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Fina2802: Investments and Portfolio Analysis Spring, 2008 Dragon Tang

Lecture 12 Managing Bond Portfolios February 28/29, 2008 Readings: Chapter 16 Practice Problem Sets: 1-13. Fina2802: Investments and Portfolio Analysis Spring, 2008 Dragon Tang.

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Fina2802: Investments and Portfolio Analysis Spring, 2008 Dragon Tang

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  1. Lecture 12 Managing Bond Portfolios February 28/29, 2008 Readings: Chapter 16 Practice Problem Sets: 1-13 Fina2802: Investments and Portfolio AnalysisSpring, 2008Dragon Tang Chapter 16: Managing Bond Portfolios

  2. You strongly believe that the yield curve is going to steepen very soon. It may be a fall in short-term rates, a rise in long-term rates, or some combination of these. What strategy should you pursue in the bond market to position yourself to profit from your beliefs? Wall Street Interview Question Chapter 16: Managing Bond Portfolios

  3. Managing Bond Portfolios • Objectives: • Analyze the features of a bond that affect the sensitivity of its price to interest rates. • Compute the duration of bonds. • Formulate fixed-income immunization strategies for various investment horizons. • Analyze the choices to be made in an actively managed fixed-income portfolio. Chapter 16: Managing Bond Portfolios

  4. Interest Rate Risk Interest rate sensitivity: • Time to maturity • Coupon rate Chapter 16: Managing Bond Portfolios

  5. Change in Bond Price as a Function of Change in Yield to Maturity Chapter 16: Managing Bond Portfolios

  6. Duration Measures the effective maturityby weighting the payments by their proportion of the bond value. where t =1, 2, 3, ... T are the times to maturity of payments y is the bond's yield to maturity (current market rate) Chapter 16: Managing Bond Portfolios

  7. Cash Flows of 8-yr Bond with 9% annual coupon and 10% YTM Chapter 16: Managing Bond Portfolios

  8. Calculating Duration • Example: What is the duration of a 6% semiannual coupon bond with par of $1,000 and maturity in two years if market interest rates are currently 5% (semi-annual)? • (1) (2) (3) (4) (5) • Time to Payment Payment Column (1) • Payment Payment discounted Weight Times • (years) Amount at 5% (3)/Sum Column (4) • 0.5 $ 30 $ 28.57 .03075 .01537 • 1.0 $ 30 $ 27.21 .02929 .02929 1.5 $ 30 $ 25.92 .02790 .04183 • 2.0 $ 1,030 $ 847.38 .91206 1.82412 • $ 929.08 1.0000 1.91061 Chapter 16: Managing Bond Portfolios

  9. Calculating Duration 5% semiannual Chapter 16: Managing Bond Portfolios

  10. Calculating Duration • Example: What is the duration of a zero-coupon bond which matures in two years if market interest rates are currently 5% (semi-annual)? • (1) (2) (3) (4) (5) • Time to Payment Column (1) • Payment Payment discounted Payment Times • (years) Amount at 5% Weight Column (4) • 0.5 $ 0 $ 0.00 .0 .0 • 1.0 $ 0 $ 0.00 .0 .0 • 1.5 $ 0 $ 0.00 .0 .0 • 2.0 $ 1000 $ 822.70 1.0 2.0 • $ 822.70 1.0 2.0 Chapter 16: Managing Bond Portfolios

  11. Spreadsheet 16.1 Calculating the Duration of Two Bonds Chapter 16: Managing Bond Portfolios

  12. A pension plan is obligated to make disbursements of $1 million, $2 million, and $1 million at the end of each of the next three years, respectively. Find the duration of the plan’s obligations if the interest rate is 10% annually. Example Chapter 16: Managing Bond Portfolios

  13. Duration Duration measure does three things: • It measures the effective average maturity of a bond. • It measures interest rate sensitivity correctly. • It provides the necessary information for immunization. Chapter 16: Managing Bond Portfolios

  14. Duration and Interest Rate Sensitivity Sensitivity of prices to interest rate changes: where y is the yield to maturity Chapter 16: Managing Bond Portfolios

  15. Duration Example: The duration for a bond, currently priced at $929.08, with a yield-to-maturity (YTM) of 10% is 1.91061 years. If interest rates rise by 0.5 percentage points (50 basis points), what will be the dollar change in the price of the bond? Chapter 16: Managing Bond Portfolios

  16. You own a fixed-income asset with a duration of five years. If the level of interest rates, which is currently 8%, goes down by 10 basis points, how much do you expect the price of the asset to go up (in percentage terms)? Example Chapter 16: Managing Bond Portfolios

  17. Bond Price Sensitivity Determinants of a bond’s price sensitivity to interest rate changes: • the time to maturity • (Duration not always increasing in time to Maturity) • the coupon rate • (Duration always decrease with high Coupon) • the yield to maturity • (Duration always decrease if YTM increase) Chapter 16: Managing Bond Portfolios

  18. Duration Rules & Results • The duration of a zero-coupon bond is equal to its time to maturity. • Other things equal, a lower coupon rate results in a higher duration. • Other things equal, a longer time to maturity increases duration (not always but usually) • Other things equal, a lower yield to maturity increases duration. • The duration of a perpetuity is equal to (1+y)/y. Chapter 16: Managing Bond Portfolios

