It’s The Volatility, Stupid
By Ron Hylton

Recently there has been a lot of interest, and some controversy, surrounding index-weighting schemes that differ from traditional capitalization weighting, with the leading alternatives being equal weighting and various flavors of weighting using so-called "fundamental" factors. Proponents claim that these schemes produce superior returns, but their explanations for why don't seem very convincing. Others point out that the strong results are all based on backtesting, which basically guarantees good results (or else you wouldn't see them), and that there's not enough real history to establish these claims. Questions also have been raised about the impact of additional trading costs due to the inherently higher turnovers associated with these schemes, as compared to traditional market-cap weighting.

I will show here that there is an extra source of positive carry for at least some of these weighting schemes, which causes them to outperform in a sideways market or when the index components move with a common trend. Moreover, I will show that the extra return has nothing to do with the schemes' ability to provide superior stock picking; instead, it is a purely mechanical phenomenon that follows from the index methodology. The name of this effect is "the monetization of volatility." It is closely connected to the phenomenon of "positive gamma," which should be familiar to options traders. This effect can cause these weighting schemes to outperform capitalization weighting in a sideways or common-trend market, but at the cost of underperforming when the index contains components moving in mixed, sustained trends.

I'm going to focus primarily on fixed-weight (including equal-weight) schemes. Fixed-weight schemes are common in the commodity arena; many commodity indexes and funds allocate a fixed (although not necessarily equal) proportion of the total investment to a number of different commodities, and then rebalance monthly or quarterly. Similarly, equal-weight equity funds (including the increasingly popular Rydex equal-weight exchange-traded funds) allocate equal weights to all the stocks in an index, and rebalance on a regular schedule. These funds are unknowingly experiencing the "monetization of volatility" in their returns.

Simple Math

Let's start simply, with a two-asset, equal-weight index composed of assets A and B. Suppose each starts at $1/share and we have $100 to invest. Our initial portfolio would include 50 shares of A and 50 shares of B. Now suppose that when the time comes to rebalance, A is worth just $0.80/share, while B is still worth $1/share. The portfolio is now worth $90, and our strategy requires us to sell $5 of B and buy $5 of A, so that we have $45 of each, and the resulting portfolio is 56.25 shares of A and 45 shares of B. Now suppose that A goes back to $1/share while B remains unchanged at $1/share. The portfolio is now worth $101.25. Hmm … the market is back to exactly where it started, and yet our index is up 1.25 percent.

Let's go back to the initial case and look at the opposite scenario, wherein A first rises to $1.25/share and then returns to $1/share, while again B does nothing. (Note: $1.20 might seem to be the correct comparison, but in a log-normal investment world, the opposite of 0.8 is 1/0.8, or 1.25.)

With A at $1.25 and the portfolio holding 50 shares of both A and B, the portfolio is worth $112.50. We need to own $56.25 of both A and B, so we must sell $6.25 of A and buy $6.25 of B. As a result, the portfolio holds 45 shares of A and 56.25 shares of B after rebalancing. If A returns to $1/share, the portfolio value becomes $101.25. Hmm … once again the market is back to exactly where it started, and yet our index is up 1.25 percent. (If we'd used $1.20 instead of $1.25, the index would still make money, but less. The diligent reader will discover that the amount we make is nearly proportional to the square of A's fluctuations.)

Looking at the two scenarios, we see that the direction of the initial fluctuation doesn't matter: Either way, the portfolio makes money.

What's going on here? It's the monetization of volatility. The fixed-weight investing scheme is forcing us to buy low and sell high at each rebalance, in response to market fluctuations. In these particular examples, we're making money off the volatility of A; B never budged, so it has no volatility and its trades are a wash. Of course, you could just swap A and B in the above scenarios and now we would be making money off the volatility of B. As hinted at above, what we're looking at is really the monetization of variance: The positive carry generated is proportional to volatility squared.

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