Fixed Income Securities Dr. Rong Chen The Department of Finance Xiamen University

Syllabus Part I basic knowledge: Fixed-income instruments, prices and yields Part II Term structure: Empirical properties and classical theories of the term structure & Deriving the zero-coupon yield curve Part III Hedging interest-rate risk with duration, convexity and other ways Part IV Investment strategies: passive, active and performance measurement. Part V: Swaps and futures Part VI: Dynamic term structure modeling Part VII: Interest-rate derivatives with options Part VIII: Securitization Copyright © Rong Chen, 2007, Finance Department, XMU2

References Lionel Martellini, Philippe Priaulet, Stephane Priaulet, 2003, Fixed-income securities: valuation, risk management and portfolio strategies, Wiley. Suresh M. Sundaresan, 1997, Fixed income markets and their derivatives, South-Western College Publishing John Hull, 2006, options, futures and other derivatives, Prentice Hall Moorad Choudhry, 2005, Fixed-income securities and derivatives handbook, Bloomberg Bond markets, 2000, Analysis and strategies, Frank J. Fabozzi,4th edition, NJ :Prentice Hall ( 美 ) 布鲁斯 · 塔克曼 (Bruce Tuckman) 著, 黄嘉斌译, 1999, 北京：宇航出版社 李奥奈尔 · 马特里尼, 菲利普 · 普里奥兰德著, 肖军译, 2002, 固定收益证券：对利率风险 进行定价和套期保值的动态方法,, 北京：机械工业出版社 谢剑平， 2003 ， 固定收益证券：投资与创新，人民大学出版社 薛立言 刘亚秋， 2006 ， 债券市场，东华书局 林清泉， 2005 ，固定收益证券, 武汉大学出版社 姚长辉 ， 2006 ，固定收益证券 : 定价与利率风险管理, 北京大学出版社 3

Internet resources ndex.jsp ndex.jsp 中国债券信息网 ew/zaxiang/shouye/index.jsp ew/zaxiang/shouye/index.jsp 中国货币网 和讯债券 Copyright © Rong Chen, 2007, Finance Department, XMU4

Fixed income securities? Relatively fixed cash flows securities — traditional fixed income instruments ： bonds/money-market instruments ( repo, T- bills) —Interest rate derivatives: futures, forwards, swaps, options (caps/ floors) —Bonds with embedded options securitization Copyright © Rong Chen, 2007, Finance Department, XMU5

Chapter 10 Interest Rate Swaps

Contents Definition Pricing and quotes Uses Other nonplain vanilla swaps Copyright © Rong Chen, 2007, Finance Department, XMU7

DESCRIPTION OF SWAPS 10.1

Definition Plain vanilla interest rate swap — a party agrees to pay cash flows equal to interest at a predetermined fixed rate on a notional principle for a number of years. In return, it receives interest at a floating rate on the same notional principal for the same period of time. —OTC derivative product Copyright © Rong Chen, 2007, Finance Department, XMU9

An Example of a “Plain Vanilla” Interest Rate Swap An agreement by Microsoft with Intel to receive 6-month LIBOR & pay a fixed rate of 5% per annum every 6 months for 3 years on a notional principal of $100 million Next slide illustrates cash flows Copyright © Rong Chen, 2007, Finance Department, XMU10

Copyright © Rong Chen, 2007, Finance Department, XMU Millions of Dollars LIBORFLOATINGFIXEDNet DateRateCash Flow Mar.5, % Sept. 5, %+2.10–2.50–0.40 Mar.5, %+2.40–2.50–0.10 Sept. 5, %+2.65– Mar.5, %+2.75– Sept. 5, %+2.80– Mar.5, %+2.95– Cash Flows to Microsoft

Terminology and conventions All swaps are traded under the legal terms and conditions fixed by the International Swap Dealer Association ( ISDA ) Terms: — Maturity — Notional amount —Fixed interest rate : fixed leg —Floating interest rate : floating leg : LIBOR — Payment dates Trade date Effective date: calculate the interest payment Payment date Day-count basis Dollars and Euro: Acutal/360 Sterling: Actual/ 365 — only the difference between the two interest payments is exchanged Copyright © Rong Chen, 2007, Finance Department, XMU12

PRICES 10.2

Pricing of Interest Rate Swaps Valuation in Terms of bonds — Interest rate swaps can be valued as the difference between the value of a fixed-rate bond and the value of a floating-rate bond : Valuation in terms of forwards —Alternatively, they can be valued as a portfolio of forward rate agreements (FRAs) Forward projection methods —On the assumption that future floating rates are equal to the forward rates Copyright © Rong Chen, 2007, Finance Department, XMU14

VALUATION IN TERMS OF BONDS

Copyright © Rong Chen, 2007, Finance Department, XMU Millions of Dollars LIBORFLOATINGFIXEDNet DateRateCash Flow Mar.5, % Sept. 5, %+2.10–2.50–0.40 Mar.5, %+2.40–2.50–0.10 Sept. 5, %+2.65– Mar.5, %+2.75– Sept. 5, %+2.80– Mar.5, %+2.95– Cash Flows to Microsoft

Valuation in Terms of Bonds Copyright © Rong Chen, 2007, Finance Department, XMU17 If a principal payments are both received and paid at the beginning and the end of the swap, this swap can be regarded as a portfolio of a fixed-rate bond and a floating-rate bond.

