Version 1.2 Copyright © 2000 by Harcourt, Inc. All rights reserved. Requests for permission to make copies of any part of the work should be mailed to: Permissions Department Harcourt, Inc Sea Harbor Drive Orlando, Florida Lecture Presentation Software to accompany Investment Analysis and Portfolio Management Sixth Edition by Frank K. Reilly & Keith C. Brown Chapter 16

Copyright © 2000 by Harcourt, Inc. All rights reserved Chapter 16 - The Analysis and Valuation of Bonds Questions to be answered: How do you determine the value of a bond based on the present value formula? What are the alternative bond yields that are important to investors?

Copyright © 2000 by Harcourt, Inc. All rights reserved Chapter 16 - The Analysis and Valuation of Bonds How do you compute the following major yields on bonds: current yield, yield to maturity, yield to call, and compound realized (horizon) yield? What are spot rates and forward rates and how do you calculate these rates from a yield to maturity curve? What is the spot rate yield curve and forward rate curve?

Copyright © 2000 by Harcourt, Inc. All rights reserved Chapter 16 - The Analysis and Valuation of Bonds How and why do you use the spot rate curve to determine the value of a bond? What are the alternative theories that attempt to explain the shape of the term structure of interest rates? What factors affect the level of bond yields at a point in time? What economic forces cause changes in bond yields over time?

Copyright © 2000 by Harcourt, Inc. All rights reserved Chapter 16 - The Analysis and Valuation of Bonds When yields change, what characteristics of a bond cause differential price changes for individual bonds? What is meant by the duration of a bond, how do you compute it, and what factors affect it? What is modified duration and what is the relationship between a bond’s modified duration and its volatility?

Copyright © 2000 by Harcourt, Inc. All rights reserved Chapter 16 - The Analysis and Valuation of Bonds What is effective duration and when is it useful? What is the convexity for a bond, how do you compute it, and what factors affect it? Under what conditions is it necessary to consider both modified duration and convexity when estimating a bond’s price volatility?

Copyright © 2000 by Harcourt, Inc. All rights reserved Chapter 16 - The Analysis and Valuation of Bonds What happens to the duration and convexity of bonds that have embedded call options?

Copyright © 2000 by Harcourt, Inc. All rights reserved The Fundamentals of Bond Valuation The present-value model Where: P m =the current market price of the bond n = the number of years to maturity C i = the annual coupon payment for bond i i = the prevailing yield to maturity for this bond issue P p =the par value of the bond

Copyright © 2000 by Harcourt, Inc. All rights reserved The Yield Model The expected yield on the bond may be computed from the market price Where: i = the discount rate that will discount the cash flows to equal the current market price of the bond

Copyright © 2000 by Harcourt, Inc. All rights reserved Computing Bond Yields Yield Measure Purpose Nominal Yield Measures the coupon rate Current yieldMeasures current income rate Promised yield to maturityMeasures expected rate of return for bond held to maturity Promised yield to callMeasures expected rate of return for bond held to first call date Realized (horizon) yield Measures expected rate of return for a bond likely to be sold prior to maturity. It considers specified reinvestment assumptions and an estimated sales price. It can also measure the actual rate of return on a bond during some past period of time.

Copyright © 2000 by Harcourt, Inc. All rights reserved Nominal Yield Measures the coupon rate that a bond investor receives as a percent of the bond’s par value

Copyright © 2000 by Harcourt, Inc. All rights reserved Current Yield Similar to dividend yield for stocks Important to income oriented investors CY = C i /P m where: CY = the current yield on a bond C i = the annual coupon payment of bond i P m = the current market price of the bond

Copyright © 2000 by Harcourt, Inc. All rights reserved Promised Yield to Maturity Widely used bond yield figure Assumes –Investor holds bond to maturity –All the bond’s cash flow is reinvested at the computed yield to maturity Solve for i that will equate the current price to all cash flows from the bond to maturity, similar to IRR

Copyright © 2000 by Harcourt, Inc. All rights reserved Computing the Promised Yield to Maturity Two methods Approximate promised yield –Easy, less accurate Present-value model –More involved, more accurate

Copyright © 2000 by Harcourt, Inc. All rights reserved Approximate Promised Yield Coupon + Annual Straight-Line Amortization of Capital Gain or Loss Average Investment =

