1. McGraw-Hill/Irwin Copyright © 2005 by The McGraw-Hill Companies, Inc. All rights reserved. Chapter 16 Managing Bond Portfolios

2. 16-2 Active strategy Trade on interest rate predictions Trade on market inefficiencies Passive strategy Control risk Balance risk and return Interest rate risk is important for both active and passive strategies Key concept - Duration Basic Strategies

3. 16-3 Inverse relationship between price and yield. An increase in a bond’s yield to maturity results in a smaller price decline than the gain associated with a decrease in yield. Long-term bonds tend to be more price sensitive than short-term bonds. Bond Pricing Relationships

4. 16-4 As maturity increases, price sensitivity increases at a decreasing rate. Price sensitivity is inversely related to a bond’s coupon rate. Price sensitivity is inversely related to the yield to maturity at which the bond is selling. Bond Pricing Relationships (cont’d)

5. 16-5

6. 16-6 A measure of the effective maturity of a bond. The weighted average of the time until each payment is received. The weights depend on contribution of each discounted cash flow to the set of cash discounted cash flows (i.e. the bond price). Duration is shorter than maturity for all bonds except zero coupon bonds. Duration is equal to maturity for zero coupon bonds. Duration

7. 16-7 Duration: Calculation Note: y is the 6 month market rate, i.e. BEY/2

8. 16-8 Duration of 9%, 8 yr, Mkt Rate=10% (Ann) tCFPVWgtt*Wgt 19081.820.0864 29074.380.07860.1571 39067.620.07140.2143 49061.470.06490.2597 59055.880.05900.2952 69050.800.05370.3220 79046.180.04880.3415 81090508.490.53714.2972 Sum946.651.005.97

9. 16-9

10. 16-10

11. 16-11 Price change is proportional to duration and not to maturity.  P/P = -D x [  (1+y) / (1+y)] D * = modified duration D * = D / (1+y) So   P/P = - D * x  y Duration/Price Relationship

12. 16-12 Duration and Slope of Price Yield Curve Footnote 4, page 526 notes that: dP/dy, is simply the slope of the price yield curve at any point, and for a given price, we simply multiply by a constant. Conclusion: The duration measure is measure of the slope of the Price Yield relationship

13. 16-13

14. 16-14 Duration/Price Change Example For the annual coupon bond used earlier, if the market rate falls to 9% (i.e. with market rate = coupon rate, the bond will now sell at par). Recall the duration is 5.97 years and price 946.65 Estimate Price Change: %chg = -5.97(-0.01/1.09) = 0.0548 Est. price = 946.65*1.0548 = 998.50

15. 16-15 Rules for Duration Rule 1 The duration of a zero-coupon bond equals its time to maturity. Rule 2 Holding maturity constant, a bond’s duration increases as the coupon rate declines. Rule 3 Holding the coupon rate constant, a bond’s duration generally increases with its time to maturity. Rule 4 Holding other factors constant, the duration of a coupon bond increases as the bond’s yield to maturity decreases.

16. 16-16 Rules for Duration (cont’d) Rules 5 The duration of a level perpetuity is equal to: Rule 6 The duration of a level annuity is equal to:

17. 16-17 Rules for Duration (cont’d) Rule 7 The duration for a corporate bond is equal to:

18. 16-18 Yield Price Duration Pricing Error from convexity Duration and Convexity

19. 16-19 Correction for Convexity Correction for Convexity:

20. 16-20 Negative Convexity for Callable Bonds

21. 16-21 Bond-Index Funds – more difficult than stock index funds because Lots of bonds in the index (> 5000) Many thinly traded New bonds getting added, and old refunded Lots of coupon income to reinvest A bond fund will often not try to exactly match the index, but carry a range of bonds that are represent the matrix of possibilities Passive Management

22. 16-22 Passive Management Immunization of interest rate risk: Net worth immunization (common for financial institutions) Duration of assets = Duration of liabilities Target date immunization (e.g. pensions) Holding Period matches Duration Holding period immunization (more general case of target date). As long as the duration of your bond portfolio matches your holding period, you are immunized.

23. 16-23 Target Date Immunization

24. 16-24

25. 16-25 Duration of a bond portfolio is the value weighted duration of the individual components D p = w 1 D 1 + w 2 D 2 + w 3 D 3 +... Rebalancing: To maintain the correct duration of a portfolio, the portfolio will usually need to be rebalanced periodically. (Recall, that as YTM change, duration changes, and as maturity changes, durations changes

26. 16-26 Cash flow matching and dedication Why not just buy a zero coupon bond to match the obligation? – This is called cash flow matching. If there are a lot of cash flows to match, you can buy a portfolio of zero coupon bonds. This is called a dedicated portfolio.

27. 16-27 Caveat For duration matching strategies to work best, the yield curve needs to be flat, as duration was computed using the current YTM of bond, while we know that each CF occurs at a different time, and in theory should be discounted at its rate on the yield curve. Even then it only works for parallel shifts in the yield curve Also – matching works best for nominal sums. If your obligation grows with inflation, these matching strategies will be less effective.

28. 16-28 Active Bond Management If you can forecast interest rates, you can make a lot of money as we know how bonds react to interest rate changes Also, if you can spot relative mispricing, you can profit

29. 16-29 Substitution swap- substitute a bond for a very similar one that seems undervalued Inter-market swap – if the spread between corporate and Treasuries is too large, may want to swap. Rate anticipation swap – need to forecast rates correctly for this to work Pure yield pickup – choose higher yielding bonds (longer term if the yield curve is normal) Tax swap – swap to exploit a tax strategy Active Management: Swapping Strategies

30. 16-30 Contingent Immunization A combination of active and passive management. The strategy involves active management with a floor rate of return. As long as the rate earned exceeds the floor, the portfolio is actively managed. Once the floor rate or trigger rate is reached, the portfolio is immunized.

31. 16-31

32. 16-32 Interest Rate Swaps Contract between two parties to exchange a series of cash flows One party pays a fixed rate and receives a variable rate One party pays a variable rate and receives a fixed rate Cheaper than refunding and issuing new bonds Payments based on notional principal

33. 16-33 Swap Example Figure 16-11 Swap Dealer Company B Company A LIBOR 7% 6.95%7.05% Company A, has a fixed rate coupon but wants to pay a variable rate. Company B has to pay LIBOR, but would rather just pay a fixed rate. After the swap each party gets what it wanted, and the Swap Dealer collects a spread.

34. 16-34 Financial Engineering Financial engineering usually refers to taking a set of cash flows and cutting it up into pieces that people want. Coupon stripping is an example Another example is using a fixed income stream and cutting it into a floating and reverse floating stream, based on what is happening to an index, such as LIBOR or Treasuries.