Over the past five years investor concern regarding capital market uncertainty has drawn attention to the VIX "fear index" and its tradable derivatives. The VIX index is a calculated "implied volatility" based on strips of near-term S&P 500 index option prices. To satisfy the demand for volatility trading certain derivative products (e.g. options and futures) have been created that reference the VIX index value. One of the most popular of these products is the VXX, or Barclay's iPath S&P 500 VIX Short-Term Futures ETN. This ETN tracks the S&P 500 VIX Short-Term Futures Index by effectively holding a blend of the first and second month VIX futures contracts. The VXX can be traded within ordinary brokerage accounts.

Over its limited lifetime, the VXX product price has performed very poorly, even after taking into account the 70% decline in the VIX index during this period. This fact raises the following questions: 1) How did this "value" leak out, and 2) How can the rate of leakage be estimated?

Most readers of this article will know that the key to answering these questions lies in defining and calculating what is called the VXX "roll yield." This is a challenge for most authors because of the temptation to take logical shortcuts and thereby oversimplify the issue. This temptation is amplified as a result of unfortunate terminological confusions. Consequently the typical cursory treatments of the roll yield are less than satisfying.

It is the goal of this article to resist this temptation and answer these questions by cutting through the confusion and presenting an abbreviated derivation of the proper roll yield for the VXX. It is shown that this roll yield is easy to compute on a daily basis.

**Do the Contango**

Because the VXX is based on futures contracts it is essential to understand the market configurations known as contango and backwardation. From Wikipedia:

Contango is the market condition wherein the price of a forward or futures contract is trading above the expected spot price at contract maturity.

The opposite market condition to contango is known asbackwardation.

For financial futures the "expected" spot price is the current spot price compounded forward to maturity at the risk-free interest rate. Under present circumstances the expected spot price for the near-term VIX futures effectively equals the current VIX spot price.

Contango is the expected price configuration within the "normal" volatility futures market. This follows from the realization that the futures contract price reflects the market's projection of a future volatility level more remote than that of the VIX index. All volatility projections must make some allowance for the possibility of so-called "black swan" events. Assuming that these unpredictable disruptive events occur at some probability per unit time, the greater the time window we define, the more likely a black swan will be seen flying through it. Since the longer-dated contracts imply a wider time window, their volatility projections will tend to possess proportionately greater premiums. Volatility longs "lose" when they pay extra for the chance to see a black swan that does not arrive.

Backwardation in the volatility market therefore exists during "abnormal" times, when the markets are in the midst of some disruptive event that is thought to be transient. At these times the VIX may be viewed as being excessively high from the perspective (and timeframe) of the futures market and the VIX index level may exceed the price of one or more of its futures contracts. This disconnect in valuations can occur because the options market that establishes the VIX level is distinct from the VIX futures market.

When a long-dated contract is priced higher than a shorter-dated counterpart it is commonly said that these two contracts are in contango. While this description is acceptable, it may nevertheless lead to some confusion since it is possible for this relative relationship to exist even when one or both contracts are in backwardation. (Such configurations were seen in the VXX components, respectively, on April 17th and 18th of this year.) In order to avoid this potential confusion I will use the term "relative contango" to describe this price relationship.

**VXX vs. VIX**

For practical purposes the VXX product is a time-weighted blend of the first and second month VIX futures contracts. This blending is designed so that the VXX mimics a synthetic VIX futures contract that is always thirty calendar days from maturity. This is accomplished by rebalancing the VXX futures portfolio at the end of each trading day.

Recall that the VIX index itself represents the implied volatility of the S&P 500 index over the upcoming thirty day interval. Since the VXX price (to within a multiplicative factor) is essentially a projection of what the VIX index level will be in thirty days, the VXX itself can be regarded as the futures market's estimate of the S&P 500's implied volatility between the upcoming thirtieth and sixtieth days. Even though the VXX and the VIX share a close conceptual connection it does not follow that their daily percentage price changes must be tightly correlated. Since September 18, 2012 the VXX has returned about 46% of the VIX daily percentage change and has tracked the VIX with a 0.83 coefficient of determination (R^{2}). The relative insensitivity of the VXX to daily change is a consequence of its more forward-looking nature.

**Rolling Confusion**

Let's now direct our attention to the meaning of the roll yield as used within the futures markets. Again, according to Wikipedia, the roll yield of a futures contract is defined as follows:

Theroll yieldis the yield that a futures investor captures when their futures contract converges to the spot price; in backwardated futures market the price rolls up to the spot price, so the roll yield is positive, whereas when the market is in contango the price rolls down to the spot price, so the roll yield is negative. The spot price can stay constant, but the investor will still earn returns from buying discounted futures contracts, which continuously roll up to the constant spot price.

Clearly the roll yield relates to the price trend of a single futures contract. Here the "roll" refers to the tendency for a given contract to converge to the spot price at contract expiration. As will become clear, the use of the word "roll" in this context is unfortunate. In retrospect a better choice for this term would have been "convergence yield," i.e. the yield that obtains from only the passage of time.

