McGraw-Hill/Irwin Copyright © 2008 by The McGraw-Hill Companies, Inc. All rights reserved. Chapter 13 Duration and Reinvestment Reinvestment Concepts Concepts

13-2 Objectives Understand that duration is a better measure of the life of a bond than maturity Understand that duration is a better measure of the life of a bond than maturity Be able to use present value techniques to compute duration Be able to use present value techniques to compute duration Explain the effect that duration has on bond price sensitivity to interest rate changes Explain the effect that duration has on bond price sensitivity to interest rate changes

13-3 Objectives continued Describe the uses of duration in protecting the value of a portfolio Describe the uses of duration in protecting the value of a portfolio Relate zero-coupon bonds to the concept of duration Relate zero-coupon bonds to the concept of duration Explain how the reinvestment rate for inflows may materially affect the final value of an investment Explain how the reinvestment rate for inflows may materially affect the final value of an investment

13-4 Duration and Reinvestment Concepts Review of Basic Bond Valuation Concepts Review of Basic Bond Valuation Concepts Duration Duration Duration and Price Sensitivity Duration and Price Sensitivity Bringing Together the Influences on Duration Bringing Together the Influences on Duration Duration and Zero-Coupon Bonds Duration and Zero-Coupon Bonds The Uses of Duration The Uses of Duration Bond Reinvestment Assumptions and Terminal Wealth Analysis Bond Reinvestment Assumptions and Terminal Wealth Analysis Appendix 13A: Modified Duration and Convexity Appendix 13A: Modified Duration and Convexity

13-5 Review of Basic Bond Valuation Concepts where: V = Market value or price of the bond n = Number of periods t = Each period C t = Coupon or interest payment for each period, t P n = Par or maturity value i = Interest rate in the market

13-6 As interest rates in the market rise The price of the bond will decline Review of Basic Bond Valuation Concepts cont.

13-7 As interest rates in the market decline The price of the bond will increase Review of Basic Bond Valuation Concepts cont.

13-8 Change in Market Prices of Bonds for Shifts in Yields to Maturity (12 percent coupon rate)

13-9 Review of Basic Bond Valuation Concepts Bonds with long-term maturities are more sensitive to changes in interest rates than short-term bonds Bonds with long-term maturities are more sensitive to changes in interest rates than short-term bonds The higher the coupon payments, the shorter the weighted average life of payout The higher the coupon payments, the shorter the weighted average life of payout

13-10 Weighted Average Life of a Bond: The time period over which the bond’s Coupon payments, & Maturity payment are recovered

13-11 Review of Basic Bond Valuation Concepts Bond price sensitivity can be more appropriately related to weighted average life than to just the maturity date

13-12 Duration Simple Weighted Average Life

13-13 Duration: Weighted average life of a bond where the weights are based on the present value of individual cash flows relative to the present value of total cash flows Weighted average life of a bond where the weights are based on the present value of individual cash flows relative to the present value of total cash flows Market rate of interest (yield to maturity) used in present value calculations Market rate of interest (yield to maturity) used in present value calculations

13-14 E.g. Five-year bond pays $80 per year for the next five years plus $1,000 at the end of five years. (Assume annual coupon payments) Weight

13-15 Duration where: D = Duration CF = Yearly cash flow for each time period PV = Present value factor for each time period, t V = Total present value or market price of bond n = Number of periods to maturity Weight Year

13-16 Duration Using the symbols from Formula 12–1, duration can also be stated as: where: D = Duration n = Number of periods t = Each period C t = Coupon or interest payment for each period, t P n = Par or maturity value i = Interest rate in the market V = Total present value or market price of bond

13-17 Duration for 8% Coupon Rate Bonds with Maturities of 1, 5, and 10 Years Discounted at 12%

13-18 Duration & Price Sensitivity Most important use of duration Most important use of duration price sensitivity of a bond price sensitivity of a bond Duration is a function of: Duration is a function of: maturitymaturity coupon ratecoupon rate market interest ratemarket interest rate show

13-19 Duration & Price Sensitivity Bonds with shorter maturities may have Longer duration Longer duration More price sensitive to interest rate changes More price sensitive to interest rate changes (than bonds with longer maturities) & be

13-20 The longer the maturity or duration, the greater the impact of a 2% change in interest rates on price Note how much more closely the percentage change in price parallels the change in duration as compared with maturity

13-21 True characteristics about the life of a bond depends on: Final date Amount of the maturity payment Pattern of coupon payments

13-22 Duration and Price Sensitivity continued Percentage change in the Change in value of a bond Duration ×interest rates approximately equals The sign is reversed because interest rates and bond prices move in opposite directions The sign is reversed because interest rates and bond prices move in opposite directions

13-23 Duration & Price Sensitivity Duration and Market Rates Interest rates and duration are inversely related Interest rates and duration are inversely related The higher the market rate of interest, the lower the duration (& vice versa) The higher the market rate of interest, the lower the duration (& vice versa) Higher market rates of interest mean lower present valuesHigher market rates of interest mean lower present values

