A Classroom Demonstration of a Relative Value Trade Using Treasury Bonds and AAA Rated Corporate Bonds
William C. Hudson
Professor of Finance
Department of Finance, Insurance, and Real Estate
G.R. Herberger School of Business
St. Cloud State University
720 4th Avenue South
St. Cloud, MN 56304
320-308-2241
Niranjan Tripathy
Professor of Finance
Department of Finance, Real Estate, Insurance and Law
UNT College of Business
1167 Union Circle
Denton, TX 76201
940-565-3045
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ABSTRACT
This paper will provide a classroom demonstration of a relative value trade using Treasury bonds and AAA rate corporate bonds as used by Long Term Capital Management. A methodology will be provided along with numerical examples.
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INTRODUCTION
In the capital markets and investment courses that we teach, there exist many opportunities to use real-life examples taken from the headlines of the financial press to illustrate class concepts. We as academics are fortunate in that the private sector has provided us with a wealth of such examples. The road to financial innovation is littered with the carcasses of trades gone afoul. A classic example is Long-Term Capital Management (LTCM). LTCM began operating at the end of February 1994 with capital of approximately $1.25 billion [Marthinsen 2009].
LTCM was heavily into spread trades. The two major types of spread trades are convergence trades and relative value trades. Convergence trades consist of taking positions in two assets whose prices must converge over time [Marthinsen 2009]. A classic example is the arbitrage of the one- the-run vs. the off-the-run 30-year Treasury bonds. The on-the-run bonds are more liquid market vs. the off-the-run bond making them more valuable. Eventually, the two bonds must converge to a roughly the same yield.
Relative value trades seek to profit from instances where the yield spread between two assets is greater than the historical mean [Marthinsen 2009]. Relative value trades can be constructed between two assets whose values have exhibited a strong historic correlation where the yield spread is currently higher than the normal range [Marthinsen 2009].
LTCM constructed numerous relative value trades using bonds issued by foreign nations, for example Argentina, Brazil, China, Korea, Mexico, Poland, Taiwan, Russia, and Venezuela
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[Marthinsen 2009]. The relative value trade this paper will discuss uses U.S. government bonds
U.S. corporate bonds.
RELATIVE VALUE TRADE
We consider a scenario where the yield spread between a T-bond and an AAA rated corporate bond exceeds what is historically normal. To profit from this spread narrowing going forward, we would take a long position in the T-bond and a short position AAA rated corporate bond. There are a number of ways the narrowing of the spread can be distributed between the two bonds of which any combination will result in a profit.
Krishnamurthy (2002) describes a manner in which LTCM constructed a convergence trade between on-the-run vs. off-the-run 30 year Treasury bonds. His method utilizes repurchase agreements which allows for the conservation of capital [Dunbar 2000].
1. Going long the Treasury- bond using a repurchase agreement [Dunbar, 2000]i
LTCM was able to go long the Treasury bond without having to use their own capital. LTCM contacts a bond dealer indicating they want to repo the Treasury bond. The dealer wires LTCM the funds and LTCM purchases the bond and delivers it to the dealer to be held as collateral. The entire process is completed electronically in a matter of seconds. LTCM still owns the bond and thus receives coupon interest payments from the issuer and, in addition, is exposed to price risk.
The dealer faces no price risk because the repo agreement calls for LTCM to repurchase the bond and pay the repo rate. When the yield spread narrows, LTCM closes out their position by repurchasing the bond from the dealer and selling it in the bond market.
Using the notation of Krishmanurthy [2002] we denote the following:
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Let tn=1 = the day Treasury bond is purchased.
Let tn=N = the day on which the spread has narrowed sufficiently and the position is closed.
θ(tn=1 ) = # of units of the Treasury bond held until time n=N.
P(tn=1) = the purchase price of the Treasury bond at time t=1 which includes accrued interest, and is for standard settlement on the next business day.
P(tn=N) = the future sale price of the Treasury bond at time n=N.
ii f(tn=N) = repurchase rate for either borrowing or lending the Treasury bond from tt=1 until tn=N. y(tn=1) = yield to maturity of the Treasury bond on the date it is repo’d.
퐷푃(푡푛=1) = 휕푃(푡푛=1)/휕푦(푡푛=1) = first derivative (duration) of Treasury bond price with respect to change in yield to maturity which indicates the sensitivity of the bond price to changes in yield to maturity.
2. Shorting the AAA corporate bond using a reverse repurchase agreement [Dunbar, 2000].
LTCM finds a buyer willing to buy the AAA corporate bond. LTCM borrows the AAA corporate bond from a dealer and delivers it to the buyer. LTCM then gives the purchase price to the dealer. The dealer pays a floating rate of interest to LTCM. To close out the position LTCM buys back the bond in the open market and delivers it to the dealer in return for the cash from the original transaction.
Using the notation of Krishmanurthy (2002) we denote the following:
휃̂(tn=1 ) = # of units of the AAA corporate bond held until time n=N.
푃̂(tn=1) = the purchase price of the AAA corporate bond at time t=1 which includes accrued interest, and is for standard settlement on the next business day.
푃̂(tn=N) = the future sale price of the AAA corporate bond at time n=N.
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푓̂(tn=N) = repurchase rate for either borrowing or lending the new bond from tt=1 until tn=N.
푦̂(tn=1) = yield to maturity of the AAA corporate bond on the date it is repo’ d.
