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Interest Rates and Swaps. Term Structure Analysis. Term-Structure. It refers to the relationships of YTM of default free bonds and their maturities Spot rate of interest: YTM on pure discount bonds spot curve Application:

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## Interest Rates and Swaps

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**Interest Rates and Swaps**Wulin Suo**Term Structure Analysis**Wulin Suo**Term-Structure**• It refers to the relationships of YTM of default free bonds and their maturities • Spot rate of interest: YTM on pure discount bonds • spot curve • Application: • it allows one to discount each cash flow separately – reasonable for default-free securities (e.g., T-securities) • Each cash flow is discounted by a rate corresponding to that maturity • It takes into consideration of reinvestment rate • Disadvantage: ignores the liquidity of a specific bond**Pure Discount Bond**• Zero-Coupon Bond (ZCB) price b(t,T): the price at time t of a bond that pays $1 at maturity time T and nothing else • ZCBs are simply called zeros • b(t,T) is essentially a discount factor: • for an amount of $C received at time T, one should pay Cb(t,T) • In realty, zero-coupon does exist • STRPIS: Separate Trading of Registered Interest and Principal Securities**ZCB**• Advantages of investing in zeros: • Assured growth (assuming one is holding to maturity) • Ideal to match liabilities • Low initial investment • Automatic compounding of interest • A wide selection of issuers, and maturities ranging from one year to 30 years • Relatively high liquidity**Building a Zero Curve**• It is the zero curve implied by the prices of coupon bonds traded in the market • The spot rates may be different from those rates implied the strips securities • The implied zero curve can be used as reference rate to check if the strips rates are out of line with the treasury security market • Unlike yield curve, which can be built the many actively traded T-securities • Zero curve are usually computed by using the bootstrapping method**Bootstrapping Method**• Example: • Write yT as the implied spot rate (or simply zero rate) with maturity T:**Bootstrapping …**• Calculating y2 and b(0,2): • For y3 and b(0,3):**Bootstrapping …**• In general, assume that for each year n, there is a coupon bond maturing in n years and paying an annual coupon of Cn, and a cash price of Pn • Step 1: • In general:**Bootstrapping …**• Semi-annual coupon payments can be handles similarly, and zero rates at semi-annual intervals can thus be obtained • What the maturities not on the annual/semi-annual intervals? • interpolation • Restriction on the zero price: for t1 < t2 < … <tn,**Par Bond Yield Curve**Question: Based on the zero curve, what is the coupon rate that makes the bond trade at par? • 1Y maturity: • 2Y maturity: • 3Y maturity: • In general:**Upward sloping**Zero Curve vs Par Yield**Continuous Compounding**• Zero rate with continuous compounding: • yT is usually simply called zero yields or spot yields • Par bond yield: • a bond is paying a coupon continuously at a rate of CT, ie, over a small interval [s, s+dt], an amount of CTdt is paid: • Par yield is defined as**Continuous Compounding …**• Instantaneous short rate is defined as • If there is no uncertainty in the short rate, then the following relationship must hold: or**Forward Rates**• Forward rate is the rate observed now that will be applied to period of time in the future • we write the forward rate as ft(T1,T2) • This rate can be achieved through Forward Rate Agreements (FRAs) • If we know all the zero rates, then**Forward Rates …**• If we know all the implied zeros, then • Implication for coupon bond pricing: • coupons are reinvested at the forward rates**If the zero curve is curve is upward sloping**If the zero curve is curve is downward sloping Forward Rates vs Zero Rates**Example**• f0(1,2): • f0(2,3): • f0(3,4): • f0(1,4):**Building Zero Curves**• Bonds with some maturities may not exist in the market • Some bonds that have similar maturities and coupons are trading at quite different yields • impossible to quantify liquidity • Which price should one use, bid or ask? • it doesn’t really matter as long as one is consistent**Building Zero Curves**• One way to overcome the lack of information corresponding to some maturities is through interpolation • interpolate yield curve • interpolate zero price • model implied zero rates directly**Futures and Forwards**• Forward contract: agreement between the buyer and the seller to settle a trade at some pre-specified (forward price) at some future date • Futures contract: similar to forward contract, but are standardized and exchange traded • Forwards, futures, options, and swaps are all in zero net supply • If the long side gains, then the short side loses: zero-sum game • Underlying can be: interest rates, bills, notes, bonds, etc**Interest Rate Futures**• Most actively traded futures in US: • 3-month T-bills: $1 million face value, IMM of CME • 3-month Eurodollar certificates of deposit: IMM of CME, London International Financial Futures Ex. • 20-year, 8% Treasury coupon bonds, $100,000 face value; CBT • 10-year, 8% Treasury note, $100,000 face value; CBT • 5-year Treasury note, $100,000 face value; CBT • Basket of 40 Muni bonds (index); CBT**Treasury Futures Contracts**• The deliverable asset to the T-bill futures is a $1 million face value T-bill that had 90 days to maturity • there is a cheapest-to-delivery option because the T-bill can a 90, 91, or 92 day T-bill • the futures price is quoted as where Yd is the discount rate on the T-bill • Example: If the quoted futures price is 93.50, then the price paid the long part at delivery is**Treasury Note and Bond Futures**• The deliverable assets to the 10- and 5-year T-note futures are, respectively: • A $100,000 face value note with maturity of 6.5 to 10 years from the delivery date • A $100,000 face value on-the-run note with original maturity of less than 5.25 years, and maturity of at least 4.25 years from delivery date • The deliverable asset to the T-bond futures is a $100,000 face value bond with maturity (or earliest call date) of at least 15 years**Treasury Note and Bond Futures ...