13-24 Duration of an 8% Coupon Rate Bond with a 16% Market Rate of Interest

13-25 Duration & Price Sensitivity Duration and Market Rates An equal change in market rates of interest have a bigger impact on duration when rates move down than when they move up An equal change in market rates of interest have a bigger impact on duration when rates move down than when they move up

13-26 Duration & Price Sensitivity Duration and Coupon Rates As the coupon rates rise, duration decreases As the coupon rates rise, duration decreases High coupon rate bonds produce higher annual cash flows before maturity and weigh duration toward the earlier or middle yearsHigh coupon rate bonds produce higher annual cash flows before maturity and weigh duration toward the earlier or middle years Low coupon rate bonds produce less annual cash flows before maturity and have less influence on durationLow coupon rate bonds produce less annual cash flows before maturity and have less influence on duration

13-27 Duration & Coupon Rates As the coupon rates rise Duration decreases Zero-coupon bonds have same maturity and duration

13-28 Relationship between Duration & Coupon Rates Duration Pick a market rate of interest in the first column and then read across the table to determine the duration at various coupon rates

13-29 Effects of Coupon Rates on Maturity DURATION Years to Maturity 45% Zero coupon bond N 4% coupon bond 8% coupon bond 12% coupon bond

13-30 Duration & Price Sensitivity Duration and Coupon Rates Investors desiring maximum price movements will look toward lower coupon rate bonds Investors desiring maximum price movements will look toward lower coupon rate bonds

13-31 Bringing Together the Influences on Duration Three factors that determine duration: Three factors that determine duration: Maturity of the bondMaturity of the bond Market rate of interestMarket rate of interest Coupon rateCoupon rate

13-32 Bringing Together the Influences on Duration Duration is positively correlated with maturity but moves in the opposite direction of market rates of interest and coupon rates Duration is positively correlated with maturity but moves in the opposite direction of market rates of interest and coupon rates

13-33 DurationMaturity DurationMaturity Positive correlation Positive correlation

13-34 Duration Market interest rates Duration Market interest rates Negative correlation Negative correlation

13-35 Duration Coupon rates DurationCoupon rates Negative correlation Negative correlation

13-36 Bringing Together the Influences on Duration From the question posed earlier in the chapter: The 8% coupon rate 20 year bond has a higher duration than the 12% coupon rate 25 year bond

13-37 Duration & Zero-Coupon Bonds Duration of a zero-coupon bond equals number of years to maturity Duration of a zero-coupon bond equals number of years to maturity For all bonds of equal risk & maturity, the zero-coupon bond has the greatest duration and therefore the greatest price sensitivity For all bonds of equal risk & maturity, the zero-coupon bond has the greatest duration and therefore the greatest price sensitivity

13-38 Column 4 indicates ratio of duration between zero-coupon and 8% coupon rate bonds Example: For a 10-year maturity period, a zero-coupon bond is almost 1.5 times as price sensitive as an 8% coupon rate bond

13-39 The Uses of Duration Duration is primarily used as a measure to judge bond price sensitivity to interest-rate changes Duration is primarily used as a measure to judge bond price sensitivity to interest-rate changes Duration includes information on several variables, it allows more accurate decisions for complex bonds strategies Duration includes information on several variables, it allows more accurate decisions for complex bonds strategies

13-40 The Uses of Duration Used as a measure to judge bond price sensitivity to interest rate changes Used as a measure to judge bond price sensitivity to interest rate changes Allows more accurate decisions for complex bond strategies Allows more accurate decisions for complex bond strategies

13-41 The Uses of Duration continued Immunization Bond strategy using timing of investment inflows to provide needed cash outlay at a known future date Bond strategy using timing of investment inflows to provide needed cash outlay at a known future date Protects portfolios against swings in interest rates Protects portfolios against swings in interest rates Problem with duration analysis: Problem with duration analysis: Assumes a parallel shift in yield curvesAssumes a parallel shift in yield curves

13-42 Bond Reinvestment Assumptions & Terminal Wealth Analysis Reinvestment Assumptions Since interest rates change daily and by large amounts over a year, what impact would a lower or higher reinvestment assumption have on the outcome of your retirement nest egg?

13-43 Bond Reinvestment Assumptions & Terminal Wealth Analysis Terminal Wealth Analysis Terminal wealth indicates the ending value of the investment at the end of each year, assuming the bond has a maturity date corresponding to that year Terminal wealth indicates the ending value of the investment at the end of each year, assuming the bond has a maturity date corresponding to that year

13-44 Terminal Wealth Table (12% coupon with 7% reinvestment rate on interest)

13-45 Summary of Highlighted Example Note that the longer the maturity period of the bond, the greater the effect the low 7% reinvestment rate has on the bond Note that the longer the maturity period of the bond, the greater the effect the low 7% reinvestment rate has on the bond

13-46 Terminal Wealth Analysis versus Realized Yield Approach

13-47 Terminal wealth Realized yield Analyzing the reinvestment assumption when bonds are held to maturity when bonds are held to maturity Assumes bonds are actively traded to take advantage of interest-rate swings Analysis Approach

