Managing Bond Portfolios

CHAPTER 11 11.1 INTEREST RATE RISK Managing Bond Portfolios Interest Rate Sensitivity Interest Rate Sensitivity (cont) Inverse relationship between price and yield An increase in a bond’ bond’s yield to maturity results in a smaller price decline than the gain associated with a decrease in yield LongLong-term bonds tend to be more price sensitive than shortshort-term bonds Sensitivity of bond prices to changes in yields increases at a decreasing rate as maturity increases Interest rate risk is inversely related to bond’ bond’s coupon rate Sensitivity of a bond’ bond’s price to a change in its yield is inversely related to the yield to maturity at which the bond currently is selling Figure 11.1 Change in Bond Price as a Function of YTM Duration A measure of the effective maturity of a bond The weighted average of the times until each payment is received, with the weights proportional to the present value of the payment Duration is shorter than maturity for all bonds except zero coupon bonds Duration is equal to maturity for zero coupon bonds 10-1 Figure 11.2 Cash Flows of 8-yr Bond with 9% annual coupon and 10% YTM Duration: Calculation t w t = [CFt (1 + y ) ] Price D = ∑ t ×wt T t =1 CF t = Cash Flow Sum t Figure 11.3 Duration as a Function of Maturity Duration Calculation 8% Time Bond years for period Payment PV of CF Weight C1 X (10%) C4 1 80 72.727 .0765 .0765 2 80 66.116 .0690 .1392 3 1080 811.420 .8539 2.5617 950.263 1.0000 2.7774 Duration/Price Relationship Price change is proportional to duration and not to maturity ∆P/P = -D x [∆ [∆y / (1+y)] D* = modified duration D* = D / (1+y) ∆P/P = - D* x ∆y 11.2 PASSIVE BOND MANAGEMENT 10-2 Immunization Figure 11.4 Growth of Invested Funds Passive management – Net worth immunization – Target date immunization Figure 11.5 Immunization Cash Flow Matching and Dedication Automatically immunizes a portfolio from interest rate movements – Cash flow from the bond and the obligation exactly offset each other Not widely pursued Sometimes not even possible Limitations of Duration 11.3 CONVEXITY Duration is only an approximation Duration asserts that the percentage price change is directly proportional to the change in the bond’ bond’s yield Underestimates the increase in bond prices when yield falls Overestimates the decline in price when the yield rises 10-3 Pricing Error from Convexity Correction for Convexity Modify the pricing equation: Price ∆P P Pricing Error from = − D × ∆ y + 1 × Convexity × ( ∆ y ) 2 2 Convexity is Equal to: Convexity 1 P × (1 + y) Duration Yield Figure 11.6 Bond Price Convexity  (t ∑  (1CF + y) N 2 t =1  t t 2  + t )  Where: CFt is the cash flow (interest and/or principal) at time t. Figure 11.7 Convexity of Two Bonds Swapping Strategies 11.4 ACTIVE BOND MANAGEMENT Substitution swap Intermarket swap Rate anticipation swap Pure yield pickup Tax swap 10-4 Horizon Analysis Contingent Immunization Analyst selects a particular investment period and predicts bond yields at the end of that period Allow the managers to actively manage until the bond portfolio falls to a threshold level Once the threshold value is hit the manager must then immunize the portfolio Active with a floor loss level Figure 11.8 Contingent Immunization 10-5