportion of that sum (as much as 15 per cent) in cash, rather than in shares and bonds. He prefers cash at the moment because he believes that stock markets are overvalued and heading for a fall. Not just a little fall, but a crash - a global calamity like the crashes of 1929 and 1987.
So far it hasn't happened, and some of Mr Dye's clients are said to be wondering about his judgment. While he has sat on the sidelines with their cash, they have missed out on the benefit of some sharp increases in share values. PDFM is said to be slipping down the fund managers' performance tables and there are suggestions that some clients will take their money elsewhere.
But Mr Dye is sticking to his guns, and what worries the markets most is that he might turn out to be another Roger Babson. Babson won fame as the man who predicted the great Wall Street Crash of 1929. As one writer put it a few years later: "Mr Roger Babson had predicted the crash for several years, which shows, among other things, that he had been very wrong for several years - before he suddenly became very right."
But today Mr Dye is not alone; his unease about stocks and in particular American stocks is now shared by a significant and growing number of his peers. There is a distinct sense that, having been very wrong for a while, he might be on the verge of becoming very right. So are we really heading for a crash? And if we are, does it matter to anybody outside the City's Square Mile?
Yes, beyond all doubt a global stock market crash is approaching. It will cause havoc in financial markets and wipe billions from the wealth of individuals and institutions alike. It will destabilise economies; it may even bring down governments. Some people will lose their livelihoods. Some will lose their lives. It will be big and it will be bad.
Unfortunately we do not know when it will happen. It may be next week. It may be in the next millennium. Fifty-eight years separated the Wall Street Crash of 1929 and the market crash in October 1987, but it would be a brave investor indeed who staked all his savings on there being no repetition before another 58 had elapsed, in 2045.
Since a crash is a function of the unexpected, by definition its arrival cannot be predicted with accuracy. What can be detected is when stock markets ought to or might crash, and that is what many are saying is the case now.
In the markets the prophets of doom are called "bears", and the current bear-pack includes not just Mr Dye, but Nick Knight, chief economist at Nomura of Japan, and Robin Aspinall of National Australia Bank. Both Mr Knight and Mr Aspinall predict a sharp "correction" (the analysts' euphemism for a crash) in New York, spreading to London, which could see the value of shares in both markets fall by 25 per cent by the end of the year - enough to wipe pounds 250bn off London shares.
Mr Knight is openly comparing 1996 with 1987. "I'm not trying to be sensationalist, but all the signs are there," he says.
Others are gloomy, but more cautious. Professor Tim Congdon, one of the Chancellor's "wise people", predicts "a correction of 20 per cent in the American market", but believes that London is not due for quite such a sharp fall. "It won't be as extreme as '87," he says.
These fears have been around since the summer (although Mr Dye has been worried for rather longer), when Wall Street began to twitch over fears that US interest rates might have to rise.
Higher interest rates are generally seen as bad news for stock markets: higher rates mean higher borrowing costs which mean lower profits for the companies represented on the stock market. That in turn means lower dividends, lower return on investment and tighter belts all round.
But why should it portend a crash? Interest rates are bound to go up - and down - every now and then, but it doesn't have to mean the end of the world. Why should a rise now be bad news? The answer lies in the broader statistics of the financial markets, and in particular in the relationship between shares and bonds, something that is held by many traders to be vital.
Bonds are government debt - you lend the government money and it pays you interest at a fixed rate in return. They are traded in the markets just like shares in companies, and their price goes up and down according to the levels of supply and demand. Big investors, shopping around the markets for the best deal, look for a good yield in bonds and shares. If the price is high and the return - the interest or the dividend - is low, then they look elsewhere. (Small private investors can have a stake in the bond market, through funds which invest there).
Bond yields and share yields - the income received by the investor as a percentage of the capital value of the share or bond - tend to have some correlation. If the difference between them becomes too great, then, according to current market thinking, some adjustment is required.
This can be expressed in figures. Past trends suggest that the trigger point comes when the ratio of average bond yields to average share yields jumps over 3:1, that is, when an investor can get three times as much income from a bond as he can from a share of the same value.
It looks like a mere statistic, but it is not. This is what happened in the US market in 1987 shortly before Black Monday, and it happened again in 1994, just before the market went into another decline. What is the ratio on Wall Street today? It stands at around 3.5:1.
In such circumstances, analysts tell us, the markets are likely to "correct" or "adjust" yields. What they do is to sell shares, driving prices down: a lower capital price by definition means a higher yield.
It sounds orderly, but it isn't. Markets, as the demise of the Exchange Rate Mechanism showed in 1992, are like big animal herds; small movements can turn into stampedes with astonishing suddenness. If the idea of a crash gets about, everyone with shares will want to sell at once, unless they are very brave or very foolish.
