Rebalancing Leveraged and Inverse Funds

Transcription

1 Rebalancing Leveraged and Inverse Funds Joanne M. Hill, PhD, and Solomon G. Teller, CFA ProFund Advisors LLC and ProShare Advisors LLC November 2009 ABSTRACT Leveraged and inverse Exchange Traded Funds (ETFs) funds designed to achieve a positive or negative multiple of index returns on a daily basis have grown to more than $30 billion in assets in the three years since their introduction in mid This article explains how periodically rebalancing a leveraged or inverse fund position can be an effective strategy for investors whose goal is to achieve returns close to the index return times the fund multiple (e.g., +2x or -2x) over time, especially during periods of high volatility. Historical analysis of unrebalanced versus rebalanced returns based on a variety of indexes across a range of market conditions provides empirical evidence of the potential effectiveness of rebalancing over 6-month holding periods. The process for monitoring and rebalancing fund positions involves measuring the gap between the index return and fund return during the holding period, and rebalancing when the gap moves beyond a pre-specified trigger percentage. (The position size is increased if the index is appreciating faster than the fund, and it is reduced if the index is lagging.) Rebalancing was most effective during high-volatility periods, when compounding has a more pronounced effect on returns. Inverse strategies and high-volatility underlying indexes generally called for more frequent rebalancing than leveraged long strategies and low-volatility indexes. Finally, we present a case study covering the first six months of 2009 that illustrates the potential effectiveness of rebalancing for a strategy providing -2x the daily return of the S&P 500. The information contained herein is for educational use only and is not meant to be investment advice. There is no guarantee that any of the strategies mentioned will result in favorable results. Trading will result in commissions and transaction costs, and will create potential tax consequences, which will have a negative impact on the investment returns. The strategies discussed do not take these factors into account. All investing involves risk, including possible loss of principal. For Financial Professional Use Only; Not for Public Distribution. 1

2 This is an edited version of the article originally published in Institutional Investor Journals, 8th Annual ETF Guide. Introduction Leveraged and inverse Exchange Traded Funds (ETFs) funds designed to achieve a positive or negative multiple of index returns on a daily basis have grown to more than $30 billion in assets since their introduction in mid These ETFs quickly gained popularity as knowledgeable investors found them useful for implementing a variety of strategies. However, the recent high-volatility environment which has raised questions about the risk and performance features of many investments has caused some concerns about leveraged and inverse ETFs. The primary area of concern is the performance of leveraged and inverse ETFs over time. Most are designed to achieve daily return objectives and it is widely recognized that they have performed well in delivering the multiple of the index return stated in their objectives (e.g., a stated daily multiple of +2x or -2x). 2 But over time, because of compounding, their performance may be greater or less than the stated multiple of the index return for the period. However, compounding does not necessarily preclude investors from using leveraged and inverse funds successfully over time. Looking at a variety of holding periods over the last 50 years, our previously published research found that, on average, the effect of compounding on leveraged and inverse strategies with daily objectives was virtually neutral over time. 3 In addition, our study showed that for broad indexes, there was a high likelihood of achieving returns close to the stated multiple over short periods. Generally, the shorter the holding period and the lower the index volatility, the higher the likelihood of getting closer to the stated multiple. In addition to putting the performance of leveraged and inverse ETFs during the recent high-volatility environment into historical perspective, our previous research showed that monitoring and rebalancing can be a valuable tool for investors whose goal is to achieve returns close to the stated multiple times the index return over time especially when the volatility of an index is high. Further research outlined in this article takes a closer look at the effectiveness of rebalancing daily objective funds in pursuing this goal. This article explains the rebalancing concept for leveraged and inverse funds, compares historical results for a variety of indexes across a range of market conditions, and analyzes the effectiveness of rebalancing. We found that: Rebalancing was an effective way to get close to the stated multiple times the index return over time. The benefits of rebalancing were most significant during highvolatility periods. Using a 5% trigger, the rebalancing frequency depended on whether the stated daily multiple was long or inverse, and on the volatility of the underlying index. Inverse strategies and high-volatility indexes generally called for more frequent rebalancing. For long-leveraged strategies (+2x), the average frequency of rebalancing needed ranged from as little as 3 to 4 times a year for lower-volatility indexes (such as the S&P 500 or an index of 20+ Year Treasurys) to as often as once a month for a more volatile index (like the NASDAQ-100). For inverse strategies, the range was from about every 2 weeks for lower-volatility indexes to as often as once a week for more volatile indexes. 2 For Financial Professional Use Only; Not for Public Distribution.

