1. 1 FIN 2802, Spring 08 - Tang Chapter 16: Managing Bond Portfolios 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

2. 2 FIN 2802, Spring 08 - Tang Chapter 16: Managing Bond Portfolios Wall Street Interview Question 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?

3. 3 FIN 2802, Spring 08 - Tang Chapter 16: Managing Bond Portfolios 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.

4. 4 FIN 2802, Spring 08 - Tang Chapter 16: Managing Bond Portfolios Interest Rate Risk Interest rate sensitivity: Time to maturity Coupon rate

5. 5 FIN 2802, Spring 08 - Tang Chapter 16: Managing Bond Portfolios Change in Bond Price as a Function of Change in Yield to Maturity

6. 6 FIN 2802, Spring 08 - Tang Chapter 16: Managing Bond Portfolios Duration Measures the effective maturity by 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)

7. 7 FIN 2802, Spring 08 - Tang Chapter 16: Managing Bond Portfolios Cash Flows of 8-yr Bond with 9% annual coupon and 10% YTM

8. 8 FIN 2802, Spring 08 - Tang Chapter 16: Managing Bond Portfolios Calculating Duration Example: What is the duration of a 6% semiannual coupon bond with par of $1,000 and maturity in two years if market interest rates are currently 5% (semi-annual)? (1) (2) (3) (4) (5) Time to Payment PaymentColumn (1) Payment PaymentdiscountedWeight Times (years)Amount at 5% (3)/SumColumn (4) 0.5$ 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

9. 9 FIN 2802, Spring 08 - Tang Chapter 16: Managing Bond Portfolios Calculating Duration 5% semiannual

10. 10 FIN 2802, Spring 08 - Tang Chapter 16: Managing Bond Portfolios 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) PaymentPaymentdiscountedPayment Times (years)Amount at 5% WeightColumn (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

11. 11 FIN 2802, Spring 08 - Tang Chapter 16: Managing Bond Portfolios Spreadsheet 16.1 Calculating the Duration of Two Bonds

12. 12 FIN 2802, Spring 08 - Tang Chapter 16: Managing Bond Portfolios Example 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.

13. 13 FIN 2802, Spring 08 - Tang Chapter 16: Managing Bond Portfolios 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.

14. 14 FIN 2802, Spring 08 - Tang Chapter 16: Managing Bond Portfolios Duration and Interest Rate Sensitivity Sensitivity of prices to interest rate changes: where y is the yield to maturity

15. 15 FIN 2802, Spring 08 - Tang Chapter 16: Managing Bond Portfolios 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?

16. 16 FIN 2802, Spring 08 - Tang Chapter 16: Managing Bond Portfolios Example 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)?

17. 17 FIN 2802, Spring 08 - Tang Chapter 16: Managing Bond Portfolios 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)

18. 18 FIN 2802, Spring 08 - Tang Chapter 16: Managing Bond Portfolios 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.

19. 19 FIN 2802, Spring 08 - Tang Chapter 16: Managing Bond Portfolios Figure 16.3 Duration as a Function of Maturity

20. 20 FIN 2802, Spring 08 - Tang Chapter 16: Managing Bond Portfolios Table 16.3 Bond Duration (Initial Bond Yield 8% APR)

21. 21 FIN 2802, Spring 08 - Tang Chapter 16: Managing Bond Portfolios Price Approximation Using Modified Duration

22. 22 FIN 2802, Spring 08 - Tang Chapter 16: Managing Bond Portfolios Figure 16.4 Bond Price Convexity (30-Year Maturity, 8% Coupon; Initial Yield to Maturity = 8%)

23. 23 FIN 2802, Spring 08 - Tang Chapter 16: Managing Bond Portfolios 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:

24. 24 FIN 2802, Spring 08 - Tang Chapter 16: Managing Bond Portfolios An Example 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%

25. 25 FIN 2802, Spring 08 - Tang Chapter 16: Managing Bond Portfolios Passive Bond Management 1.Net Worth Immunization (Present) (e.g. Banks: Asset/Liability Management) 2. Target Date Immunization (Future) (e.g. Pension Funds: meet future obligations) Takes prices as given and tries to control the risk of the fixed-income portfolio. Measures:

26. 26 FIN 2802, Spring 08 - Tang Chapter 16: Managing Bond Portfolios Main Idea Behind Immunization 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.

27. 27 FIN 2802, Spring 08 - Tang Chapter 16: Managing Bond Portfolios Target Date Immunization 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% ?

28. 28 FIN 2802, Spring 08 - Tang Chapter 16: Managing Bond Portfolios Table 16.4 Terminal value of a Bond Portfolio After 5 Years (All Proceeds Reinvested)

29. 29 FIN 2802, Spring 08 - Tang Chapter 16: Managing Bond Portfolios Target Date Immunization Reinvestment Value of Coupon Bond Obligation D*=5yrs Value of Coupon Bond r = 8% Value of Coupon Bond r = 9% Time $10,000

30. 30 FIN 2802, Spring 08 - Tang Chapter 16: Managing Bond Portfolios Target Date Immunization 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

31. 31 FIN 2802, Spring 08 - Tang Chapter 16: Managing Bond Portfolios Example 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?

32. 32 FIN 2802, Spring 08 - Tang Chapter 16: Managing Bond Portfolios Net Worth Immunization Given a liability currently worth L and with duration D L Match it with an asset currently worth L and with duration D L. This guarantees that, for small changes in the interest rate the net worth will always be approximately zero.

33. 33 FIN 2802, Spring 08 - Tang Chapter 16: Managing Bond Portfolios Net Worth Immunization Current Value of Asset and Liabilities Interest rate Current of Coupon Bond (Asset) (YTM=8%) Present Value of CIG (Liability) (YTM=8%) 8%=YTM

34. 34 FIN 2802, Spring 08 - Tang Chapter 16: Managing Bond Portfolios Figure 16.12 Contingent Immunization

35. 35 FIN 2802, Spring 08 - Tang Chapter 16: Managing Bond Portfolios Active Bond Management Sources of potential profits: Interest rate forecasts Identification of mispriced bonds

36. 36 FIN 2802, Spring 08 - Tang Chapter 16: Managing Bond Portfolios Maturity Yield to Maturity % 3 mon 6 mon 9 mon 1.5 1.25.75 Yield Curve Ride

37. 37 FIN 2802, Spring 08 - Tang Chapter 16: Managing Bond Portfolios Current “Hot” Strategies 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

38. 38 FIN 2802, Spring 08 - Tang Chapter 16: Managing Bond Portfolios 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