The yield ratio is a popular "warning light" for investors, but it is not the only one. Nor, unfortunately, is it the only one that is flashing at the moment. Some traders, for example, believe they can foresee trouble by adjusting the FT all-share index to take account of inflation. Using that measure, the index appears to have a natural "ceiling", which it reached in 1968, 1973, 1987 and 1994, all of which were followed by declines. Again, if you do the sums, the figures show that ceiling is now being tested.
The extent to which a crash might affect us all is, perhaps surprisingly, a matter of some debate. Few economists today separate the 1929 crash from the Depression, but opinion is divided on how far the 1987 crash was connected to the recession that followed soon afterwards. A lot of City yuppies may have been sacked, and ended up carrying their office belongings in bin-bags to their Porches while an unsympathetic nation sniggered, but that did not in itself cause the economic slowdown.
On that occasion, governments in the US and Britain were quick to react, cutting interest rates and increasing liquidity in the markets - giving investors something to spend. Economists believe this delayed, rather than prompted the recession.
But circumstances have changed. "The last time we had a large stock market crash the economic impact was positive because government immediately boosted the economy," said Andrew Sentance, professor of economics at London Business School and a former adviser to the Chancellor. "This time the authorities may hold back on cutting interest rates and stimulating the banking system. In a climate of low inflation and not very rapid growth, this could cause problems."
Bridget Rosewell, chairman of Business Strategies, an economic consultancy, and another of the Chancellor's "wise people", warns that business confidence is immediately eroded by a market crash. "The value of businesses goes down in a crash, which distracts business people and makes them look over their shoulders," she said.
Nick Knight, at Nomura, thinks that this time around a crash will have a direct impact on people's wealth - and their inclination to spend money. "Nearly everyone now has holdings in the market in one way or another, even if it is just via pensions and PEPs. In the US, mutual funds [the American equivalent of unit trusts] have become the preferred way to keep savings. So nowadays it would be flippant to say that a crash didn't matter to the man in the street."
A new crash, in other words, would probably wreck whatever feel-good factor there may be in the wider economy today. People with share investments would lose money, while the rest of us would slip back into the cautious, underspending mentality of the recession. Economic recovery would be dented, or worse.
Of course, it may not happen. The market's preparedness to crash was tested last July, when there were fears that US interest rates were set to rise. Higher interest rates meant higher bond yields; higher bond yields meant a wider disparity between bonds and shares and that in turn meant "correction".
Shares fell quite sharply. The Dow Jones index of leading stocks shed nearly 400 points, or around 7 per cent of its value, in July alone. London, as is so often the case, caught the cold represented by the abrupt Wall Street sneeze.
For a few weeks there was fever, and the pundits became extremely voluble and gloomy. The market was overvalued, they said; the flow of funds from small investors which had fuelled share price rises would dry up; the differential between yields on bonds and yields on shares was at danger point; the bubble represented by the dramatic rise in the share prices of high-technology companies would burst.
Elaine Garzarelli, a stock market guru who shot to fame when she correctly and accurately predicted the 1987 crash, was predicting on 21 July that the Dow Jones index would rise by about 1,000 points. Two days later, as alarm bells rang, she was saying it would fall by about 1,000 points.
The reality has been somewhat different. Shares bounced back. The flow of funds from new investors, which fell to $490m (pounds 313m) the last week in July, has bounced back and reached $3.5bn in the second week in September. The high-tech bubble has not burst, with some of the leading shares such as Microsoft and Oracle reaching new highs. Recent economic data has suggested that US inflation is not out of control, and at a meeting last week the US central bank decided that it did not need to raise interest rates.
Mr Dye and his fellow bears, however, are not impressed and they may yet have their moment.
Inflation remains a concern for central bankers on both sides of the Atlantic. So far Eddie George, Governor of the Bank of England, and Alan Greenspan, his opposite number at the US Federal Reserve Board, may not have pressed for increases in interest rates, but questions are being asked about their reasons.
Mr George does not have the right to adjust interest rates; that is the prerogative of the Chancellor and with an election on the way the last thing the Government needs is a rise in interest rates. In the US there is a presidential election in November. Although the Fed can act independently of the Clinton administration there is an argument that it is better for the Fed to keep a low profile ahead of that election.
The worry, then, is that interest rates are being held artificially low for political reasons. This would imply, some traders believe, that once the elections are out of the way the battle against inflation can begin again, with a rise in interest rates which, having been deferred, might have to be sharper than an earlier, more genteel adjustment would have been. That could be enough to set the big bear running.
Should you sell now? Business, page 16