yield curve analysis and fixed-income arbitrage

33
Yield Curve Analysis and Fixed-income Arbitrage Seng Yuen Leung HKUST February 23, 2006 Yield Curve Analysis & Fixed-income Arbitrage 2 Outline Part I What is yield curve? Forward rate A series of increases in short-term rate v.s. A low long-term yield Part II Fixed-income Arbitrage Strategies Empirical studies on the risk and return characteristics

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Page 1: Yield Curve Analysis and Fixed-Income Arbitrage

Yield Curve Analysis and

Fixed-income Arbitrage

Seng Yuen LeungHKUST February 23, 2006

Yield Curve Analysis & Fixed-income Arbitrage

2

Outline

Part I

• What is yield curve?

• Forward rate

• A series of increases in short-term rate v.s. A low long-term yield

Part II

• Fixed-income Arbitrage Strategies

• Empirical studies on the risk and return characteristics

Page 2: Yield Curve Analysis and Fixed-Income Arbitrage

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3

Bonds

• Bond is a debt used for borrowing money and is one type of fixed-income securities

• Bonds can be mainly distinguished by – issuers (government, corporates)– maturity (short/long-term)– Coupon (interest paid by issuer at payment dates)

• U.S. government bonds are called treasury– Bills (maturity less than 1 year)– Notes (maturity between 1 – 10 years)– Bonds (maturity longer than 10 years)

• Par/Face value is the amount that investors will get back per bond at maturity (usually $1000)

Yield Curve Analysis & Fixed-income Arbitrage

4

Bonds (cont’d)

• Accrual interest: If an investor wants to buy the bond before the next coupon dates, he needs to pay the previous holder the amount of interest accrued during his ownership

• Clean price: Bond price with a whole coupon period – Accrued interest

• Dirty price: Clean price + Accrued interest

• Bonds are usually quoted on a clean basis but settled on a dirty basis

• Bond prices depends on day-count basis and frequency

• Day-count convention: A/360, A/365, 30/360 (Eg. Mar 15, 2005 to Jun 15, 05 is counted as 92/360=0.2556 in A/360 basis)

• Frequency: quarter, semi-annual, annual (Most bonds have semi-annual coupon)

Page 3: Yield Curve Analysis and Fixed-Income Arbitrage

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World Bond Markets (source: Bloomberg LP)

Yield Curve Analysis & Fixed-income Arbitrage

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USD Market Instruments (source: Bloomberg LP)

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HKD Market Instruments (source: Bloomberg LP)

Yield Curve Analysis & Fixed-income Arbitrage

8

Yield-at-maturity

• Yield-to-maturity (YTM) is the most frequently used measure of return from holding a bond

• YTM can be used to compute bond price, so it is often used as a market quote (per annum)

• Inverse relationship: A higher yield, a lower bond price or vice versa

• YTM is equivalent to the internal rate of return on the bond, i.e., the rate that equals the value of the discounted cash flows (coupons C and principle F) on the bond to its current (dirty) price

• Coupon = coupon rate x day-count basis and m = 1 (annual), 2 (semi-annual), or 4 (quarterly), and N = number of total payments

NYFC

mYC

mYCP

)()/()/( ++

+++

++

=111 2 L

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9

Coupon & Yield-to-maturity

• Coupon rate is an obligated (fixed) return rate and does not change during the life of the bond

• YTM is NOT fixed and changes over time as long as the price changes

• Although both can be regarded as the measure of the return rate of a bond, they must be interpreted differently:– Coupon rate is the return rate of holding a bond if you purchase at par, i.e,

bond price = face value, and hold up to maturity – YTM is the return rate of a bond at the time you purchase

• Both coupon rate and YTM are the “average rates” since cash flows (coupon payments) in a bond are received at different dates and should be discounted at corresponding rates

• Analogy: 3-month and 6-month deposit rates are usually different

Yield Curve Analysis & Fixed-income Arbitrage

10

Credit Risk

• Three main determinants of the YTM of a bond: – Credit risk– Interest rate risk– Supply/demand force (liquidity risk)

