Beta Mutual Funds: Do They Make Cents?

BACKGROUND
The concept for these beta mutual funds is that the investing vehicle will offer a leverage of ROC (rate of change of daily return) against a given underlying index. For example, Rydex's RYCTX Titan Fund has a goal of multiplying the everyday up day positive ROC, or down day negative ROC by a factor of 2 (thus establishing a beta goal of 2.0). The index to be leveraged, SP-CP (FastTrack's designation for the Standard & Poor's 500), has a daily standard deviation (Std.) of 1.15% and an average up day ROC of 0.8%. Thus, the goal of RYCTX is to double the Std. to 2.3% and the up day ROC to 1.6% (and unfortunately, it doubles the down day ROC to -1.6% as well).

Let's take a simple case for the annual return of the S&P 500 and see what the effect would be of doubling all daily returns. Suppose the up day (average) ROC is +1% and the down day (average) ROC is -1%, and over a year, there are 131 up days and 121 down days (with an up day bias 52%), yielding a net 10 up days (a typical value for index mutual funds), then the annualized gain (ANN) is:

ANN (index) = (1.01)^10 = 1.1046 or 10.46%
ANN (beta=2) = (1.02)^10 = 1.2190 or 21.9%
But 1.1046 ^ 2 = 1.220 = 22.0% (close to the 21.9% above)

Thus, doubling by a beta of 2 has the effect of raising the (perfect) beta fund ANN by the square (which is greater than a double). So instead of investing in the S&P 500 (as you can in Vanguard using VFINX), you can (in principle) more than double your gain by buying RYCTX. In fact, Rydex will tell you its goal is to double the everyday ROC, but of course they do not promise a square rate increase in ANN. This is understandable, as they have a difficult task to replicate; at the end of each day, a ROC value that changes randomly day by day. They attempt to do this by buying options, futures, and swaps to try to properly leverage the daily ROCs to the goal value.

This sophisticated approach is complicated, since every day, they have to settle the overall equity as investors are constantly buying and selling, and there are costs associated with all such turnovers. These three companies also state that increasing beta increases the standard deviation, thus increasing the risk, although they do not quantify that risk. In my mind, this makes the true danger associated with such funds obscure to the investor, a task I will try to make clear.

FIGURE 1: BETA MUTUAL FUND SUMMARY OF DATA

I set two basic goals, one to replicate an up return beta (for example, doubling the ROC) every up day and, of course, every down day as well. In practice, it is not possible to do that, so some up days are "lost" and even reversed, such that the net up days for a given year are not perfectly preserved (in fact, they are generally lowered). A measure of this is the correlation term, which indicates how the beta fund replicates the number of up and down days. Even though the correlation has, apparently, a very high correlation (the order of 98% or more), this small change from 100% can lead to a deterioration of the desired ANN (as will be explored subsequently).

A second, and even more difficult requirement, is to try to replicate the up and down ROCs values as precisely as possible every day. In my Working-Money.com article (October 7, 2005) entitled "The Effects Of Tilt On Mutual Fund Investing," I defined "tilt" (the difference between the up day ROC minus the down day ROC) and showed that, in general, it is of negative value leading to a significant reduction in the desired (no tilt) ANN. Here, I will show in the subsequent data for the beta funds that the beta process exacerbates this problem and is the major reason why the beta ANN falls well short of the theoretical square rate enhancement goal. As an aside, these two goals can be thought of as analogous to a high-fidelity amplifier that would need to have a high level of amplitude as well as good reproduction of frequency (of time). If this didn't occur, distortion would be introduced.

FIGURE 2: BETA MUTUAL FUND PERFORMANCE

At this point, I should point out a key difficulty associated with beta funds. It is mathematically true that with 100% correlation and exact replication of all ROCs that the ANN for a beta of 2 will increase as the square (as previously explained), but, probably, unknown to beta fund investors, the MDD will also increase at a square rate. For example, if the index has an MDD of -50% (0.5 of equity), then a "perfect" beta 2 fund would have an MDD of -75% (0.5^2 = 0.25 of equity). This is something that the mutual fund people do not make clear: the beta fund could have more than twice the standard deviation (implying over twice the risk), but that does not convey the more worrisome message that the MDD could deteriorate at a square rate (in fact, as can be seen from the data, the actual funds always exhibit more than the theoretical MDD).