3 Of course, not every investor s goal is to match the stated multiple over time, so it is important to note that rebalancing serves to remove both the negative and potentially positive effects of compounding. Compounding reduces losses and increases returns in a trending market, so the returns from a rebalancing strategy can be lower than those of an unrebalanced strategy in such situations. In practice, trading costs and tax consequences, which have a negative impact on returns, also need to be considered. Rebalancing Involves Monitoring and Adjusting the Fund Position: Mind the Gap Rebalancing involves monitoring and regularly adjusting the fund position over time. For leveraged and inverse funds, our analysis shows that the most straightforward approach is to monitor the value of the position one would have had in a traditional unleveraged index strategy and compare this to the value of one s current position in the leveraged or inverse daily objective fund. Exhibit 1 illustrates the general guidelines in the rebalancing process: Add to the fund position if its current value is less than you would have had in an unleveraged long index strategy, and decrease the position if its current value is higher than you would have had in an unleveraged long strategy. Exhibit 1: Rebalancing Process Fund Return Less Than Index Return Increase Fund Exposure Rebalancing Process Decrease Fund Exposure Fund Return Greater Than Index Return For illustrative purposes only. For Financial Professional Use Only; Not for Public Distribution. 3

4 In a rebalancing strategy, the investor adds to or reduces the fund position, realigning exposure to seek returns close to the stated multiple of the index return. Exhibit 2 provides an example of the rebalancing process and rebalancing amounts for a +2x and a -2x fund for a period when the index returned +5%, and the +2x and -2x funds returned +9% and -9%, respectively. 4 Exhibit 2: A. Rebalancing After Index Returns +5% and +2x Fund is Up 9% Underlying Index $109 +2x Before Rebalance +2x After Rebalance Ending Value $105 Initial Value +5% $100 +9% $100 +9% } Sold $4 Index Value Fund Position Value Fund Position Value For illustrative purposes only. B. Rebalancing After Index Returns +5% and -2x Fund is Down 9% Ending Value Initial Value Underlying Index +5% -2x Before Rebalance $105-2x After Rebalance $100 $100-9% -9% Bought $14 $91 Index Value Fund Position Value Fund Position Value For illustrative purposes only. 4 For Financial Professional Use Only; Not for Public Distribution.

5 Note that both the +2x and -2x funds started with the same asset levels as an unleveraged long index strategy. But, at the end of the period, the +2x position size of $109 gives the investor a larger amount of exposure relative to the index than he or she started with: $109 to $105 (vs. initial $100 to $100). Rebalancing reduces the size of the position to $105, which means that the exposure is equal to $210. Inverse funds typically require more frequent trading and larger trades to keep exposure in line, because fund value moves in opposition to the index. This emphasis on maintaining the exposure of the leverage or inverse fund holding consistent with an unleveraged index leads the investor to a process for monitoring and rebalancing the fund positions: Monitor the gaps between the index return and fund return during the holding period, and rebalance when the gap moves beyond a pre-specified trigger percentage. (The position size is increased if the index is appreciating faster than the fund, and it is reduced if it is lagging.) To measure the gap, investors subtract their current fund position size from their initial fund position size increased by the index return amount. The size of the rebalance trade in dollars can be calculated as : 5 Initial $ Invested x (1 + Index Return) Current $ Assets in Position For instance, in Exhibit 2, the initial $100 x (1+ the 5% index return) = $105. Subtracting the $109 ending value of the +2x fund from $105 indicated a rebalancing trade of -$4, or selling $4. How often is rebalancing needed? The method an investor chooses to design a rebalancing strategy depends on the desired level of monitoring and the extent to which he or she wants to customize the rebalancing strategy to market conditions or to the volatility level of the index on which the fund is based. Rebalancing rules typically fall into one of three categories: Calendar-based: Adjustments are made at regular intervals, such as weekly or monthly. Fixed-percentage triggers: Rebalancing is triggered when the variance between index and fund return reaches a certain level, such as 5%. Dynamic-percentage triggers: The trigger percentage adjustment is larger in more volatile market conditions and smaller in lower-volatility markets. Percentage-return triggers are commonly used for asset-mix rebalancing and were selected as the methodological basis for this study. Percentage triggers have the advantage of being more adaptive to market conditions, leading to more frequent rebalancing in high-volatility periods and less in more trending markets. This approach is designed to be consistent with the compounding effects on leveraged and inverse funds that the rebalancing process is seeking to reduce. For percentage-trigger rebalancing, time intervals between trades vary depending on the leverage multiple and prevailing market conditions (that is, volatility of returns) of the index on which the fund is based. For Financial Professional Use Only; Not for Public Distribution. 5