• Other (secondary) risks include inflation risk and political/economic risk

• If the issuer of a bond fails to fulfil his obligation (default or bankruptcy), the investors are subject to credit risk

• Fixed-income managers are very concern on credit rating (a measure of the risk level of a bond) because are often restricted to invest in low-rating bonds

• Rating are typically categorized into two large classes: – Investment grade (S&P: BBB or above, Moody’s: Baa or above)– Non-investment/Speculative grade (junk/high-yield bonds)

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Interest Rate Risk

• Holding a default-free bond (bond without credit risk) DOESN’T MEAN investors bearing no risk

• Current level of interest rate and market expectation would change YTM, i.e, a higher interest rate, a higher yield and vice versa

• Eg. If the coupon of a bond is 4.5%, but the current market rate moves up by 1%, 4.5% coupon is not longer attractive to investors unless there is a discount on bond price, resulting a higher yield

• If an investors buy a bond and sell it before maturity, he may incur a loss due to interest risk

• Eg. Suppose an investor bought a bond 1 year ago at $100 with coupon of 4% p.a. (annual). After one year, the bond price drops to $94. So his net position is $94 + $4 - $100 = -$2 (a loss!)

Yield Curve Analysis & Fixed-income Arbitrage

12

What is Yield Curve?

• Yield curve (or term structure of yield) is a graph that indicates the relationship between the yield of a bond and its time-to-maturity

• It is a snapshot of the current level of yields in the market

• A yield curve is only an indicator of current interest rate level, and it does evolve over time

• But it is an important indicator and knowledge source of the state of a debt capital market

Page 7: Yield Curve Analysis and Fixed-Income Arbitrage

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What is Yield Curve? (cont’d)

• Yields usually increase over time, i.e., a longer maturity, a higher yield

• This is mainly due to two reasons:– Inflation: The value of a bond will be eroded over a longer term– Credit risk: Lender needs to bear a higher risk for holding a long-term

bond than short-term bond issued by the same issuer (i.e. same credit rating)

• This gives a upward-sloping shape yield curve

• But the curve does not continue to slope upwards, all the way to 30-year mark (why?)

• Supply/demand effect: 30-year bond is usually in a great demand from institutional investors (pension funds and insurance companies) to meet their long-term liabilities and this outstrip supply

• As a result, the price of a long-term bond is forced upwards, and this moves the yield down to below what it should be

Yield Curve Analysis & Fixed-income Arbitrage

14

US Treasury Yield Curve (source: Bloomberg LP)

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Applications of Yield Curve (1)

• Yield curve tells us what the bond market is trading now

• So it implies the level of trading for the future (or at least what the market thinks will be happening in the future)

• All market participants in the debt capital markets are interested in the current shape and level of the yield curve and use it to decide their investment decisions

Yield Curve Analysis & Fixed-income Arbitrage

16

Applications of Yield Curve (2)

Four main uses of yield curve:

(1) Benchmark for all debt instruments• The yield of government bonds from the shortest maturity to the

longest set the benchmark for yields for all other debt instruments• Eg. If a U.S. 5Y treasury is trading at 5%, all other five bonds (corporate or

sovereign) will be issued at a yield over 5% because U.S. treasuries are generally considered to be “default-free” instrument -- since they are backed by the full faith and credit of the U.S. government -- so credit spread is required to compensate credit risk

(2) Indicator of Forward Yield• Yield curve takes certain shapes to reflect market expectation of

future interest rates• Bond market participants analyze the current term structure of the

yield curve and determine the implications regarding the direction of market interest rates

• Bond traders decide their trading strategies • Central banks analyze the yield curve for its implied information

(forward interest rates and inflation level)

Page 9: Yield Curve Analysis and Fixed-Income Arbitrage

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Applications of Yield Curve (3)

