Every day, at about 4:20 pm, I tweet the contango based on the settlement values of the VIX futures. Contango as I use the term is the percent by which the second month VIX future is higher than the first month VIX future. For example, if contango is 7% it means that the second month future is 7% higher than the first month future.

I tweet the contango because it is what predominantly determines in the long run the value of VXX. Yes, this is counterintuitive. The long term value of VXX does not depend on the actual values of the futures or the VIX index. It depends on the relation between the first and second month futures.

If you look at a feather floating in the air being buffeted by strong and changing winds, you would not be faulted for believing that it is the winds that determine the height of the feather off the ground. Yet, you can be assured that even if the feather encounters hurricane force winds or a tornado, it will eventually find itself on the ground because eventually storms dissipate and wind speeds revert to their long term average. What is obscured by the winds is the fact that gravity is constantly at work pulling the feather down and that given enough time, gravity will win. In this imperfect analogy the height of the feather is the value of VXX, the winds are the values of the VIX futures and gravity represents contango. Just as in the analogy, the long term value of VXX depends only on the contango.

In this article I will show why the above is true. Unfortunately, I cannot do so without utilizing a heavy dose of high school level algebra. VXX tracks an index that is somewhat complex and words alone are not sufficient to accurately describe its long term behavior.

In order to understand the rest of this article, please first read Vance Harwood's blog post How Does VXX s Daily Roll Work?

Vance's post is a great introduction to computing VXX and I will not repeat this essential information here. In addition, I will be using Vance's equations and notations as a starting point.

For convenience, I provide here the definition of the variables and the equations as Vance describes them in his post:

**The Variables**

Lower case "t" stands for the current trading day, "t-1" stands for the previous trading day.

The index level for today (IndexTR_{t} ), is equal to yesterday's index (IndexTR_{t-1}) multiplied by one plus a complex ratio plus the Treasury Bill Return TBR_{t.} 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 (first month) contract is designated by "dr" and the total number of trading days on the M1 contract is "dt."

M1 and M2 are the daily mark-to market settlement values of the futures, not the close values of the VIX futures.

**The Equations**

Equation 1

When dr is not equal to dt :

Equation 2

When dr = dt (the day the previous M1 expires):

Using Vance's notation and equations above, let's begin the analysis.

The first point to make is that TBR_{t} is negligible. At the current interest rate of 0.02% on the 90 day T-bill, and taking into account the worst case scenario of 3 calendar days between the last two business days, it is equal to 0.00000167. Even if the interest on the T-Bill would go up to 3%, it would only go up to 0.00025, still negligible. Furthermore, the yearly fee charged by VXX is 0.89%. On an average day, the expense per day is over 60 times larger than the contribution of TBR_{t}

In what follows TBR_{t} will be treated as being 0. We can then simplify Vance's equations to the following.

When dr is not equal to dt:

(1) |

When dr = dt (the day the previous M1 expires):

(2) |

Let's define *c*_{t} as the contango at time *t*:

(3) |

The contango is the difference in percent between the two futures.

Rearranging equation (3) we obtain:

(4) |

We use the relation in equation (4) and substitute for *M2*_{t} and *M2*_{t-1}in (1) to obtain:

(5) |

Rearranging equation (5) we obtain:

(6) |

Going over to the case where dr = dt we again use the relation in equation (4) and substitute for M2_{t-1} in (2) and rearrange to obtain:

(7) |

To summarize what we have done so far: We derived (6) from (1) and (7) from (2). (6) applies when dr is not equal dt and (7) applies when dr equals dt.

Now, let's assume that contango is constant over time. This is of course is not the actual case but it is a useful assumption because it helps to discuss the behavior of the index by simplifying the algebra. After we discuss this example, we will return to discussing the realistic case where contango changes over time but we will see that conceptually there is not much difference. Under the constant contango assumption equation (6) simplifies to:

(8) |

Since we are assuming that contango is constant over time we can remove the time subscript from its notation and write equation (7) as:

(9) |

Based on the above let's derive the index at time *t* based on the value of the index at some initial time 0. We will denote that initial value of the index as *IndexTR _{0}*

Let's begin by assuming that *t* is less than 20 and falls within one roll period and therefore only equation (8) is applicable. Based on equation (8) we can write:

(10) |

This is just a rewrite of the same equation; it shows the relation between the index values at time *t-1* and *t-2* instead of at times *t*and *t-1*.

We then replace the value of the index at time *t-1* in equation (8) with the expression on the right hand side of equation (10):

(11) |

And we repeat the process by replacing the value of the index at time *t-2* with its calculation based on the index at time *t-3*:

(12) |

If we simplify (12) we obtain:

(13) |

And as can be seen if we would have continued this process we would have obtained the following formula:

(14) |

Now, let's relax the assumption that the time period is all within one roll period and let's assume it spans 2 roll periods. In that case, we need to use equation (9) one time for the substitution instead of using equation (8) all the time. Therefore equation (14) changes to:

(15) |

And in the general case where t spans *n* roll periods (n>1) we can write:

(16) |

This is quite a surprising result. The index at time t only depends on the value of the first future at time *t* and time *0* and on the contango. The value the futures take between the start and end period do not matter at all for computing the index change. All that matters is the relation between the values of the first and second future. Since volatility mean reverts, for investors who are long or short VXX for substantial periods of time equation (16) can be simplified further to:

(17) |

Over time, the futures will revisit or come close to the values at time 0 and as *n* grows, the contango term of equation (16) becomes more dominant. For example, even if volatility rises 50% and remains at elevated levels, 8 months of 5% contango will wipe out any gain in the index. 7% contango would wipe out these gains in 5.5 months. Contango of 15% would wipe out a 50% volatility rise in 2.5 months.

Let's return to the general case in which the contango is not constant and changes over time. We can repeat the process above but now only the terms with the futures cancel out and the ones with the contango remain. I will not bother you with the math anymore, but even though contango is not constant, over a long period what predominantly matters is the average contango over that period.

The bottom line is that investors in VXX and similar products must religiously follow the contango and be aware how it is degrading their investment. In periods of high contango investors should look for other alternatives, such as buying options, in order to go long volatility or hedge their portfolio.