Valuation in Terms of Bonds (Cont.) The fixed rate bond is valued in the usual way The floating rate bond is valued by noting that it is worth par immediately after the next payment date Also called “zero-coupon method” Copyright © Rong Chen, 2007, Finance Department, XMU18

Example 10.1 Suppose that a financial institution has agreed to pay 6-month LIBOR and receive 8% per annum (with semiannual compounding) on a notional principal of $100 million. The swap has a remaining life of 1.25 years. The LIBOR rates with continuous compounding for 3-month, 9- mon and 15-mon maturities are 10%, 10.5% and 11%. The 6-mon LIBOR rate at the last payment date was 10.2% Copyright © Rong Chen, 2007, Finance Department, XMU19

Solutions: — — V Swap =98.24 － =-$ 4.27 Copyright © Rong Chen, 2007, Finance Department, XMU20

VALUATION IN TERMS OF FRAS

Valuation in Terms of FRAs Each exchange of payments in an interest rate swap is an FRA The FRAs can be valued on the assumption that today’s forward rates are realized Steps: —Find forward rates —Calculate cash flows of each FRA on the assumption that the LIBOR rates will equal the forward rates —The sum of all the discounted cash flows is the value of the swap Copyright © Rong Chen, 2007, Finance Department, XMU22

Copyright © Rong Chen, 2007, Finance Department, XMU Millions of Dollars LIBORFLOATINGFIXEDNet DateRateCash Flow Mar.5, % Sept. 5, %+2.10–2.50–0.40 Mar.5, %+2.40–2.50–0.10 Sept. 5, %+2.65– Mar.5, %+2.75– Sept. 5, %+2.80– Mar.5, %+2.95– Cash Flows to Microsoft

Example 10.1 Suppose that a financial institution has agreed to pay 6-month LIBOR and receive 8% per annum (with semiannual compounding) on a notional principal of $100 million. The swap has a remaining life of 1.25 years. The LIBOR rates with continuous compounding for 3-month, 9- mon and 15-mon maturities are 10%, 10.5% and 11%. The 6-mon LIBOR rate at the last payment date was 10.2% Copyright © Rong Chen, 2007, Finance Department, XMU24

1. 2. (1) (2) (3) 3. (1) (2) (3) =-4.27 Copyright © Rong Chen, 2007, Finance Department, XMU25

Valuation in Terms of FRAs The result agrees with the result of the method in terms of bonds---the forward rates are based on the term structure. The zero value of a swap initially doesn’t mean that each FRA is equal to zero. Copyright © Rong Chen, 2007, Finance Department, XMU26

FORWARD PROJECTION METHOD

Forward projection method This method is also based on the assumption that the future floating rates of the floating leg are equal to the forward rates. Actually it is based on the idea of cash flows Copyright © Rong Chen, 2007, Finance Department, XMU28

Example 10.1 Suppose that a financial institution has agreed to pay 6-month LIBOR and receive 8% per annum (with semiannual compounding) on a notional principal of $100 million. The swap has a remaining life of 1.25 years. The LIBOR rates with continuous compounding for 3-month, 9- mon and 15-mon maturities are 10%, 10.5% and 11%. The 6-mon LIBOR rate at the last payment date was 10.2% Copyright © Rong Chen, 2007, Finance Department, XMU29

Copyright © Rong Chen, 2007, Finance Department, XMU30

Copyright © Rong Chen, 2007, Finance Department, XMU31 Forward projection method This method agrees with the previous two methods for plain vanilla swaps. They are equivalence in essence. — Forward projection method / FRA method —Forward projection method / bond method This method is more general than the other two and is the standard pricing approach used by the market.

