All volatility Exchange Traded Products (ETPs) use indexes that track a mix of two or more months of the CBOE’s VIX Futures. Calculating this mix is not trivial and has resulted in a lot of bleary eyes—including my own. My intent with this post is to help you understand, and if you desire accurately compute the key indexes used in VXX and other short term volatility funds using Excel or similar tools.
Why do we need a roll anyway?
If we could directly buy the CBOE’s VIX® index none of this would be necessary. Unfortunately no one has figured out a cost effective approach so we are forced to use the next best thing—VIX Futures. Like options, VIX futures have fixed expiration dates so volatility indexes need a process of rotating their inventory of futures in order to have consistent exposure to volatility. This rotation process is evident in the open interest chart below—the next to expire futures being closed out and the next month of futures being opened.
Before we dive into the details of how this rotation is dealt with, I’d like to address one source of confusion. ETP’s are not obligated to follow the approach detailed in the indexes. They are allowed to use other approaches (e.g., over-the-counter swaps) in their efforts to track their indexes. When ETPs are working properly, their prices closely track the index they specify in their prospectus minus their fees that are deducted on a daily basis.
Because indexes are theoretical constructs they can ignore some practical realities. For example they implicitly assume fractional VIX futures contracts exist and that the next day’s position can be put in place at market close—even though calculating that position requires market close information. I’m sure these issues cause headaches for the fund managers, but to their credit the funds usually closely track their index.
The Index Calculation
The details for the index (ticker SPVXSTR) that VXX tracks are detailed in VXX’s prospectus, pages PS-21 through PS-22. The math is general enough that it covers both the short term index that VXX uses and the midterm index VXZ uses—which adds to its complexity. The equations use Sigma notation, which probably makes it challenging for people that haven’t studied college level mathematics. I will present the math below using high school level algebra.
Except for interest calculations all references to days are trading days, excluding market holidays and weekends.
The volatility indexes used by short term volatility ETPs (list of all USA volatility ETPs) utilize the same roll algorithm—at the end of each trading day they systematically reduce the portion of the overall portfolio allocated to the nearest to expiration contracts (which I call M1) and increase the number of the next month’s contracts (M2).
The mix percentages are set by the number of trading days remaining on the M1 contract and the total number of days the current M1 will be the next to expire contract (varies between 16 and 25 days). So if there are 10 days before expiration of the M1 contract out of a total of 21 the mix ratio for M1 will be 10/21 and 11/21 for M2. At close on the Tuesday before the Wednesday morning M1 expiration there’s no mix because 100% of the portfolio is invested in M2 contracts.
It’s important to understand that the mix is managed as a portfolio dollar value, not by the number of futures contracts. For example, assume the value at market close of a VIX futures portfolio was $2,020,000, and it was composed of 75 M1 contracts valued at 12 and 80 M2 contracts at 14 (VIX futures contracts have a notional value of $1K times the trading value). To shift that portfolio to a 9/21 mix for M1 and 12/21 for M2 you should take the entire value of the portfolio and multiply it by 9/21 to get the new dollar allocation for M1, $865,714 (72.14 contracts) and 12/21 times the entire portfolio value to get the dollar allocation for M2, $1,154,286 (82.45 contracts).
Value weighting gives the index a consistent volatility horizon (e.g., 30 calendar days)—otherwise higher valued futures would be disproportionately weighted.
The next section is for people that want to compute the index themselves. Yes, there are people that do that. If you are interested in the supposed “buy high, sell low” theory of roll loss you should check out the “Contango Losses” topic at the bottom of this post.
Lower case “t” stands for the current trading day, “t-1” stands for the previous trading day.
The index level for today ( IndexTRt ) is equal to yesterday’s index (IndexTRt-1) multiplied by a one plus a complex ratio plus the Treasury Bill Return TBRt. The index creators arbitrarily set the starting value of the index to be 100,000 on December 20th, 2005.
