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Mutual Fund = 
Index Fund + Hedge Fund


Ross M. Miller
Miller Risk Advisors
September 12, 2005

[This commentary  appears in the September/October 2005 issue of Financial Engineering News. This is the raw, unedited version of that commentary.]

Hedge funds are evil and anyone who invests in them is seriously misguided. That is what the financial media have been telling us for some time. Fortune slammed hedge funds back in August of 2001 with no lasting effect. The money keeps flooding in along with the disparagement.

"Outrageous" fees—typically 2% of invested funds per year plus 20% of all positive returns—are behind most of the gripes. While critics make a big deal of the fact that there are no givebacks for negative returns, they neglect to mention that no one is forced to own hedge funds. Compared with the typical, big-name mutual fund, the 2% cut alone would appear excessive. Hedge funds must be doing pretty well, certainly better than the efficient-market crowd would lead us to expect, to justify such inflated fees. Or are mutual funds more expensive than they appear?

Consider two major no-load mutual funds. The top fund by assets under management is an index fund that tracks the S&P 500 almost precisely and has an expense ratio (expenses divided by assets under management) of a mere 18 basis points (bps) per year. (I will neither name names nor delve into messy details here—interested readers can visit the related working paper, "Measuring the True Cost of Active Management by Mutual Funds."

The second largest no-load fund, which was for a long time the largest fund overall, also tracks the S&P 500, if less precisely. It had an expense ratio of 70 bps at the end of 2004. Like a hedge fund, this fund's fees are linked to its performance, albeit to a far lesser degree.

Now for the "equation" in the title to this piece, which is an equation more in a literary sense than in a mathematically one. The typical non-index mutual fund can be separated into two distinct two parts. The first part, known as the systematic or passive part, behaves like an index fund that tracks the stated benchmark for the fund. The second part, known as the active part, behaves like a hedge fund whose returns are uncorrelated with those of the benchmark index. (This is originally how hedge funds worked and how they got that name, but now virtually all unregulated investment vehicles are so designated.)

Investors in mutual funds, however, cannot buy the two parts separately—they face what economists call "bundling." To get the active part, one also has to take the passive part. This is like taking your car into the dealer for repairs and being required to purchase parts from him at a premium to what they might cost at an auto supply store. The true cost of the dealer's labor as an economist would see it is the billed labor plus the implicit "overcharge" for the bundled parts. The "portfolio alpha" strategies that have become popular with institutional investors are a way of substituting passive and active investments that can be mixed and matched for traditional, bundled investments.

While the title equation has not made into investment textbooks, a variance-based version of it has. Page 320 of the sixth edition of Investments by Bodie, Kane, and Marcus shows how the variance of a fund's returns can be separated into a passive component (the variance "explained" by the index) and a residual component (the variance uncorrelated with the index and hence not explained by it). The passive component is nothing more than a beta-adjusted version of the index known as the fund's tracking portfolio.

This variance decomposition comes from regressing the fund's returns against those of the index. The beta of this regression tells how much leverage is required to construct the tracking portfolio and R-squared indicates how much of the fund's variance is explained by the index. These fund statistics are available for free on the Internet. The index fund mentioned earlier has an R-squared of 100%, while the other fund has an R-squared of 99%. (Both numbers are from Morningstar at the end of 2004 and are rounded to the nearest percent.)

Comparing these two funds, the second fund is "overcharging" for indexing by 52 bps in return for active management that contributes just 1% to the variance in the fund's returns. It appears that for each $100 invested in this fund, $99 is being indexed and only $1 is being actively managed. Of that dollar, 52 cents is eaten up by expenses.

The problem with this back-of-the-envelope analysis is that mutual fund shares are denominated in dollars, not variance. Making a rigorous leap from variance to dollars in order to work out the true cost of the active part, however, requires an assumption concerning the fund's variance.

It seems natural to assume that because both the index fund and the hedge fund are derived from the same mutual fund assets, they should have roughly the same variance. (This is not an innocuous assumption and raises issues that the aforementioned working paper addresses.) Doing a bit of algebra, the 99 to 1 split in variance translates into a 90.87 to 9.13 split in assets. Properly allocating the extra 52 bps on 90.87% of the investment to the other 9.13% yields what I call the fund's active expense ratio of 5.87%.

Whether a fee of 5.87% for active management is outrageous is a matter of personal opinion. It is worth noting, however, that the fund in question sported an alpha (a standard measure of fund performance that adjusts for the passive component) of -2.67% relative to the S&P 500 for the three-year period ending December 31, 2004. Considering that only 9.13% of its funds were implicitly being managed actively, outrageous may only begin to describe an active alpha of -27.45% over a three-year period.

Okay, you might think, so this fund is a bad apple. In fact, as large-cap funds go, it may be a poor performer, but it is relatively cheap. The average active expense ratio for no-load and low-load large-cap funds is 7% and the mean active alpha is -9%. Funds in other categories fare a bit better—a rough average taken over the Morningstar universe of funds gives a mean active expense ratio of over 5% and a mean active alpha of under -3%.

This exercise in financial engineering resembles (and was inspired by) the standard decomposition of an option into equity and debt parts. While these two decompositions come about very differently, they do share the property that beta never enters into either of them. The division of assets between passive and active management is independent of the amount of leverage (as captured by beta) applied to them.

Another common feature is that the underlying mathematics boils down to a single equation. For options, the equation comes courtesy of heat-diffusion; for mutual funds, we have Pythagoras and Euclid to thank. And although option pricing formulas have yet to become standard built-in calculator functions, any dollar-store calculator with a square-root key can be used to calculate a fund's active expense ratio and active alpha.

Whether or not hedge funds are worth their high fees, especially with more funds fighting for less alpha, remains open to debate. If you are looking for active management, however, most mutual funds look to be no bargain.

Copyright 2005 by Miller Risk Advisors and Financial Engineering News.