Copyright © 2000 by Harcourt, Inc. All rights reserved Present-Value Model

Copyright © 2000 by Harcourt, Inc. All rights reserved Promised Yield to Call Approximation May be less than yield to maturity Reflects return to investor if bond is called and cannot be held to maturity Where: AYC = approximate yield to call (YTC) P c = call price of the bond P m = market price of the bond C t = annual coupon payment nc = the number of years to first call date

Copyright © 2000 by Harcourt, Inc. All rights reserved Promised Yield to Call Present-Value Method Where: P m = market price of the bond C i = annual coupon payment nc = number of years to first call P c = call price of the bond

Copyright © 2000 by Harcourt, Inc. All rights reserved Realized Yield Approximation Where: ARY = approximate realized yield to call (YTC) P f = estimated future selling price of the bond Ci = annual coupon payment hp = the number of years in holding period of the bond

Copyright © 2000 by Harcourt, Inc. All rights reserved Realized Yield Present-Value Method

Copyright © 2000 by Harcourt, Inc. All rights reserved Calculating Future Bond Prices Where: P f = estimated future price of the bond C i = annual coupon payment n = number of years to maturity hp = holding period of the bond in years i = expected semiannual rate at the end of the holding period

Copyright © 2000 by Harcourt, Inc. All rights reserved Yield Adjustments for Tax-Exempt Bonds Where: T = amount and type of tax exemption

Copyright © 2000 by Harcourt, Inc. All rights reserved What Determines Interest Rates Inverse relationship with bond prices Forecasting interest rates Fundamental determinants of interest rates i = RFR + I + RP where: –RFR = real risk-free rate of interest – I = expected rate of inflation – RP = risk premium

Copyright © 2000 by Harcourt, Inc. All rights reserved What Determines Interest Rates Effect of economic factors –real growth rate –tightness or ease of capital market –expected inflation –or supply and demand of loanable funds Impact of bond characteristics –credit quality –term to maturity –indenture provisions –foreign bond risk including exchange rate risk and country risk

Copyright © 2000 by Harcourt, Inc. All rights reserved What Determines Interest Rates Term structure of interest rates Expectations hypothesis Liquidity preference hypothesis Segmented market hypothesis Trading implications of the term structure

Copyright © 2000 by Harcourt, Inc. All rights reserved Yield Spreads Segments: government bonds, agency bonds, and corporate bonds Sectors: prime-grade municipal bonds versus good-grade municipal bonds, AA utilities versus BBB utilities Coupons or seasoning within a segment or sector Maturities within a given market segment or sector

Copyright © 2000 by Harcourt, Inc. All rights reserved Yield Spreads Magnitudes and direction of yield spreads can change over time

Copyright © 2000 by Harcourt, Inc. All rights reserved What Determines the Price Volatility for Bonds Bond price change is measured as the percentage change in the price of the bond Where: EPB = the ending price of the bond BPB = the beginning price of the bond

Copyright © 2000 by Harcourt, Inc. All rights reserved What Determines the Price Volatility for Bonds Four Factors 1. Par value 2. Coupon 3. Years to maturity 4. Prevailing market interest rate

Copyright © 2000 by Harcourt, Inc. All rights reserved What Determines the Price Volatility for Bonds Five observed behaviors 1. Bond prices move inversely to bond yields (interest rates) 2. For a given change in yields, longer maturity bonds post larger price changes, thus bond price volatility is directly related to maturity 3. Price volatility increases at a diminishing rate as term to maturity increases 4. Price movements resulting from equal absolute increases or decreases in yield are not symmetrical 5. Higher coupon issues show smaller percentage price fluctuation for a given change in yield, thus bond price volatility is inversely related to coupon

Copyright © 2000 by Harcourt, Inc. All rights reserved What Determines the Price Volatility for Bonds The maturity effect The coupon effect The yield level effect Some trading strategies

Copyright © 2000 by Harcourt, Inc. All rights reserved The Duration Measure Since price volatility of a bond varies inversely with its coupon and directly with its term to maturity, it is necessary to determine the best combination of these two variables to achieve your objective A composite measure considering both coupon and maturity would be beneficial

Copyright © 2000 by Harcourt, Inc. All rights reserved The Duration Measure Developed by Frederick R. Macaulay, 1938 Where: t = time period in which the coupon or principal payment occurs C t = interest or principal payment that occurs in period t i = yield to maturity on the bond