The number of trading days remaining on the M1 contract is designated by “dr”. The “dt” is the total number of trading days that the M1 contracts are the next to expire futures contract.
M1 and M2 are the daily mark-to market settlement values, not the close values of the VIX futures. The CBOE provides historical data on VIX futures back to 2004 here.
When dr is not equal to dt:
When dr = dt (the day the previous M1 expires) we have a different equation:
Yes, this 2nd equation could be simplified because dr/dt = 1, but then it wouldn’t fit as nicely into the equation below which uses a little logic to combine both cases:
The equation assumes that the entire index value is invested in treasury bills.
- An interesting special case occurs when you assume that the M1 and M2 prices are completely stable and in a contango term structure for multiple days—for example, M1 at 17 and M2 at 18. In that situation the equation simplifies to:
- This special case illustrates that there is no erosion of the index value just because it’s selling lower price futures and buying higher priced futures—in fact it goes up because of T-bill interest. It’s the equivalent of exchanging two nickels for a dime—no money is lost. For more on this see: The Cost of Contango.
For more information:
It’s expensive to buy securities that track volatility. Their holding costs are so high that your timing has to be exquisite in order to end up with a profit. However, if you’re hedging a short volatility position, or poised to jump into the general market at a possible transition point a long volatility position might make sense.
If SPX breaks through the trendline it’s likely volatility will really spike. Alternately if the market rallies then volatility will quickly fade, so an asymmetric bet (e.g., call options) is attractive. If volatility spikes you benefit from the rapid run-up, but if it’s a false alarm your losses are limited.
The next question is to determine what underlying volatility product is best for this hedge and how large a position is needed to balance the risk in your general market position. Investing in the CBOE’s VIX® would be ideal, but unfortunately there’s no way to directly invest in the VIX, so we’re left with a set of compromised choices—volatility Exchange Traded Products (ETPs) like TVIX, VXX, or VIXM (see volatility tickers for the complete list), or VIX futures. Later in this post I’ll analyze how three specific investments would have performed during an actual correction, but first I’ll examine a key issue—how much will the volatility products move up if the market drops.
The chart below shows how the volatility ETPs have historically reacted during negative S&P 500 (e.g., SPY) market moves. The data uses simulations of ETP prices from 2004 until their inceptions and actual data after that.
The median value of these ratios stays fairly stable over a wide range of percentage moves. For example, the median percentage moves of 1X short term ETPs like VXX will consistently cluster around negative 2.25 times the percentage moves in the S&P. A daily -1% move in SPY typically results in a VXX positive move of around 2.25%.
These ratios aren’t guaranteed—they’re statistics. In fact 20% of the time the volatility products move in the same direction as the S&P 500. Fortunately, when the market is dropping the distribution of ratios tightens up
The chart below shows the historical distribution of VXX percentage moves compared to SPY moves of > -0.1% and > -1%. SPY moves of less than +-0.1% are excluded because they can generate high ratios that aren’t meaningful.
When the S&P makes a 1% or larger negative move the median doesn’t shift much, but the number of results on the positive side drops from 21% of the total down to under 5%.
Since these ratios are relatively stable regardless of the size of the market moves we can view these ratios vs. the various ETPs / indexes.
Remember these are one day relative % ratio numbers. While TVIX & UVXY ratios are close to the VIX’s on this metric, the contango losses in holding these ETPs other than during a market downswing are ruinous. The 1X short term ETPs (e.g., VXX) aren’t much better.
So far I’ve only discussed the CBOE’s indexes and some of the volatility ETPs. There are also VIX futures that have various sensitivities to the moves of the S&P 500. These products differ from the indexes and ETPs in that they have expiration dates like options.
As these futures get closer to expiration their sensitivity increases. Interestingly, a simple natural log relationship (shown on the chart) gives a good match to the data.
There are also VIX weekly futures based on the CBOE’s 9 day VXST index, but I’ll discuss those in a different post.
