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It is no secret that Julian Robertson is not a huge fan of long-dated bonds. In his recentCNBC interview he had some downright nasty words about the back-end of the UST curve,especially if the "downside contingency" case of foreign purchases ceasing, were to pass.However, while many have known about his propensity for the bond steepener trade, hislatest trade position is the so called Constant Maturity Swap trade. Moving away from an

outright steepener makes sense as it can now only profit from a tail end widening, since thefront end of the curve is at zero. Unless Bernanke follows Sweden into negative ratesterritory, the steepener upside potential has just been mechanically limited by 50%. As forhis current preferred iteration of expressing Treasury bearishness, CMS, here is some recentcommentary from JR on the topic:

"The insurance policy I would buy is called a CMS Rate Cap, which is theequivalent of buying puts on long-term Treasuries. If inflation happens the wayit could, long-term Treasuries are just going to explode. Less than 30 yearsago, long-term interest rates got to 20%. I can envision that seeming like avery low interest rate compared to what might occur in the future."

No surprise then, that Morgan Stanley's Govvy desk has started pimping this trade(including some hedged and Knock Out variants) to anyone who wants to imitate thatoriginal Tiger. In a recent version of their Interest Rate Strategist, the key proffered trade isprecisely Shorting back-end rates, with the following summary recommendations:

Buy a 5 year Cap on 30 year CMS struck ATM (5.38%) for 105 bps Buy a 5 year Cap on 30 year CMS struck ATM. Sell a 5 year Cap on 30

year CMS struck at 8.38%, for a net cost of 65 bps. Buy a 5 year Cap on 30 year CMS struck at 5.38% that knocks out in 1

year if 30 year CMS is above 5.38% for 62 bps.

Here is MS' entire modest proposal:

In the past month, longer-dated volatility has declined and longer-dated rateshave rallied (Exhibit 1). Getting short back-end rates with defined downside is becoming increasingly attractive. We maintain our long-held belief that, in thelong run, the curve will steepen significantly, and we continue to believe thatlong-dated rate caps will benefit from the higher rates and higher volatility thatwill come from increased Treasury issuance and an end to the public stimulusprograms.

Specifically, we propose a selection of the following trades: Buy a 5y cap on 30y CMS struck ATM (5.38%) for 105bp

Buy a 5y cap on 30y CMS struck at 5.38%, Sell a 5y cap on 30y CMSstruck at 8.38%, for a net cost of 65bp Buy a 5y cap on 30y CMS struck at 5.38% that knocks out in 1y if 30y

CMS is above 5.38%, for 62bpInflation and long-end supply remain substantial concerns, particularly forlonger maturities. Our economists expect 10y UST gross issuance to more thandouble from 2008 to 2009, and for 30y UST gross issuance to more than triple.After 2009, we also project 10y UST issuance to increase by $40 billion peryear, and long bond gross issuance by approximately $50 billion per year

http://www.zerohedge.com/article/julian-robertson-discusses-us-debt-and-upcoming-inflation-expectationshttp://www.zerohedge.com/article/julian-robertson-discusses-us-debt-and-upcoming-inflation-expectationshttp://www.zerohedge.com/article/julian-robertson-discusses-us-debt-and-upcoming-inflation-expectationshttp://www.zerohedge.com/article/julian-robertson-discusses-us-debt-and-upcoming-inflation-expectations

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(Exhibit 2). This is while the Fed is projected to keep short-term rates on hold inorder to stimulate the economy and maintain a steep curve.

We aim to target 30y rates. This is because we project 30y UST gross issuance

to keep increasing at a faster pace than 10y gross issuance (Exhibit 2).Moreover, we expect the curve to steepen in periods of high inflation.

We also target longer expiries (3-5y). Two reasons behind this: first, we havebeen in a secular downward trend in longer-term rates since the mid 1980s(Exhibit 3). This is a trade for us to break out of that range we expect such ashift to occur over a longer period of time as opposed to in the next year.Second, flows out of lower yielding money market funds into the belly of thecurve are expected to keep longer rates bid, at least for the next couple of

months, in our view. This is something that we can exploit by entering into aknock-out cap.

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In order to play for higher rates, we suggest a 5y cap on 30y rates struck ATM(5.38%) for 105bp. As opposed to a payor swaption, the payout of a cap is

linear with respect to rates. For example, if 30y rates in 5y are at 8.38%, thenthe investor will make 300bp, multiplied by the notional on the cap.

Investors looking to decrease the upfront cost of the option can accomplish thisin one of two ways: either by limiting their upside, or by playing the timing ofthe sell-off in longer-dated rates.Limiting the upside would involve selling an OTM cap against the ATM cap thatthe investor is long. For instance, if the investor sells a 300bp OTM cap againstbuying long an ATM cap, this cheapens the upfront cost of the option to 65bp,or by 38%. Note that OTM skew on longer tails has richened substantially overthe past three months. Exhibit 4 graphs the spread between 100bp OTM 5y30ypayors and 100bp OTM receivers, normalized by the level of at-the-money vol the higher this spread, the more expensive payor skew is relative to receiver

skew. Over the past three months, OTM payor skew has become increasinglyexpensive. This is why we prefer monetizing and selling it as opposed to movingthe strike of the CMS cap that were long further out of the money.

