How Much Leverage Is Dangerous? It depends on where you put it.

Averages are among the most dangerous yardsticks in investing. After all, people regularly drown in rivers with an *average *depth of six inches. Similarly, average risk can be quite misleading – if your standard deviation is 20%.

But survivors of the Great Liquidity Crisis of 2008- 2009 know that. The question now is how to gauge risk and cage it appropriately. For most hedge funds, a major risk is leverage – and a major cause of bankruptcy. But what is acceptable leverage – leverage that can be controlled – and what is not?

Ditch the arithmetic and go for the “harmonic” – *means *that is. An arithmetic mean takes all the numbers and treats them equally. A “harmonic mean” segments them. That’s why hedge funds of funds may present a lower risk than multi-strategy hedge funds – or bank prop desks.

Here’s an example: “…take a FoF with equal investments in 100 separate funds. In the first structure (A), assume each fund has leverage of 2. In the second structure (B), all funds have leverage of 1, except for the last one, which has leverage of 100. How do we compare the leverage of these two structures? ” For the arithmetic moment, both funds appear to have the same leverage.

That example is posed Philippe Jorion, professor at the Paul Merage School of Business, University of California at Irvine and Mayer Cherem, his associate at Pacific Alternative Asset Management Company (PAAMCO), Irvine, CA, in their paper “Limited Liability Leverage (L3): A New Measure of Leverage,”

But do they have the same leverage? It depends on the harmonic mean for leverage, which they name L3. But why a harmonic mean? Multi-strat funds can be compared to the big investment banks that led us into the financial crisis of 2008-2009-2010 (if anyone’s still counting). The prop desks aggregate risk across all trades, which leads to an average risk position that can be fed into a value at risk schema.

“For an institution with multiple trading units or desks,” Jorion and Cherem say, “balance-sheet leverage is equivalent to average leverage, which is computed by adding the leverage of each unit multiplied by its portfolio weight. This is appropriate when losses are absorbed by a common capital cushion, as in the case of large commercial and investment banks, as well as for multi-strategy (MS) hedge funds, where capital is shared across traders in different sectors.”

It was certainly appropriate for Long-Term Capital Management and Amaranth, where “losses on some positions propagate throughout the entire portfolio, threatening the viability of the entire institution.”

But not all funds are structured that way. In a fund of funds,

“all individual investments, or hedge funds, are kept separate from each other because they are structured as investments in separate limited partnerships or corporations…. Due to the segregation of funds, investors in each individual fund cannot lose more than their investment in each fund, so that the failure of one fund cannot spill over onto the others.”

So what does “L3” – the harmonic mean – tell us? Let’s go back to example of FoF A and FoF B. Using L3, FoF B actually has a leverage of 1.01 – not the leverage of 2 produced by the arithmetic mean. “Therefore, this measure of leverage correctly signals that the structure for B is much less risky than that for A,” write Jorion and Cherem.

But we’re talking about risk, not reward. How useful is a L3 as a downside measure – as a risk dampener? In some sense, it involves the difference between an option on a portfolio versus a portfolio of options. It thus becomes a matter of payoff functions – and break points.

“The question is whether there is a single measure of leverage that could make the profile of the [FoF] portfolio and single-fund structure comparable,” the paper notes. “To do so, we need to decrease the slope, or leverage, of the single-fund line. Exhibit 5 illustrates the result. The two lines now cross each other at point C. Because the break points are different across structures, however, it is impossible to have a perfect fit. To the left of this point, the FoF structure dominates. The gain is measured by triangle G, which is the area between the two lines. To the right of this point and up to the origin, the FoF structure loses relative to the single-fund. The loss is measured by triangle L.”

When liquidity declines and rivers of capital dry up, hedge funds often sink or swim depending on their leverage. So they may be well advised to look beyond averages.