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April 04, 2009 – Comments (12)

I know that you guys hate leveraged ETFs, but in this case a commodity ETF is much different.

Usually commodities move in the same direction for a sustained period, unlike stocks.

Therefore the expected returns on this ETF should be exponential (if the bet is correct). Just look at DEE, its performance is very good eventhough it is a leveraged ETF.

DYY should dublicate that on the upside if we assume that price movement is similair.

#1) On April 04, 2009 at 6:37 AM, portefeuille (98.88) wrote:

It is not the leverage (factor of 2 for this ETF) that makes the returns (rather the price-function) "exponential". It just makes the exponent larger by a factor of 2, if indeed you have steady returns "for a sustained period".

1% per day for 100 days for the underlying index gives you a return of (1+1/100)^100 = 2.705, close to e = ca. 2.718, whereas (1+2/100)^100 =ca. 7.245 close to e^2.

In this example the return for the index would thus be around 170.5% for the index and 624.5% for the (2x)leveraged ETF.

The exponential nature is not due to the leverage but due to the steady growth.

If you are right the leveraged ETF would certainly enhance your profit quite nicely.

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#2) On April 04, 2009 at 7:27 AM, kaskoosek (30.18) wrote:

Portefeulle

I was doing some calculations yesterday, and I beleive that in fact leverage makes the price function exponential even if the returns are constant.

If we start with a stock that is 100\$. Each day the price increases by 1 dollar (not 1% per day).

The returns in this case are constant with no leverage. 1\$/day

But when there is leverage the returns are no longer constant, and the price function becomes exponential.

That is why we some times see huge spikes for leveraged ETF when the market moves in the same direction for a sustained period.

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#3) On April 04, 2009 at 10:53 AM, portefeuille (98.88) wrote:

in that case -  yes, it would look exponential.

with l :=  leverage, x_n := index after n days, y_n = leveraged ETF after n days you get

y_n = y_(n-1) * (1 + l / (x_0 + n - 1))

x_n = x_0 + n

For your leveraged ETF (l =2) and for x_0 = y_0 = 100 you get

x_n = 100 + n and

y_n = y_(n-1) * (1 + 2/(99+n))

here some (rounded) numbers:

y_1 = 102,

y_10 = 120.89 (-> return of 20.89% still close to twice the 10% return for the underlying),

y_20 = 143.76 (43.76% vs. 2*20% = 40%)

y_50 = 224.26 (124.26% vs. 100%)

y_100 = 398.02 (298.02% vs. 200%)

y_1000 = 11,947.54 (11847.54% vs. 2000%) (i.e. better than what might be "naïvely expected" by a factor larger than 5 after 1000 days).

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#4) On April 04, 2009 at 12:43 PM, kaskoosek (30.18) wrote:

y_n = y_(n-1) * (1 + 2/(99+n))

Yup correct.

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#5) On April 04, 2009 at 4:03 PM, portefeuille (98.88) wrote:

correction: y_1000 = 11,991.09 (y_998 = 11,947.54)

The solution to the resulting differential equation is something like

y(n) = c * (n+100.5)^l , where c = ca. 0.0099009883 and l = 2 in your example.

So the function y(n) is polynomial in n (NOT exponential!).

Hope that is any help to you ...

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#6) On April 04, 2009 at 11:27 PM, kaskoosek (30.18) wrote:

For any exponentially growing quantity, the larger the quantity gets, the faster it grows.

A polynomial equation is exponential.

n^(1.5) is also exponential, since ithe derivative is upward sloping.

Anyway, I am just curious how did you get the complexity?

Isn't it of the form (n+(2/n))^n

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#7) On April 04, 2009 at 11:42 PM, kaskoosek (30.18) wrote:

"Isn't it of the form (n+(2/n))^n"

Wrong, I mean (1+2/n)^n.

I am definitely wrong some where, because this is e^2.

It should be an infinite product and not the way I wrote it, because n is changing.

101

102

103

...