This example of gain associated with a 1% ROC illustrates that the buy & hold ANN is proportional (actually resulting in a square rate output) to the net up days in a year. How well do the beta funds replicate the net up days? Fund 1 does well, missing only one day over a year (accounting for its very nice 0.998 correlation to SP-CP), but as each day, on average, has a ROC of 1.6% that represents a loss of about -1.6% to the ANN. For fund 3, a net difference of 4.4 days with an average ROC of 1.5% results in a much greater loss of -6.6% compared to theoretical. It is now obvious that a non-100% correlation leads to a reduction in the beta fund ANN.

Another difficulty is illustrated with fund 1, exhibiting a tilt of -0.164%. This would not be a problem if the theoretical fund had the same tilt, but beta SP-CP has only a -0.116% tilt. This means that the beta fund has created an additional beta tilt of -0.048%, which (shown later) results in another -5.6% loss in ANN over the theoretical. These are the two main effects I have found leading to the poor ANN performance of the beta funds, resulting in a deterioration ranging from -7% to -18.4% ANN for the seven funds studied. This performance is so poor that we would be better off staying in the non-beta index funds, particularly when MDD is taken into account.

With our emphasis now on MDD, look at fund 7 (UOPIX), which has the inevitable task of replicating the NASDAQ (dominated by volatile high-tech stocks), a devastating MDD of -98.4%, a result that would clearly destroy any portfolio unfortunate enough to hold such a fund over the period March 27, 2000, to October 7, 2002, the recent bear market period. Even though this fund has an additional five years more than the MDD period, it still could not achieve a positive ANN, even though its beta surrogate does. The reason, an excessive tilt of -0.126%, which spread over 117 down days per year, results in a -14% loss compared to the surrogate gain of 12.4%, a most distressing result.

FIGURE 3: EFFECTS OF TILT AND NET UP DAYS ON ANN

One reason for the generation of negative tilt is the effect of fund expense on both positive and negative ROC. Suppose the positive day beta (up day performance only) is exactly 2.0 and the negative day beta (down day performance only) is also 2.0. There should be no additional tilt term introduced. But now, at the end of each day, the mutual fund company imposes its operating expense (applied to equity), which has the effect of reducing the positive ROC by a small amount and increasing the negative ROC by a similar small amount. This effect can be thought of as decreasing the positive beta and increasing the negative beta, generating a net negative beta differential. This effect is either added to or slightly suppressed by the variable daily beta, resulting in the final resulting net tilt. It is little wonder this net tilt is usually negative.

Undoubtedly, the recent bear market has contributed to the difficulties that the beta funds encountered over the period studied. I can imagine a huge number of sell orders causing all kinds of chaos (expense) within those companies. To take a more favorable period, I looked at a post-bear market period from October 9, 2002, to January 30, 2006, for Rydex's Nova fund, RYNVX, with improved results for ANN/MDD:

RYNVX 25.2%/-22.1%
RYNVX2 25.6%/-21.3%
SP-CP 16.4%/-14.7%

Here there is an improvement in ANN, with a higher but tolerable MDD. This illustrates that, with caution (and perhaps such luck), you could invest in such a beta fund as long as you don't go off to Hawaii for long periods and lose sight of the investment.

CONCLUSIONS
The seven long-term beta mutual funds studied all had a rather disappointing performance with suppressed ANN. and increased MDD compared to their theoretical models. It was to be expected, theoretically, that the ANN increase would exceed the beta ratio, whereas all funds failed to do so. The reason for this was their failure to more accurately replicate the net up days (nonperfect correlation) and to exacerbate the negative tilt (already inherent) within the surrogate funds. Just as distressing is the excessive MDDs exhibited by all beta funds exceeded even the theoretical models. This is an important parameter to any investor, and the beta fund companies do not make this danger clear. They just comment that "increased risk is to expected," yielding little insight into what can occur in practice. In fairness, if you were able to pick a more favorable time period, enhanced ANN could be obtained with the inevitable increase in MDD.

SUGGESTED READING
Brown, Norman J. [2005]. "The Effect Of Tilt On Mutual Fund Investing," Working-Money.com, October 7.

-Rydex (rydexfunds.com)
-Potomac (potomacfunds.com)
-Profunds (profunds.com)
-Investor FastTrack software (Fasttrack.net)