Circling back to the trend chart at the beginning of this post—what would be a volatility hedge that would protect you if you bet on a 5th upward bounce?
There’re a lot of moving parts here (e.g., security, strike price, expiration date) and a lot of different strategies. I’ll pick one general approach, and work through the details if the hedge had been applied during the 30-July-2014 through 8-Aug-2014 period.
- $100K invested in the SPY (betting that the market will start climbing again)
- One percent of the market investment ($1K) invested in a volatility hedge—call options expiring around 16-Aug-2014. It’s very likely the market will have gone one way or the other by then.
- Goal of breaking even (losses in SPY & cost of the options offset by profits) if the market drops 3% or more.
I’ll review the results from three different trades—buying calls on UVXY (2x Short term), August VIX calls (based on next to expire VIX futures or M1 futures), and VXX (1X Short term).
So, in spite of the underlying volatility instruments moving around 2X more than expected, the $1K spent on hedges did not achieve the goal of break even with a 3% decline in the S&P 500—although UVXY was pretty close. During this period the VIX ramped from 13.33 to 15.77—an increase of 18.3% (the expected move was 15%). If the correction had continued volatility would have probably increased rapidly (the intraday option prices spiked > 50% on the 8th –when the VIX climbed to 17.09), so the hedges probably would have worked well protecting the S&P 500 position against further declines.
One of the challenges of trading is wrestling with strategies that work until they don’t. With short term volatility hedges you can bet on the market going up—without paying too much for insurance in case you’re wrong.
Volatility based Exchange Traded Funds and Notes (ETF / ETN) have only been on the market for a few years (see volatility tickers for the full list of USA based funds). The oldest one, Barclays’ VXX only started trading in late January 2009. Because of their relative youth we don’t have actual trade data on how they would have performed through critical periods—for example the 2008/2009 crash. Fortunately the CBOE provides historical data starting in March 2004 for the VIX futures that underlie the VXX, so it’s possible to objectively simulate how it would have performed from that point forward.
Some aspects of the VXX simulation are tricky. For example some VIX future expiration months did not trade in the 2004 to 2008 time frame so those values need to be interpolated / extrapolated. The prospectus does not spell out whether closing or settlement values of the futures are used for the index calculations (they use settlement values), and the calculation using the daily rolled and rebalanced futures is not straightforward. Even the final step, figuring out the daily fees is not a trivial exercise.
The chart below shows the reverse split adjusted results (as of Dec 2013).
Clearly VXX would have performed horribly over the 2004 to 2013 time frame with a brief respite in 2008/2009. If you had invested $1000 in VXX in March 2004 you would now have $1.80 left of your initial investment—a 99.8% decline. The long volatility funds have a structural tendency to decline because they hold VIX futures that are historically in contango 70% to 80% of the time, for more on this process see How Does VXX Work? and The Cost of Contango.
If you are interested in purchasing the results of the VXX simulation back to March 2004 I have made a spreadsheet available for purchase (see bottom of post) that includes the simulated close values with the annual fee (0.89%). The maximum deviation in my results from the Barclays’ published closing indicative values since the product started trading is less than +-0.04%.
Seperately I have done a simulation of VXX open / high / low values over that same period. Those results are inherently less accurate, but still should be useful for testing strategies that are sensitive to intraday values. For more information see this post. That spreadsheet is also available for purchase at the bottom of this post.
The chart below shows VXX’s performance (in black) relative to a few other volatility based Exchange Traded Products.
Among the long funds, the 2X leveraged short term TVIX from VelocityShares fares even worse than VXX, declining 99.99999%. Barclays’ medium term VXZ only declines 78%. The daily inverse funds do better with VelocityShares’ medium term ZIV going up 43% and its short term XIV going up 13 fold—however not without some serious dips along the way.