To achieve its purpose the VXX product must reduce its front month exposure and increment its end month position on a daily basis. The supposed impact of these "rolling" transactions is often (erroneously) referred to as the VXX roll yield. These transactions are usually described as the source of the VXX value leakage whenever the underlying contracts are in relative contango, i.e. when the portfolio has to pay more for the end month contract than it receives from the sale of the front month contract. This "contango problem" is said to almost certainly vex the inveterate VXX holder.

The source of this confusion can be traced back to Barclay's VXX prospectus. On page S-104 of this document we find:

Roll yield is generated as a result of holding futures contracts. When longer-dated contracts are priced lower than the nearer contract and spot prices, the market is in "backwardation", and positive roll yield may be generated when higher-priced near-term futures contracts are "sold" to "buy" and hold lower priced longer-dated contracts. When the opposite is true and longer-dated contracts are priced higher than the nearer contracts and spot prices, the market is in "contango", and negative roll yields may result from the "sale" of lower priced near-term futures contracts to "buy" and hold higher priced longer-dated contracts.

This explanation for the roll yield phenomenon has been promoted in various popular discussions such as here and here. It has a natural appeal because it seems easy to understand. In fact it is technically correct for the case of monthly rebalancing, where only the current month contract is held to expiration. At that point the expiring position is exchanged for an equivalent position in the subsequent month's contract. This practice would naturally lead to the computation of a monthly roll yield (MRY). Assuming a fixed VIX spot price (V_{S}) the MRY is approximately:

Here V_{1}(n_{0}) and V_{2}(n_{0}) refer to the prices of the futures contracts (of the first and second months respectively) at the time of their purchase (with n_{0} being the total number of trading days in the respective monthly cycles). In this static example these prices are assumed to be equal. Also V_{S} = V_{1}(0) is true by definition, as the spot price is the contract price at maturity.

The final link in this mathematical chain is the source of the conceptual confusion. By the time the front month contract has expired the roll yield has already exerted itself on the position. The purchase of the replacement contract is in no way the cause of the roll yield-it is merely a reflection of it. The fact that V_{2}(n_{0}) = V_{1}(n_{0}) holds in this ideal example creates the mathematical illusion that the roll yield is determined by the relative contango percentage at any given time. It will be shown that the MRY calculation is a limiting case of a general roll yield equation that defines the roll yield over each rebalancing interval. In more technical terms the "price sampling" frequency (for the roll yield calculation) is set equal to the rebalancing frequency. The more frequently the roll yield is computed the more accurately its long term impact on a futures portfolio can be assessed.

With daily portfolio rebalancing it makes sense to use daily closing prices to compute a daily roll yield (DRY). The appropriate roll yield for the VXX must then be the DRY. If a MRY is desired it can be determined retrospectively by compounding the DRY values over (n_{0}) trading days. Serious error can result, however, if the VXX DRY is estimated based on the degree of relative contango.

The key to understanding the VXX roll yield comes back to quantifying its impact on a daily basis. An accurate approximation for the DRY must take into account the change in the VXX portfolio structure at each day within its monthly cycle. Now an easily computed approximate expression for the VXX DRY can be presented.

**The VXX DRY: A Simple Approximation**

In mathematical terms the roll yield of a portfolio of futures contracts is the partial logarithmic derivative of the portfolio's notional value taken with respect to time. The DRY approximates this ideal roll yield. As a partial derivative we intentionally ignore the typically dominant notional value variations induced by the daily change in the VIX index level.

The total notional value [F] of the VXX is just the number of synthetic contracts held [N] multiplied by the price [U] of each contract:

As indicated the total notional value is also proportional to the current VXX price [V_{X}]. Here [M] is the fixed number of VXX units outstanding over the period of interest.

Using only the first order terms the DRY is approximately the sum of the daily percentage changes in [N] and (U]:

Note that the differences expressed here and subsequently are only "partial" time differences, i.e. differences taken over a hypothetical day in which the VIX index level does not change.

The price of the synthetic contract is a weighted blend of the front (V_{1}] and end [V_{2}] month contract prices:

The weights [w_{1}] and [w_{2}] are determined by the number trading days [n] remaining in the monthly cycle. Again, [n_{0}] is the total number of trading days between the current monthly VIX futures settlement dates.

Now expressions for ΔU and ΔN are needed. The derivation of ΔU is straightforward and yields:

The derivation of ΔN is based on the condition that the notional funds from the sale of the front month contracts are expended entirely on the purchase of the end month contracts in a way that maintains the intended portfolio balance. The notional value decrement in the front month position must equal the notional value increment in the end month position. This condition leads to:

The definitions for the daily value changes ΔV_{1} and ΔV_{2} must now be specified. Here I assume that the front month contract price drifts towards the VIX spot price [V_{S}] at a rate inversely proportional to the remaining trading days. Similarly the end month contract price is taken to approach the current synthetic contract price [U] in a parallel manner. Specifically,

Combining these results leads to my approximate expression for the VXX DRY:

**The VXX DRY Examined**

The DRY approximation tells us everything we need to know about the VXX roll yield in terms of easily accessible values and prices. The monthly cycle is defined by the VIX futures settlement date. According to the CBOE website:

The settlement date for VIX futures is the Wednesday that is thirty days prior to the third Friday of the calendar month immediately following the month in which the contract expires.