(3) Comparing returns across maturity spectrum• Yield curve indicates the returns that are available at different

maturity points • Portfolio managers use the yield curve to assess the relative value of

investments across the maturity spectrum• Fixed-income managers can use it to assist them to assess which

point of the curve offers the best returns relative to other

(4) Relative Value between different bonds of similar maturity• Yield curve can be analyzed to indicate which bonds are cheap or

expensive to the curve • Eg. If the yield of a bond is traded at a level below the yield curve, the

bond is expensive

Yield Curve Analysis & Fixed-income Arbitrage

18

Shape of Term Structure

• Market exhibits that a yield curve can have four basic shapes:

– Normal or conventional: yields are at “average” levels and the curve slopes gently upwards over maturity

– Upward-sloping: yields are at historically low levels, with long rates substantially greater than short rates

– Downward-sloping, inverted: yield levels are very high by historical standards, but long-term yields are significantly lower than short rates

– Humped: yields are high with the curve rising to a peak in the medium-term maturity, and then sloping downwards at longer maturities

• Sometimes yield curves will incorporate a mixture of the these features

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Flat Yield Curve

• Flat yield curve do emerge in response to particular economic circumstances but this phenomenon is quite rare

• In the conventional thinking, a flat curve is not tenable because investors should have no incentive to hold long-term bonds over shorter-term bonds when there is no yield premium

• So bond investors would sell off long-term bonds and the yield at long end should increase, producing an upward-sloping shape again

• A flat curve is usually more influenced by supply and demand than anything else

Yield Curve Analysis & Fixed-income Arbitrage

20

Problem on Yield Curve

• Yield curve does not distinguish between different coupons

• A low-coupon bonds pay a higher portion of their cash flows at later date than high-coupon bonds of the same maturity

• Given the same prices and same maturity, a low-coupon bond has a higher yield (at discount) while the high-coupon bond has a lower yield (at premium)

• Cash flows are not discounted at the “appropriate rate” for the bonds in the group being used to construct the curve

• High-low coupon effect could distort the shape of yield curve and this could lead to a wrong implication

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Coupon Yield Curve

• In general, high-coupon bonds trade at higher yields relative to low-coupon bonds because of reinvestment risk and tax reasons

• To compensate for this, some bond analysts construct a coupon yield curve, i.e., yield curve constructed from a group of bonds with the same coupon

• In practice, it is unusual to observe bonds with same coupon along the whole term structure so this type of curve is rare

Yield Curve Analysis & Fixed-income Arbitrage

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Par Yield Curve

• A bond is traded at par if the yield is equal to coupon rate, or, the bond price equals face value

• If bonds in the market are trading substantially away from par, the resulting yield curve will be distorted

• Par yield curve can be constructed directly from bond yields when bonds are trading at or near par

• But it is rare to encounter bonds trading at par for any maturity

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Zero (Spot) Yield Curve

• Spot rate is today’s interest rate for a given maturity

• Spot rate can be regarded as the rate on a zero-coupon bond (“zero”)

• Zero yield curve is the graph of zero rates against time-to-maturity

• In many debt markets, zeros are not traded

• Theoretically, a zero yield curve can be derived from the conventional bonds

• If zeros are traded in the market, the observed zero curve could be different from theoretical one

Yield Curve Analysis & Fixed-income Arbitrage

24

Forward Yield

• Forward rate to a bond is the spot bond yield at the future date

• It is the yield of a zero that is purchased at the forward date

• Relationship between spot rates and forward rates (no-arbitrage principle):

• Based on this relationship, bond price can be expressed in terms of forward rates:

• Note that R(n) = F(0,n) for any n > 0

( ) ( ) ( )( )( )

11

111

111111

1

−+

++=+

+++=+++

+

T

T

TT

TRTRTTF

TTFTRTR

)()(),(

),()()(

( )( ) ( ) ( )),(),(),(),()( TTFFFC

FFC

,FCP

11101211101101 −+++

++++

++

=L

L

Page 13: Yield Curve Analysis and Fixed-Income Arbitrage

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Interpretation of Forward Rate (1)