6 Historical Analysis of Rebalancing Effectiveness: Methodology To develop meaningful insights into the effect of rebalancing, we examined how a 5% trigger-rebalancing strategy would have worked across a variety of market conditions using a long history of index returns. Our study focused on a 6-month investment horizon for strategies designed to deliver +2x and -2x the daily performance of an index (Daily Objective Strategy). To determine the degree of effectiveness of rebalancing, we compared the returns from these rebalanced leveraged and inverse index positions with the same stated multiples (+2x and -2x) of the index return for each 6-month period. The indexes analyzed included the S&P 500 Index over the 50 years between 1959 and 2009, Barclays Capital 20+ Year U.S. Treasury Index (since 1994), NASDAQ-100 Index (since 1985) and the Dow Jones U.S. Financials and Dow Jones U.S. Oil & Gas Indexes (since 1992). 6 To assess the impact from rebalancing, we examined unrebalanced returns along with returns based on a 5% rebalancing trigger for each 6-month holding period starting at the end of each calendar quarter. 7 This gave us a sample size that ranged from 200 holding periods for the S&P 500 to 60 periods for the 20+ Year U.S. Treasury Index. 8 Rebalancing occurred when the gap between the value of the assets based on the index return over the holding period was more than 5% above or below the value of the assets of the investor s position prior to rebalancing. This means that when the index return for the holding period was above the Daily Objective Strategy s return since the time the position was rebalanced, the rebalancing trade resulted in an increase in the size of the position. Similarly, the position size was reduced when the index return was below that of the Daily Objective Strategy. We assumed that the investor had ready capital to add to the position (at no financing cost), and earned no interest on proceeds from reductions or sales of the position. 9 To examine the effectiveness of the 5% trigger rebalancing strategy for each 6-month period for both the rebalanced and unrebalanced return series, we calculated the following measures for each index studied: The differences between the returns of the Daily Objective Strategy and the 6-month period index return times the same stated multiple. The realized leverage ratio or multiple over each 6-month period, calculated by dividing the Daily Objective Strategy returns by the index return over the same period with and without rebalancing. The frequency that these realized leverage ratios or multiples fell within a specific range of the stated multiple (with and without rebalancing): That is, for a +2x strategy, we looked at the range of + and 0.50 around the stated multiple. The average frequency of trading 10 and trade size 11 with rebalancing across all 6-month periods for each index. 6 For Financial Professional Use Only; Not for Public Distribution.

7 Comparison of Unrebalanced and Rebalanced S&P 500 Leveraged and Inverse Index Returns Rebalancing effectively reduced the largest performance differences We would not expect rebalancing to have much of an effect on the average difference between a +2x and -2x S&P 500 Daily Objective Strategy and their period index returns multiplied by +2x and -2x, respectively. On the other hand, we would expect rebalancing to reduce the difference in those periods in which compounding had a larger positive or negative effect. The average and median differences remained close to zero with and without rebalancing over the 50-year history of market conditions. For the +2x strategy, the average difference between the leveraged Daily Objective Strategy and the index period return times two was 0.4% unrebalanced and 0.1% rebalanced; for the -2x strategy, the average difference of 0.5% unrebalanced was reduced to 0.2% with rebalancing. 12 What was more significant was how much rebalancing affected the more extreme periods those with the most positive and negative differences between the +2x and -2x Daily Objective Strategy, and the period index returns times the same multiples. Exhibit 3 illustrates the significant impact of rebalancing for individual holding periods. The chart shows the sequence of differences for the last 30 years of our sample between the -2x S&P 500 Daily Objective Strategy return and -2 times the 6-month index return, both with and without 5% trigger rebalancing. From this chart we see: The differences were largest in volatile years: 1982, 1986, 1987, 2002 and Rebalancing was effective in reducing the size of these differences, with the biggest impact on those years in which volatility was large in magnitude. For Financial Professional Use Only; Not for Public Distribution. 7

8 Exhibit 3: Rebalancing at a 5% Trigger Reduced Differences for S&P 500 Inverse Strategies: -2x Daily Objective Return Minus -2 times the 6-Month Index Return 20% 15% 10% 5% 0% -5% -10% -15% -20% -25% -30% -35% Unrebalanced -2x Differences Rebalanced -2x Differences Note: Differences between -2x Daily Objective Strategy and -2 times S&P 500 period return (with and without 5% rebalancing trigger) over consecutive (non-overlapping) 6-month periods between 12/31/1978 and 12/31/2008. Source: Bloomberg. See Appendix A for complete data set. For illustrative purposes only. Rebalancing increased the percentage of returns that were within close ranges We also calculated the ratio of the Daily Objective Strategy returns to the underlying index returns over each 6-month period with and without rebalancing. Across these periods, the percentage of realized multiples that fell within a tight range around the stated multiple increased with rebalancing. Exhibit 4 shows the frequency of these periods within each of three ranges: +/-0.25, +/-0.50, and +/-1.0 around the stated daily multiple of or For Financial Professional Use Only; Not for Public Distribution.

