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					Performance comparison of the leveraged
inverse ETF Proshares Ultrashort 20+
Year Treasury and the underlying
Barclays Treasury Index
March 17, 2012

Keywords:        Proshares, TBT, Barclays Capital 20+ year U.S. Treasury Index, lever-
aged ETF, inverse ETF, swaps, Matlab


The behaviour of an Exchange-traded-fund (ETF) was investigated vis-à-vis the un-
derlying benchmark, i.e. Proshares Ultrashort 20+ Year Treasury (ticker: TBT) and
Barclays Capital 20+ Year U.S. Treasury Index (referred to as Barclays Index) re-
spectively. An ETF which tracks the Barclays Index, gross of fees, is iShares Barclays
20+ Year Treasury Bond Fund (ticker: TLT ). It was found that volatility in the un-
derlying index harms the buy-and-hold investor . In fact, using the historical volatility
of the Barclays Index, it was found that buy-and-hold investors of TBT should ex-
pect their investment sum to erode fastly over time. What is more, this destructive
dynamic comes on top of signicant friction costs and fees. This paper illustrates the
friction costs by providing a graph which compares the aims of the TBT fund with its
real performance, gross of fees. Finally, some additional risks were discussed.


Barclays Index Barclays Capital 20+ Year U.S. Treasury Index

TBT                  Proshares Ultrashort 20+ Year Treasury

TLT                  iShares Barclays 20+ Year Treasury Bond Fund

1        Introduction

The ETF industry has grown exponentially the last decade, as shown in Figure
1, from [1]. In H1 2011, the ETF industry had approximately 1.45 trillion USD

    1   Which is the case for all leveraged ETF's, as widely covered in literature.

1 Introduction                                                                  2

             Fig. 1: Global growth in AUM in the ETF industry [1]

under management. The rst ETF's oered conservative products akin to tra-
ditional mutual funds. However, in the years leading up to the nancial crisis,
there was a rise in ETF's which invested in derivative contracts. The fast-pace
change in this industry led to the absence of an accomodating regulation frame-
work (e.g. because derivative-based ETF's trade on the stock exchange, a lot
of countries still treat them as equity instruments with respect to capital gains,
dividends and etc.).   In section 2, it will be shown that the rise of leveraged
ETF's destabilises the market increasingly.

This paper compares a swap-based ETF, i.e.       Proshares Ultrashort 20+ Year
Treasury (ticker:NYSE:TBT) with the underlying index of which it aims to
replicate twice the inverse daily performance, i.e.   the Barclays Capital 20+
Year U.S. Treasury Index. Firstly, an analysis on the mathematics of long-term
returns for TBT will be conducted. Secondly, friction costs will be examined.
Lastly, some other characteristics of TBT will be scrutinised, specically those
which have an impact on whether or not the fund performance tracking aims
are achieved.
2 The mathematics of long term returns for TBT, excluding costs                       3

Fig. 2: An intuitive example as to why buy-and-hold strategies in leveraged

           ETF's destroy value over time (S and A represent the underlying in-
           dex and double-leveraged ETF value resp.; rt0,t1 and rt1,t2 represent two
           consecutive daily returns), from [2]

2        The mathematics of long term returns for TBT,
         excluding costs

First of all, a performance comparison was made between TBT and the under-
lying Barclays Index for the period since inception of TBT, i.e. May 2008. A
proxy was used to calculate the performance of the Barclays Index, i.e.             the
ETF TLT. TLT tracks the Barclays Index performance, gross of fees. In this
paper, the Barclays Index performance was obtained by adding back annual fees
incurred for TLT.

Ever since leveraged ETF's emerged, there has been confusion and controversy
about their suitability for buy-and-hold investors. However, this controversy is
rooted in ignorance.        For example, [2] provides a detailed analysis on the be-
haviour of leveraged ETF's. An important conclusion from [2] is that leveraged
ETF's which aim to return a multiple of daily returns are expected to destroy
buy-and-hold investor value over time if the underlying index is volatile. Figure
2, from [2], illustrates this intuitively.

As derived in [2], over time, the gross return of a leveraged ETF returning
a multiple         x on the daily performance of the underlying, is the gross return
of the underlying in that period to the x           power, then multiplied by the scalar
             2 2
exp((x-x ) tN /2), with the annualised volatility and tN the number of years.

For example, I found that the Barclays Index had an annualised volatility of
18.4% in the period May 2008 - March 2012. For the sake of illustration: assume
the Barclays Index followed a geometric brownian motion with no drift in this
period, it was found that the buy-and-hold investor should then expect to nish
with 12% of his original sum invested. This is an annualised expected loss of

An alternative intuitive explanation for these economics are oered by [2]. Since

    2   For TBT, x = -2
3 Frictions incurred                                                                        4

TBT aims to return double the inverse daily returns of the Barclays index, it
rebalances its position every day depending on the performance of the Barclays
index.      If it were statically leveraged (i.e.   it were not to change its position
over time), the (-2) return multiple it starts out with would change over time.
For example, if changes are favourable for TBT investors on day 1, the stati-
cally leveraged TBT fund would track the inverse performance of the Barclays
index less then twice on day 2. Therefore, the fund has to increase its expo-
sure when the underlying index goes up and vice versa. Eectively, the fund
is short gamma . The economics for the fund holder are akin to writing an
option and consequently delta hedging it: If volatility turns out to be low, the
option premium covers the hedging costs and an economic prot. If volatility
is unexpectedly high, a net loss is realised. Likewise, for the TBT fund holder,
the short gamma disadvantage comes with an advantage, i.e. compounding is