Playing the timing of the sell-off in longer-dated rates would cheapen theupfront cost of the cap by selling a shorter-expiry option against the longer-expiry cap. Flows out of money market funds into the belly of the curve arelikely to keep the long end somewhat bid in the near term, in our view.Investors can monetize this by entering into a 5y cap on 30y CMS rates thatknocks out in 1y if 30y CMS is above a strike of their choosing. For instance, a5y cap on 30y CMS struck ATM (5.38%) that knocks out in 1y if 30y CMS isabove 5.38% has an upfront cost of 62bp; if investors move the strike of theknock-out to 6%, the cost increases to 79bp. Note that a 2y knockout cheapens

the cap even more than the 1y knockout. The principal risks to the outright CMSCap are either that rates continue to rally, or that vol falls. Note that both ofthese risks are mitigated with a 1x1 cap spread, or with a knock-out cap. Ineach of the three trades, however, the maximum downside for investors isequal to the initial premium invested.

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If last week's pounding of the 30 year is any indication, Robertson may just be on the righttrade yet again. The demonstratory selling of 30 years both into and after the Auction wasobviously agenda driven, and it is doubtful it bypassed Bernanke's, and the PD's attention.Yet as China is increasingly boxed and realizes fully well it needs to buy some Treasuries(lest it sends the world a signal that it is willing to write off its $2.5 trillion in dollarreserves), it is conceivable that going forward it will merely focus in the 1-3/5 Year Tenorrange, as it leaves anything 10 years and out to other, Fed financed chums. Some deskshave in fact argued that what the ABX trade was for subprime, and CMBX was for CRE, theCMS trade will end up being for Treasuries. Although be careful: while your opponents in thefirst two were subprime borrowers and Cohen & Steers respectively (hardly admirableopponents), in the last trade you are taking on the Federal Reserve and the full faith andcredit of the US head on. For if the Fed losses control of the 10-30 year span, it might aswell go home, since that means 30 Year mortgages will skyrocket,maybe all the way into

double digit territory, thus destroying all hopes of inflating the GSE bubble.Yet as Soros showed in the 90's, Central Banks can lose. All that needs to happen to toppleBen, is for the bond vigilantes to come out in force and support Robertson's fatalistic viewon USTs. Not even the worlds most overheating printing press can take on the combinedpower of all the bond vigilantes in the world. Although, it is arguable if one can take on theFed via passive strategies such as CMS. Someone with real guts would have to be the firstto go all out and short the back-end. If substantiated by a sufficient number of syntheticbearish positions, at that point it will be merely a matter of time before Bernanke is finallyforced to fold his endless deck of Aces.

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Swaps: Constant maturity swaps (CMS) and constant maturity

Treasury (CMT) swaps

A Constant Maturity Swap (CMS) swap is a swap where one of the legs pays

(respectively receives) a swap rate of a fixed maturity, while the other leg

receives (respectively pays) fixed (most common) or floating. A CMT swap is

very similar to a CMS swap, with the exception that one pays the par yield of

a Treasury bond, note or bill instead of the swap rate.

More generally, one calls Constant Maturity Swap and Constant Maturity

Treasury derivatives, derivatives that refer to a swap rate of a given maturity

or a pay yield of a bond, note or bill with a constant maturity. Since most

likely, treasury issued on the market will not exactly match the maturity of the

reference rate, one needs to interpolate market yield. (rates published by the

British Banker Association in Europe and by the Federal Reserve Bank of

New York)

MARKETING OF THESE PRODUCTS

CMT and CMS swaps provide a flexible and market efficient access to long

dated interest rates. On the liability side, CMS and CMT swaps offer the ability

to hedge long-dated positions. Great clients have been life insurers as they

are heavily indebted in long dated payment obligations. Generous insurance

policies need to be hedged against the sharp rise of the back end of the

interest rate curve. Typical trade is a swap where they received the swap rate.

On the asset side, corporate and other financial institutions have heavily

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invested in CMS market to enjoy yield enhancement and diversified funding.

In a very steep curve environment, swaps paying CMS look very attractive to

clients that think that the swap rates would not go as high as the market (and

the forward curve) is pricing. Alternatively, in a flat yield curve environment,

swaps receiving CMS look very attractive to market participants thinking that

swap rates would rise in the futures as a consequence of the steepening of

the curve. In a swap where one pays Libor plus a spread versus receiving

CMS 10 year, the structure is mainly sensitive to the slope of the interest rate

yield curve and is almost immunized against any parallel shift of the interest

rate yield curve.

For all these reasons, it is not surprising that the CMS markets and the CMS

options markets now trade in large quantities, both interbank and between

corporates and financial institutions.

Pricing

Because of the increasing size of the CMS market, the market has seen its

margin eroding. Banks have developed more and more advanced models to

account for the smile, resulting in first a more pronounced smile and also an

increasingly spread between CMS swap and their swaption hedge.

