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Fixed Income

Fixed Income. Zvi Wiener. 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. Plan. Non-US Bonds Mortgage Loans

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Fixed Income

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  1. Fixed Income Zvi Wiener

  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. -PV 5 5 5 5 105 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

  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. price yield

  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 = 1.042-1=8.16% Effective annual yield = 1.024-1=8.24%

  22. Bond selling at Relationship Par Coupon rate=current yield=YTM Discount Coupon rate<current yield<YTM Premium Coupon 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 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?

  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 Bond duration price impact of +1% YTM A 3 yr B 1 yr C 10 yr D 20 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. $ r Meaning of Duration

  35. $ r Convexity

  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: KeR*(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 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!

  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+r2)=(1+r1)(1+f12)

  46. Forward Rates (1+r3)=(1+r1)(1+f13)= (1+r1)(1+f12)(1+f13) 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. 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?

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