If volatility is low, the eect of daily compounding is magnied and the, for
example, yearly returns of TBT will outperform the yearly inverse returns of
the underlying multiplied by two.          This is the case for both gains and losses.
However, if volatility is high, the short gamma exposure will dwarf the ben-
ets of the magnication of compounding, the TBT return will underperform.
This trade-o is depicted in Figure 3. If history is any guide to the future w.r.t.
volatility and magnitude of returns, TBT falls in the shaded area, destroying
value for its long-term holders.

3        Frictions incurred

Rather than repeat the derivation of the relationship of performance between
levered funds and their underlying indices, found in [2], we proceed by compar-
ing the fund performance of TBT (gross of fees) with an imaginary index which
tracked twice the inverse daily return of the Barclays Index perfectly (the TBT
fund aim). We then observe and investigate the dierence with TBT. This can
be seen in Figure 4. Stock market days were used. After almost four years, the
perfect replication is quoting 17.7% higher. This is an annualised cost of 4.4%
for the buy-and-hold TBT investor, excluding 0.93% of fees!

Because the fund needs to readjust its exposure on a daily basis, it aims to
nd counterparties in time. This results in additional brokerage costs and pos-
sible borrowing costs (to obtain exposure).

    3   An actor which is short gamma exacerbates market movements and thus destabilises
the market. This destabilising eect might not seem signicant when looked at individually,
but section 1 already provided evidence that there is an increasing centralisation of leverage
in leveraged ETF's with increasing assets under management.
3 Frictions incurred                                                    5

            Fig. 3: TBT Fund returns versus the underlying index, [3]
4 Additional risks                                                                     6

Fig. 4: Perfect replication of twice the inverse daily performance of the Barclays

             Index versus the performance of TBT (x-axis in stock market days)

4          Additional risks

Although there is an extensive list of risks legally disclosed in the TBT prospec-
tus [3], I will discuss the most important ones. Quoting the TBT prospectus

              Additionally, with respect to the use of swap agreements, if the
           Index has a dramatic intraday move in value that causes a material
           decline in the Fund's NAV, the terms of the swap agreement between
           the Fund and its counterparty may permit the counterparty to im-
           mediately close out the transaction with the Fund. In that event,
           the Fund may be unable to enter into another swap agreement or
           invest in other derivatives to achieve the desired exposure consistent
           with the Fund's objective.

Although the historical correlation of the fund versus its aims was 99% accord-
ing to the TBT fact sheet [4], there exists a risk that the fund will be unable to
replicate the fund's aim in the case of extreme events. However, this possibility
is not limited to the occurance of the event quoted.

Additional risks in the case of extreme events are:

       •   the fund might be forced to liquidate a position at re-sale prices (liquidity

       •   the counterparty might default (counterparty risk)

       •   a lack of counterparties
5 Conclusion                                                                    7

    •   high nancing costs in case of a distressed money market (liquidity risk)

5       Conclusion

ETF's which deploy leverage on daily returns of an underlying asset are suitable
for traders wishing to trade intraday. True, for retail investors, the mechani-
cal replication of (negative) performance tracking on some underlying indices
might not be available. Still, being long twice the daily performance of an asset
is by no means a guarantee that the long-term performance characteristics of
the ETF versus the underlying will be alike.     Indeed, the performance of the
former is dependent on the path travelled and the volatility of the latter. For
Proshares Ultrashort 20+ Year Treasury, the costs to express a long-term bear
view on high duration bonds are signicant.

Firstly, I showed that long-term returns for the leveraged ETF's discussed un-
derperform if the volatility for the underlying instrument is high. It was shown
that the relevant volatility for Proshares Ultrashort 20+ Year Treasury is indeed
high, which is destructive for buy-and-hold investors. Secondly, by performing
an analysis on the historical performance of Proshares Ultrashort 20+ Year
Treasury, I showed that the average annualised friction costs were 4.4% for the
period 2008-2012. Thirdly, an annual fee of 0.93% is charged. Lastly, in section 4
I showed that for leveraged ETF's there are substantial additional risks involved.

Buy-and-hold investors seeking to take a position in order to express their long-
term views on interest rates might be better o not using TBT. After all, the
saying A bird in the hand is worth two in the bush holds for loss avoidance
as well: in the long-term we must be humble in forecasting by avoiding large
deterministic costs.


[1] BlackRock. ETF Landscape: Industry Review H1 2011. Technical report,

[2] Minder Cheng and Ananth Madhavan. The dynamics of leveraged and in-
    verse exchange-traded funds. 7(4), 2009.

[3] ProShares. Prospectus ProShares UltraShort 20 + Year Treasury, 2011.

[4] ProShares. Fact Sheet ProShares UltraShort 20+ Year Treasury, 2012.


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