9 Exhibit 4: Frequency (%) of Realized Multiples for S&P x and -2x Daily Objective Strategies (6-Month Periods) Within a Range of the Stated Daily Multiple +2x Multiple -2x Multiple Ranges Unrebalanced Rebalanced Ranges Unrebalanced Rebalanced 1.75 to % 95.0% to % 92.0% 1.5 to % 99.5% -1.5 to % 95.0% 1.0 to % 99.5% -1.0 to % 98.0% < 0 0.5% 0.5% > 0 2.0% 0.0% Note: 6-month time periods for S&P 500 Daily Objective Strategy analysis begin each calendar quarter over January 1959 March 2009 study time frame. Source: Bloomberg. For illustrative purposes only. The potential benefits of rebalancing for investors seeking to get closer to the stated multiple over periods of 6 months were clear from this analysis. For the rebalanced strategies, more than 90% of the realized multiples fell within the tightest range shown (that is, within + or of the stated multiple). For the +2x S&P 500 Daily Objective Strategy, the unrebalanced strategy was within the 1.50 and 2.50 realized multiple range 93.5% of the time, but this frequency increased even further to 99.5% with rebalancing. For the -2x S&P 500 Daily Objective Strategy, the percentage in the range of to (within 0.50 of the -2x multiple) increased from 71.5% without rebalancing to 95.0% with rebalancing. It is worth noting that the frequency of reversals was also reduced for the -2x Daily Objective Strategy (a reversal is a period in which the multiple was negative when the stated daily multiple was positive, or positive when the stated daily multiple was negative). 13 Inverse strategies called for more frequent rebalancing than long strategies How often investors would need to rebalance positions depended on whether the stated daily multiple was long or inverse, and on the volatility of the underlying index. For the +2x Daily Objective Strategy held for 6 months, the average number of days between rebalancing triggers was 88.4, or about once a quarter. For the inverse -2x Daily Strategy, rebalancing was triggered more frequently, averaging 12.1 days, or just under two weeks. Even though the trading frequencies were much higher for the -2x S&P 500 Daily Strategy, the size of each rebalancing trade was similar 5.3% on average for the +2x leveraged strategy, and 6.7% for the -2x inverse strategy. We would expect a trade size of at least 5% given that this is the size of the trigger for initiating a rebalance trade. For Financial Professional Use Only; Not for Public Distribution. 9

10 To more closely analyze the variation in the adjustments needed for effective rebalancing, Exhibit 5 shows the frequency of rebalancing for a -2x S&P 500 Daily Strategy for each 6-month period in our historical sample. The data is displayed in two ways bars representing the average number of days between rebalancing for each 6-month holding period and a line showing the number of rebalances per period. (Note: Within a given 6-month period studied, the actual number of days between each rebalancing varied depending on the time interval required to meet the 5% trigger.) Exhibit 5: Frequency of Rebalancing -2x S&P 500 Daily Objective Strategy Varies Over the Holding Periods Analyzed ( ) AVERAGE NUMBER OF DAYS BETWEEN REBALANCES Average Number of Days Between Rebalances Number of Rebalances Per Period NUMBER OF REBALANCES PER PERIOD Note: Frequency of rebalancing -2x S&P 500 Daily Objective Strategy with 5% band between 12/31/1958 and 3/31/2009. Source: Bloomberg. For illustrative purposes only. The frequency of rebalancing trades tracks index volatility very closely for this -2x S&P 500 Daily Strategy. Across the 50 years of index history studied, the average number of days between rebalances varied greatly. Between 2003 and mid-2007, a period of relatively low volatility for the S&P 500, it was common to average 15 to 20 days between rebalances. More recently, a 5% trigger would have led to rebalancing an average of every 3 to 5 days per period. Consistent with record volatility levels experienced in 2008, rebalancing frequencies in the 6-month holding periods for that year were the highest studied in the 50-year history of the S&P For Financial Professional Use Only; Not for Public Distribution.