퐷푃̂ (푡푛=1) = 휕푃̂(푡푛=1)/휕푦̂(푡푛=1) = first derivative (duration) of AAA corporate bond price with respect to changes in yield to maturity which indicates the sensitivity of the bond price to changes in yield to maturity.
C. The Relative Value Trade
Adapting from Krishnamurthy [2002], the profit from going long the Treasury bond by financing the purchase through a repurchase agreement at time t=1 and later unwinding the position at time t=N is as follows:
(푡 − 푡 ) 휃(푡 ) [푃(푡 ) − 푃(푡 ) − 푃(푡 )푓(푡 ) ( 푛=푁 푛=1 )] (1) 푛=1 푛=푁 푛=1 푛=1 푛=푁 360
That is, the difference in bond prices minus the cost of financing the purchase of the old bond at the repurchase rate all multiplied by the number of bonds in the position.
The profit from shorting the AAA corporate bond using a repurchase agreement at time t=1 and later unwinding the position at time t=N is as follows:
푡 − 푡 휃̂(푡 ) [(푃̂( 푡 ) − 푃̂(푡 ) + 푃̂(푡 )푓̂(푡 ) ( 푛=푁 푛=1)] (2) 푛=1 푛=1 푛=푁 푛=1 푛=푁 360
Profit would therefore be the difference in bond prices plus the financing revenue from the reverse repurchase agreement. Total profit π(tn=N )from convergence trade would be equal to the sum of equations 1 and 2 as follows:iii
(푡 −푡 ) 휋(푡 ) = 휃(푡 ) [푃(푡 ) − 푃(푡 ) − 푃(푡 )푓(푡 )( 푛=푁 푛=1 )]+ 푛=푁 푛=1 푛=푁 푛=1 푛=1 푛=푁 360
푡 − 푡 휃̂(푡 ) [푃̂(푡 − 푃̂(푡 ) + 푃̂(푡 )푓̂(푡 )( 푛=푁 푛=1)] (3) 푛=1 푛=1) 푛=푁 푛=1 푛=푁 360
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If the yield spread narrows as expected at time t=N, a number of scenarios can occur:
1. The yield on the T-bond remains constant while the yield on the AAA corporate bond declines. In this case, the short position in the AAA corporate bond will generate a profit based on the decline in bond price in addition to a positive cash flow from the reverse repurchase agreement, while the long position in the Treasury bond will generate no profit and a small loss due to the cost of the repurchase agreement. The overall position will make a profit if the sum of the gain and loss is still positive.
2. The yield of the Treasury bond increases while the yield of the AAA corporate bond decreases. That is the Treasury bond increases in price while the AAA corporate bond declines in price. In such as case, the short position in the AAA corporate bond would provide a positive payoff as the bond declines in price. The long position in the Treasury bond may generate a positive cash flow if the profit generated from the increase in bond price is enough to offset the cost of financing the repurchase agreement.
3. The yields of both bonds increase, while the spread narrows. In this case the short position in the AAA corporate bond will create a gain which will be larger than the loss from the long position in the Treasury bond.
4. The yields of both bonds decrease, while the spread narrows. In this case the long position in the Treasury bond will create a gain which will be larger than the loss from short position in the
AAA corporate bond.
The Treasury bond and the AAA corporate bond will differ slightly in maturity, coupon rate and of course yield to maturity. Therefore, we would expect the bonds to have different price sensitivities to changes in market interest rates which may occur over the time during which the
7 positions are held. In establishing these types of trades the relative positions in each bond should be selected such that profits are invariant to a level change in the yield of each bond, but only sensitive to a change in yield spreads Krishnamurthy [2002]. Thus, the following condition is invoked:
휃̂(푡푛=1)푃̂푡푛=1퐷푃̂ (푡푛=1) = 휃(푡푛=1)푃(푡푛=1)퐷푃(푡푛=1) (4)
This condition can be satisfied by selecting relative position sizes such that profits are independent of equal changes in bond yields. That is, we will adjust our holding weights by the duration of each bond.
A NUMERICAL EXAMPLE:
In this section we will provide two numerical examples with real prices taken from the Wall
Street Journal. First, we will provide an example of a relative value trade between a Treasury bond and an AAA corporate bond where the yield spread declines and illustrate the resulting profit. Second, we will provide an example of such a trade in the context of the Russian debt default on August 17, 1998. At this time the spreads between such bonds actually widened as investors sold riskier bonds and bought safer bonds in what is typically known as a flight to quality. We will illustrate the losses from our position that would have occurred in this scenario.
i We follow the notation used by Krishnamurthy (2002). ii Typically repurchase agreements have very short mature ties (e.g. overnight). LTCM was able to negotiate term repurchase agreements having six-month to twelve-month maturities (Marthinsen p. 238). iii LTCM was able to negotiate repo agreements that allowed them to borrow the full value of the bonds being purchased rather than being required to pay for a portion of the bond with their own equity, known as a “haircut.” LTCM and a small number of select hedge funds were not required to pay “haircuts” and, thus, were able to enter into these positions at zero value without posting any collateral. Thus, with no capital at stake, LTCM had virtually infinite leverage in these types of trades [Dunbar 2000, pages 210-211].
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REFERENCES
Dunbar, Nicholas. Inventing Money, The Story of Long-Term Capital Management and the Legends Behind It, John Wiley & Sons, 2000.
Krishnamurthy, Arvind. “The Bond/Old-Bond Spread”, Journal of Financial Economics, 66, (November –December 2002), 463-506.
Marthinsen, John. Risk Takers, Uses and Abuses of Financial Derivatives, Pearson Prentice Hall, 2009.
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