• Forward rates that exist at one time point reflect everything that is known in the market up to that point (efficient market hypothesis)

• Forward rate curve is a forecast of the future spot rate curve

• The derivation of forward rates reflects all currently known market information

• This DOES NOT mean that forward rates are a prediction

• The instant after they have been calculated, new market knowledge may become available that alters the markets view of future interest rate

Yield Curve Analysis & Fixed-income Arbitrage

26

Interpretation of Forward Rate (2)

• But forward rates are still important because they are required to make prices for trading today but settling at a future date

• For example, a bank may wish to fix today the interest rate payable on a loan that begins in one year from now (payable in arrear)

• Forward rate is used by market makers to quote prices for dealing today, and is the best expectation of future interest rates , given everything that is known in the market up to now

• If traders happen not to agree with this market view, they will decide trading strategy accordingly

• Forward rates can serve as a tool for making investment decision

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Interpretation of Forward Rate (3)

• Spot and forward rates are calculated from the current market rates based on the no-arbitrage principle, i.e., no profit opportunity arises if the actual spot rate at future date equals forward rate

• Forward rates are somehow a “market-view” rate that will be (or should be!) in the future

• If we compare the today 3-month forward curve with the yield curve 3 months from now, they are certainly different (market participants cannot predict future interest rate 100% correct!)

• When constructing a forward curve, we are using current term structure (all information includes market view, political and economic factors)

• As mentioned before, the market evolves over time with the latest (unpredictable) information and this is why forward rates are not agree with future interest rates

Yield Curve Analysis & Fixed-income Arbitrage

28

Concern on Yield Curve

• Our discussion on yield curve is based on the perfect market assumption:– Perfect information– Bullet bonds (call without embedded option) with all maturities– No taxes– No transaction cost

• In practice, markets are not completely perfect (arbitrage opportunity exists), but these assumptions makes us easier to deal with spot and forward rates

• When analyzing yield curves, bear in mind that they DO NOT contain complete market information and it is frequently to observe anomalies not explained by conventional theories

Page 15: Yield Curve Analysis and Fixed-Income Arbitrage

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Fed Funds Rate (source: Bloomberg LP)

Yield Curve Analysis & Fixed-income Arbitrage

30

Treasury Yield Spread: 2Y v.s. 5Y (source: Bloomberg LP)

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Why Have Long-term Rates Hardly Budged?

• It is quite strange that long-term yield have not risen much in the face of 14 straight increases (from 1% to 4.5%, raise 0.25% each time) in the Fed funds rate

• Many explanations have been offered to explain the conundrum

• Historically, longer-term interest rates usually have been higher that short-term rates because investors required compensation for expected future inflation (likely to be volatile)

• Some analysts suggest that calculus has changed because of the Federal Reserve's success in keeping core inflation both low and less volatile

• They argue flattening of the yield curve over the past 19 months (since June 2004) isn't signaling a significant slowdown in U.S. economic growth, much less a near-term recession

Yield Curve Analysis & Fixed-income Arbitrage

32

Why Have Long-term Rates Hardly Budged? (cont’d)

• The bond risk premium on the 10-year Treasury note is unusually low around zero -- compare the 10-year yield to expected future short-term interest rates

• But the collapse in the volatility of inflation relative to the volatility of real rates is a much more compelling explanation

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Inflation Volatility

• Over the past 15 years, volatility of the core personal consumption price index consistently has been much lower than that of inflation-adjusted short-term interest rates

• Investing in longer-dated fixed-income assets becomes more attractive, i.e, investors can lock in a stable real rate of return and are subject to little inflation risk

• On the other hand, buying a series of short-term securities isn't likely to provide a similarly stable real rate of return, i.e., possibility of raising or lowering short-term rate to keep inflation in “comfort zone”

• Pension funds, insurance companies or long-term investors can benefit from locking in a steady real rate of return by investing in long-duration fixed-income assets -- these investment decisions are built upon their confidence on the Fed’s credibility to continue controlling inflation

Yield Curve Analysis & Fixed-income Arbitrage

34

Is Asian Money Financing U.S. Spending?

