CHAPTER 16. Managing Bond Portfolios INVESTMENTS BODIE, KANE, MARCUS. Copyright 2011 by The McGraw-Hill Companies, Inc. All rights reserved.

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1 CHAPTER 16 Managing Bond Portfolios INVESTMENTS BODIE, KANE, MARCUS McGraw-Hill/Irwin Copyright 2011 by The McGraw-Hill Companies, Inc. All rights reserved.

2 INVESTMENTS BODIE, KANE, MARCUS 16-2 Bond Pricing Relationships Bond prices and yields are inversely related. An increase in a bond s yield to maturity results in a smaller price change than a decrease of equal magnitude. Long-term bonds tend to be more price sensitive than short-term bonds, but price sensitivity increases at a decreasing rate. Interest rate risk is higher for lower bond s coupon rates. Price sensitivity is inversely related to the bond s yield to maturity.

3 INVESTMENTS BODIE, KANE, MARCUS Coupon, Yield, and Price Check your intuition about Coupons and Yield. Which of the following bonds are par, discount or premium bonds? a) Coupon rate > current yield > YTM b) Coupon rate = current yield = YTM c) Coupon rate < current yield < YTM

4 Figure 16.1 Change in Bond Price as a Function of Change in Yield to Maturity INVESTMENTS BODIE, KANE, MARCUS 16-4

5 Table 16.1,2 Prices of 8% semiannual coupon bond, and a Zero Coupon Bond INVESTMENTS BODIE, KANE, MARCUS 16-5

6 INVESTMENTS BODIE, KANE, MARCUS Duration and effective life A measure of the effective maturity of a bond The weighted average of the times until each payment is received; weights are proportional to the present value of the payment Duration is obviously equal to maturity for zero coupon bonds (one cash flow only!) Duration is shorter than maturity for all bonds, except zero coupon bonds

7 INVESTMENTS BODIE, KANE, MARCUS 16-7 Duration: Calculation w t PV CF CF 1 y Price t t Price t t D T t 1 wt CF t = cash flow at time t t Q. What are the units of measure of D?

8 INVESTMENTS BODIE, KANE, MARCUS Key Duration Relationship Duration is important because it leads to the following key relationship between the change in the yield on the bond, and the change in its price (notice the sign): P P D * y Think of Δy as a change to interest rates

9 Duration/Price Relationship - Derivation Compute price sensitivity w.r.t. yield y: 1 P P y = 1 P = 1 P = 1 P t y t t C t 1 + y t+1 t = D 1 t C t 1 + y t 1 + y C t 1 + y t y = D INVESTMENTS BODIE, KANE, MARCUS 16-9

10 INVESTMENTS BODIE, KANE, MARCUS Duration/Price Relationship Price change is proportional to duration (not to maturity). Notice the sign. D* = modified duration = D/(1 + y) [note: Δ(1 + y) = Δy] Therefore: P P D (1 (1 y) y) P D * P y

11 INVESTMENTS BODIE, KANE, MARCUS Modified Duration When the yield y is expressed with compounding m times per year P P D y 1 y m The modified duration becomes: D 1 y m

12 INVESTMENTS BODIE, KANE, MARCUS Example 16.1 Duration Two bonds have duration of years: 1. A 2-year, 8% semiannual coupon bond with YTM=10% 2. zero coupon bond with maturity = years Duration of both bonds is x 2 = semiannual periods Remember: semiannual yield y= 10%/2 = 5% Modified D* = (1+0.05) = periods

13 INVESTMENTS BODIE, KANE, MARCUS Example 16.1 Duration Suppose the semiannual interest rate increases by 0.01%. Bond prices fall by: P D * y P ΔP/P = x 0.01% = % Bonds with equal D have the same interest rate sensitivity

14 INVESTMENTS BODIE, KANE, MARCUS Example 16.1 Duration Coupon Bond The coupon bond, initially sells at $ it falls to $ when its yield increases to 5.01% Percentage decline is % Zero coupon bond The zero-coupon bond initially sells for $1,000/(1.05) = = $ At higher yield, it sells for $1,000/(1.05) = = $ This price also falls by %

15 INVESTMENTS BODIE, KANE, MARCUS Duration of a Portfolio The duration for a portfolio is the weighted average duration of the instruments in the portfolio with weights proportional to PVs The key duration relationship for a portfolio describes the effect of small parallel shifts in the yield curve What exposures remain if the duration of the portfolio assets equals the duration of the portfolio liabilities?

16 INVESTMENTS BODIE, KANE, MARCUS Duration Check your intuition How does each of these changes affect duration? Having no coupon payments Decreasing the coupon rate Increasing the time to maturity Decreasing the yield-to-maturity

17 INVESTMENTS BODIE, KANE, MARCUS Pictorial look at duration Cash flows of a 7 year par 12% bond Shaded area of each box is PV of cash flow Duration Duration, measured as time, is the position of the center of mass of the shaded areas

18 INVESTMENTS BODIE, KANE, MARCUS Lower Coupon Duration is similar to the distance to the fulcrum Lower coupons shift the center of mass to the right. Higher coupons shift the center of mass to the left Duration

19 INVESTMENTS BODIE, KANE, MARCUS Higher Coupon Duration is similar to the distance to the fulcrum Lower coupons shift the center of mass to the right. Higher coupons shift the center of mass to the left Duration

20 INVESTMENTS BODIE, KANE, MARCUS Example of the coupon effect Consider the durations of a 5-year and 20- year bond with varying coupon rates (semiannual coupon payments): 5 year bond 20 year bond Zero coupon % coupon % coupon

21 INVESTMENTS BODIE, KANE, MARCUS Effect of maturity on duration Duration increases with increased maturity Example: add one period Duration Duration

