Over the past decade, from 2000 to 2009, the S&P 500 produced positive calendar returns about 60% of the time (based on four negative calendar-year returns in 2000, 2001, 2002 and, of course, 2008). On the surface, that seems to be a pretty impressive batting average.

The problem is that negative returns have an ugly mathematical property. Very simply, a negative return requires a larger positive return to restore a portfolio to breakeven condition. As shown in "Ugly Math," on page 126, the larger the loss, the larger the gain needed to get back to the starting point. For example, a 35% loss requires a 53.8% gain to break even.

Over the past 10 years, Vanguard 500 Index (an S&P 500 index fund) actually lost ground despite a batting average of 0.600. A $10,000 investment on Jan. 1, 2000, was worth $9,015 (with reinvested dividends, but not accounting for taxes or inflation) as of Dec. 31, 2009. Even though the S&P 500 had more positive years than negative years (six to four), the down years overpowered the up years because of the brutal realities of ugly math.

Big losses require extraordinary gains to make up for them-and that hasn't been the case over the past 10 years. The 10-year average annualized return of Vanguard 500 Index (VFINX) from 2000 to 2009 was -1%.

Is there any way to protect a U.S. equity portfolio against such large losses? One idea is to use an inverse fund that seeks to behave opposite to a particular index, such as the S&P 500.

To test this theory, this analysis will use Rydex Inverse S&P 500 (RYURX, formerly called the Rydex Ursa Fund) and Vanguard 500 Index to represent the S&P 500. The time frame for the analysis is from 1995 to 2009, a 15-year period, and the raw data in this study comes from the Morningstar Principia database.

OPPOSITES ATTRACT

In theory, an inverse fund should move in the opposite direction from the index it is attempting to mirror. For example, if the index fund goes up 20%, the inverse fund is supposed to go down 20%.

In general, inverse funds tend to behave as expected. But it's important to note that their "inverse-ness" is not perfect, as "Short of Inverse," on page 127, shows.

During the bull run of the late 1990s, the inverse fund performed poorly while the index fund's performance was stellar. Five years of great returns for the S&P 500 from 1995 to 1999 could only spell trouble for an S&P 500 index inverse fund.

However, notice that in 1995, VFINX was up 37.5%, while its inverse counterpart RYURX was only down 3.89%. In theory, the inverse fund should have had a return of around -37%. The following year, VFINX was up nearly 23% and the RYURX was only down 12.44%.

Two performance measures point out this lack of perfect mirrored performance: a 15-year correlation between VFINX and RYURX of -0.92 and a 15-year beta coefficient between the two of 0.8. The correlation coefficient indicates that the performance of these two funds moves in opposite directions 92% of the time. This is not perfect; a perfect inverse relationship between two funds would have a correlation coefficient of -1.

The beta coefficient of 0.8 indicates that over the past 15 years, the inverse fund has been about 80% as volatile as VFINX. This relatively low beta is a direct product of RYURX failing to be fully inverse during times when the S&P 500 has a positive return.

Simply put, the inverse fund has demonstrated downside resistance, which it should not do, according to its designed behavior. Understandably, most investors wouldn't complain bitterly that this particular inverse fund (the only one in the Morningstar Principia database with 15 years of performance history) has a high beta when the benchmark index is down and a low beta when the index is up.

The high beta effect is visible in the years when VFINX had negative returns (2000, 2001, 2002 and 2009). Notice that RYURX has a positive return that is reciprocally larger than the corresponding negative return for the index.

For example, in 2000, VFINX lost 9.1%, but RYURX gained 21.6%. In theory, RYURX should have had about a 9% positive return in 2000. The respective returns of each fund in 1997 illustrate this low beta behavior. VFINX was up 33.2%, but its supposedly mirror image RYURX was only down 18.1%.

The long-term performance of the inverse fund has been poor (a 15-year average annualized return of -1.3% from 1995 to 2009). That is not surprising because in the long run the U.S. equity market tends to produce more positive returns than negative returns.