PRODUCTS PRODUCTS Figure 2: A “Bearish” Leveraged ETF* This was soon followed by actions by other major U.S. bro- LEVERAGED ETFs: kers, like Morgan Stanley. FINRA then issued a notice in June 2009 requiring its members to disclose fully the risks inherent in these products.3 Subsequently, in August, the Securities and All You Wanted to Know Manager of short Exchange Commission issued an Alert notice. LETF To better understand the suitability issues, let’s consider, for instance, UYG and its underlying ETF, the IYF. Figure 3 (be- Short $200 low) compares the price history of UYG and that of a static but Were Afraid to Ask in assets leverage-2 position in IYF, with both funds starting with the Determine number same investment in August 2008. Authorized of ETFs outstand- Leveraged exchange-traded funds present a variety or risks, particularly for ing, daily Market Participants Figure 3: UYG vs. 2X IYF, 1 Year* buy-and-hold investors. To manage these risks effectively, investors must 120 – Invest proceeds in MM 100 – understand the impact of different factors – including compounding, daily 80 – 60 – 40 – balancing and volatility – on returns. Money 20 – market B Y MARCO AVELLANEDA AND STANLEY JIAN ZHANG 0 – -20 – -40 – everaged exchange-traded-funds (LETFs) Figure 1: A “Bullish” Leveraged ETF* *Figure 2 is a schematic representation of the management of a bear- 1/4/2009 7/4/2009 2/4/2009 4/4/2009 8/4/2008 9/4/2008 3/4/2009 5/4/2009 6/4/2009 11/4/2008 12/4/2008 are garnering considerable amount of 10/4/2008 ish leveraged ETF for a double-short fund. In this case, the manager interest from investors, active traders adjusts his position daily by shorting the underlying asset with double *Figure 3 depicts the difference between the performance of UYG and and professional portfolio managers. leverage. a static double-leveraged position in IYF. Notice, in particular, that UYG From the point of view of traders sub- Manager of long Let’s consider the following example: ProShares’ Ultra Fi- does not grow like the double-leveraged fund in the March 2009 rally. ject Reg T margin, these instruments LETF nancial ETF (ticker symbol: UYG) offers investors an expo- provide a simple way of doubling or tri- sure to twice the total daily return of the Dow Jones Financial Clearly, the two charts (above) do not coincide. In particu- pling exposure to an index while using Index. The same index is also tracked (without leverage) by lar, UYG clearly underperforms the twice leveraged IYF dur- the same amount of capital.1 Invest $200 I-Shares Financial Sector ETF, which has symbol IYF. Thus, in assets ing the rally of the Spring 2009. The corresponding chart for Active traders can use reverse LETFs as a substitute for UYG offers investors a daily return equal to twice the daily the reverse double-leveraged fund (SKF versus -2 times IYF), short-selling the underlying asset when the latter is hard to Determine number L of ETFs outstand- return of IYF. shown in Figure 4 (below), is even more dramatic. Authorized borrow. For instance, many traders took long positions in ing, daily Market Another example is the ProShares UltraShort Financial Participants SKF, a financial bearish fund, toward the end of 2008, when ETF (ticker symbol: SKF). This product offers investors a daily Figure 4: SKF vs. –2X IYF* financial stocks were difficult or even impossible to short. return of equal to –2 times the daily return of the Dow Jones Long-short hedge-fund managers often use LETFs as a way Borrow $100 Financial Index, or-2 times the return of IYF. SKF is an ex- 250 – of hedging their portfolios. ample of a reverse LETF. 200 – Classical ETFs track an index or basket in a one-for-one 150 – fashion; they are essentially passively managed. In contrast, Money Buy-and-Holders LETFs require active management: this involves borrow- market Beware! 100 – ing funds to purchase additional shares (bullish LETFs), or 50 – short-selling (bearish LETFs) and rebalancing the position Throughout 2009, issues have been raised in the marketplace 0 – on a daily basis. Managers sometimes simplify the hedging regarding the suitability of leveraged ETFs for long-term investors seeking to replicate a multiple of an index perfor- 1/4/2009 – 7/4/2009 – 2/4/2009 – 4/4/2009 – 8/4/2008 – 9/4/2008 – 3/4/2009 – 5/4/2009 – 6/4/2009 – 11/4/2008 – 11/4/2008 12/4/2008 – 12/4/2008 of LETFs by entering into daily resetting of total-return *Figure 1 is a schematic representation of the management of a bullish – 10/4/2008 swaps with qualified counterparties. Figures 1 and 2 describe double-leveraged ETF. The manager finds out how many new shares mance. UBS AG announced in July 2009 that it would cease have been created or redeemed. Based on this, the manager adjusts graphically the management of leveraged ETFs. marketing LETFs “because such products do not conform to *Figure 4 shows a comparison between the performances of SKF and a daily the exposure to the underlying index using leverage. 