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Http: \www. tfii. org Fixed Income Zvi Wiener 1 Http: \www. tfii. org Fixed Income Zvi Wiener 1

Plan • • Pricing of Bonds Measuring yield Bond Price Volatility Factors Affecting Yields Plan • • Pricing of Bonds Measuring yield Bond Price Volatility Factors Affecting Yields and the Term Structure of IR • Treasury and Agency Securities Markets • Corporate Debt Instruments • Municipals Http: \www. tfii. org 2

Plan • • Non-US Bonds Mortgage Loans Mortgage Pass-Through Securities CMO and Stripped MBS Plan • • Non-US Bonds Mortgage Loans Mortgage Pass-Through Securities CMO and Stripped MBS ABS Bonds with Embedded Options Analysis of MBS Analysis of Convertible Bonds Http: \www. tfii. org 3

Plan • • Active Bond Portfolio Management Indexing Liability Funding Strategies Bond Performance Measurement Plan • • Active Bond Portfolio Management Indexing Liability Funding Strategies Bond Performance Measurement Interest Rate Futures Interest Rate Options Interest Rate Swaps, Caps, Floors Http: \www. tfii. org 4

Characteristics of a Bond • • Issuer Time to maturity Coupon rate, type and Characteristics of a Bond • • Issuer Time to maturity Coupon rate, type and frequency Linkage Embedded options Indentures Guarantees or collateral Http: \www. tfii. org 5

Sources • Fabozzi, “Bond Markets, Analysis and Strategies”, Prentice Hall. • P. Wilmott, Derivatives, Sources • Fabozzi, “Bond Markets, Analysis and Strategies”, Prentice Hall. • P. Wilmott, Derivatives, Wiley. • Hull, White, Manuscript. Http: \www. tfii. org 6

Sectors • • Treasury sector: bills, notes, bonds Agency sector: debentures (no collateral) Municipal Sectors • • Treasury sector: bills, notes, bonds Agency sector: debentures (no collateral) Municipal sector: tax exempt Corporate sector: US and Yankee issues – bonds, notes, structured notes, CP – investment grade and noninvestment grade • Asset-backed securities sector • MBS sector Http: \www. tfii. org 7

Basic terms • • Principal Coupon, discount and premium bonds Zero coupon bonds Floating Basic terms • • Principal Coupon, discount and premium bonds Zero coupon bonds Floating rate bonds Inverse floaters Deferred coupon bonds Amortization schedule Convertible bonds Http: \www. tfii. org 8

Basic Terms • The Money Market Account • LIBOR = London Interbank Offer Rate, Basic Terms • The Money Market Account • LIBOR = London Interbank Offer Rate, see BBA Internet site • FRA = Forward Rate Agreement • Repos, reverse repos • Strips = Separate Trading of Registeres Interest and Principal of Securities Http: \www. tfii. org 9

Basic Terms • gilts (bonds issued by the UK government) • JGB = Japanese Basic Terms • gilts (bonds issued by the UK government) • JGB = Japanese Government Bonds • Yen denominated issued by non-Japanese institutions are called Samurai bonds Http: \www. tfii. org 10

Major risks • Interest rate risk • Default risk • Reinvestment risk • Currency Major risks • Interest rate risk • Default risk • Reinvestment risk • Currency risk • Liquidity risk Http: \www. tfii. org 11

Time Value of Money • present value PV = CFt/(1+r)t • Future value FV Time Value of Money • present value PV = CFt/(1+r)t • Future value FV = CFt(1+r)t • Net present value NPV = sum of all PV -PV 5 5 105 Http: \www. tfii. org 12

Term structure of interest rates Yield = IRR How do we know that there Term structure of interest rates Yield = IRR How do we know that there is a solution? Http: \www. tfii. org 13

Price-Yield Relationship • Price and yield (of a straight bond) move in opposite directions. Price-Yield Relationship • Price and yield (of a straight bond) move in opposite directions. price yield Http: \www. tfii. org 14