• U.S. budget and current-account deficits both reach a record high and this would normally make the dollar depreciate and result a high inflation

• But the situation now is that these unsustainable deficits appear sustainable due to great demand of U.S. treasuries from Asian central banks (holding more than $1 trillion treasuries)

• Due to the great demand from Asian central banks, the shortage has raised the price and reduced the yield on long-term bonds

• This keeps bond yields low, the dollar up and enables U.S. customers to live beyond their means

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Shortage of Long-term Bonds

• Another force to push up the yield is that the Treasury Department has reduced the supply of long-term bonds since Oct 2001 by ending the sale of 30-year bonds* and financing more of the growing budget deficit with shorter-term bonds

• This shortage has raised the price and reduced the yield on long-term bonds

• Furthermore, export is the main driver of a growth economy in most Asian nations, so they can’t afford to sell off dollar-denominated bonds and to rise in interest rates or appreciate their currencies in order to keep their goods attractive

* The $14 billion sale of long-term bond is resumed on Feb 9, 2006, giving the U.S. government a new tools to finance the expanding deficit and responding the resisted calls from Wall Street

Yield Curve Analysis & Fixed-income Arbitrage

36

Recession Predictor

• Traditionally, the relationship between the shape of the yield curve acts as a predictor of recessions

• If the bond risk premium is zero, then the curve would be flat when Fed policy is in a “neutral'' position (neither stimulating nor restraining economic activity)

• Some supporters of the above views argue that, in the past, when inverted yield curves did foretell recessions because it took a tight monetary policy to invert the yield curve, but that isn't the case today

• Their points is that even if core inflation were to accelerate , the bond risk premium wouldn't necessarily increase so long as investors remained confident that the inflation rise would be limited and temporary

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Another View

• Some analysts rather believe that U.S. is currently in a low-inflation environment, but it does not imply that inflation is gone

• As the economy continues to expand and reaches capacity constraint in labor and product markets, inflation will show its head again

• Consequently, they think that long-term rates are too low based on economic fundamentals

• As core inflation rises above the Fed's `”comfort zone,” the Fed would raise its current overnight lending rate and this implies a sell-off at the long-end of the curve, so that the yield curve will steepen once again

• The difference in these two views ultimately is a matter of Fed’s credibility and capability of controlling inflation, so it is hard to judge which view is correct at this moment (But time will tell!)

Yield Curve Analysis & Fixed-income Arbitrage

38

Hedge Fund

• A loosely regulated investment pool for affluent and institutions with less transparency

• Typically, hedge funds charge 2% of the assets per year and 20% of the profits

• Performance is measured in absolute terms, i.e., independent of market direction and uncorrelated to any benchmark

• Unlike mutual funds, hedge funds can use a wide range of investment strategies: – Holding derivatives, futures, swaps, etc– Financial leverage– Short selling

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Hedge Fund Performance

• Hedge funds had a good time to earn huge profit over the past decade, but the days of easy money are over

• Hedge funds have returned more than 10% in one of the last five year, compared with four of the five years through 1999 (source: CSFB/Tremont Hedge Fund Index)

• The average fund worldwide was up 7.4% in 2005, but the return declined to 5.99% while the average return of 31% was posted in the golden years like 1999 (source: Hedge Fund Research Inc.)

• Generating a standout profits will get tougher as the proliferation of hedge funds and the emergence of “mega-managers” such as D.E. Shaw & Co. and Bridgewater Associates Inc.

