Why Leveraged ETF Returns Diverge: Hedging Costs and Daily Compounding

# Leveraged and inverse leveraged ETFs - what is the exact defintion?


# Leveraged and inverse leveraged ETFs - what is the exact defintion?












I just had a quick look at the daily returns for a few pairs of leveraged ETFs - it appears that the percent daily returns do not match perfectly. For instance, looking at FAS/FAZ, the returns for the two can diverge as much as 15 bps per day.

Reading the summary, the language appears very clear:

FAZ:The investment seeks daily investment results, before fees and expenses, of 300% of the inverse (or opposite) of the performance of the Russell 1000® Financial Services Index.

FAS:The investment seeks daily investment results, before fees and expenses, of 300% of the performance of the Russell 1000® Financial Services Index.

Seems to me that the process has to be fairly simple - every day, managers of these ETFs submit MOC orders that will make the notional exposure to the underlying index is equal to the leverage.

- I am missing something in understanding how these products function like mandate does not have to followed exactly and the manager uses his discretion in matching the returns (e.g. he does not need to match the close to close returns)?

- Is the divergence is daily returns driven by the managers inability to execute order directly at MOC or some sort of expences (e.g. comissions/borrow/fees or maybe execution mismatches) are passed into the ETF price?

- I am understanding the concept of "300% of the performance" incorrectly (I assume that means R(ETF) = R(Index) * Leverage, so R(ETF 3x) == -R(ETF -3x) by definition? I assume that under performance they mean percentage return of the index.

- It's a problem with the data - I used Yahoo data for this little study. Can I assume that the closes should be correct (I know opens, high/lows are sometimes not).

PS. I do know that the re-balancing process creates negative exposure to the variance (by virtue of buying on the up days and selling at the close of the down days) but the daily percentage returns should not be affected, at least that's how I understand the replication process.

## Answer by chrisaycock (score 1)

https://quant.stackexchange.com/a/7354

There are three prices to consider when discussing an ETF:

- the ideal price as represented by the index,

- the NAV of the fund based on that day's holdings, and

- the market value traded on a stock exchange.

This third price is what you see.

In your example, Russell has calculated a cap-weighted value based on the annual membership. Direxion then determines daily holdings to mimic the index returns as best as possible. As you point-out, there is transaction cost associated with the daily rebalance, plus Direxion is subtracting their management fee. This slippage is reflected in the holdings; Direxion may liquidate some assets and claim a portion of the cash in order to satisfy these costs.

Once Direxion releases the daily holdings, market markers will provide liquidity with the goal of making a profit. They do this by quoting prices that give them a slight arbitrage to the underlying assets.

So the effect you see isn't a misunderstanding or the result of nefarious activity; it's merely the by-product of how ETFs work.

## Answer by Matt Wolf (score 1)

https://quant.stackexchange.com/a/7355

The key reason why you observe divergent performance patterns is related mostly to the following:

- The biggest reason is the different cost to hedge those products. The costs to implement and especially maintain the hedge on the long vs short side can be very different. Either the hedge is implemented through an index replication in which case the manager has to buy/sell stocks (different costs involved in borrowing shares vs simply buying the shares outright) or through derivatives contracts in which case the manager is subject to opposing effects of being long vs. short contango/normal backwardation, different costs of the roll. So, even though some ETFs may attempt to track an index (many actually track the futures traded on the underlying, such as VXX), what matters for the performance is the hedging costs as well, and if the hedge is done through futures contracts then there is a direct link between the question at hand and my mentioning futures.

- Even for simple stock investments the costs associated with long and short positions are vastly different. Long non-margined holdings do not incur any cost. However, short positions incur borrowing cost plus the cost inherent in the risk of borrowed shares being called. Even if both products' underlyings are margined there is a difference between the costs on the long and short side as borrow and loan rates are different.

In summary, the simple answer is different cost of hedging are mainly responsible for the divergence in performance. Another more trivial reason is that each ETF's price is subject to supply demand. The answer thus is different costs of hedging under which no-arbitrage arguments still hold.

## Answer by cdcaveman (score 0)

https://quant.stackexchange.com/a/7358

to more concise i will answer the question.. There is a known mathematical decay from volatility. For this reason companies like Proshares will state that these investments are not good long term investments. i'll go over an example below. (extrapolate this... 50% gain on 100 = 150, 50% loss on 150 = 75, there for to get back to 100 from 150 you have to lose 66.6666..%) = fact... empirically.. just something to note.. on to my example

lets say index A goes up 10 percent on day and then drops 9.09 percent the next day. for a two day return of zero percent.. Simultaneously a leveraged short fund based on the index would go up 20 percent ,then on day two would jump down 18.18 percent... (second days index drop in percentage times 2) but because the rise would be from a base equaling to 80 percent of your original investment you would now have less then 95 percentage of what you had put up.. so even though the index is sitting right where it started out at your sitting on a loss of like 5.46% ...

simply math... index day one: 100(1.1)=120 (ten percent gain) index day two: 120(.9090..)= 100 thats a loss of approx 9.09 percent

now the leveraged fun short ETF day one: 100(.80)= 80 (this is the 80 base mentioned above) short ETF day two: 80(1.1818)=94.54 (at this point your sitting on the 5.46% los)

this happens outside of any other fees and or replication intervention