# SKF: Inverse Construction and Volatility

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I previously explained why market returns should be lognormally distributed with positive daily expectation (not continuous). However, imagine a security that is artificially constructed to make daily returns opposite of what it is based on. Then it should have negative daily expectation. This is the Ultrashort Financial Sector ETF, SKF, with a dash of leverage added in.

If you look at the chart of UYG and SKF over one year:

it is obvious that the mean of the two return paths is way less than 0%. UYG is the 2x leveraged financial sector ETF and SKF is the inverse of that. SKF is at +32.99% and UYG is at -89.54% from where they were a year ago. Intuitively you probably expect that the sum of the returns is equal to 0%. If not, it should be possible to constuct a pairs trading strategy which shorts both of them and makes a high return, e.g. -1*(32.99-89.54)/2= 28.75% annualized, with very, very low risk (since they move opposite each other daily).

First of all, how are SKF’s returns engineered?

Some kind of swap- basically a bet on the direction of the financial sector index put out by Dow Jones.

However I find it more intuitive put another way. Consider this thought experiment:

You think…

I previously explained why market returns should be lognormally distributed with positive daily expectation (not continuous). However, imagine a security that is artificially constructed to make daily returns opposite of what it is based on. Then it should have negative daily expectation. This is the Ultrashort Financial Sector ETF, SKF, with a dash of leverage added in.

If you look at the chart of UYG and SKF over one year:

it is obvious that the mean of the two return paths is way less than 0%. UYG is the 2x leveraged financial sector ETF and SKF is the inverse of that. SKF is at +32.99% and UYG is at -89.54% from where they were a year ago. Intuitively you probably expect that the sum of the returns is equal to 0%. If not, it should be possible to constuct a pairs trading strategy which shorts both of them and makes a high return, e.g. -1*(32.99-89.54)/2= 28.75% annualized, with very, very low risk (since they move opposite each other daily).

First of all, how are SKF’s returns engineered?

Some kind of swap- basically a bet on the direction of the financial sector index put out by Dow Jones.

However I find it more intuitive put another way. Consider this thought experiment:

You think the financial sector is going to drop more over the next 3 days- how will you replicate the daily returns of UYG, inverted? For the sake of example, imagine the price of UYG is currently at $100. Your first inclination is to short UYG and then just stay in that position for 3 days. The first day it falls 10%, you now have $10/10% of available, uninvested capital. The next day it is down another 10%, i.e. -9$ = ($100-$10)*-10%. But this only translates to you having made 9%! Next day, another 10% i.e. $8.1 = ($90-$9)*10%. Now it’s way off, only 8.1% when you aimed for 10%. The total is 127.1

Obviously the problem is that you had uninvested profits sitting on the sideline at the beginning of each day. If you cover the short at the end of the first day and then use all your money, $110, to open a new short position on UYG for the next day, when it falls 10% on day 2 you will make 11$. And the next day, covering and reinvesting in a similar fashion you will be up to $133.1 total. The trick is compounding the short position by reinvesting. It’s very, very risky because you essentially buy high and sell low to to match the daily returns (remember- buy low and sell high is supposed to be how to make money).

Take a look at p. 18-20 (20-22 of the pdf doc) of Statement of Additional Information for Proshares Trust. The colored tables show exactly how volatility and expected return interact, which I explored in the previous note. It’s quite well hidden, even the watered down version has only a tiny little link embedded on the SKF product webpage:

Another “problem” with SKF is its excessive leverage. Using data for the year up to 2/9/09, this Excel sheet shows that the optimal leverage would be .569 . Anything less than one means it’s overleveraged. I used Excel’s ‘solver’ add-in to find how much leverage maximized ending wealth, but feel free to test different numbers, including less than 0, de-inverting it. The cell you modify is in orange and the effect on final price can be seen in blue. (fyi spreadsheet methodology: leveraged returns are the daily closing price ratios, minus one, times the leverage multiplier. Finally this is turned back into a stream of prices, with the oldest price on 2/12/08 being the basis- the formulas are simple) Basically SKF is inappropriate for anyone who wants to hold it for a long time because it goes over the optimal Kelly leverage.

I like the Ultrashorts because I’m too young to open my own margin account, but it’s hard to look past the steady historical downtrend of the Ultrashort ETFs. However they make for interesting studies in financial engineering and position sizing and probability. I doubt most investors understand exactly what they are getting. I’ve been doing quite a bit of trading (compared to fundamental long-only “investing”) to enjoy the volatility of the past year and the Ultrashorts can be profitable. It’s nice having a short position that cannot lose over 100% no matter what, unlike a normal short.

Unfortunately it doesn’t look like I’ve found an arbitrage opportunity. Poor performance is just the result of uncommon volatility. Please leave a comment if you have any ideas related to this or anything else – I always may have missed something.

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