“Zero-coupon method” When the difference between the measurement date and the payment date is equal to the maturity of the reference index: — Forward price: —Forward rate: Copyright © Rong Chen, 2007, Finance Department, XMU32

Copyright © Rong Chen, 2007, Finance Department, XMU33 Equivalent to the bond method Only dependent on zero –coupon bonds

Some understandings Initially, the value of a swap should be zero so that it is a fair deal. Later on, prices can differ depending on the evolution of the term structure. The fixed leg has longer duration and therefore is more sensitive to the change of the interest rate than the floating leg. The advantage of a plain vanilla swap compared to a coupon-bearing bond is that its price is very much lower than that of the latter while it has almost the same sensitivity to rate changes. Copyright © Rong Chen, 2007, Finance Department, XMU34

Quotes of swaps The fixed rate which makes the initial value of the swap equal to zero is swap rate. The floating rate is usually LIBOR. — e.g. The bid price quoted by the market maker is 6% to pay the fixed rate as the ask price to receive the fixed rate 6.05% A swap could also be quoted as a swap spread -- - the difference between the fixed rate of the swap and the treasury benchmark bond yield of the same maturity. —E.g. a 7 year 3-month Libor swap, 45-50: paying 45 points above the 7-year benchmark bond yield and receiving the 3-month Libor, or receiving fixed 50 basis points above the 7-year bond yield and paying the Libor. Copyright © Rong Chen, 2007, Finance Department, XMU35

USES OF SWAPS 10.3

Optimizing the financial conditions of a debt : Comparative Advantage AAACorp wants to borrow floating BBBCorp wants to borrow fixed 32 FixedFloating AAACorp10.00%6-month LIBOR % BBBCorp11.20%6-month LIBOR %

The Swap 33 AAA BBB LIBOR LIBOR+1% 9.95% 10%

The Swap when a Financial Institution is Involved 39 AAAF.I. BBB 10% LIBOR LIBOR+1% 9.93% 9.97%

Converting the financial conditions of a debt or an asset Fixed rate floating rate: debts or assets E.g. To finance their needs, most firms issue long-term fixed-coupon bonds because of the large liquidity of these bonds. Sometimes they hope to transform their debts into a floating-rate debt. E.g. To optimize the matching of assets and liabilities: A bank has an asset of 4- year bond with a semiannually-paid 7% fixed rate and $10 million principal value and a CD at the 6-month Libor rate+0.2%. Copyright © Rong Chen, 2007, Finance Department, XMU40

Creating new assets using swaps An asset swap: —An investor believes CAD rates will rise over the medium term. They would like to purchase CAD 50million 5yr Floating Rate Notes. There are no 5yr FRNs available in the market in sufficient size. The investor is aware of XYZ Ltd 5yr 6.0% annual fixed coupon Bonds currently trading at a yield of 5.0%. The bonds are currently priced at The investor can purchase CAD 50million Fixed Rate Bonds in the market for a total consideration of CAD 51,955,000 plus any accrued interest. They can then enter a 5 year Interest Rate Swap (paying fixed) with the Bank as follows: Copyright © Rong Chen, 2007, Finance Department, XMU41

Copyright © Rong Chen, 2007, Finance Department, XMU42

Flexibility Customised to match underlying securities Can be reversed at any time Can be traded as a package or separately Copyright © Rong Chen, 2007, Finance Department, XMU43

Hedging interest rate risk using swaps Duration hedge Duration/ convexity hedge Note that hedging interest-rate risk of a bond portfolio with swaps is an efficient way when they have exactly the same default risk. If not, a default risk still exists that is not hedged. Copyright © Rong Chen, 2007, Finance Department, XMU44

Duration hedge Copyright © Rong Chen, 2007, Finance Department, XMU45

Duration/ convexity hedge Copyright © Rong Chen, 2007, Finance Department, XMU46

NONPLAIN VANILLA SWAPS 10.4

Accrediting, amortizing and roller coaster swaps Bullet swap: the notional principal remains unchanged Accrediting swap: the notional amount increases overtime Amortizing swap: the notional amount decreases in a predetermined way over the life of the swap. roller coaster swaps: the notional amount may rise or fall from one period to another. Copyright © Rong Chen, 2007, Finance Department, XMU48

Basis swap A basis swap is a floating-for-floating interest-rate swap that exchange the floating rates of two different market or/and different maturities. Copyright © Rong Chen, 2007, Finance Department, XMU49

Constant maturity swap and constant maturity treasury swap CMS: LIBOR—a particular swap rate CMT: LIBOR—a particular treasury-bond rate CMS-CMT —A firm pays quarterly to a bank the 10-year CMT rate+20 bps and receives quarterly from the bank the 10-year CMS rate. —Assuming there is a positive correlation between the evolution of the spread CMS-CMT and the spread between the yield of risky bonds and default-free treasury bonds. It is a valid hedging instrument to the credit spread. Copyright © Rong Chen, 2007, Finance Department, XMU50

Other swaps Forward-starting swap: a swap starting in the future Inflation-linked swap: could be used by issuers of inflation-linked bonds Libor in arrears swap: the floating rate is set and paid in arrears. Yield-curve swap: bet on the difference between interest rates at two points on a given yield curve. Zero-coupon swap: exchange a fixed or floating index that delivers regular coupons for an index that delivers only one coupon at the beginning or at the end of the swap. Copyright © Rong Chen, 2007, Finance Department, XMU51