The Volatility Index
Stefan Iacono
University System of Maryland Foundation 28 May, 2014
The Volatility Index
Introduction
The CBOE’s VIX, often called the market “fear gauge,” measures investor sentiment. The VIX is calculated through the number of puts and calls on the S&P500.
Purchases of call options are interpreted as positive investor sentiment whereas purchases of put options indicate negative sentiment. The indicator is inversely related to the S&P500; the value of the S&P500 will decrease as the VIX increases, and vice versa.
Whereas most market indicators are based on historical information, the VIX is especially useful in predicting the future. This is due to the forward-‐looking nature of puts and calls. This paper will address the VIX’s use to investors, how it
statistically relates to the broader market, and how VIX-‐based ETFs compare to each other.
Use to Investors
Investors assess the VIX in order to predict the future values of its underlying asset, the S&P500. The following graph shows the relationship between EOD values of the VIX and S&P500.
Figure 1: One-‐year historical data from 5/9/2013 to 5/9/2014
Figure 1 indicates a 7-‐10 day lag between movements of the VIX and changes in the S&P500 price. Thus, an investor that is wary of the VIX will have a window of opportunity to take a position on the S&P500.
Statistical Perspective
The relationship can be modeled by regressing the returns of the VIX (the explanatory variable) and juxtaposing them to the return of the S&P500 (the response variable).
Some investors argue that the VIX is simply the inverse of the S&P500; however, there is economic reason to believe that such an inverted relationship does not exist in a one-‐to-‐one fashion. If this is the case, a simple moving average of the S&P500 could tell you how to take a position on the market.
The reason why the VIX does have higher predictive power than simple technical indicators is due to the VIX’s use of option contracts used in its calculation. Such forward-‐looking instruments offer higher predicative value for the following two reasons: 1) options are inherently forward-‐looking instruments, and 2) options are sophisticated products and are used by more experienced investors.
Regression Model
A quantitative model was developed by regressing VIX returns with those of the S&P500. To better fit the S&P500 data, returns were manipulated to better fit a normal distribution. To do this, the log of the returns was used as the response variable. This was done through the following equation:
log(P
t/P
t-‐1)
= log(P
t) -‐ log(P
t-‐1) = S&P500
r
Mechanically, the regression model was formatted as the following, with t (time) and number of observations (k):
S&P500
r= B
0+B
1VIX
t-‐k+ e
tA lagging factor of t-‐k was used when finding the VIX coefficients due to the
observation from Figure 1 in that significant changes in the response variable were observed roughly a week after changes in the explanatory variable. Since the data was pulled in weekly intervals, a one-‐unit time difference in the VIX adequately captured the lag effect. The following output was generated from weekly returns over the period of January 1, 2011 – May 22, 2014.
The fitted regression model, calculated through Minitab software, is:
S&P Percentage Returns = 0.00326 -‐ 0.123 VIX Return
The VIX variable is highly significant, given its P-‐value of 0.00 and high T-‐statistic of -‐17.30 (seen in Appendix A).
Figure 2: Residual graphs of the VIX regression; all calculations done on Minitab software
Four assumptions must be made before declaring a statistical relationship between the VIX and S&P500: Assumption 1: Relationship is linear, Assumption 2: Errors are normally distributed, Assumption 3: Errors have constant variance, Assumption 4: Errors do not display obvious ‘patterns’
Figure 2 answers the above assumptions. The versus fits graph (shown above) tests linearity, constant variance, and randomness of errors. There is a slight pattern in the graph, but, given a liberal interpretation, the relationship is linear, errors have a constant variance, and the errors do not show any obvious patterns.
Since all four statistical assumptions pass, there is statistical support that movements in the VIX (or changes in the purchases of puts and calls) relate to returns of the S&P500 during the following week.
VIX-‐Based ETFs
So far, this paper has discussed the CBOE VIX itself, not the value of an ETF centered on the VIX. The VIX itself cannot be traded. ETFs require a slightly different way of thinking due to the fact that VIX-‐based funds are traded on the index’s futures and not its spot price.
Many investors use VIX-‐based ETFs to hedge their portfolios, since many of these products have betas near -‐1. Various ETF liquidity, structure, and true tracking error was compared and listed below.
Ticker Name Issuer Futures Timeframe
VXX S&P500 VIX Short-‐Term Futures ETN Barclays iPath Short-‐term VIXY VIX Short-‐Term Futures ETF ProShares Short-‐term VIIX VIX Short-‐Term ETN VelocityShares Short-‐term VIXM VIX Mid-‐Term Futures ETF ProShares Mid-‐term VXZ S&P500 VIX Mid-‐Term Futures ETN Barclays iPath Mid-‐term VIIZ VIX Medium-‐Term ETN VelocityShares Mid-‐term
Liquidity
Liquidity was measured by looking at average trading volume. The volume of trades was summed over a three-‐month range, then divided by number of trading days.