Yield Curve Analysis & Fixed-income Arbitrage

40

Hedge Fund Industry Today

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Hedge Fund Industry Today (cont’d)

Yield Curve Analysis & Fixed-income Arbitrage

42

Hedge Fund Strategies

• Four major styles and sub-categories of investment strategies:

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Hedge Fund Ranking (source: Bloomberg LP)

Yield Curve Analysis & Fixed-income Arbitrage

44

HF Ranking by Strategies (source: Bloomberg LP)

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Fixed-income Arbitrage

• Fixed-income arbitrage (FIA): the strategy that involves exploiting the interest rate spread between related fixed income instruments

• FI arbitrageurs take a long position on a higher yielding fixed income instrument and a short position on the lower yielding instruments

• This can be done between different maturities on the fixed income yield curve, or between different types of fixed income instruments

• FIA managers tend to employ considerable leverage to magnify returns

• Some most widely-used FIA in the markets:– Swap spread arbitrage– Volatility arbitrage– Yield spread arbitrage– Capital structure arbitrage

Yield Curve Analysis & Fixed-income Arbitrage

46

Swap Spread Arbitrage

• Taking positions on swap spread and floating spread

• Enter into a par swap and receive a fixed coupon rate CMS and floating LIBOR Lt

• Short a par treasury bond with the same maturity as the swap: pay coupon rate CMT and invest the proceeds in a margin account to earn repo rate rt

• This trade is structured with the view:Swap spread paid (CMS – CMT) > Floating spread received (Lt - rt )

• The difference (swap spread – floating spread) is quite historically stable and positive, so swap spread arbitrage has traditionally been one of the most popular FIA strategies

• Indirect default risk: It can be negative when the banking sector has increasing default risk, resulting a significantly high LIBOR

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Swap Spread Arbitrage: Implementation

• Collect a set of swap and treasury data and fit a mean-reverting process to the floating spread

• Determine each month whether swap spread differs from the expected value of floating spread over the life of strategy

• Enter a trade if the difference exceeds a pre-specified range (say 15 bps)

• Close out the trade if the swap spread converges to the floating spread or until the maturity of the swap

• If the difference becomes widen, should the trader unwind his position (cut loss) or wait for the convergence? It is simply based on the trader’s confidence on his view and market condition

• This strategy requires little on complex model for implementation

Yield Curve Analysis & Fixed-income Arbitrage

48

Volatility Arbitrage

• In its simplest form, volatility arbitrage is often implemented by selling options and then delta-hedging the exposure to the underlying

• This strategy aims at earning a profit from the volatility of the underlying -- an excess return proportional to the gamma of the option times the difference between the implied variance and the realized variance of the underlying

• Let V(F,t) be any function of the futures price and time. By Ito’s lemma, the P&L from delta hedging with a constant volatility in a stochastic volatility market is given by

• When selling an option with a constant volatility σ , the instantaneous P&L from delta hedging over (t,T) is half the dollar gamma weighted-average of the difference between the hedge variance and the true variance

dssF

sFVFeT

t

sTr ))(();,(2

L&P 222

22)( σσσ

−∂

∂= ∫ −

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49

Volatility Arbitrage: Implementation

• This strategy usually focuses on selling cap and delta hedging the position using Eurodollar futures

• In practice, it is implemented in a slightly different way -- holding a series of short-term variance swaps -- to avoid a number of technical problems

• Payoff of variance swap: σ2(caplet) - σ2(futures)

• These two strategies have identical P&L with the notional scaled by F2 x gamma / 2

• This strategy is essentially model-independent

Yield Curve Analysis & Fixed-income Arbitrage

50

Yield Spread Arbitrage (1)

• Yield spread arbitrage is not market-directional trading, but the view on the shape of a yield curve or the spread between two particular points on the curve

• Generally, there is no analytical relationship between changes in a specific yield spread and changes in the general level of interest rates

• The yield curve may be flattening or steeping when rates are both falling or rising

• Spread trade is structured so that P&L is made as a result of a change in the spread, and not due to any change in overall yield curve levels

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Yield Spread Arbitrage (2)

• Suppose a trader believes that the yield curve is going to flatten, but has no particular strong view on whether this flattening would occur in falling or rising interest rates