Earlier in this post I mentioned that computing VXX’s fee was surprisingly difficult. The appropriate equation is not present in any ETP prospectus I have seen—instead you are treated to prose that would make an IRS agent proud.
Exchange Traded Products typically state their fees on an annual basis (e.g., 0.89%), but in practice they deduct a fee each day from the assets under management. In computing the fees it’s tempting to start with the daily value of the underlying index (SPVXSTR in the case of VXX) but the actual calculation starts with the final indicative value of the ETP from the previous trading day. It multiplies the previous value by the index gain at close for the current day (one if it is a non-trading day) and then applies the fee. The applicable formula is:
While it’s interesting to simulate how a security would have behaved in the past it’s only one of many possible outcomes. If VXX had existed in 2004, it’s likely the VIX futures that underlie it would have been affected—at least in small ways. Looking forward the uncertainties multiply—there’s no guarantee that VIX futures will behave the same way through upcoming corrections and market crashes and with the open interest on VIX futures growing 40% a year we can anticipate that someday they, and indirectly VXX will be influencing the behavior of the S&P 500 itself.
For more information on the VXX simulation spreadsheet see this readme.
If you purchase the spreadsheet you will be eventually be directed to paypal where you can pay via your paypal account or a credit card. When you successfully complete the paypal portion you will be shown a “Return to Six Figure Investing“ link. Click on this link to reach the page where can download the spreadsheet. Please email me at [email protected] if you have problems, questions, or requests.
I have generated the end of day trading day values for the most popular long and short volatility Exchange Traded Products (ETPs) for March 26th, 2004 through February 28th, 2017
These ETP histories are required if you want to backtest various volatility strategies through the quiet times from 2004 to 2007, or the 2008/2009 crash. The chart below shows the simulated values with a logarithmic vertical axis so that you can see a reasonable amount of information for each fund.
The algorithms for generating these ETPs values are documented in the prospectuses for the various volatility ETNs and ETFs. Barclays’ VXX/VXZ fund prospectus is a good example. See Volatility tickers for the current universe of USA based volatility ETPs and their associated reference indexes. The futures settlement data required for these calculations is available on this CBOE website—in the form of 100+ separate spreadsheets. To make the calculation of the indexes underlying the ETPs tractable I created a master spreadsheet that integrates the futures settlement data into a single sheet. See this post for more information about that spreadsheet.
With the exception of TVIX—which has had severe tracking problems since early 2012 my simulated values very closely track the published indicative values (IV) of the funds. Barclays provides a full set of IV values for VXX and VXZ—my simulation tracks them within +-0.04% and +-0.025% respectively. Sampled IV values for the other funds give error terms of +-0.2% for Proshares UVXY, and for VelocityShares XIV and ZIV +-0.2% and +- 0.01% respectively. My TVIX simulation tracks sampled IV values within +2%/-4%.
If you need simulated intraday open, high, low values also check out this post.
These ETP prices reflect the contribution of 91 day treasury bills on their overall performance. Thirteen-week Treasuries yields averaged 0.05% in 2013, but in February 2007 they yielded over 5%— things have changed a bit… The simulated ETP values do include applicable fees which vary from fund to fund. The fee calculation is surprisingly difficult. For more on that see Backtest on VXX Including Annual Fees
I am making these 6 simulation spreadsheets (values only, no formulas) available for purchase, individually, or as a complete package. The VXX package is also available here. If you cannot see purchase information immediately below then please click this link to the stand-alone post and look at the bottom of the page.
For more information on the spreadsheets see readme.
If you purchase the spreadsheet you will be directed to paypal within a few minutes where you can pay via your paypal account or a credit card. When you successfully complete the paypal portion you will be shown a “Return to Six Figure Investing” link. Click on this link to reach the page where can download the spreadsheet. Please email me at [email protected] if you have problems, questions, or requests. It’s easy to miss the “Return to Six Figure Investing” link. If you don’t get it / can’t find it please email me.
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