How Well Do Leveraged ETFs Track Their Benchmarks?

There's been some debate on and off about how well various ETFs track their underlying benchmarks. Debate has focused on commodity, currency, and leveraged ETFs. For example, CNY - the recently issued ETN for the Renminbi - has gone down recently as the RMB continued to revalue relative to the USD. The reason is that the ETN is tracking the RMB futures traded on the Chicago Mercantile Exchange. Those futures also declined in value during the relevant period. People have also questioned whether QID, QLD, SSO and other leveraged ETFs track their underlying indices. For example, this discussion on Roger Nusbaum's blog about SSO, which attempts to return twice the S&P 500's return. I made some comments but I was getting confused and so I decided to do a statistical analysis to answer this question definitively.

I downloaded daily prices for SPY and and SSO from Yahoo Finance and computed the daily percentage returns since 21 June 2006 for both ETFs (N=469). I used Yahoo's adjusted prices, which account for distributions paid. I then regressed the daily returns for SSO on those for SPY. The results: the slope coefficient (equal to CAPM beta) = 1.99 and the intercept coefficient = -0.000225. Converting this daily intercept to an annual intercept results in a value of -5.4%. The R-squared in the regression is 0.98:



So in terms of reproducing two times the index, this fund is very good with a beta of 1.99. But why does it have an intercept of -5.4%? One factor is the higher expense ratio of SSO - 0.95% vs. 0.08% for SPY. The remaining intercept term is -4.53%. This appears to be because I didn't subtract a risk-free rate from the returns as is usually done for CAPM regressions and it represents the implicit borrowing costs of the fund. The basic CAPM equation is:



where r is the rate of return (of SSO in this case), f the risk free rate, and m the market rate of return (here SPY). Rearranging we get:



For SSO beta is 2 and, therefore, the intercept term in my regression is equal to the sum of the true CAPM alpha (here negative due to the higher expense ratio) and the negative of the risk free rate. 4.5% sounds about right for the average risk free rate over this period.

The bottom-line: SSO appears to track SPY leveraged two times as well as can be expected taking into account higher expenses and borrowing costs.