• If he thinks that the flattening would be most pronounced in the 2-year and 10-year buckets, he can take short position on 2-year bond and long position on 10-year bond

• He needs to use the price value of a basis point (PVBP) to determine the weight in this spread trade (duration-hedging)

• PVBP is the change of the bond price if the yield changes by 1 basis point

∆Y∆P

1001

=PVBP

Yield Curve Analysis & Fixed-income Arbitrage

52

Yield Spread Arbitrage (3)

• By the definition of PVBP,

• This quantity is the change of 10-year bond with respect to the change in the change of 2-year bond, so it can be used to hedge against parallel shift risk, i.e., both 2-year yield and 10-year yield change in the same direction

• For example, if ΔY10 = ΔY2 = 0.01% (no change in spread size), the change in 10-year bond price is ΔP10

• If the trader holds ΔP10 / ΔP2 units of 2-year bond, then the change is

• This implies that there is no P&L (offset the delta risk) if 2-year and 10-year yields have parallel shift

2

10

2

10 1001002

10PP

PY

YP

∆∆

=∆∆

⋅∆

∆=

Y)PVBP(Y)PVBP(

1022

10 PPPP

∆=∆⋅∆∆

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53

Exotic Spread Arbitrage

• There are more exotic type of yield spread trading

• Butterfly trade: Short position (sell) in one T2-year bond and long position (hold) in two bonds with maturities T1 (< T2) and T3 (> T2)

• Traders would use this trading strategy if he believes that, for example, 2-year bond would outperform 3-year and 5-year bonds

• His view reflects that the short end of the curve would steepen relative to the “middle” of the curve while the long end would flatten

• It is important noting that duration-hedging is used to offset the parallel shift risk, but two positions could behave differently for given changes in the yield curve due to liquidity and other reasons (not completely get rid of this risk!)

Yield Curve Analysis & Fixed-income Arbitrage

54

Yield Spread Arbitrage: Implementation

• Intellectual Capital: Require a yield curve model to identify points that are either “rich” or “cheap”, i.e., to explore arbitrage opportunities (model risk!)

• Fit a yield curve model to a swap/bond yield curve by matching exactly some pre-specified points (Eg. 1Y and 10Y) each month

• If the 2Y swap is more than a certain amount (say 5 or 10 bps) above the fitted 2Y point, a trade is structured by receiving fixed on a 2Y swap and shorting a 1Y and 10Y swaps for delta-neutral

• The trade will be held for 12 months or until the 2Y swap converges to the model value

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Capital Structure Arbitrage

• A strategy that involves take long and short positions in different instruments of a company’s capital structure

• It includes a wide range of strategies between equity, debt, and credit instruments of a given company, sector or industry

• The number of underlying securities and its option leads to a massive number of possible combining trading strategies

• Trading strategies are typically based on the investment view of hedge fund manager and the availability of market instruments

• Some popular types of capital structure arbitrage strategies:– Convertible arbitrage: long bond and short equity– Basis arbitrage: long bond and short CDS– Credit arbitrage: long CDS and short equity

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Convertible Arbitrage

• It is one of the most popular capital structure arbitrage strategies

• The strategy that involves buying a portfolio of convertible bonds and offsetting or hedging these long positions by selling short the underlying stocks

• Intuitively, if the stock increases in price, the bonds will appreciate and if the stock falls, the short position will make money

• Convertible bond arbitrageurs favor equity market volatility

• They take advantage of stock price movements to adjust their short stock hedge positions -- maintain a market neutral position to capture profits

• A reverse strategy -- short a CB and long the underlying stock -- is so-called Chinese hedge

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Risks in Convertible Arbitrage

• A convertible bond (CB) is a hybrid security that is characterized by a fixed-income part as well as an equity part, resulting a diverse risk factors

• Equity risk: At higher share price, the price of a CB behaves like pure equity; at low share price, the value of CB falls to a lower level and flattens out to a constant level and CB is likely to redeem at maturity

• Interest rate risk: This risk is usually hedged with treasury futures or interest rate swaps

• Credit risk (omicron): A lower credit rating leads to a lower equity price and a widening in credit spread, resulting a lower price of a CB and is hedged with longing CDS/shorting a plain bond or equity (a structural model is needed!)