2 its emphasis on long-term investing.” double-leveraged short position in IYF. The discrepancy is remarkable. 54 RISK PROFESSIONAL FEBRUARY 2010 www.garp.com www.garp.com FEBRUARY 2010 RISK PROFESSIONAL 55 PRODUCTS PRODUCTS Another interesting observation can be made by considering Figure 7: UYG/SKF (Since Inception)* dL dS the price history of a pair consisting of a leveraged ETF and t = t + (1 - ) rdt - fdt, (1) Bearish LETFs have more exposure the corresponding reverse product with same leverage (e.g., L t St UYG as of 18-Sep-2009 to volatility than bullish LETFs UYG and SKF). Typically, the price charts should be “mirror +200% where r is the funding rate and f is the expense ratio of the images” of each other, at least over short periods of time. +100% LETF. We model the underlying ETF price as an Itô process, 0% However, as we increase the time horizon, we see clearly as below: with the same leverage ratio. that the graphs are no longer mirror images, and the correla- dS tion between the two products breaks down as time passes (see t = tdZt + tdt, (2) perfectly correlated. Leveraged ETFs are dynamically hedged figures 5, 6 and 7). Mathematically speaking, the returns have St to replicate daily returns (and therefore do not replicate over long-term horizons), and correlations with the reference ETF correlation -1 for short periods of time, but the correlation di- -100% where Zt is a Brownian motion and t and t are stochastic tend to zero across time. minishes as the horizon increases. 400.0 processes that are non-anticipative with respect to Z . 300.0 t 200.0 Cheng and Madhavan (2009) and Avellaneda and Zhang Millions 100.0 Figure 5: SKF/UYG (Three-Month Period)* 0.0 (2009) showed that this leads to an explicit formula that re- Empirical Validation Source: http://finance.yahoo.com/ lates the prices of LETFs to their underlying index or ETF, as We analyzed the returns of a broad class of double-and triple- ULTRA FINANCIALS PRO as of 18-Sep-2009 +100% follows: leveraged ETFs (bullish and bearish). For double-leveraged t *Figure 7 compares the charts of UYG and SKF from their inception ETFs (issued by ProShares), we considered the period from L 2 +50% to September 18, 2009. There is little correlation between long-term t St – 2 =( )exp (1–) rt–ft– s ds (3) February 2008 to March 2009. In the case of tripe-leveraged returns. L 0 S0 [ 2 ∫ ] ETFs (issued by Direxion), we took the histories of these prod- 0% 0 ucts since their inception until March 2009. Tracking daily returns is not the same as tracking long-term St The interpretation of the formula is simple: the factor ( S ) In all cases, we calculated the difference between the right- returns. As we shall see, there are two primary differences. 0 corresponds to the “fast” compounding of returns alluded to hand and left-hand sides of equation (3) and plotted the average -50% First, there is the issue of compounding. 80.0 earlier, which will produce a divergence from simply-com- error, as well as the standard deviation of the standard error. 60.0 We all know the difference between daily compounded and 40.0 pounded returns over long periods of time. We note that if Tables 1 through 4 display the results. The agreement be- Millions 20.0 annually compounded interest rates. This effect is sometimes 4 0.0 || > 1 or = -1, the function f(s) =s is convex. tween theory and observation is striking. Notice that the aver- Source: http://finance.yahoo.com/ referred to as convexity. In fixed income, compounding is tan- tamount to investing at higher returns more frequently, which The tendency would be thus to overperform simply com- age and standard deviations of tracking errors are inferior to *Figure 5 depicts a comparison of the charts of UYG and SKF for a implies a faster growth of capital. pounded returns over the same time horizon. However, this is the daily volatility of the price of the ETFs, which shows that period of three months, ending on September 18, 2009. Notice that not the case empirically: several LETFs underperformed the the formula matches very well the observations. the two charts are “mirror images” of one another, with a slight over- In the case of LETFs, there is a second important reason performance by UYG. for the discrepancies seen in the charts: volatility.