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Jack Bogle and Burton Malkiel are the arch-prophets of indexation. Nobody has done more to ram home public understanding of the rationale for passive investment, or to make it a reality by offering index funds to the masses.

Mr Malkiel, an economics professor at Princeton – where Bogle wrote his undergraduate thesis on mutual funds – even sat on the board of Vanguard, the fund management company Mr Bogle founded, and which still dominates the market for index-linked mutual funds.

Mr Malkiel’s A Random Walk Down Wall Street has sold more than a million copies, and has made more than one generation of investors aware that it is prohibitively difficult to beat the market with consistency, and virtually impossible to do it if you charge the normal fees for active management.

But the two men are on opposite sides of one of the biggest debates in the world of fund management. Mr Bogle is against exchange-traded funds, and Mr Malkiel is for them.

ETFs seem to be rapidly taking over as the vehicle of choice for index funds in the US. Over the 12 months to the end of April, US equity mutual funds suffered significant net outflows, while the assets held in US equity ETFs increased their assets by $71.8bn, or 30 per cent – vastly more than can be explained by the rise in the market, as the S&P 500 rose only 13.6 per cent over the same period.

But Mr Bogle has set himself against the trend. In his latest book, The Little Book of Common Sense Investing, he declares: “The ETF is a trader to the cause of classic investing. Surely using index funds as trading vehicles can only be described as short-term speculation.”

This is the crux of Mr Bogle’s objection. Short-term trading is a mug’s game, as he and Mr Malkiel have shown. So why produce an index fund that can trade minute by minute on an exchange?

ETFs cannot make the classic index fund’s guarantee to investors that they will earn “their fair share of the stock market’s return”, he claims. In fact, he says, “after all the selection challenges, the timing risks, the extra costs, and the added taxes, typical ETF investors have absolutely no idea what relationship their investment return will have to the return earned by the stock market”.

The growth, or “stampede” as he calls it, to launch ETFs has seen particular interest in concentrated sector funds whose costs, he says, “can run three to six times the level of the lowest-cost all-market index funds”. He suggests ETFs should come with a warning label, either “Handle with Care”, or “Caution: Performance-Chasing at Work”.

He is, of course, on strong ground when he says Wall Street is pushing ETFs hard, and they are being used for possibly excessive speculation. But what of the example of someone who already has a brokerage account (so the extra expenses are limited), who makes a few disciplined purchases of ETFs and keeps hold of them for the long term?

He admits there is a possibility that the widest “total market” ETFs could match index mutual funds on this basis.

Mr Malkiel, meanwhile, is an enthusiast. “I’ve been a fan of index funds since before index funds existed, so how could I not like ETFs?” he asks.

Mr Malkiel was speaking at a breakfast hosted in New York by Barclays Global Investors, which has embraced ETFs far more enthusiastically than Vanguard. Why the enthusiasm? First, he sees ETFs’ ability to avoid paying tax on sales of appreciated stock as an “inherent advantage” (although he conceded that a mutual fund would need huge outflows before this became a problem).

He also sees a bright side to the way ETFs encourage trading.

“There are those who really believe they can time the market and trade in and out – I don’t think they can, but there are those who believe that. If they do that within a mutual fund framework, they create costs for people sitting in the mutual fund.”

In an ETF, short-termists are free to do their own thing, without damaging fellow shareholders who want to “buy and hold”, he argues. For those who want to time their way in and out of sectors, he again sees little harm in allowing those who think they can do it to try.

Indeed, Mr Malkiel admits to trying a variation on this himself. The core of his personal portfolio is in broad indices, but he attempts something more interesting with satellites. “That strikes me as a very sensible way to proceed – it’s the way institutional investors in general proceed.”

Building on his experience as a company director in the biotechnology industry, he describes biotech as “a crapshoot”. “Even biotech companies themselves don’t know which one is going to make it.”

But there is obvious huge money to be made by the biotech company that makes a big medical breakthrough – so he suggests buying a concentrated sectoral ETF. “Many of them will fail and some of them will discover the new great drug. It strikes me as an excellent way to index and a great use of the exchange-traded fund to do it.”

Mr Malkiel and Mr Bogle still seem to be friends. However, Mr Bogle has taken to “sending me everything negative that’s ever written about ETFs”, says Mr Malkiel.

As for Mr Malkiel’s forthcoming book, which will advocate investing in China through ETFs – “Jack doesn’t know about it yet, but I’m sure he won’t like it”.

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