Financial Engineering is a multidisciplinary field involving finance and economics, mathematics, statistics, engineering and computational methods. The emphasis of FE & RM Part II will be on the use of simple stochastic models to (i) solve portfolio optimization problems (ii) price derivative securities in various asset classes including equities and credit and (iii) consider some advanced applications of financial engineering including algorithmic trading and the pricing of real options. We will also consider the role that financial engineering played during the financial crisis. We hope that students who complete the course and the prerequisite course (FE & RM Part I) will have a good understanding of the "rocket science" behind financial engineering. But perhaps more importantly, we hope they will also understand the limitations of this theory in practice and why financial models should always be treated with a healthy degree of skepticism.
Negative Exposures and Leveraged ETFs
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来自 Columbia University 的课程
金融工程与风险管理，第 2 部分
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从本节课中
Practical Issues in Implementing Mean Variance
Problems with mean-variance analysis; ETFs and leveraged ETFs; VaR and CVaR for asset allocation; survivorship bias, performance evaluation and other statistical pitfalls.
与讲师见面
Martin HaughCo-Director, Center for Financial Engineering
Industrial Engineering & Operations Research
Garud IyengarProfessor
Industrial Engineering and Operations Research Department
In this module we're going to talk about how to get negative exposures in the
context of mean variance portfolio selection.
And in doing so, we'll introduce the idea of exchange traded funds and leveraged
exchange traded funds. Short positions can often result in very
high returns. But short positions are very risky.
A long position has a limited downside. The lowest price an asset can take is 0
and therefore the largest amount of money that one is exposed to lose is the amount
invested. Short positions, on the other side, have
unlimited downsides. Short positions are created by selling
assets borrowed from a broker and these have to be repurchased and returned to the
broker at a later date. Since the price of an asset can become
arbitrarily large, the potential loss from a short position can also be arbitrarily
large. So we need a product that has negative
exposure, meaning it gives me, it behaves almost as if it was a short position, but
has limited liability, limited downside. One such product is something called an
exchange traded fund. Exchange traded funds are products that
track the returns on stock indices, bond indices, commodities, currencies, etc
cetera. This table is something that I took from
bloomberg.com and which gives you some idea of ETF's that are tracking various
indices in the world. So SPDR S&P 500 tracks the S&P 500 index.
Ishares Russell 2000 is another one which tracks the FTSE.
The QQQ, Japan, and so on. You can have exchange traded funds on any
number of different things. Commodities, currencies, and so on.
If you go to Bloomberg and look for ETFs, you will see lots of different ETFs.
So these ETFs that are shown on this particular slide track the daily return on
the underlying index. So the daily return on the S, spiral, S&P
500 would be the tracking the daily return on the SMP500 index.
Etfs also allow you to leverage the returns indices.
Leveraged ETFs produce daily returns that are multiple of the daily returns on the
index. Bull ETFs will return beta times the daily
return on the index and beta could have values 2 and 3 and so on.
So po, pro ultra share as in P500 is a bull, ETF on the under, underlying index
which is acid P500. Pro SHORT is another ETF which is a
leveraged ETF, but it turns a negative number times the daily return.
So an inverse EDF returns minus beta times the daily return and beta can then take
values 1, 2, 3, and so on. An inverse EDF is a product that can a
negative exposure to the index ...with a limited liability.
And why is the liability limited? It's because you tend, you're only exposed
to the amount that you bought. You bought a 100 dollars worth of an
inverse ETF, you can only lose a 100 dollars and yet at the same time get a
negative exposure. But one has to be very careful when one
uses leveraged ETFs in constructing a portfolio.
In the next few slides, I'll go over what kind of risks are associated with
leveraged ETF's? Before we get down to explaining, the
source of the trouble with leveraged ETF's lets understand what happens, to the
returns on an ETF. And return on an ETF is the return on the
underlying index compounded daily. So on a long ETF, it's simply a, the
return, the cumulative return or the gross return on an ETF will be a product of each
of the daily returns, just like a stock. The return on a leveraged ETF will be the
daily return multiplied by beta. So if you have a beta ETF and beta can
take values plus 2 and 3 or minus 1, minus 2, minus 3, The way you compute the gross
return is you take beta times the daily return on the underlying index, that gives
you the daily return. On the ETF, you multiply it all the way
through and you end up getting the gross return over TPS.
The return itself is actually constructed by investing in the derivatives which are
on the index rather than buying the stock s in the index.
This daily compounding has consequences that are not immediately obvious.
On the next slide, I'm going to work through and explain to you, how, depending
upon whether the ETF is simply a long ETF. Where it a leverage EDF often have an
intuition about daily returns can go wrong.
So, here's a simple example. Suppose you have an index.
The index value today, let's time t is equal to 0 is 100.
At time t equal to 1, let's say day 1, t equal to 1.
It goes up to 105, and at time t equal to 2 which is day 2, it comes back to 100.
So if you look at what happened over this period, nothing changed.
You went up or, and then came down. So it really there was no level shift.
The only thing that you experienced over this period is volatility.
It went up and down. So if you had bought the index you would
have bought the index at 100. The value would have gone up to 105 and
then come down. If you had bought a long ETF what would
have happened? Suppose the current value of the ETF or
the index or Is 100. Then on day 1, the price of the index
would have been 100 times 1 plus R 1, which is 105.
Perfectly tracking what is going to happen, so it should be 100 here, it would
have gone up to 105 and what happens on day 2?
The ETF will go down by 105, which is the current price times the return on day 2
and you will drop back to 100. So, so far when you're looking at what
happens to just a plain vanilla long EDF and the index they're tracking.