**• There are some option embedded in the T-note and T-bond futures (for the short part) • the delivery instrument is is not unique, and it can also be a basket of qualified securities with total face value of $100,000. A conversion factor is used to determine quantity of the eligible securities to be delivered • time option: underlying can be delivered any day during the delivery month • wild-card option • end-of-month option: contract cease to trade seven business days prior to the last business day of the delivery month, although delivery can be made until the last business day**Treasury Note and Bond Futures ...**• Although most of the contracts are settled before maturity, a significant amount is settled by delivery • unlike futures on equity, which settled by cash, futures on T-notes and T-bonds are settled by delivery • how to close out a futures position before maturity? • Delivery tends to take place at the end of the month when the spot curve is upward sloped (why?). However, when the spot curve is downward sloping, the delivery pattern may be mixed • With all those options for the short side, does it mean the short side has an advantage?**Conversion Factor**• For the futures contract on T-bonds Cash price received by party with short position = Quoted futures price × Conversion factor + Accrued interest • The conversion factor for a bond is computed in the following way: it is the value the bond on the first day of the delivery month on the assumption that the interest rate for all maturities equals 8% per annum (with semiannual compounding). The bond maturity and the times to the coupon payment dates are rounded down to the nearest three months for the purpose of the calculation. If, after rounding, the bond lasts an exact number of half years, the first coupon is assumed to be paid in six months. Otherwise the first coupon is assumed to be paid after three months and accrued interest is subtracted.**Conversion Factor ...**• Example: Consider a 14% coupon bond with 20 years and 2 months to maturity. For the calculation of the conversion factor, the bond is assumed to last exactly 20 years, and the first coupon is paid in six months time. The value of the bond is so the conversion factor for this bond is 1.5938**Conversion Factor ...**Example: Consider a 14% coupon bond with 18 years and 4 months to maturity. For the purpose of calculating the conversion factor, the bond is assumed to have exactly 18 years and 3 months to maturity, and the first coupon is assumed to paid in three month. Discounting all the payments back to a point in time three months from today: Interest rate for a three-month period: PV of the bond is 163.72/1.019804=160.55. Subtracting the accrued interest of 3.5, it becomes 157.05, so the conversion factor is 1.5705**Cheapest to Deliver Bond**• Party with short position in the future contract receives • The cost of purchasing the bond: • The cheapest to delivery bond is the one for which is the least • This number is usually referred to as the basis of the bond • Obviously, during the delivery month, the basis of a bond has to be positive**Spot-Forward Parity Condition**• In the following, we assume there is only one delivery security and hence the BAC should be zero to rule out the arbitrage opportunities • Consider the case where the underlying instrument is a ZCB, and the futures contract matures at T=1 • Consider the following strategies • buy one bond: now –P; at T=1: P’ • borrow Pf/(1+y1): now Pf/(1+y1), at T=1: -Pf • net: now Pf/(1+y1) – P, at T=1: P’- Pf which is equivalent to a long forward, so the cost should be zero:**Spot-Forward Parity ...**• In general, the spot-forward parity relation for a zero is • Similarly, if the forward is written on a coupon bond and the contract matures at t=2, consider the following strategies • buy one bond • borrow • borrow • the net result is the same as a long forward**Spot-Forward Parity ...**• In general • In fact, consider a forward contract for delivery of an instrument which today has n+t periods until maturity • Assume the maturity of the contract is n**Futures vs. Forward Contracts**• Although the spot-forward parity relation is sometimes applied to futures contract, this is incorrect: the daily resettlement of a futures contract leads to random cash flows throughout the life of the contract • For most of the futures contracts, the resulting cash flows are small and the maturity is short, so discounting CFs as it would be appropriate does not change the prices much • Long position is the futures contract: CF < 0 when y increases CF > 0 when y decreases it realizes a lose when cost of financing is high, and profit when reinvestment is low**Futures vs. Forward Contracts ...