There exist two different methodologies for pricing CMS swaps:

Parametric computation of the CMS convexity correction (See Hull(200),

Benhamou (1999) and (2000)). In this approach, one assumes a model

and uses some (smart) approximation methods to compute the expected

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swap rate under the forward measure. Non parametric computation of the

swap rates. This approach assumes

Non parametric computation of the CMS rates. This approach tries to

minimize the amount of hypothesis between the computation of the CMS

rate (see the works of Amblard, Lebuchoux (2000), Pugachevsky (2001)).

Note also that practitioners focus heavily on the computation of the forward

CMS as they use these modified forwards and the volatility read from

swaption market to compute simple options on CMS (CMS cap and floor,

CMS swaption). This practice is justified by the fact that the first order effect

comes mainly from the convexity corrected forwards as opposed to modified

volatility assumptions. Using the same vol is therefore right at first order

approximation, and strictly right in a Black Scholes setting.

Let use derive shortly the sketch lines of the two methods mentioned above.

First, one can rapidly see that pricing a CMS swap boils down to price a

simple swap rate received at time T. This can be done under the forward

measure forward neutral measure TQ , leading to compute:

( )[ ]nQ

TTTSwE T ,...,, 1 , (1.1)

where [ ]TQ

E is the expectation under the forward neutral measure TQ ,

and ( )nTTTSw ,...,, 1 the value at time T of the swap rate with fixed payment

datesnTT,...,1 .

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We can then use standard change of numeraire technique to change the

expression above. The natural numeraire for the swap rate is the annuity (also

called level or dvo1, defined as the pv of one basis points paid over the life of

the forward swap rate) of the swap rate, denoted by ( )TLVL . This leads to:

( )[ ]( )

( )

( )

( )( )

= n

LVL

n

QTTTSw

TB

LVL

TLVL

TTBETTTSwE TT ,...,,*

,0

0*

,,...,, 11 (1.2)

since

( )( )

( )( )TB

LVL

TLVL

TTB

dQ

dQ

TLVL

T

,0

0*

,=

. (1.3)

This shows that the CMS rate is equal to the swap rate plus an extra term

function of the covariance under the annuity measure between the forward

swap rate and the forward annuity:

( )[ ]

( )( ) ( )

( ) ( )( )

+= n

LVL

n

n

Q

TTTSwTBTLVL

TTBLVLCovQTTSw

TTTSwE

T

T

,...,,,,0

,0,...,,0

,...,,

11

1

(1.4)

As a result, the CMS rate depends on the following three components:

The yield curve via the swap rate and the annuity.

The volatility of the forward annuity and the forward swap rate.

The correlation between the forward annuity and the forward swap rate.

The first method relies on deriving an approximation for the covariance terms.

There are many ways of doing this, in particular, using one factor

approximation with lognormal assumptions, Wiener chaos expansion or

simply martingale theory . To be more specific, let us examine the lognormal

case. It assumes a lognormal martingale diffusion for the swap rate under the

annuity measure:

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( )( ) ttn

n dWTTTS

TTTdS=

,...,,

,...,,

1

1 (1.5)

The one factor approximation relies on assuming that the level can be

represented as a function of the swap rate (which is rigorously true for cash

settled swaptions). This leads to

( ) ( )( )nTTTSfTLVL ,...,, 1= (1.6)

One can show that the adjustment is given by:

( ) ( )

( ) ( )( )

( )( )

( )

( )( )

( )( )( )

TTTSwTTSwf

TTSwf

TB

LVLTTSw

TTTSwTBTLVL

TTBLVLCovQ

nn

n

n

n

LVLT

,...,,0,...,,0

,...,,0'exp

,0

0,...,,0

,...,,,,0

,0

1

2

1

1

1

1

(1.7)

The second approach relies on the fact that in the one factor approximation;

the computation boils down to computing:

( )

( )( )

n

nQ

TTSwf

TTTSwE

TLVL

,...,,0

,...,,

1

1 (1.8)

But we know that any function of only the swap rate can be evaluated as a

portfolio of swaptions1. This comes from the fact that an expectation can be

translated into an integral of the integrand times the density function of the

swap rate. We can therefore evaluate the CMS swap rate as a portfolio of

swaptions. As a matter of fact, replicating CMS with cash settled swaptions is

accurate, while one needs to make a one factor approximation to extend the

replication argument to physical settled swaptions. Using regression ideas,

one can also extend the ideas of CMS replication to deferred payment CMS

structures.

1see Breeden Litzenberger (1979) result on the fact that the second order derivatives of a call price

with respect to the strike is simply the density function, hence the result

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References

Amblard G. Lebuchoux J (2000), Model For CMS Swaps, Risk,September.

Benhamou E. (1999), A Martingale Result for the Convexity Adjustment inthe Black Pricing Model, London School of Economics, Working Paper.

Benhamou E. (2000), Pricing Convexity Adjustment with Wiener Chaos,Financial Markets Group, London School of Economics, FMG Discussion

Paper DP 351.

Hull, John C, Options, Futures, and Other Derivatives, Fourth Edition,Prentice-Hall, 2000.

Pugachevsky, D. (2001), Adjustments for Forward CMS Rates, RiskMagazine, December.