Chapter 11Slide 2Interest Rate SensitivityInterest Rate Sensitivity (cont)Summary of Interest Rate SensitivityChange in Bond Price as a Function of YTMDurationSlide 8Duration FormulaSlide 10Using Excel to Calculate DurationMore on DurationSlide 13Figure 11.2 Duration as a Function of MaturityDuration/Price RelationshipSlide 16Interest Rate RiskImmunizationTerminal Value of an Immunized Portfolio over a 5 year HorizonFigure 11.3 Growth of Invested FundsSlide 21Figure 11.4 ImmunizationCash Flow Matching and DedicationProblems with ImmunizationSlide 25The Need for ConvexityPricing Error Due to ConvexityConvexity: Definition and UsageBond Price ConvexityConvexity of Two BondsPrediction Improvement With ConvexitySlide 32Swapping StrategiesSlide 34Horizon AnalysisChapter 11Managing Bond Portfolios Copyright © 2010 by The McGraw-Hill Companies, Inc. All rights reserved.McGraw-Hill/Irwin12-211.1 Interest Rate Risk12-3Interest Rate Sensitivity1. Inverse relationship between bond price and interest rates (or yields)2. Long-term bonds are more price sensitive than short-term bonds3. Sensitivity of bond prices to changes in yields increases at a decreasing rate as maturity increases12-4Interest Rate Sensitivity (cont)4. A bond’s price sensitivity is inversely related to the bond’s coupon5. Sensitivity of a bond’s price to a change in its yield is inversely related to the yield to maturity at which the bond currently is selling6. An increase in a bond’s yield to maturity results in a smaller price decline than the gain associated with a decrease in yield12-5Summary of Interest Rate SensitivityThe concept: •Any security that gives an investor more money back sooner (as a % of your investment) will have lower price volatility when interest rates change.•Maturity is a major determinant of bond price sensitivity to interest rate changes, but•It is not the only factor; in particular the coupon rate and the current ytm are also major determinants.12-6Change in Bond Price as a Function of YTM12-7Duration1 2 3 4 5$100 $100 $100 $100 $1100Consider the following 5 year 10% coupon annual payment corporate bond:•Because the bond pays cash prior to maturity it has an “effective” maturity less than 5 years.•We can think of this bond as a portfolio of 5 zero coupon bonds with the given maturities.•The average maturity of the five zeros would be the coupon bond’s effective maturity.•We need a way to calculate the effective maturity.12-8Duration•Duration is the term for the effective maturity of a bond•Time value of money tells us we must calculate the present value of each of the five zero coupon bonds to construct an average.•We then need to take the present value of each zero and divide it by the price of the coupon bond. This tells us what percentage of our money we get back each year.•We can now construct the weighted average of the times until each payment is received.12-9Duration FormulaicePr)ytm1(CFWN1ttttN1tttWDurWt = Weight of time t, present value of the cash flow earned in time t as a percent of the amount investedCFt = Cash Flow in Time t, coupon in all periods except terminal period when it is the sum of the coupon and the principalytm = yield to maturity; Price = bond’s priceDur = Duration12-10Calculating the duration of a 9% coupon, 8% ytm, 4 year annual payment bond priced at $1033.12, $1,033.12100.00%3.5396 yrs $ 83.3377.1671.45$801.188.06%7.47% 6.92% 77.55%0.08060.14940.20763.1020Duration = 3.5396 yearsicePr)ytm1(CFWN1ttttN1tttWDur12-11Using Excel to Calculate DurationExcel can be used to calculate a bond’s duration.Usage notes:•The dates should be entered using the formulas given•If you don’t know the actual settlement date and maturity date, set the 6th term in the duration formulae to 0 as shown and pick a maturity date with the same month and day as the settlement date and the correct number of years after the settlement date.•The par is not needed12-12More on Duration1. Duration increases with maturity2. A higher coupon results in a lower duration3. Duration is shorter than maturity for all bonds except zero coupon bonds4. Duration is equal to maturity for zero coupon bonds5. All else equal, duration is shorter at higher interest rates12-13More on Duration5. The duration of a level payment perpetuity isytmy ;yy1Dperpetuity12-14Figure 11.2 Duration as a Function of Maturity12-15Duration/Price Relationship•Price change is proportional to duration and not to maturityP/P = -D x [y / (1+y)]D* = modified durationD* = D / (1+y)P/P = - D* x yD = Duration12-1611.2 Passive Bond Management12-17Interest Rate RiskInterest rate risk is the possibility that an investor does not earn the promised ytm because of interest rate changes.A bond investor faces two types of interest rate risk:1.Price risk: The risk that an investor cannot sell the bond for as much as anticipated. An increase in interest rates reduces the sale price.2.Reinvestment risk: The risk that the investor will not be able to reinvest the coupons at the promised yield rate. A decrease in interest rates reduces the future value of the reinvested coupons.The two types of risk are potentially offsetting.12-18Immunization•Immunization: An investment strategy designed to ensure the investor earns the promised ytm. •A form of passive management, two versions1. Target date immunization•Attempt to earn the promised yield on the bond over the investment horizon.•Accomplished by matching duration of the bond to the investment horizon12-19Terminal Value of an Immunized Portfolio over a 5 year Horizon12-20Figure 11.3 Growth of Invested Funds12-21Immunization2. Net worth immunization•The equity of an institution can be immunized by matching the duration of the assets to the duration of the liabilities.12-22Figure 11.4 Immunization12-23Cash Flow Matching and Dedication•Cash flow from the bond and the obligation exactly offset each other–Automatically immunizes a portfolio from interest rate movements•Not widely pursued, too limiting in terms of choice of bonds•May not be feasible due to lack of availability of investments needed12-24Problems with Immunization1. May be a suboptimal strategy2. Does not work as well for complex portfolios with option components, nor for large interest rate changes3. Requires rebalancing of the portfolio periodically, which then incurs transaction costs–Rebalancing
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