• Liquidity risk: Long position not being liquid as expected or short position being called in/short squeezed and this risk cannot be hedged away

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Impact of Stock Price on a Convertible Bond

Source: Deutsch Bank

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Delta Hedging

• The most popular CB arbitrage strategy is the delta-neutral hedging: long a CB and short the underlying stock at the appropriate delta -- fully hedged for very small movements in stock price

• In this hedge, delta risk is neutralized but not the interest rate risk and the long position in vega exposure

• If the implied volatility level decreases or stays the same, the position will benefit just from the income flow from convertible’s yield and the short interest rate’s rebate

• Because of long vega exposure, a higher implied volatility, a more earning the position gets

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Risk Analysis

• Delta risk: CB moves towards the ITM position as delta tends to 1 and moves towards the OTM position as delta tends to 0

• Gamma risk: Deep OTM/ITM results a lower gamma and ATM gives thelargest gamma, so gamma is associated with the rebalancing frequency of a delta-hedged portfolio

• Vega risk: ITM/OTM possesses lower vega and ATM presents higher vega

• Theta risk: A lower theta when ITM/OTM (less conversion premium to lose) and a higher theta when ATM

• Rho risk: CB reaches its maximum rho when OTM and minimum when ITM

• Omicron risk: A higher omicron when OTM and a lower omicron when deep ITM, and OTM CB is more influenced by omicron than any other greek

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Criteria to Identify Convertible Arbitrage Opportunities

• A higher value for vega than for omicron

• A low conversion premium

• A high gamma

• A lower liquidity risk

• Convertible’s yield is higher than LIBOR

• Short selling stock is possible

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Other Risks Taken into Consideration

• A decreasing in implied volatility

• A widening of credit spreads

• Call risk from the issuer if provided this feature in CB

• Model risk from pricing/decomposing option and fixed-income component and from risk management (greek calculations)

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Open Issues on Fixed-income Arbitrage

• Is FIA truly arbitrage?

• Is it merely a strategy that earns small profit most of the time, but occasionally experiences dramatic losses?

• Is it really market-neutral?

• Is there a link between hedge fund returns and hedge fund capital?

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Empirical Studies on Fixed-income Arbitrage

• Durate, Longstaff and Yu (2005) provides a comprehensive study on the return and the risk characteristics of 5 popular FIA strategies: (1) swap spread arbitrage, (2) yield spread arbitrage, (3) volatility arbitrage, (4) mortgage arbitrage, and (5) capital structure arbitrage: CDS and equity

• Empirical findings:

– A majority of the strategies produce significant excess returns– The annualized Sharpe ratio lies between 0.3 and 0.9– With the exception of volatility arbitrage, most of the monthly returns are

positively skewed -- contrary to the common wisdom that risk arbitrage generates small positive returns most of the time, but experiences infrequent large losses

– No significant autocorrelation– The amount of capital required to obtain an annual volatility of 10% varies

across strategies– Returns on many arbitrage strategies are sensitive to equity, bond and

credit market risk factors

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Things to be aware

• Implementation of some FIA strategies (yield spread and capital structure) requires a high level of intellectual capital, resulting an excess return for model risk even after adjusting for market risks (Eg. LTCM)

• It is questionable whether these results reflects the actual behavior of return rate as hedge fund managers have their own models and views to decide the trading strategies

• Models only tell you the time to enter into a trade, but don’t give you any hint when to terminate if the market situation is against hedge fund managers’ view

• Decision of cut-loss/holding portfolio would result a substantial change in return rates

• As the number of hedge funds and the asset under management with similar styles across the world are rapidly growing, it would heighten financial system instability (create another risk!)