22 INVESTMENTS BODIE, KANE, MARCUS Lower Yield Higher Yield discounts more heavily longer dated cash flows and shift the center of mass to the left Duration

23 INVESTMENTS BODIE, KANE, MARCUS Higher Yield Higher Yield discounts more heavily longer dated cash flows and shift the center of mass to the left Duration

24 INVESTMENTS BODIE, KANE, MARCUS Rules of Thumb for Bond Duration The duration of a zero-coupon bond equals its time to maturity Holding maturity constant, duration is higher when the coupon rate is lower Holding coupon rate constant, duration generally increases with time to maturity Holding other factors constant, duration is higher (longer) when YTM is lower The minute after a coupon is paid, duration jumps up, as that cash flows disappears

25 Figure 16.2 Bond Duration versus Bond Maturity INVESTMENTS BODIE, KANE, MARCUS 16-25

26 Table 16.3 Bond Durations (YTM = 8%; Semiannual Coupons) INVESTMENTS BODIE, KANE, MARCUS 16-26

27 INVESTMENTS BODIE, KANE, MARCUS Industry calc. of Rate Sensitivity: dv01 Traders in practice use dv01: dollar value of 1bp increase in rates Shock interest rates by +1bp and compute dollar impact dv01 Also compute bucketed dv01 by shocking interest rates by 1bp at various tenor buckets, and then compute dollar impact

28 INVESTMENTS BODIE, KANE, MARCUS Convexity The relationship between bond prices and yields is not linear. Duration rule is a good approximation for only small changes in bond yields. Bonds with greater convexity have more curvature in the price-yield relationship.

29 Figure 16.3 Bond Price Convexity: 30-Year Maturity, 8% Coupon; Initial YTM = 8% INVESTMENTS BODIE, KANE, MARCUS 16-29

30 INVESTMENTS BODIE, KANE, MARCUS Convexity 1 Convexity P (1 Correction for Convexity: n CF t ( t 2 t) y) 2 (1 y) t t 1 P P D y Convexity y

31 INVESTMENTS BODIE, KANE, MARCUS Figure 16.4 Convexity of Two Bonds

32 INVESTMENTS BODIE, KANE, MARCUS Why do Investors Like Convexity? Bonds with greater curvature gain more in price when yields fall than they lose when yields rise. The more volatile interest rates, the more attractive this asymmetry. Bonds with greater convexity tend to have higher prices and/or lower yields, all else equal.

33 INVESTMENTS BODIE, KANE, MARCUS Callable Bonds As rates fall, there is a ceiling on the bond s market price, which cannot rise above the call price. Negative convexity Use effective duration: Effective Duration = P/ P r

34 fig 16.5 Price Yield Curve for a Callable Bond INVESTMENTS BODIE, KANE, MARCUS 16-34

35 INVESTMENTS BODIE, KANE, MARCUS Mortgage-Backed Securities The number of outstanding callable corporate bonds has declined, but the MBS market has grown rapidly MBS are based on a portfolio of callable amortizing loans Homeowners have the right to repay their loans at any time MBS have negative convexity

36 INVESTMENTS BODIE, KANE, MARCUS Mortgage-Backed Securities Often sell for more than their principal balance Homeowners do not refinance as soon as rates drop, so implicit call price is not quite a firm ceiling on MBS value Tranches the underlying mortgage pool is divided into a set of derivative securities

37 Figure 16.6 Price-Yield Curve for a Mortgage- Backed Security INVESTMENTS BODIE, KANE, MARCUS 16-37

38 Figure 16.7 Cash Flows to Whole Mortgage Pool; Cash Flows to Three Tranches INVESTMENTS BODIE, KANE, MARCUS 16-38

39 INVESTMENTS BODIE, KANE, MARCUS Passive Management Two passive bond portfolio strategies: Indexing Immunization Both strategies see market prices as being correct, but the strategies have very different risks.

40 INVESTMENTS BODIE, KANE, MARCUS Bond Index Funds Bond indices contain thousands of issues, many of which are traded infrequently Bond indices turn over more than stock indices as the bonds mature Therefore, bond index funds hold only a representative sample of the bonds in the actual index

41 INVESTMENTS BODIE, KANE, MARCUS Figure 16.8 Stratification of Bonds into Cells

42 INVESTMENTS BODIE, KANE, MARCUS Immunization Immunization is a way to mitigate interest rate risk Widely used by pension funds, insurance companies, and banks Requires deep understanding of duration and convexity of your portfolio

43 INVESTMENTS BODIE, KANE, MARCUS Immunization Immunize a portfolio by matching the interest rate exposure of assets and liabilities Match the duration of the assets and liabilities Price risk and reinvestment rate risk cancel out for small interest rate movements Result: Value of assets will track the value of liabilities whether rates rise or fall (for small movements, need to rebalance)

44 INVESTMENTS BODIE, KANE, MARCUS Table 16.4 Terminal value of bond after 5 yrs

45 INVESTMENTS BODIE, KANE, MARCUS Table 16.5 Market Value Balance Sheet

46 INVESTMENTS BODIE, KANE, MARCUS Figure Immunization

47 INVESTMENTS BODIE, KANE, MARCUS Cash Flow Matching and Dedication Cash flow matching = automatic immunization Cash flow matching is a dedication strategy Not widely used because of constraints associated with bond choices

48 INVESTMENTS BODIE, KANE, MARCUS Active Management: Swapping Strategies Substitution swap Intermarket swap Rate anticipation swap Pure yield pickup Tax swap

49 INVESTMENTS BODIE, KANE, MARCUS Horizon Analysis Select a particular holding period and predict the yield curve at end of period Given a bond s time to maturity at the end of the holding period, its yield can be read from the predicted yield curve and the end-of-period price can be calculated