11 Rebalancing Was Effective Across a Number of Indexes We also measured the impact of rebalancing for other indexes, including some with historically lower volatility than the S&P 500, such as a 20+ Year U.S. Treasury Index return, and some with higher volatility. The higher-volatility indexes analyzed included the NASDAQ-100 Index, the Dow Jones U.S. Financials Index and the Dow Jones U.S. Oil & Gas Index. The NASDAQ-100 had a daily return volatility of 28.6% (annualized) over the sample period of , as compared to the S&P 500, which measured 18.3% over the same period. The sector indexes also had higher volatility measures than we have seen for the S&P 500 the Dow Jones U.S. Financials at 24.85%, and the Dow Jones U.S. Oil & Gas Index at 24.80% over the period. Exhibit 6 summarizes the frequency of realized multiples falling into a range of to for a +2x Daily Objective Strategy and to for a -2x Daily Objective Strategy, both with and without rebalancing. The effect of rebalancing was very consistent across indexes, whether they exhibited low, moderate or high return volatility. The percentage of realized multiples after rebalancing in the range of +/ around the stated daily multiple is above 90% for both +2x and -2x Daily Objective Strategies. This was observed even for the higher-risk sector indexes where the -2x Daily Strategy showed only a 59% and 60%, respectively, of frequency of realized multiples in the to range without rebalancing. Exhibit 6: Frequency (%) of Realized Multiples Across Indexes for +2x and -2x Daily Objective Strategies (6-Month Periods) Within a +/ Range of the Stated Daily Multiple +2x Daily Objective Strategy* -2x Daily Objective Strategy* Unrebalanced Rebalanced Unrebalanced Rebalanced Barclays 20+ Year U.S. Treasury 91.7% 96.7% 75.0% 95.0% S&P % 99.5% 71.5% 95.0% NASDAQ % 97.8% 51.1% 96.7% Dow Jones U.S. Financials 88.2% 97.1% 60.3% 91.2% Dow Jones U.S. Oil & Gas 79.4% 94.1% 58.8% 97.1% Note: *Frequencies are for the range of +1.5 to +2.5 for +2x Daily Objective Strategy and -1.5 to -2.5 for -2x Daily Objective Strategy. 6-month time periods begin each quarter over the following dates for each index: S&P 500 (1/59-3/09), NASDAQ-100: (10/85-12/08), Barclays 20+ U.S. Treasury: (4/94-6/09), Dow Jones U.S. Oil & Gas and Dow Jones U.S. Financials: (1/92-3/09). Source: Bloomberg. For illustrative purposes only. For Financial Professional Use Only; Not for Public Distribution. 11

12 Exhibit 7 shows the average days between rebalances across the periods studied for each index. 14 The amount of rebalancing required was a function of the volatility of the index. The long-term U.S. Treasury +2x Daily Objective Strategy was rebalanced the least, on average, with 127 days between rebalancing trades, and 17% of the periods not triggering any rebalancing trades at all. For the +2x index strategies, the NASDAQ-100 triggered the most frequent rebalancing, averaging every 35 days across 6-month holding periods. Exhibit 7: Average Days Between Rebalances (5% Trigger) Across 6-Month Periods +2x Daily Objective Strategy -2x Daily Objective Strategy Barclays 20+ Year U.S. Treasury S&P NASDAQ Dow Jones U.S. Financials Dow Jones U.S. Oil & Gas Note: Data show average of the mean days between rebalancing trades for each period. 6-month time periods for index analysis begin each quarter over the following index histories: S&P 500 (1/59-3/09), NASDAQ-100: (10/85-12/08), Barclays 20+ Year U.S. Treasury: (4/94-6/09), Dow Jones U.S. Oil & Gas and Dow Jones U.S. Financials: (1/92-3/09). Source: Bloomberg. For illustrative purposes only. As we observed with the S&P 500-2x Daily Objective Strategy, the average number of days between rebalancing was significantly lower than the +2x version of the strategy. The 20+ Year U.S. Treasury Index -2x Daily Objective Strategy had an average of 16 days between rebalances, while the NASDAQ-100 the most volatile index had the most frequent rebalancing of once every 6 days, on average. A Look at the effectiveness of Rebalancing a -2x S&P 500 daily Strategy in early 2009 In the study, we showed the value of rebalancing over a long history of returns for several indexes. By looking at a specific, recent 6-month period, we see the results even more clearly in a particularly volatile market environment. We tracked the cumulative returns of a -2x Daily Objective Strategy based on the S&P 500 index over the first half of 2009, a period when the index had levels of far-above-normal return volatility. 15 Exhibit 8 shows the cumulative return based on closing index values of an unrebalanced -2x Daily Strategy, along with a rebalancing strategy using the 5% trigger, assuming all rebalancing was done based on closing index levels and ignoring transaction costs For Financial Professional Use Only; Not for Public Distribution.