**• So everything else being equal, one prefers a long position in a forward contract: • Exact relationship can be calculated when specifying an explicit model of interest rates Note: under the risk-neutral probability, the forward price is expected future spot price, while the futures price is a martingale**Eurodollar Futures Contract**• Traded on IMM (Chicago), SIMMEX (Singapore), and the LIFFE (London) • The futures price is quoted as and the contract is settled in cash for a price equal to where LIBOR is a money market rate quoted on an annualized basis • The face amount of the contract is $1 million dollars • Unlike other options, the underlying is based on an interest rate, not a security price • There are no flexibilities in Eurodollar futures contract**Eurodollar Futures Contract ...**• The most actively traded contracts are for three and six month LIBOR • Contracts are settled on the 2nd business day before the 3rd Wednesday of the maturity month • Example: Consider the three month Eurodollar futures price quoted on 01/02/87 for maturity 03/16/87 of 93.95, the implied LIBOR rate on the contract is Suppose I take a short position in the contract. On 03/16/87 the 3-month LIBOR was 6.50 for a cash price of 93.50. Hence I receive the futures price and I pay the cash price of**Eurodollar Futures Contract ...**The net cash flow is • Obviously, in practice, this is the accumulated payment because the change is settled daily • The price sensitivity of the contract can be measured by its PVBP: given that the futures price is linear in the underlying LIBOR rate, we do not need to take derivatives. The change in the futures price for one basis point change in the LIBOR rate can be easily calculated. • In the case of the 3-month LIBOR contract**How to Calculate LIBOR Rate?**• Let b(t,s) be the principal amount the LIBOR rate , at the date t for the maturity date s is quoted on, and is the number of days between t and s, then we have so • How is the LIBOR rate determined in the futures contract on settlement date?**FRAs**• Then Forward Rate Agreement (FRA) market is the OTC equivalent of the exchanged-traded Eurodollar futures • The liquid and easily accessible sector of the FRA market is for 3- and 6-month LIBOR, 1-month forward • they are referred to as 1x4 and 1x7 contracts, respectively • Contracts for delivery of 2, 3, 4, 5, and 6-month forward are also available • On the delivery date, the buyer of the contract receives**FRAs ...**• Consider a 1x4, $100 million FRA at 11%. In one month, if the three-month reference rate, say LIBOR, is above the forward rate, then the seller must pay the payer the discounted difference between the two rates times the principal $100 million. For example, if the 3-month rate is 11.5%, then the payment will equal • When the FRA is first initiated, what should the rate in the contract be?**Floaters**• Floating-rate notes, or floaters, are debt securities with coupons based on a short-term index, such as the prime rate or the 3-month T-bill rate, and that are reset for more than once a year • Big impetus to the market: $650 million issue of floating-rate notes issued on July 30, 1974 by Citicorp • Characteristics of the issue: • coupon rate to be adjusted semi-annually (every june and Dec) at 100bp above T-bill rate • Beginning on June 1976, and on every reset date thereafter, the notes were puttable at par • A floor of 7.7% on the coupon was established for the first year**Floaters ...**• This market mushroomed after 1982 • at the end of 1992: 221 issues for 26.7 billion of floating-rate corporate debt outstanding in US • How to price such a debt? (ignore the option and credit risk) • Consider a coupon bond whose coupon rate is set equal to the one-period spot rate y1 at the beginning of every period. Maturity value: M • one period before maturity, the price of the floater • two periods before maturity**Floaters ...**• Hence, at any reset date, the price of the floater equals its maturity value. In between reset dates, the price is the floater is the same as a zero with maturity value M+Coupon, for example, ½ period before maturity • Duration • at reset: • in between: same as zero**Inverse-Floaters**• An inverse floater is a bond whose coupon payment is inversely related to some index level of interest rate • Consider a coupon bond whose coupon rate is set equal to • A long position in an inverse floater generates the same cash flows as • being long in a coupon bond with c=c’ • being short in a simple floater with c=y1 • being long in a zero where all bonds have the same maturity and principal**Inverse-Floaters ...**• Hence the price can be written as • Duration: • Note that the Macaulay duration of an inverse floater can exceed the time to maturity. The value of the bond is negatively affected by interest rate through two sources: coupon rate, and discount rate • In general, an inverse floater comes from a floor on the coupon payment to prevent the coupon from fall below zero**The General Case**• In general, we can set the coupon rate equal to • A long position in such a bond generates the same cash flow as • being long in a coupon bond with c=c’ • being long in k simple floaters with c=y1 • being short in a k zeros where all bonds have the same maturity and principal • Similarly**Adjustable-Rate Notes**• Adjustable-rate notes, or variable-rate notes, are debt securities with coupon based on a longer-term index For example, the base rate may be the 2-year treasury yield. The coupon is reset every two years to reflect the new level of the treasury security • A plain vanilla adjustable-rate note trades at par at reset, and trades like a coupon bond with maturity equal to the time until the next reset between reset dates

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