Triple leveraged ETFs marketed by Direxion have been all the rage lately. The fund management company says that they do not recommend buying and holding these ETFs. But is there any mathematical justification for this caution?

Before I answer this, it is interesting to note that these ETFs (e.g. BGU is 3x Russell 1000, TNA is 3x Russell 2000) are managed as constant rebalanced portfolios, a concept I discussed before. In other words, the fund manager has to sell stocks (or futures) when there is a loss, and buy stocks (or futures) when there is a gain in the market value of the portfolio, in order to maintain a constant leverage ratio of 3. This is also identical to what Kelly formula would prescribe, a methodology discussed extensively in my book, if the optimal leverage f were indeed 3.

However, the optimal f for such market indices are quite a bit lower than 3. Both Russell 1000 and 2000 have f at about 1.8. This means that since the funds are leveraged at 3, there is a real possibility that sustained losses could ruin the funds (i.e. NAV going to zero unless new capital is injected, which, er..., reminds me of a Ponzi scheme). So I would argue that not only should an investor not hold these funds for the long term, the funds themselves should not be leveraged at this level. Otherwise, it is a disaster waiting to happen.

## 28 comments:

@Ernie: Cheng and Madhavan from BGI recently published a nice research article which mathematical explains (including nice continuous-time models) why caution is indeed justified, and indeed the long-time NAV is zero. I analyzed the article in a recent post, including their effects quantitative strategies:

http://quantivity.wordpress.com/2009/07/28/leveraged-etfs-market-close/

quantivity,

Thanks for the reference ... certainly interesting and make a good point that these ETF's will create market momentum as well.

Ernie

Hi Ernie

This is a general comment having just purchased your book and reading it !

Excellent book, enjoying it.

The question i have is : are the pairs strategies, like the ones you describe in the backtesting chapter of your book, codable in Excel ? I dont have Matlab. Getting the sharpe ratio, the way you have shown it in Excel is easy. I was wondering whether it would be similar in Excel.

Another thing : it is not possible to get access to your website without a password. Where in your book is it ? Thanks, fx.

Hi Anonymous,

Thanks for your kind words.

Except for determining cointegration, it is easy to backtest the pairs strategies in Excel. However, automating the trading of pairs strategies in Excel is not easy.

Please see http://epchan.com/mybook.html for login instructions.

Ernie

Thanks for yr coments Ernie.

I trade forex profitably with Interactive Brokers using a discretionary MA based strategy.

I used Tradestation last year but got tired of (no time)learning Easylanguage - and as you say in your book, it binds you to TS only.

I might make more time for Matlab.

Incidentally, if you`re interested, try a 21 and 50 ema for GBPUSD in the 1 and 4 hour time frames for both momentum and mean reverting strategies. ( I use the terms loosely here). I find the 21 period MA acts as an excellent mean reverting tool upto the 5 min time frame. But upto 15min, both 21 and 50 work. You can see this by looking at a chart of Cable today.

If I get Matlab - how long do you think it takes a complete novice to start using - say your examples ?

Apologies for the long message.

Ta

Fxspasm.

Hi shamik,

Thank you for the tips about FX ... will explore them!

It will probably take a month to get to the point where you can understand and modify the code examples I gave in my book. Of course, it took me a year or 2 to find a way to code elegantly and maximize the benefit of array processing, but that's not strictly required to get started.

Ernie

shamik, having previous programming experience helps a lot.

Assuming a pair of assets ( stocks, fx commodities) are cointegrated, would the following plan work for pair testing in excel ? :

1) Take daily close prices of 2 stocks over a 5 year period

2)For each calculate a ratio : todays price/yesterdays price

3) Calculate the spread : ratio of stock 1 - ratio of stock 2

4)FInd out the SD of the spread ( and do a graph)

5) Take positions when spread > 3SD.

Many thanks for any comments.

shamik,

Your scheme would not work very well. What you need to do is to use linear regression (available in Excel I think, but not in a continuously rolling calculation) to determine the hedge ratio. Linear regression is a least square fit, quite different from just dividing the two prices and averaging them.

Ernie

Ernie,

Tried to email you but it bounced back. I have your book - liked it very much, by the way - and am trying to login into the premium area of your site using the password from the book. However, the site also asks for a username, which I don't have. Am I doing something wrong?

Best

Eric

Hi

When you do pairs trading, do you analysis min data, hour data or daily data?

And when you do cointergration, do you always analysis data from the very beginning?

thx in advance.

