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Kindly see the emboldened sentence below. Is it hyperbole? How can fund managers charge infinity as a fee?!!?!

      But even this recalculation substantially understates the real cost of active beat-the-market investment manage-ment. Here's why: Broad-based index funds and exchange- traded funds that reliably produce the market rate of return—with no more than market risk—are now avail- able with even lower fees than when the first edition of Elements was published. Today, market-matching returns

Page 141 below.

are available to all investors at low "commodity" prices, such as 5 basis points (5/100 of 1 percent). Therefore, investors should consider the fees charged by active man- agers not as a percentage of total returns, but as the incre-mental fee for active management as a percentage of the incremental risk-adjusted returns above the market index.
      Thus correctly stated, management fees are quite high. Leaving aside several interesting quibbles—such as that, for individual investors, taxes on short-term gains with 100 percent portfolio turnover can be significant—fees are remarkably high. Incremental fees are somewhere between 50 percent of incremental returns and, amazingly, infinity for the majority of fund managers who do not beat the market. And fees are generally closer to the high end than the low end of that range. Are fees for any other services of any kind at such a high proportion of value?

The Elements of Investing: Easy Lessons for Every Investor 2013 Updated Edition. By Burton Malkiel, Charles Ellis. p 141.

  • If the incremental return is zero and you still has to pay a fee $f$, you could say you have to pay "$\frac{f}{0}=\infty$" percent incremental fee. But I don't know if this is the correct interpretation. – garondal Jan 30 '21 at 22:44
  • @garondal Isn't any number divided by 0 undefined? That's what I remember from school. –  Jan 30 '21 at 22:52
  • Yes, therefore I used the " ... ". Mathematically it's wrong. See David's answer below. – garondal Jan 30 '21 at 22:53
  • @garondal "Yes, therefore I used the " ... ". What do you mean? I don't see ... in your comment. –  Jan 30 '21 at 22:54
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    I think garondal was referring to the quotation marks, not to three dots. – David K Jan 30 '21 at 22:55

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Fund managers set their fees as a percentage of total investment, at most a few percent. The writer is asking what value they produce for you in return for the fee, and says that is what you should compare the fee to. Moreover the only value they produce is the amount by which they can do better than the index funds. If they don't beat the index funds, the value they have delivered to you is zero, so the author claims that any fee they charge is infinitely larger (in proportion) than what you got in return.

That's not a mathematical statement; mathematically, you can cause a lot of trouble by trying to say that a positive number divided by zero is infinity. But the question (as framed by this author) is much the same as this question: if you try to buy beef from a butcher, and the butcher charges you ten dollars but does not give you any beef, what is the butcher's price per pound for beef?

David K
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