13 Exhibit 8: Rebalancing Significantly Tightens Performance Difference for a -2x S&P 500 Daily Objective Strategy for First 6 Months of % 60% Unrebalanced -2x Strategy 50% CUMULATIVE RETURN 40% 30% 20% 10% 0% -10% 5% Trigger Rebalanced -2x Strategy 1.78% -3.62% -20% S&P 500 Index % -30% Dec 08 Jan 09 Feb 09 Mar 09 Apr 09 May 09 Jun 09 Note: Cumulative Return on S&P 500 index, unrebalanced and rebalanced -2x Daily Objective Strategies (using 5% rebalanced trigger) for period from December 31, 2008, through June 30, Source: Bloomberg. For illustrative purposes only. Through 70% early March 2009, the index had fallen more than 20%, but then reversed to end the 6-month period up 1.78%. From year-end 2008 through June 30, 2009, there was 60% a large difference between the % change in the unrebalanced -2x Daily Objective Unrebalanced -2x Strategy Strategy 50% and -3.60% (that is, close to -2 times the 5% 1.78% Trigger index Rebalanced return -2x Strategy for the period). Using a 5% rebalancing trigger, the 6-month return S&P for 500 the Index -2x Daily Objective Strategy 40% was -3.62% percent, very close to the theoretical -3.60% goal and consistent with the results of the 30% historical study detailed earlier. CUMULATIVE RETURN On 20% March 5, about two months into this 6-month period in 2009, the S&P 500 index reached its low. Notably, at this point, the unrebalanced -2x S&P 500 Daily Objective Strategy position 10% showed a large gain relative to the rebalanced position, benefiting from the compounding effect in the downward-trending market. This shows that rebalancing (like 0% 1.78% compounding) can reduce as well as increase returns depending on the degree of trend -3.62% and volatility -10% in the marketplace during the holding period. Later in the period, after the S&P 500 reversed direction and began to rise and move in a trading range, the unrebalanced -20% % strategy began to lag the value of the rebalanced -2x Daily Strategy position, and it was over -30% this period that the rebalancing of the position reduced the gap between -2x times the Dec 08 Jan 09 Feb 09 Mar 09 Apr 09 May 09 Jun 09 period index return and that of the -2x Daily Strategy. For Financial Professional Use Only; Not for Public Distribution. 13

14 Rebalancing Helps Narrow the Gap The performance of leveraged and inverse ETFs is affected by compounding which means that, over time, their performance may be greater or less than the fund s multiple times the index return. Prior research has shown that generally for holding periods up to about 30 days, leveraged and inverse Daily Objective Strategies based on broad indexes had a high likelihood of achieving returns close to the stated multiple. The lower the volatility of the index and the shorter the time period, the higher the frequency of returns within a tight range to the stated multiple. Rebalancing can be an effective tool to help investors pursue returns closer to the stated multiple. Over time, the research shown here indicates that a buy-monitor-rebalance strategy with a 5% trigger achieved returns closer to the stated multiple times the index return than an unrebalanced strategy. For long leveraged (+2x) Daily Objective Strategies, on average, the frequency of rebalancing was as little as about once per quarter for low-volatility indexes and about once a month for more volatile indexes. For inverse (-2x) Daily Objective Strategies, the frequency of rebalancing was higher on average, but varied quite a bit depending on the level of market and index volatility associated with the strategy. For the indexes studied, the highest frequency was about once a week for the NASDAQ-100, and the lowest was once every two weeks for the S&P 500 and Treasury indexes. We did not include transaction costs in our analysis. Investors must carefully evaluate the benefits of the reduction in the compounding effect against the trading costs they might incur to achieve this reduction. Rebalancing is most likely to be effective when the market is experiencing both a high degree of volatility and is not trending; these are times when investors should also consider monitoring and reevaluating changes to other investment strategies in their portfolios, rebalancing their asset mix and monitoring any hedging or risk-reduction strategies. The authors would like to thank George Foster for his contributions to this article. 14 For Financial Professional Use Only; Not for Public Distribution.