Erni,

I have question about hedge ratio when using with not dollar currency pairs. How we should count it ? eg. how we should calculate hedge ratio for EURUSD and USDCHF or eg EURJPY and EURUSD ?

I have Your book and it's great, You describe in simple words (and code) topics I've try understand using some research papers and had some trouble ;)

Regards

Eric,

The username is the same as the password.

Ernie

Anonymous,

I analyze pair trading using daily data.

Cointegration can be studied either from the beginning of data, or for a rolling 1 to 3-year lookback period.

Ernie

Andrzej,

I am no expert in Forex, but I believe you would need to convert the cross-rates to a base currency before you can calculate the hedge ratio.

Ernie

Ernie

for quant naive people like me a beginners book would beuseful to properly understand some of the stuff you describe in your book - like PCA.

Could you recommned one ?

Cheers

Shamik.

Shamik,

For PCA, you can read the Active Portfolio Management book on the right sidebar of my recommended books. For other topics, Market Models (also in the recommended list) is more suitable. These and other books actually involve more complicated math than is contained in my book though.

Ernie

This article says that triple leverage is probably a bit high but double leverage is OK for long term holding

http://www.ddnum.com/articles/leveragedETFs.php

Why not simply rebalance the the leveraged ETFs to undo the fund's rebalancing? When the leveraged ETF return exceeds the underlying index by a trigger point, you sell. When the leveraged ETF return is less than the index, buy more.

Fund management is essentially buying more as the fund as goes up and selling as the fund falls in value. That pretty much guarantees poor long term performance.

Américain à Paris,

Daily rebalancing (buy more when the market value goes up) is not the problem. In fact, the fund must daily rebalance to optimize long-term performance, as is proven by Kelly formula (and explained in my book and Ed Thorp's article). It is the leverage that will kill the fund. But of course, you can easily undo the leverage by using 1/3 of your equity to buy this fund. This may save your account from ruin, but alas, it won't save the fund from ruin.

Ernie

hi ernie,

I think what Américain à Paris is pointing out is that we can replicate exactly twice or thrice the underlying index by rebalancing the fund and thus achieve the target return. Leveraged etfs have the problem of not meeting its objective over long term. If you look on a monthly basis, you can see that if an index had returned 2%, a triple leveraged etf will not be exactly 6%, but would rather depend on the PATH it took to reach the 2%. So it depends on the realized variance. what Américain à Paris points out is that an investor can get exactly 6% by rebalancing regularly. So if the index is at 1% and the triple leveraged etf is greater than 3% by a certain trigger point, then you sell the fund and vice versa. Hope its clear.

Hi Anonymous,

I don't disagree that you can improve your returns by daily-rebalance an unleveraged ETF. But I think Américain à Paris is trying to rebalance a leveraged ETF. As I explained, no matter how one wants to trade a leveraged ETF, it doesn't alter the fact that the fund itself may go to zero at some point in the future. So unless you are short the ETF, it is an instrument that you would want to hold for long.

Ernie

You mean NOT want to hold?

Thanks to the author and all commenting participants for sharing their reflections on leveraged ETF's.

Aside from all the caveats, I succesfully trade several of the 3x bull ETF's. I'd short the bears if their stock were easy to borrow. Not always the case.

A few papers have been published over this last year on the math behind these instruments.

I beleive my success is not from having a complete handle of the underlying math behind them, but simply utilizing volatility as a trade filter within my strategy to avoid the aforementioned deterioration over time.

I also factor in the existing leverage of the ETF in my position sizing algo, so as to avoid over exposure and downside risk after the ETF has trended powerfully at times.

I witnessed an associate lose several hundred thousands of dollars trading the SSO/SDS during late 2008 early 2009, when volatility was at its highest. Not for the faint of heart my friends.

Last but not least, consider that certain industries/sectors in the 3x class suffered deterioration much more than others.

DRN, TYH, MWJ, & EDC have not experienced such drastic deterioration as the others, thus far...

Anonymous,

Right, it should read "you would NOT want to hold for long".

Thanks.

Ernie

Hi!Is there anyone that have a good matlab file where u can se the effect of different volatility etc. on leveraged etfs?

Hi Anon,

You may need to define your request better ... what exactly do you mean by "effects"?

Ernie

Hi Anon,

You may need to define your request better ... what exactly do you mean by "effects"?

Ernie

I mean that is it posible to have a matlab file that shows the ways of leveraged etfs? And if we change the indata of volatility (sigma?) than we se a different result, the effects of volatility. great volatility vs great trend ect...

And maby its posible to change between 2X(-2) and 3X (-3) leveraged and se how they respond to the volatility

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