General pricing formula Http: \www. tfii. org 15 General pricing formula Http: \www. tfii. org 15

Accrued Interest Accrued interest = interest due in full period* (number of days since Accrued Interest Accrued interest = interest due in full period* (number of days since last coupon)/ (number of days in period between coupon payments) Http: \www. tfii. org 16

Day Count Convention Actual/Actual - true number of days 30/360 - assume that there Day Count Convention Actual/Actual - true number of days 30/360 - assume that there are 30 days in each month and 360 days in a year. Actual/360 Http: \www. tfii. org 17

Floater The coupon rate of a floater is equal to a reference rate plus Floater The coupon rate of a floater is equal to a reference rate plus a spread. For example LIBOR + 50 bp. Sometimes it has a cap or a floor. Http: \www. tfii. org 18

Inverse Floater Is usually created from a fixed rate security. Floater coupon = LIBOR Inverse Floater Is usually created from a fixed rate security. Floater coupon = LIBOR + 1% Inverse Floater coupon = 10% - LIBOR Note that the sum is a fixed rate security. If LIBOR>10% there is typically a floor. Http: \www. tfii. org 19

Price Quotes and Accrued Interest Assume that the par value of a bond is Price Quotes and Accrued Interest Assume that the par value of a bond is $1, 000. Price quote is in % of par + accrued interest the accrued interest must compensate the seller for the next coupon. Http: \www. tfii. org 20

Annualizing Yield Effective annual yield = (1+periodic rate)m-1 examples Effective annual yield = 1. Annualizing Yield Effective annual yield = (1+periodic rate)m-1 examples Effective annual yield = 1. 042 -1=8. 16% Effective annual yield = 1. 024 -1=8. 24% Http: \www. tfii. org 21

Bond selling at Relationship Par Coupon rate=current yield=YTM Discount Coupon rate<current yield<YTM Premium Coupon Bond selling at Relationship Par Coupon rate=current yield=YTM Discount Coupon ratecurrent yield>YTM Yield to call uses the first call as cashflow. Yield of a portfolio is calculated with the total cashflow. Http: \www. tfii. org 22

YTM and Reinvestment Risk • YTM assumes that all coupon (and amortizing) payments will YTM and Reinvestment Risk • YTM assumes that all coupon (and amortizing) payments will be invested at the same yield. Http: \www. tfii. org 23

YTM and Reinvestment Risk • An investor has a 5 years horizon Bond Coupon YTM and Reinvestment Risk • An investor has a 5 years horizon Bond Coupon Maturity YTM A 5% 3 9. 0% B 6% 20 8. 6% C 11% 15 9. 2% D 8% 5 8. 0% What is the best choice? Http: \www. tfii. org 24

Bond Price Volatility Consider only IR as a risk factor Longer TTM means higher Bond Price Volatility Consider only IR as a risk factor Longer TTM means higher volatility Lower coupons means higher volatility Floaters have a very low price volatility Price is also affected by coupon payments Price value of a Basis Point = price change resulting from a change of 0. 01% in the yield. Http: \www. tfii. org 25

Duration and IR sensitivity Http: \www. tfii. org 26 Duration and IR sensitivity Http: \www. tfii. org 26

Duration Http: \www. tfii. org 27 Duration Http: \www. tfii. org 27

Duration Http: \www. tfii. org 28 Duration Http: \www. tfii. org 28

Duration Bond duration A 3 yr B 1 yr C 10 yr D 20 Duration Bond duration A 3 yr B 1 yr C 10 yr D 20 yr price impact of +1% YTM -3% -10% -20% Http: \www. tfii. org 29

Measuring Price Change Http: \www. tfii. org 30 Measuring Price Change Http: \www. tfii. org 30

The Yield to Maturity The yield to maturity of a fixed coupon bond y The Yield to Maturity The yield to maturity of a fixed coupon bond y is given by Http: \www. tfii. org 31

