Blog Cms

It is no secret that Julian Robertson is not a huge fan of long-dated bonds. In his recent CNBC 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, his latest 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 the front end of the curve is at zero. Unless Bernanke follows Sweden into negative rates territory, the steepener upside potential has just been mechanically limited by 50%. As for his current preferred iteration of expressing Treasury bearishness, CMS, here is some recent commentary from JR on the topic: "The insurance policy I would buy is called a CMS Rate Cap, which is the equivalent of buying puts on long-term Treasuries. If inflation happens the way it could, long-term Treasuries are just going to explode. Less than 30 years ago, long-term interest rates got to 20%. I can envision that seeming like a very 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 that original Tiger. In a recent version of their Interest Rate Strategist, the key proffered trade is precisely 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 rates have rallied (Exhibit 1). Getting short back-end rates – with defined downside – is becoming increasingly attractive. We maintain our long-held belief that, in the long run, the curve will steepen significantly, and we continue to believe that long-dated rate caps will benefit from the higher rates and higher volatility that will come from increased Treasury issuance and an end to the public stimulus programs. 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 CMS struck 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 62bp Inflation and long-end supply remain substantial concerns, particularly for longer maturities. Our economists expect 10y UST gross issuance to more than double 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 per year, and long bond gross issuance by approximately $50 billion per year

(Exhibit 2). This is while the Fed is projected to keep short-term rates on hold in order 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 have been 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 a shift 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 the curve 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 a knock-out cap.

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%, then the investor will make 300bp, multiplied by the notional on the cap. Investors looking to decrease the upfront cost of the option can accomplish this in one of two ways: either by limiting their upside, or by playing the timing of the sell-off in longer-dated rates. Limiting the upside would involve selling an OTM cap against the ATM cap that the investor is long. For instance, if the investor sells a 300bp OTM cap against buying 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 over the past three months. Exhibit 4 graphs the spread between 100bp OTM 5y30y payors 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 increasingly expensive. This is why we prefer monetizing and selling it as opposed to moving the strike of the CMS cap that we’re long further out of the money. Playing the timing of the sell-off in longer-dated rates would cheapen the upfront cost of the cap by selling a shorter-expiry option against the longerexpiry cap. Flows out of money market funds into the belly of the curve are likely 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 that knocks out in 1y if 30y CMS is above a strike of their choosing. For instance, a 5y cap on 30y CMS struck ATM (5.38%) that knocks out in 1y if 30y CMS is above 5.38% has an upfront cost of 62bp; if investors move the strike of the knock-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 CMS Cap are either that rates continue to rally, or that vol falls. Note that both of these risks are mitigated with a 1x1 cap spread, or with a knock-out cap. In each of the three trades, however, the maximum downside for investors is equal to the initial premium invested.

If last week's pounding of the 30 year is any indication, Robertson may just be on the right trade yet again. The demonstratory selling of 30 years both into and after the Auction was obviously 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 dollar reserves), it is conceivable that going forward it will merely focus in the 1-3/5 Year Tenor range, as it leaves anything 10 years and out to other, Fed financed chums. Some desks have in fact argued that what the ABX trade was for subprime, and CMBX was for CRE, the CMS trade will end up being for Treasuries. Although be careful: while your opponents in the first two were subprime borrowers and Cohen & Steers respectively (hardly admirable opponents), in the last trade you are taking on the Federal Reserve and the full faith and credit of the US head on. For if the Fed losses control of the 10-30 year span, it might as well 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 topple Ben, is for the bond vigilantes to come out in force and support Robertson's fatalistic view on USTs. Not even the worlds most overheating printing press can take on the combined power of all the bond vigilantes in the world. Although, it is arguable if one can take on the Fed via passive strategies such as CMS. Someone with real guts would have to be the first to go all out and short the back-end. If substantiated by a sufficient number of synthetic bearish positions, at that point it will be merely a matter of time before Bernanke is finally forced to fold his endless deck of Aces.

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

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

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 QT , leading to compute:

E QT [Sw(T , T1 ,..., Tn )], where

E QT [

(1.1)

] is the expectation under the forward neutral measure Q , T

and Sw(T , T1 ,..., Tn ) the value at time T of the swap rate with fixed payment dates T1 ,..., Tn .

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

LVL(T ) . This leads to:

 B(T , T ) LVL(0)  E QT [Sw(T , T1 ,...,Tn )] = E LVLT  * * Sw(T , T1 ,...,Tn ) (1.2)  LVL(T ) B(0, T ) 

since

dQT B(T , T ) LVL(0) = * LVLT LVL(T ) B(0, T ) . dQ

(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:

E QT [Sw(T , T1 ,..., Tn )]

 LVL(0 )B(T , T )  = Sw(0, T1 ,..., Tn ) + CovQ LVLT  , Sw(T , T1 ,..., Tn )  LVL(T )B(0, T ) 

(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:

dS (T , T1 ,..., Tn ) = σ t dWt S (T , T1 ,..., Tn )

(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

LVL(T ) = f (S (T , T1 ,..., Tn ))

(1.6)

One can show that the adjustment is given by:  LVL(0 )B(T , T )  CovQ LVLT  , Sw(T , T1 ,..., Tn )  LVL(T )B(0, T ) 

 f ' (Sw(0, T1 ,..., Tn )) 2  LVL(0 ) ≈ Sw(0, T1 ,..., Tn ) exp − σ Sw(0, T1 ,..., Tn )T  B(0, T )  f (Sw(0, T1 ,..., Tn )) 

(1.7)

The second approach relies on the fact that in the one factor approximation; the computation boils down to computing:

EQ

LVLT

 Sw(T , T1 ,..., Tn )     f (Sw(0, T1 ,..., Tn )) 

(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. 1

see 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

References •

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

•

Benhamou E. (1999), A Martingale Result for the Convexity Adjustment in the 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, Risk Magazine, December.