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#8) On April 05, 2009 at 9:43 AM, portefeuille (98.88) wrote:

You might want to have a look here. IV§1 is about the exponential function, that might be a good starting point (polynomials are mentioned earlier (f(x) = a_0 + a_1 * x + a_2 * x^2 + ... + a_n * x^n is a polynomial in x).

Now starting from the recursion formula

y_n = y_(n-1) * (1 + l / (x_0 + n - 1)),

you turn this into a difference equation and solve the resulting ordinary differential equation. I chose the parameters to have a nice fit for n between 1 and 1000 (n is the "day" in your example).

You can have a look at the abovementioned solution, it fits quite nicely (use a spreadsheet if you like, you might want to produce a graph of the solution).

In a first semester German "analysis" course this goes by the name "Exponentialfunktion schlägt alles tot" (something like "exponential function kills everything", have a look at slide 10/13 here, that presentation might be a nice quick introduction to the exponential function.

If someone is interested I can "fill in the blanks" (I might produce a nice tex-document and deposit that on google) ...

Anyone still thinking exponential and polynomial is all the same have a look here (the lecture is nice - AND you have a shot at winning a million dollars) or have a look at the other millennium problems.

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#9) On April 07, 2009 at 10:37 PM, anchak (99.91) wrote:

portefeuille : Are you the RA/TA , Student or the Prof?

I am surprised you didn't start with the fact that the exponential number - e itself is an approximation thru an infinite polynomial expansion ....

lim ( 1 +r/n)^n = e^r .....which is what you showed thru the example.....

Thus in reality ( especially general parlance) - exponential function ( ie of the form e^x) and the compounding interest expansion ( which is polynomial) is interchangeable - right?

Of course this is a specific example - ie all exponential functions are not polynomials and vice-versa.

However, they are

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#10) On April 07, 2009 at 10:41 PM, anchak (99.91) wrote:

The only issue is of course timing.....while commodities have large unidirectional movements - but during uncertain times ( like now) - they are also prone to this yo-yo behavior

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#11) On April 07, 2009 at 11:31 PM, portefeuille (98.88) wrote:

I am sorry, I do not quite understand what you are saying.

"I am surprised you didn't start with the fact that the exponential number - e itself is an approximation thru an infinite polynomial expansion ....

lim ( 1 +r/n)^n = e^r .....which is what you showed thru the example....."

This I believe is in response to comment, which was my response to "Therefore the expected returns on this ETF should be exponential".

I think kaskoosek wanted to make the point that if "commodities move in the same direction for a sustained period" the leveraged ETF DYY (whose underlying is the "Deutsche Bank Liquid Commodity Index - Optimum Yield") should display an "exponential" price function whereas the underlying does not. Since he did not specify what exactly he meant by "move in the same direction for a sustained period" I simply assumed he meant something like "1% per day".

All I wanted to say in #1 was that in that case the exponential character would already be seen in the underlying and was not due to the "leverage". I thought it is obvious that e = lim_n (1+1/n)^n (I even linked the wikipedia article for e where that formula is really hard to overlook.

Then you say:

"Thus in reality ( especially general parlance) - exponential function ( ie of the form e^x) and the compounding interest expansion ( which is polynomial) is interchangeable - right?"

The exponential function is the limit of a series of partial sums and each partial sum is a polynomial function, but the exponential function is not. In fact "any exponential function ... eventually outgrows any polynomial function"  (taken from slide 10/13 of the presentation I mention in #8).

I did not mean to sound like a professor, sorry if I did!

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#12) On April 08, 2009 at 8:10 AM, anchak (99.91) wrote:

portefeuille : My friend you took it in the wrong connotation....I would have liked you even more if you actually taught that course!

What's your background incidentally? ( You really intrigue me.... you write about cricket...and by the looks of it you are from Germany)

Bang on! :  "any exponential function ... eventually outgrows any polynomial function"

I guess people forget the distinction of an Infinite expansion limit( which the exponential function is of a polynomial form) as opposed to a finite polynomial.

Its very nice to have a conversation like this.

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