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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9 comments:
Hi Max,
I don't think the issue with your thought experiment is that you have unrealized gains, since you can't do what you've proposed with the unrealized gains from a long position, but I know what you're getting at... you have available uninvested capital.
Also, that wouldn't be very, very risky because you're buying high and selling low, since that's what you should be doing for short positions.
Best,
Josh
You're right Josh, I've edited it.
R is great by the way- I learned apply, sapply, and merge yesterday. Very handy.
Regards,
Max
Hi Max,
Your post is quite thought-provoking.
As I said in my comment to your previous post, I know next to nothing about UYG and SKF. I will only give you my explanation from a pure mathematical point of view.
"However, imagine a security that is artificially constructed to make daily returns opposite of what it is based on."
I don't believe this is possible, as I will explain later.
First of all, when one talks about the "return" of a short position, there is always the question what is used as the denominator in the calculation. Let's ignore the ambiguity here, and assume that we have securities A and B -- the daily return of A is always the negative of that of B ...
Say, the daily return of A is either -2% or 1%, with equal probabilities. Then after 200 days, a very possible outcome is a return of
(0.98^100)*(1.01^100)-1 = -64.13%
The corresponding return of B is
(1.02^100)*(0.99^100)-1 = 165.18%
So the sum of the returns not only can be negative (as in UYG and SKF), but also can be positive.
Now let's change the numbers. Let's say that the daily return of A is either -10% or 9% with equal probabilities. After 200 days, a likely outcome for A is
(0.9^100)*(1.09^100)-1 = -85.31%
The corresponding return for B is
(1.1^100)*(0.91^100)-1 = 10.51%
Is it possible to construct a pairs trading strategy which shorts both of them and makes a high return with very very low risk? I don't think so.
Consider the not unlikely event that in 105 of the 200 days A's value goes down. The return of A after 200 days will be
(0.9^105)*(1.09^95)-1 = -94.36%
The corresponding return of B will be
(1.1^105)*(0.91^95)-1 = 185.21%
The strategy of shorting both would have been disastrous.
What went wrong? In order to achieve a very very low risk, one has to re-balance the portfolio each day so that A and B are equally weighted. But then the daily return of the portfolio would have always been zero. If one doesn't re-balance the portfolio, after a while, the portfolio might be heavily bias toward A (or B). But then the risk would be very high considering the high volatility of A (or B).
Similar to the coin flipping example in your previous post, it's the expectation of the outcome that matters. Suppose we have a $200 portfolio with $100 invested in A and $100 invested in B. What is the expected return after 200 days? After 200 days the expected value of A is $100*((0.9+1.09)/2)^200=$36.70. The expected value of B is $100*((1.1+0.91)/2)^200=$271.15. The total expected value of the portfolio after 200 days is $307.85. That's a return of more than 50%!
You could try to use different numbers, but you will always get a positive return because of the mathematical fact
(1+x)^n+(1-x)^n > 1 if 0< x<1.
It turns out that the profitable strategy is not to short both A and B, but to long both A and B. How could this be possible? I believe the answer is that such A and B do not normally exist. Otherwise, there would be exploitable opportunities.
What happens if we let A = long a share of a stock, and B = short a share of the same stock? This is probably similar to the UYG and SKF situation. Wouldn't the expected return of a portfolio of A and B always be zero? I will leave it to you to figure out the differences ...
Xun
An after thought ... If we already know the distribution of A and B, long B only would have been profitable. So it shouldn't be surprising that a portfolio of A and B has positive return. Hmm ...
Xun,
After our analysis I don't think there's an arbitrage opportunity. Maybe the pair strategy allows you to bet on volatility rather than price though.
Long both A and B is clearly not the same as long SKF and UYG since if you had bought both of them you would be down.
Regards,
Max
Xun said..
"However, imagine a security that is artificially constructed to make daily returns opposite of what it is based on."
"I don't believe this is possible, as I will explain later."
Maybe you have a better insight into this than I do (I've been asking around about this one). I have been trying to replicate the risk profile of an ultrashort ETF, and the closest I can come is a deep ITM option with considerable time till expiration. Clearly, such a vehicle does not "make daily returns opposite of what it is based on."
My question (as an aside to this running thread) to Max and the respondees then, is A. Do you concur on my assesment of the risk profile (I have strategies that involve ultrashort ETFs, and I may need to employ them in the future sans the ETFs, and hence require their replication) and B. Do you think there is a better way to replicate the risk profile of these creatures?
(great blog site btw, many *very* interesting and insightful threads on here, thanks!)
Anon,
To replicate an ultrashort ETF, short an index with 2X leverage/margin and releverage the position every day near the close. Using options this should be similar, the key is to rebalance each day like you have to when using short positions. There is some speculation that these leveraged ETFs might incur heavy slippage from their unidirectional rebalancing, which would be impossible to replicate. So depending on your goals in replicating the ETFs, it may or may not be feasible.
Regards,
Max
Sorry, didn't intentionally mean to post that last one as "Anon." So what you are saying (and I have not considered) is that for a 2X ultrashort, I want puts whose delta sums to -2.0, and would rebalance daily for this (makes sense). Since I am rebalancing daily, this makes me rethink that it matters not what the expiration or strike price is (actually, however, I want to keep that delta as close to -2.0 through the trading day as possible, without rebalancing until the following day; there is some other greek here that I am not accounting for I think?) -Ralph Vince
Ralph,
I think that's right but I know very little about the intricacies of options. You can replicate the inverse ETF by shorting an ETF of the underlying index and rebalancing daily.
After some conversations with people at Credit Suisse AES and Proshares and Direxion, I learned that they do not rebalance throughout the day, but only once per day near the close (within the last 30min). So I don't know if you should rebalance throughout the day to keep the delta at -2.0 or only at the end of the day. I don't have experience trading options so I don't know exactly how they differ.
Regards,
Max
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