  19. Figure 16.3 Duration as a Function of Maturity Chapter 16: Managing Bond Portfolios

  20. Table 16.3 Bond Duration (Initial Bond Yield 8% APR) Chapter 16: Managing Bond Portfolios

  21. Price Approximation Using Modified Duration Chapter 16: Managing Bond Portfolios

  22. Figure 16.4 Bond Price Convexity (30-Year Maturity, 8% Coupon; Initial Yield to Maturity = 8%) Chapter 16: Managing Bond Portfolios

  23. Correction for Convexity • All else equal, a higher coupon corresponds to a smaller convexity • All else equal, a longer maturity entails a larger convexity • All else equal, convexity is larger at a lower yield • Correction for Convexity: Chapter 16: Managing Bond Portfolios

  24. 18-year 12% coupon bond @ 9% YTM, priced at 126 ½. Modified duration = 8.38, convexity = 107.70 1% decline in yield (price increase 8.92%) Percentage increase in price due to duration: 8.38% Percentage increase in price due to convexity: 0.54% 3% increase in yield (price decline 20.56%) Percentage decline in price due to duration: 25.41% Percentage increase in price due to convexity: 4.85% An Example Chapter 16: Managing Bond Portfolios

  25. Passive Bond Management Takes prices as given and tries to control the risk of the fixed-income portfolio. Measures: • Net Worth Immunization(Present) • (e.g. Banks: Asset/Liability Management) • 2. Target Date Immunization (Future) • (e.g. Pension Funds: meet future obligations) Chapter 16: Managing Bond Portfolios

  26. Net Worth Immunization: Match duration of asset and liabilities by adjusting their maturity structure (Gap Management) Target Date Immunization: Set the duration of a portfolio equal to the target date. This guarantees that at this date reinvestment risk and price risk exactly cancel out. Main Idea Behind Immunization Chapter 16: Managing Bond Portfolios

  27. Example. An insurance company issue a 5-years Guaranteed investment contract (GIC) at 8%, nominal value $10,000. The insurance company decides to meet this obligation by investing $10,000 in 8% annual coupon bonds with maturity in 6yrs. Can the firm meets its obligation at time 5? What if interest rate drops to 7% ? What if interest rate increases to 9% ? Target Date Immunization Chapter 16: Managing Bond Portfolios

  28. Table 16.4 Terminal value of a Bond Portfolio After 5 Years (All Proceeds Reinvested) Chapter 16: Managing Bond Portfolios

  29. Target Date Immunization Reinvestment Value of Coupon Bond Obligation Value of Coupon Bond r = 8% $10,000 Value of Coupon Bond r = 9% Time D*=5yrs Chapter 16: Managing Bond Portfolios

  30. Given a future obligation X to be met in D* years Match it with a portfolio with Duration D* and worth X at time D* This guarantees that the value of the portfolio at time D* will be always be approximately X for any relatively small change in the interest rate Target Date Immunization Chapter 16: Managing Bond Portfolios

  31. You are managing a portfolio of $1 million. Your target duration is 10 years, and you can choose from two bonds: a zero-coupon bond with maturity 5 years, and a perpetuity, each currently yielding 5%. a. How much of each bond will you hold in your portfolio? b. How will these fractions change next year if target duration is now nine years? Example Chapter 16: Managing Bond Portfolios

  32. Given a liability currently worth L and with duration DL Match it with an asset currently worth L and with duration DL. This guarantees that, for small changes in the interest rate the net worth will always be approximately zero. Net Worth Immunization Chapter 16: Managing Bond Portfolios

  33. Net Worth Immunization Current Value of Asset and Liabilities Current of Coupon Bond (Asset) (YTM=8%) Present Value of CIG (Liability) (YTM=8%) 8%=YTM Interest rate Chapter 16: Managing Bond Portfolios

  34. Figure 16.12 Contingent Immunization Chapter 16: Managing Bond Portfolios

  35. Active Bond Management Sources of potential profits: • Interest rate forecasts • Identification of mispriced bonds Chapter 16: Managing Bond Portfolios

  36. Yield Curve Ride Yield to Maturity % 1.5 1.25 .75 Maturity 3 mon 6 mon 9 mon Chapter 16: Managing Bond Portfolios

  37. Convertible arbitrage Sell stock, buy convertible bond of the same company Ken Griffin, founder of Citadel, made a fortune as a sophomore Capital structure arbitrage Trade stock and bond in opposite direction Hurt by the GM/Ford event Current “Hot” Strategies Chapter 16: Managing Bond Portfolios

  38. Summary • Interest rate risk and default risk • Duration as a measure of the average life of a bond • Sensitivity of a bond's price to changes in yield • Passive Bond Management • Immunization (Net Worth and Target date) makes the individual or firm immune from interest rate movements • Portfolio must be rebalanced periodically • Active Bond Management • Adjusting portfolio based on interest rate forecasts • Next Class: Equity Valuation Chapter 16: Managing Bond Portfolios

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