According to the VXX prospectus, the monthly cycle begins (i.e. n=n_{0}) on the day before the futures settlement date. The cycle length [n_{0}] is the number of trading days from this day to the Tuesday before the next settlement date.

The first point to consider is that the role of the relative contango percentage (RCP), i.e.

is muted within the DRY approximation. Using the RCP to estimate the DRY is not recommended. In fact let's look at the special case where RCP=0 (i.e. V_{2} = V_{1} = V). For this case the DRY approximation simplifies to:

This means that the roll yield can be significantly negative for V> V_{S} (contango) and significantly positive for V< V_{S} (backwardation). There is nothing particularly "seamless" in this instance as there is no guard against loss if the equally priced contracts are in contango.

Now consider the situation at the midpoint of the trading cycle, when n = n_{0} /2. At this point the DRY approximation becomes:

Here the price in the denominator is the simple average of the two contract prices. This is the clearest demonstration that the VXX DRY is more sensitive to the sum of the component futures prices than to their difference.

The following diagram plots the relative contributions of the two components of the VXX DRY for the idealized case where the VIX index level is fixed and the front and end month contract prices follow the daily changes prescribed over the monthly cycle. In this example the VIX index is held at the value of 15 and the front and end month contract prices are set initially to 18 and 20 respectively. This deep contango configuration leads to a very negative roll yield.

Note that the contribution of ΔN/N dominates over that of ΔU/U, especially near the end (n≈1) of the trading cycle. The DRY itself is rather insensitive to the phase within this idealized cycle.

Only daily closing prices are used within the VXX roll yield approximation. If you wanted to create your own futures portfolio then you could select any rebalancing frequency. Rebalancing every other day, for example, would mean setting the cycle length to n_{0}/2 and a two-day roll yield would result. Of special interest would be a monthly rebalancing, where the roll yield formula (with n=n_{0}=1) would correspond to a monthly roll yield where only the front month contract would be held:

This result is identical to my original definition of the monthly roll yield.

Finally, it is important to emphasize that the roll yield, for whatever rebalancing frequency, is a backward-looking calculation. The roll yield is obtained for the time interval just passed, and it has no predictive power regarding the roll yield in subsequent intervals. This means that roll yield estimates are most useful for evaluating their impact on historical VXX prices. There is no guarantee that future roll yield distributions will reflect those seen in the past.

**Is the VXX DRY All Wet?**

Unfortunately the VXX DRY approximation cannot be compared (easily) to VXX price trends on a daily basis. Most of the daily VXX price variation is caused by changes in the underlying futures prices, which in turn are connected to variations in the VIX index level. As with any security it would be unusual for the VIX to be unchanged over a trading day.

As a simple sanity check it is possible to compare the VXX price from two widely separated days for which the VIX index had essentially the same value. This condition occurred over a six month period between 9/21/12 and 3/21/13 where the listed closing VXX price fell from 34.80 to 21.02 as the VIX index on these days closed at 13.98 and 13.99 respectively. The compounded impact of the DRY approximation over this period implies a VXX price ratio of 0.620 between these two dates. This result agrees reasonably well with the actual price ratio of 0.604. The expected price ratio falls to 0.618 once the (0.15% annualized) three month Treasury yield return and the 0.89% annual VXX holding fee are included over this half-year interval.

Fortunately a component of the VXX DRY approximation can be tested on a daily basis. To see this let's define the ratio [r] between the closing prices of the VXX [V_{X}] and the synthetic contract [U]. The daily percentage change in this ratio is:

Consequently an estimate of the price ratio [r] can be computed over time by compounding its daily percentage change from the component contract prices once the first price ratio has been initialized. These derived ratio values can then be compared with the directly observed ratios. The following plot compares these two data sets taken over the most recent seven months:

The model ratio values closely track the actual ratio values absent the latter's "noise." This result lends credence to the assumed VXX framework and by extension to the VXX DRY approximation.

**Conclusion**

The roll yield of a futures contract follows from the nature of the contract. It exists whether one holds a single month contract or a portfolio of contracts. In a continually rebalanced portfolio such as the VXX the roll yield should be defined over the rebalancing interval. For the VXX this means that the daily roll yield is of paramount interest.

A useful approximation to the VXX daily roll yield has been derived and justified based on the published information regarding VIX futures and the VXX ETN. Compounding this roll yield on a daily basis can account for nearly all of the recent VXX "value" leakage.

**Disclosure: **I have no positions in any stocks mentioned, and no plans to initiate any positions within the next 72 hours. I wrote this article myself, and it expresses my own opinions. I am not receiving compensation for it (other than from Seeking Alpha). I have no business relationship with any company whose stock is mentioned in this article.