Macaulay Duration Definition of duration, assuming t=0. Http: \www. tfii. org 32 Macaulay Duration Definition of duration, assuming t=0. Http: \www. tfii. org 32

Macaulay Duration A weighted sum of times to maturities of each coupon. What is Macaulay Duration A weighted sum of times to maturities of each coupon. What is the duration of a zero coupon bond? Http: \www. tfii. org 33

Meaning of Duration $ r. Http: \www. tfii. org 34 Meaning of Duration $ r. Http: \www. tfii. org 34

Convexity $ r. Http: \www. tfii. org 35 Convexity $ r. Http: \www. tfii. org 35

FRA Forward Rate Agreement A contract entered at t=0, where the parties (a lender FRA Forward Rate Agreement A contract entered at t=0, where the parties (a lender and a borrower) agree to let a certain interest rate R*, act on a prespecified principal, K, over some future time period [S, T]. Assuming continuous compounding we have at time S: -K at time T: Ke. R*(T-S) Calculate the FRA rate R* which makes PV=0 hint: it is equal to forward rate Http: \www. tfii. org 36

ALM Duration • Does NOT work! • Wrong units of measurement • Division by ALM Duration • Does NOT work! • Wrong units of measurement • Division by a small number Http: \www. tfii. org 37

ALM Duration A similar problem with measuring yield Http: \www. tfii. org 38 ALM Duration A similar problem with measuring yield Http: \www. tfii. org 38

Do not think of duration as a measure of time! Http: \www. tfii. org Do not think of duration as a measure of time! Http: \www. tfii. org 39

• Key rate duration • Principal component duration • Partial duration Http: \www. • Key rate duration • Principal component duration • Partial duration Http: \www. tfii. org 40

Factors affecting Bond yields and TS • Base interest rate - benchmark interest rate Factors affecting Bond yields and TS • Base interest rate - benchmark interest rate • Risk Premium - spread • Expected liquidity • Market forces - Demand supply Http: \www. tfii. org 41

Taxability of interest • qualified municipal bonds are exempts from federal taxes. After tax Taxability of interest • qualified municipal bonds are exempts from federal taxes. After tax yield = pretax yield (1 - marginal tax rate) Http: \www. tfii. org 42

Do not use yield curve to price bonds Period A B 1 -9 $6 Do not use yield curve to price bonds Period A B 1 -9 $6 $1 10 $106 $101 They can not be priced by discounting cashflow with the same yield because of different structure of CF. Use spot rates (yield on zero-coupon Treasuries) instead! Http: \www. tfii. org 43

On-the-run Treasury issues Off-the-run Treasury issues Special securities Lending Repos and reverse repos Http: On-the-run Treasury issues Off-the-run Treasury issues Special securities Lending Repos and reverse repos Http: \www. tfii. org 44

Forward Rates Buy a two years bond Buy a one year bond and then Forward Rates Buy a two years bond Buy a one year bond and then use the money to buy another bond (the price can be fixed today). (1+r 2)=(1+r 1)(1+f 12) Http: \www. tfii. org 45

Forward Rates (1+r 3)=(1+r 1)(1+f 13)= (1+r 1)(1+f 12)(1+f 13) Term structure of instantaneous Forward Rates (1+r 3)=(1+r 1)(1+f 13)= (1+r 1)(1+f 12)(1+f 13) Term structure of instantaneous forward rates. Http: \www. tfii. org 46

Determinants of the Term Structure Expectation theory Market segmentation theory Liquidity theory Mathematical models: Determinants of the Term Structure Expectation theory Market segmentation theory Liquidity theory Mathematical models: Ho-Lee, Vasichek, Hull -White, HJM, etc. Http: \www. tfii. org 47

Home Assignment • What is the duration of a floater? • What is the Home Assignment • What is the duration of a floater? • What is the duration of an inverse floater? • How coupon payments affect duration? • Why modified duration is better than Macaulay duration? • How duration can be used for hedging? Http: \www. tfii. org 48