13-48 Zero-Coupon Bonds & Terminal Wealth Benefit of zero-coupon bonds: Benefit of zero-coupon bonds: No coupon payments to be reinvested during life of the bond No coupon payments to be reinvested during life of the bond Originally quoted rates holds throughout if held to maturity Originally quoted rates holds throughout if held to maturity Lock in a compound rate of return (or reinvestment rate) for the life of the bond if held to maturity

13-49 Zero-Coupon Bonds &Terminal Wealth Disadvantage of zero-coupon bonds: If sold before maturity, there could be large swings in the sales price of the bond because of high duration characteristics

13-50 Appendix 13A: Modified Duration & Convexity To more accurately measure bond price sensitivity resulting from the impact of a change in interest rates, use To more accurately measure bond price sensitivity resulting from the impact of a change in interest rates, use modified duration modified duration which is Macaulay duration divided by which is Macaulay duration divided by (1 plus the yield to maturity) (1 plus the yield to maturity) Modified duration (D*) = Macaulay duration (D)/(1 + i) where i= yield to maturity

13-51 Appendix 13A: Modified Duration & Convexity Reason for calculating modified duration: Reason for calculating modified duration: More accurate than Macaulay duration in measuring change in the price of the bond for given change in interest rate More accurate than Macaulay duration in measuring change in the price of the bond for given change in interest rate

13-52 Appendix 13A: Modified Duration & Convexity

13-53 Appendix 13A: Modified Duration & Convexity As was true of Macaulay duration, the value using modified duration gets less accurate as the term of the bond is extended. The reason for the loss of accuracy is predicting the change in the bond’s price comes from the issue of convexity As was true of Macaulay duration, the value using modified duration gets less accurate as the term of the bond is extended. The reason for the loss of accuracy is predicting the change in the bond’s price comes from the issue of convexity

13-54 Appendix 13A: Modified Duration & Convexity Convexity The approximation formula used with the modified duration generates a linear relationship in bond price changes when in fact actual bond price changes are not linear The approximation formula used with the modified duration generates a linear relationship in bond price changes when in fact actual bond price changes are not linear

13-55 Appendix 13A: Modified Duration & Convexity Convexity While convexity makes duration a less than perfect measure for predicting bond price changes, it does not negate its value to the analyst in predicting bond price changes in individual bonds or bond portfolios While convexity makes duration a less than perfect measure for predicting bond price changes, it does not negate its value to the analyst in predicting bond price changes in individual bonds or bond portfolios

13-56Convexity Yield to Maturity (i) Bond Price (v) + V + V D - V - V D i V - i + i B

13-57 WebsiteComment Fee-based source of information on convertible bond issues Provides calculator for convertible bonds cbs.marketwatch.com Personal finance section contains general information on Provides information about convertible bonds

13-58 Summary Duration’s primary use is in explaining price volatility, but it also has applications in the insurance industry and other areas of investments where interest-rate risk can be reduced by matching duration with predictable cash outflows in a process called immunization

13-59 Summary Each year is weighted by the present value of the cash flow as a proportion of the present value of the bond and then summed Each year is weighted by the present value of the cash flow as a proportion of the present value of the bond and then summed Duration, a number captures three variable: Maturity Maturity Coupon rate Coupon rate Market rate of interest Market rate of interest to indicate the price sensitivities to indicate the price sensitivities

13-60 Summary Bond duration increases with the increase in number of years to maturity. Duration also increases as coupon rates decline to zero, and finally, duration declines as market interest rates increase Bond duration increases with the increase in number of years to maturity. Duration also increases as coupon rates decline to zero, and finally, duration declines as market interest rates increase The higher the duration, the more sensitive the bond price is to a change in interest rates The higher the duration, the more sensitive the bond price is to a change in interest rates

13-61 Summary The terminal wealth analysis assumes that the investment is held to maturity and that all proceeds over the life of the bond are reinvested at the reinvestment rate The terminal wealth analysis assumes that the investment is held to maturity and that all proceeds over the life of the bond are reinvested at the reinvestment rate In general, the longer the maturity, the more total annualized return approaches the reinvestment rate In general, the longer the maturity, the more total annualized return approaches the reinvestment rate

13-62 Summary Zero coupon bonds are the most price sensitive of bonds to a change in market interest rates, and comparisons are made between zero- coupon bonds and coupon bonds Zero coupon bonds are the most price sensitive of bonds to a change in market interest rates, and comparisons are made between zero- coupon bonds and coupon bonds

13-63 Summary When the reinvestment rate is significantly different from the coupon rate, the annualized return can differ greatly from the coupon rate in as little as five years When the reinvestment rate is significantly different from the coupon rate, the annualized return can differ greatly from the coupon rate in as little as five years Duration is the number of years, on a present-value basis, that it takes to recover an initial investment in a bond Duration is the number of years, on a present-value basis, that it takes to recover an initial investment in a bond