1 Zvi Wiener Fixed Income

2 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

3 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

4 Plan Active Bond Portfolio Management Indexing Liability Funding Strategies Bond Performance Measurement Interest Rate Futures Interest Rate Options Interest Rate Swaps, Caps, Floors

5 Characteristics of a Bond Issuer Time to maturity Coupon rate, type and frequency Linkage Embedded options Indentures Guarantees or collateral

6 Sources Fabozzi, “Bond Markets, Analysis and Strategies”, Prentice Hall. P. Wilmott, Derivatives, Wiley. Hull, White, Manuscript.

7 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

8 Basic terms Principal Coupon, discount and premium bonds Zero coupon bonds Floating rate bonds Inverse floaters Deferred coupon bonds Amortization schedule Convertible bonds

9 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

10 Basic Terms gilts (bonds issued by the UK government) JGB = Japanese Government Bonds Yen denominated issued by non-Japanese institutions are called Samurai bonds

11 Major risks Interest rate risk Default risk Reinvestment risk Currency risk Liquidity risk

12 Time Value of Money present value PV = CF t /(1+r) t Future value FV = CF t (1+r) t Net present value NPV = sum of all PV -PV

13 Term structure of interest rates Yield = IRR How do we know that there is a solution?

14 Price-Yield Relationship Price and yield (of a straight bond) move in opposite directions. yield price

15 General pricing formula

16 Accrued Interest Accrued interest = interest due in full period* (number of days since last coupon)/ (number of days in period between coupon payments)

17 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

18 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.

19 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.

20 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.

21 Annualizing Yield Effective annual yield = (1+periodic rate) m -1 examples Effective annual yield = =8.16% Effective annual yield = =8.24%

22 Bond selling atRelationship ParCoupon rate=current yield=YTM DiscountCoupon rate<current yield<YTM PremiumCoupon rate>current yield>YTM Yield to call uses the first call as cashflow. Yield of a portfolio is calculated with the total cashflow.

23 YTM and Reinvestment Risk YTM assumes that all coupon (and amortizing) payments will be invested at the same yield.

24 YTM and Reinvestment Risk An investor has a 5 years horizon BondCouponMaturityYTM A5%39.0% B6%208.6% C11%159.2% D8%58.0% What is the best choice?

25 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.

26 Duration and IR sensitivity

27 Duration

28 Duration

29 Duration Bonddurationprice impact of +1% YTM A3 yr B1 yr C10 yr D20 yr -3% -1% -10% -20%

30 Measuring Price Change

31 The Yield to Maturity The yield to maturity of a fixed coupon bond y is given by

32 Macaulay Duration Definition of duration, assuming t=0.

33 Macaulay Duration What is the duration of a zero coupon bond? A weighted sum of times to maturities of each coupon.

34 Meaning of Duration r $

35 Convexity r $

36 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

37 ALM Duration Does NOT work! Wrong units of measurement Division by a small number

38 ALM Duration A similar problem with measuring yield

39 Do not think of duration as a measure of time!

40 Key rate duration Principal component duration Partial duration

41 Factors affecting Bond yields and TS Base interest rate - benchmark interest rate Risk Premium - spread Expected liquidity Market forces - Demand and supply

42 Taxability of interest qualified municipal bonds are exempts from federal taxes. After tax yield = pretax yield (1- marginal tax rate)

43 Do not use yield curve to price bonds PeriodAB 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!

44 On-the-run Treasury issues Off-the-run Treasury issues Special securities Lending Repos and reverse repos

45 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 )

46 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.

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

48 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? Home Assignment