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# Kalman Smoother/Filter Estimation Of XIV's Dynamic Beta And Alpha

|Includes:

There has been increasing interest in volatility ETFs/ETNs and many people are thinking the beta of XIV is close to 3. Instead of relying on OLS (ordinary least square) to get a single beta or moving window OLS to get a "dynamic" beta, I am going to use Kalman Filter/ Smoother to illustrate the time-varying beta and alpha of XIV.

Basically we are assuming the beta and alpha are stochastic processes and we use all or part of the data to estimate the realizations of such stochastic processes. It has much less delay than the moving window OLS and the results are less susceptible to "outliers". Since I am mainly interested in demonstrating the time-varying beta/alpha and some of the interesting features I will use Kalman Smoother which employs all the information from 11/30/2010 to 09/06/2013 to estimate the historical beta/alpha. It cannot directly be used for trading purposes since it is non-causal but the smooth curves of beta and alpha can give us some hints. On the other hand real-time Kalman Filter might serve the trading purposes. The mathematic details are skipped here for simplicity. The data are coming from Yahoo Finance. I simply used the daily percentage return of XIV and SP500 (w/o dividend) in the equation. I ignored the risk free rate for simplicity and it would not change the results much as the risk free rate is close to 0 with the easy money policies. The coding was done in R, a free statistical software.

At first, the OLS results:

lm(formula = XIV ~ SPX)

Coefficients:

Estimate Std. Error t value Pr(>|t|)

(Intercept) 0.05244 0.08972 0.584 0.559

SPX 3.15541 0.08364 37.727 <2e-16 ***

Residual standard error: 2.364 on 694 degrees of freedom

Multiple R-squared: 0.6722, Adjusted R-squared: 0.6718

F-statistic: 1423 on 1 and 694 DF, p-value: < 2.2e-16

The OLS beta is 3.16 and the intercept(alpha) is 0.05 but not significant. The R^2 is pretty high which indicates such a simple CAPM model can explain a lot of XIV actions.

Secondly, the Kalman Smoother Alpha and Beta:

There are several distinctive features from the two graphs.

1.Beta and Alpha are far from constant.

2.Alpha was very negative in the second half of 2011 and that happens to be when XIV was slammed by the US debt downgrade and Euro crisis.

3.Alpha became positive for almost all 2012 during which XIV more than doubled.

4.Alpha became negative in 2013 until the July rally. Meanwhile Beta increased significantly. Thus I will venture here to say the profit made in XIV this year is mainly due to its excessive beta in a bull market but not its alpha.

At last, thank you Prof. Donoho for your lectures on time series, you have brought me to the wonderland of time series analysis. BTW, I do not own any of the stocks (volatility ETPs and SPY) mentioned above.

Disclosure: I have no positions in any stocks mentioned, and no plans to initiate any positions within the next 72 hours.