15 References Marco Avellaneda, Stanley Zhang. Path-Dependence of Leveraged ETF Returns, Working Paper. Courant Institute of Mathematical Sciences, New York University; Finance Concepts, Richard Co, Leveraged ETFs vs. Futures: Where Is the Missing Performance? CME Group Research & Product Development, February 9, Joanne Hill and George Foster. Understanding Returns of Leveraged and Inverse Funds, Journal of Indexes, September/October Matthew Hougan, How Long Can You Hold Leveraged ETFs? Journal of Indexes, March/ April Lei Lu, Jun Wang, Ge Zhang. Long-Term Performance of Leveraged ETFs, Working Paper. Shanghai University of Finance and Economics, Jason Ruspini, Making Leveraged ETFs Work, Seeking Alpha, July 7, Yesim Tokat, Portfolio Rebalancing in Theory and Practice, Vanguard Investment Counseling & Research, For Financial Professional Use Only; Not for Public Distribution. 15

16 Endnotes 1. Source: Bloomberg, as of August 31, Matt Hougan. IndexUniverse.com, Getting Leverage. Going Short. Webinar, May 14, Joanne Hill and George Foster. Understanding Returns of Leveraged and Inverse Funds. Journal of Indexes, September/October The time frame is not specified but could be days, weeks or months. The daily objective leveraged and inverse returns are chosen to provide an illustration of returns one might find as a result of compounding effects to highlight the direction and size of the rebalancing trades. The returns are not intended to predict fund performance levels in particular market conditions. 5. The rebalancing trade required in shares or units can be found by dividing the dollar amount by the estimated closing price or NAV of the ETF or mutual fund. 6. This 50-year period for the S&P 500 is consistent with that used in the Journal of Indexes article referenced above. The other dates are based on the inceptions of the indexes. 7. In this article, we focus on rebalancing with a 5% trigger but have also conducted analyses using calendar-based rebalancing and percentage triggers ranging from 1.5% to 10% on the same sample of index-return history. Results for these alternative triggers are available from the authors, but in all cases they support our conclusions that rebalancing was an effective way to get close to the stated multiple times the index return over a 6-month holding period. Inverse strategies and high-volatility indexes generally called for relatively more frequent rebalancing with all triggers analyzed. 8. There was an approximate 50% overlap between consecutive 6-month periods. 9. Summary of Assumptions: For the unrebalanced strategy return, the daily +2x and -2x exposure is reset daily. Therefore, the Daily Objective Strategy return is exactly the daily multiple times the daily index price return each and every day of the holding periods. For the equity indexes, dividends are ignored in calculating the rebalance timing as well as in the realized leverage and return differences. The Barclays 20+ Treasury Index returns are total returns. All return calculations excluded fees, financing, interest and expenses. There were no capital constraints (that is, losses could have exceeded -100%). Rebalancing was based on end-of-day index prices and implemented at those prices on the same day. No transaction fees or taxes were incurred in connection with rebalancing the Daily Objective Strategy position. For 6-month periods where no rebalancing was triggered, the full period length of 183 days was used as the number of days between rebalances. The 5% trigger criteria for this particular study were measured by using a ratio of index values to stated Daily Objective Strategy values. Differences could also be used to establish a trigger, which would yield slightly different results. 10. The average number of days between rebalances and number of trades in each 6-month period. 11. The average trade size as a percentage of the rebalanced closing value of the position. 12. The median differences for +2x were -0.1% unrebalanced and 0.1% rebalanced; for -2x the median differences were -0.4% unrebalanced and 0.2% rebalanced. 13. Note that reversals also occurred when the observed returns are close to 0.0, so that a small return difference can lead to a sign switch. 14. These statistics are based on the average days between rebalances within each period. 15. The annualized standard deviation of S&P 500 daily returns was 34.8% for the period 12/31/08 through 6/30/09. Source: Bloomberg. 16. Our study used no transaction costs since they can vary. To get an idea of the difference these costs might make, we also did this analysis assuming a 10 basis-point transaction cost per rebalance. The 5% trigger rebalancing strategy would have had a return of -4.09% (0.47% lower than without transaction costs). 16 For Financial Professional Use Only; Not for Public Distribution.