Now, let's see what happens to a bull ETF. A bull ETF let's say in this particular
case we take beta to be equal to 2. Twice bull 8 ETF.
The current price of the index is 100. Over the 1st day the index returned 5%, so
the bull ETF will go up 10%. So it now, on Day 1 it's value will be
110. It would have been increased by 10%
instead of 5%. On Day 2, the index went down 4.76%, so R2
was minus 4.76% and therefore, the, the amount by which it's going to go down,
this should be 110. The price, the ETF will go down by 110
times 1 plus 2R2. So this time it's going to drop down and
the net value that you'll end up getting here is 99.52.
So we'll, you will not come back 100, you will go below it.
So the bull ETF, over the 2 day period, actually lost money.
Now, if you look at the mechanics, it's not the case that bull ETF is returning
something that they did not promise. But if you were to interpret as if a bull
ETF was holding a twice leveraged position, then if the index went up from
100 to 105 and came back to 100, you expect to come back again.
And that intuition is wrong, you end up losing money because in every day you're
compounding by twice. So if you make money and next time you
lose double, you end up losing much more than what you made over the first.
Now let's see what happens to a inverse ETF.
So in an inverse ETF, you start form 100. On the first day, you end up making money
on the index. So the r one was 5%.
And therefore, the inverse ETF will lose money, so it'll come down to 95, by the
same percentage but now it's come down. On day 2, the return is minus 4.76 %, and
therefore the inverse ETF is going to make money, it's going to be 95 times 1 minus R
2. You'll end up getting to 99.52.
Again, you have lost money. You started from 100, you ended up to only
99.52. Moreover, the two times ETF and the
negative ETF both ended up at the same position.
Intuitively, we would have expected something which is twice the ETF.
And something which is negative ETF should not end up at a different position.
One of them making money, the other one losing money and so on.
But here we end up getting to the same position.
The index did not lose money. The long ETF did not lose money.
But the leveraged ETF as well as the inverse ETF Ended up losing money.
All of this happens because the daily compounding leads to some unintuitive
results. All the computations that I've shown on
the slide are correct in according to the rules of what the EDF returns are, but
they do not conform to the intuition that we have for these products.
In this slide, we are going to work through the mathematical expression for
the return of a leverage ETF. The gross return on the static leverage
position in the underlying index is simply going to be beta times s capital T minus
beta minus 1 as 0 times 1 plus rt, so this entire expression.
It's coming because we have to borrow beta minus as zero amount of money and time 0
and the funding rate is R so this is the amount of money that I have to return at
time capital T. So, that's the amount that I have to
remove from what I would gain from selling my positions in the underlying text.
Therefore, the return that I end up getting is approximately beta times.
St over SC. Now, that should think of almost as a
simple interest rate analog. So when I invest in a beta leveraged ETF,
I know that the returns that I'm going to get on this ETF is going to be compounded
daily. So I expect to see some sort of a
compounding effect. So it will be S1 over S0.
So I should expect a term like S1 over S0 to be the power of beta.
S2 over S1 to the power beta and so on. And therefore all of these terms should
cancel and it will end up getting approximately S capital T over S 0 to the
power beta. This is what I should expect based on the
daily return. Now, what truly happens with EDFs is that
you get a term which corresponds to this compounding effect, which is S T over S 0
to the power of beta. You get a term that corresponds to the
expense ratio which is sort of similar to this term up there.
You get another term which is an expense associated with the ETF itself which is FT
and then you get a third term which corresponds to, the volatility, of the wa,
of the returns over the period 0 to T. Si over Si plus 1 squared, sum it from i
equals 0 to n minus 1. And the reason I'm using N here and not
capital T is because N is the number of daily observations between time zero and
time capital T. So this term over here corresponds to the
volatility. And unless the term beta squared minus
beta is equal to 0. This extra term which corresponds to the
volatility drags the return down. So, there's a minus sign in front and that
minus sign drags the return down on the leverage EDF.
Effectively, the leverage EDF is short on volatility and so in markets that are
associated with high volatility, we expect that the performance of EDF will be much
worse than the compounding rate that we expect take and therefore we think we
should expect that in markets with high volatility leveraged ETF's are not going
to preform very well. Etf's are generally designed for short
terms plays on an index or a sector and should be used in that way.
Over long periods, leveraged ETFs do not work as one may expect, especially in
volatile markets. In this slide I'm just going to show you a
simple example illustrating this point that leveraged ETFs during volatile
markets do not perform as we expect them to do.
In the first half of this slide, I'm showing you the returns on pro ultra
shares. Oil and gas which is at twice ETF on the
underlying oil and natural gas index and DUG which is a Pro UltraShort oil and na,
natural gas which is minus 2 times leveraged ETF.
So if you look at what happened to the daily returns on these 5 days, they are
roughly opposite of each other. And there's a little bit of a gap between
them because of the expense ratio. Because of the funding rate and so on.
But roughly they're going opposite of each other.
And so if the intuition over data returns carries over, then we expect that these
ETF should give me opposite returns over a large period of time as well.
But if you look at what happened to the returns on these or the price of the
returns on these EDFs from September 2008 to February 2009, the blue line
corresponds to DUG which is the same thing up here.
The red line, or brown line, corresponds to DIG, which is the same thing up here.
Up here, DIG and DUG are running opposite each other.
Down here because of the volatility, both dig and dug get dragged down.
The price of these, both of these ETFs get dragged down.
And this is because ETFs, particularly leveraged ETFs, are short volatility.
They do not perform very well when the volatility is large.