A second aspect to measuring liquidity is to look at trading volume of the ETF’s components. According to Paul Wisbruch, Director of Sales at RevenueShares Investor Services, ETF liquidity ought to be measured by the trading volume of the fund components rather than the trading volume of the ETF itself. However, accurate data on the number of outstanding short or mid-‐term futures contracts were not found. Regardless, investors should not experience too much difficulty if they must liquidate their positions; trading volume on S&P500 products is high relative to other market products.
Structure
All of the discussed ETFs are based from futures contracts on the VIX. The difference in the funds is that their respective futures contracts have different maturities, either, short or mid-‐term.
True Tracking Error
The value of an ETF does not move in perfect sequence with its underlying asset. The extensiveness of this error is known as an ETF’s true tracking error.
True tracking error was calculated by taking the difference of percentage returns between an ETF and the net asset value (NAV). More specifically, the EOD price of the underlying was compared to the EOD price of the ETF. From there, the absolute difference was found between EOD prices. After finding the errors from the same month, the standard deviation of differences was calculated to arrive at true
tracking error. The lower the error, the better the “match” between changes in ETF and underlying asset values.
ETF Findings
The following data was taken from Fidelity to assess the performance of three short-‐ term and three mid-‐term ETFs. Figure 3 shows the various metrics used to measure liquidity, structure, and true tracking error.
Ticker Trading Volume Tracking Error Net Assets Futures
VXX 15,129,658 0.88 $1.1B Short-‐term VIXY 935,644 0.83 $110.5M Short-‐term VIIX 107,258 0.86 $9.1M Short-‐term VIXM 81,172 0.75 $51.4M Mid-‐term VXZ 987,170 0.67 $69.6M Mid-‐term VIIZ 10,305 0.64 $1.8M Mid-‐term
Figure 3: Data from Fidelity
Returns were negative across the board, with short-‐term futures funds suffering much more heavily. Figure 4 compares the various YTD returns.
Ticker NAV Return Market Return S&P 500 Index
VXX -20.75% -20.40% 4.30% VIXY -21.03% -21.07% 4.30% VIIX -20.74% -20.75% 4.30% VIXM -11.76% -11.98% 4.30% VXZ -11.62% -11.64% 4.30% VIIZ -12.30% -12.41% 4.30%
Figure 4: Returns from Fidelity
VXZ is arguably the best product to hedge a portfolio: it produced the best return, has high trading volume and produced the second-‐lowest true tracking error.
Appendix A
Regression Analysis: S&P Percentage Returns versus VIX Return The regression equation is
S&P Percentage Returns = 0.00326 - 0.123 VIX Return Predictor Coef SE Coef T P Constant 0.0032634 0.0009427 3.46 0.001 VIX Return -0.122722 0.007093 -17.30 0.000 S = 0.0124910 R-Sq = 63.2% R-Sq(adj) = 63.0% Analysis of Variance Source DF SS MS F P Regression 1 0.046706 0.046706 299.35 0.000 Residual Error 174 0.027148 0.000156 Total 175 0.073855 Unusual Observations S&P Percentage
Obs VIX Return Returns Fit SE Fit Residual St Resid 14 -0.143 -0.006400 0.020776 0.001417 -0.027176 -2.19R 29 0.441 -0.039200 -0.050881 0.003223 0.011681 0.97 X 30 0.267 -0.071900 -0.029540 0.002075 -0.042360 -3.44R 32 0.184 -0.046900 -0.019317 0.001572 -0.027583 -2.23R 36 -0.196 0.053500 0.027280 0.001716 0.026220 2.12R 37 0.331 -0.065400 -0.037419 0.002490 -0.027981 -2.29RX 40 -0.220 0.059800 0.030250 0.001862 0.029550 2.39R 45 0.065 -0.038100 -0.004738 0.001029 -0.033362 -2.68R 46 0.077 -0.046900 -0.006211 0.001067 -0.040689 -3.27R 47 -0.202 0.073900 0.028004 0.001751 0.045896 3.71R 49 -0.079 -0.028300 0.012983 0.001121 -0.041283 -3.32R 76 -0.142 -0.005800 0.020702 0.001414 -0.026502 -2.14R 97 -0.118 -0.014500 0.017769 0.001292 -0.032269 -2.60R 104 -0.365 0.045700 0.048094 0.002801 -0.002394 -0.20 X 159 0.437 -0.026300 -0.050403 0.003197 0.024103 2.00 X R denotes an observation with a large standardized residual.
X denotes an observation whose X value gives it large leverage.
