The dynamics of leveraged ETFs returns: a panel data study

Abstract

Leveraged exchange-traded funds (LETFs) are limited liability securities that allow investors to take daily constant leverage bets on a reference index. This work proposes a new empirical design to investigate the dynamics of quarterly LETFs returns. Rather than relying on fund-by-fund overlapping regressions, as in existing literature, the paper exploits a large panel of non-overlapping data covering the whole universe of Proshares, the US primary LETFs provider. Overall, it is found that the variables prescribed by theory broadly explain cross-sectional variability. It is also found that inverse LETFs and more generally, leveraged funds operating in asset classes like international equity, bonds and commodities underperform theoretical predictions. This underperformance is mainly attributed to frictions in the process of implementing the required daily leverage.

Keywords:

• Leveraged ETFs
• Equity return swap
• Panel data

JEL Classification:

• G0
• G12
• G13

Notes

¹ $\mathit{\lambda }$ is usually referred to as the leverage multiple. When $\mathit{\lambda }=+2,+3$ this is a long Leveraged ETF. When $\mathit{\lambda }=-1,-2,-3,$ we are dealing with a short or Inverse Leveraged ETF.

² In the paper, we use both (simple) returns and log returns. Simple returns are expressed as $R=\left(\frac{P_1}{P_0}\right)-1$, whereas log returns are expressed as $\Re =ln\left(\frac{P_1}{P_0}\right)$ where P₀ is the initial price and $P_1$ is the terminal price (including accumulated dividend) of a security over some holding period.

³ ‘Due to the compounding of daily returns, ProShares’ returns over periods other than one day will likely differ in amount and possibly direction from the target return for the same period’. ProShares product information.

⁴ For ease of exposition and because it is convenient for empirical work, we mostly rely on the discrete time approach of Avallenada and Zhang (2010). However, both Avallenada and Zhang (2010) and Jarrow (2010) have developed continuous time models allowing for general diffusion processes to drive the underlying index. Cheng and Madhavan (2009) also have proposed a valuation model. However, as pointed out by Jarrow (2010), their model does not incorporate explicitly the interest rate required to implement the leverage strategy.

⁵ Usually slopes are tested separately and no joint-test is conducted.

⁶ A well-known quadratic approximation of log returns by simple returns is given by: $\Re =ln\left(1+R\right)\approx R-\frac{1}{2}{R}^2$. Tang and Xu (2013, p. 317) drop the quadratic term to obtain empirical specifications. Unfortunately, from Ito’s rule, first-order approximation is not appropriate for stochastic returns as the quadratic term is not asymptotically negligible.

⁷ Since there are about D = 252 business days per year, for quarterly sampled data, $Q=4$ hence m = 63 or about 63 business days per quarter.

⁸ With daily reset, the interest income term $R_f\left(t_d\right)$ is of order ${\mathrm{\Delta }}_d\approx \frac{1}{252}\approx 0.004$, while the index return term $R_s\left(t_d\right)$ is of order $\sqrt{{\mathrm{\Delta }}_d}\approx 0.063$, about 15 times larger. Jarrow (2010, p. 137) elaborates on this point.

⁹ The world’s largest provider of geared (leveraged and inverse) according to Lipper, based on a worldwide analysis of all known providers of funds in these categories; the analysis covered ETFs and ETNs by the number of funds and assets, as of 30 June 2013.

¹⁰ This figure was obtained by aggregating the Datastream statistics Market Value (or ‘MV’) across all LETFs in the sample on 31 March 2014.

¹¹ Datastream data-type ‘RI’; ‘total return index’ proxies for the theoretical growth in value of a share holding over a specified period, assuming that dividends are reinvested to purchase additional units of equity at the closing price applicable on the ex-dividend date. RI is provided for all LETFs (which are treated as equity) and initialized, usually at a nominal value of 100, at inception.

¹² Observe that measurement errors in the independent variables, usually refer to as error in variables, is a far more serious econometric complication because it leads to inconsistent parameter estimates. In this work, the indices’ log returns on the right-hand side of the estimated model are assumed to be correctly measured as they do not, a priori, suffer from the illiquidity issue that plagues the LETFs’ log returns on the left-hand side.

¹³ From the technical standpoint, those statements may be precisely stated by comparing the functional form of the OLS and robust (White) and clustered standard errors. See http://www.stata.com/support/faqs/statistics/standard-errors-and-vce-cluster-option/ for a simple introduction.

¹⁴ For ease of notation, in the empirical work, we abbreviate t_q into$t$, but it should be understood that it applies to quarterly data.

¹⁵ The plots displayed here are for the estimation of the long model that covers the full data-set.

¹⁶ It does more so in the cross-sectional dimension on the far hand-side of the scatterplot, but those LETFs are the latest to be introduced so the entity averages are noisy estimates calculated from few quarters of available data.

¹⁷ Petersen (2009, p. 458) and Thompson (2011, p. 2) discuss applications of double-clustered standard errors in Finance. Double standard-clustered standard account for both entities effects (i.e. pricing errors are correlated within each LETF but uncorrelated otherwise) and time effects (i.e. pricing errors are correlated within each quarter but uncorrelated across quarters otherwise).

¹⁸ Recall that a F distribution with 1 numerator degree of freedom is identical to a t² distribution. So, for example, a F(12,544) = 195.93 corresponds to a $\sqrt{195.93}\approx 14$, t distribution.

¹⁹ An alternative view developed in Bessembinder et al. (2014) is that ETFs hedging demands are an instance of ‘sunshine trading’ and as a consequence are likely to be absorbed by liquidity providers who may be less concerned that those trades are information based.

²⁰ Engle and Sarkar (2006) distinguish the ‘true’ economic value from the net asset value which is a noisy proxy for the former.

²¹ Alternatively, we estimated distinct model’s coefficients for each investment category. However, this is only possible for double LETFs as the single and triples are not represented outside of Category 1. Estimated coefficients are qualitatively similar for each investment category but standard errors are substantially larger for less capitalized categories.