17 Appendix A: Data Displayed in Exhibit 3 on page 8 Differences between -2x Daily Objective Strategy and -2 Times S&P 500 period return (with and without 5% rebalancing trigger). Starting Date for -2x Daily Objective Strategy Differences Actual S&P 500 Performance 6-month Periods Unrebalanced Rebalanced Index Return Index Volatility 12/31/78 0.1% 0.1% 7.1% 9.8% 06/30/79-1.2% 0.5% 4.9% 11.9% 12/31/79-2.7% 0.1% 5.8% 16.8% 06/30/80 5.7% 0.2% 18.8% 16.2% 12/31/80-2.2% -0.4% -3.4% 12.7% 06/30/81-2.1% 0.6% -6.6% 14.3% 12/31/81 0.1% 0.3% -10.6% 14.3% 06/30/ % 0.5% 28.3% 21.7% 12/31/82 6.8% -0.2% 19.5% 14.6% 06/30/83-2.1% 0.0% -1.9% 12.0% 12/31/83-1.0% 0.3% -7.1% 12.5% 06/30/84 0.1% -0.5% 9.2% 12.9% 12/31/84 4.2% 0.3% 14.7% 10.2% 06/30/85 1.4% -0.1% 10.1% 10.3% 12/31/85 6.3% -0.2% 18.7% 14.0% 06/30/86-3.3% 0.0% -3.5% 15.4% 12/31/ % 0.8% 25.5% 16.4% 06/30/ % -0.9% -18.7% 42.3% 12/31/87-2.0% -0.3% 10.7% 20.6% 06/30/88-2.3% -0.1% 1.5% 12.7% 12/31/88 3.7% 0.1% 14.5% 11.7% 06/30/89 0.8% 0.5% 11.1% 14.3% 12/31/89-2.4% 0.7% 1.3% 13.2% 06/30/90-3.8% 0.5% -7.8% 18.3% 12/31/90 1.2% -0.3% 12.4% 15.4% 06/30/91 1.9% 0.2% 12.4% 13.2% 12/31/91-1.5% -0.2% -2.1% 10.4% 06/30/92 0.2% 0.1% 6.8% 9.0% 12/31/92-1.0% 0.0% 3.4% 10.0% 06/30/93-0.3% 0.1% 3.5% 7.1% 12/31/93-1.0% -0.3% -4.8% 10.3% 06/30/94-0.9% -0.5% 3.4% 9.3% 12/31/94 7.6% 0.0% 18.6% 7.9% 06/30/95 3.6% 0.3% 13.1% 7.9% 12/31/95 0.4% 0.2% 8.9% 11.8% 06/30/96 1.1% 1.0% 10.5% 11.9% 12/31/96 6.6% 0.2% 19.5% 15.4% 06/30/97-2.7% 0.9% 9.6% 20.6% 12/31/97 4.9% 0.3% 16.8% 13.6% 06/30/98-5.9% 1.0% 8.4% 25.2% 12/31/98-0.8% -0.3% 11.7% 19.2% 06/30/99-2.4% 0.7% 7.0% 16.9% 12/31/99-8.9% 0.7% -1.0% 24.7% 06/30/00-3.8% 0.1% -9.2% 19.4% 12/31/00-7.0% 0.4% -7.3% 23.0% 06/30/01-5.2% 0.0% -6.2% 19.7% 12/31/01 0.3% -0.6% -13.8% 18.5% 06/30/ % 0.9% -11.1% 31.8% 12/31/02-2.0% 0.0% 10.8% 20.7% 06/30/03 3.2% -0.7% 14.1% 12.6% 12/31/03-1.7% 0.3% 2.6% 11.5% 06/30/04-0.4% 0.2% 6.2% 10.7% 12/31/04-1.7% 0.5% -1.7% 10.8% 06/30/05-0.7% -0.5% 4.8% 9.8% 12/31/05-1.7% 0.1% 1.8% 11.2% 06/30/06 2.6% -0.2% 11.7% 8.7% 12/31/06-0.7% -0.5% 6.0% 11.3% 06/30/07-5.7% -0.6% -2.3% 19.5% 12/31/07-2.6% -0.6% -12.8% 21.2% 06/30/ % -0.9% -29.4% 53.9% For Financial Professional Use Only; Not for Public Distribution. 17

18 Important Information Joanne Hill and Solomon Teller are associated with ProFund and ProShare Advisors LLC. This material is for financial professional use only. It is not for the public. It is not to be republished without the express consent of ProShare Advisors LLC. This material is not designed to represent the performance of any specific investment or make a recommendation for purchase. Information contained herein has been obtained from sources believed to be reliable, but the accuracy of the information cannot be guaranteed. This material is not designed to represent the performance of any specific investment or to make a recommendation for purchase. Any projections and/or forwardlooking statements regarding estimated investment outcomes are based on assumptions that we believe are reasonable at this time. However, actual results may vary materially from stated expectations, and we make no guarantees about specific investment results. 18 For Financial Professional Use Only; Not for Public Distribution.

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