Copyright © 2000 by Harcourt, Inc. All rights reserved Characteristics of Duration Duration of a bond with coupons is always less than its term to maturity because duration gives weight to these interim payments –A zero-coupon bond’s duration equals its maturity An inverse relation between duration and coupon A positive relation between term to maturity and duration, but duration increases at a decreasing rate with maturity An inverse relation between YTM and duration Sinking funds and call provisions can have a dramatic effect on a bond’s duration

Copyright © 2000 by Harcourt, Inc. All rights reserved Duration and Bond Price Volatility An adjusted measure of duration can be used to approximate the price volatility of a bond Where: m = number of payments a year YTM = nominal YTM

Copyright © 2000 by Harcourt, Inc. All rights reserved Duration and Bond Price Volatility Bond price movements will vary proportionally with modified duration for small changes in yields An estimate of the percentage change in bond prices equals the change in yield time modified duration Where:  P = change in price for the bond P = beginning price for the bond D mod = the modified duration of the bond  i = yield change in basis points divided by 100

Copyright © 2000 by Harcourt, Inc. All rights reserved Trading Strategies Using Duration Longest-duration security provides the maximum price variation If you expect a decline in interest rates, increase the average duration of your bond portfolio to experience maximum price volatility If you expect an increase in interest rates, reduce the average duration to minimize your price decline Note that the duration of your portfolio is the market-value- weighted average of the duration of the individual bonds in the portfolio

Copyright © 2000 by Harcourt, Inc. All rights reserved Bond Duration in Years for Bonds Yielding 6 Percent Under Different Terms

Copyright © 2000 by Harcourt, Inc. All rights reserved Bond Convexity Equation is a linear approximation of bond price change for small changes in market yields

Copyright © 2000 by Harcourt, Inc. All rights reserved Bond Convexity Modified duration is a linear approximation of bond price change for small changes in market yields Price changes are not linear, but a curvilinear (convex) function

Copyright © 2000 by Harcourt, Inc. All rights reserved Price-Yield Relationship for Bonds The graph of prices relative to yields is not a straight line, but a curvilinear relationship This can be applied to a single bond, a portfolio of bonds, or any stream of future cash flows The convex price-yield relationship will differ among bonds or other cash flow streams depending on the coupon and maturity The convexity of the price-yield relationship declines slower as the yield increases Modified duration is the percentage change in price for a nominal change in yield

Copyright © 2000 by Harcourt, Inc. All rights reserved Modified Duration For small changes this will give a good estimate, but this is a linear estimate on the tangent line

Copyright © 2000 by Harcourt, Inc. All rights reserved Determinants of Convexity The convexity is the measure of the curvature and is the second derivative of price with resect to yield (d 2 P/di 2 ) Convexity is the percentage change in dP/di for a given change in yield

Copyright © 2000 by Harcourt, Inc. All rights reserved Determinants of Convexity Inverse relationship between coupon and convexity Direct relationship between maturity and convexity Inverse relationship between yield and convexity

Copyright © 2000 by Harcourt, Inc. All rights reserved Modified Duration-Convexity Effects Changes in a bond’s price resulting from a change in yield are due to: –Bond’s modified duration –Bond’s convexity Relative effect of these two factors depends on the characteristics of the bond (its convexity) and the size of the yield change Convexity is desirable

Copyright © 2000 by Harcourt, Inc. All rights reserved Duration and Convexity for Callable Bonds Issuer has option to call bond and pay off with proceeds from a new issue sold at a lower yield Embedded option Difference in duration to maturity and duration to first call Combination of a noncallable bond plus a call option that was sold to the issuer Any increase in value of the call option reduces the value of the callable bond

Copyright © 2000 by Harcourt, Inc. All rights reserved Option Adjusted Duration Based on the probability that the issuing firm will exercise its call option –Duration of the non-callable bond –Duration of the call option

Copyright © 2000 by Harcourt, Inc. All rights reserved Convexity of Callable Bonds Noncallable bond has positive convexity Callable bond has negative convexity

Copyright © 2000 by Harcourt, Inc. All rights reserved Limitations of Macaulay and Modified Duration Percentage change estimates using modified duration only are good for small-yield changes Difficult to determine the interest-rate sensitivity of a portfolio of bonds when there is a change in interest rates and the yield curve experiences a nonparallel shift Callable bonds duration depends on market conditions

Copyright © 2000 by Harcourt, Inc. All rights reserved Effective Duration Measure the interest rate sensitivity of an asset Use a pricing model to estimate the market prices surrounding a change in interest rates Effective DurationEffective Convexity P- = the estimated price after a downward shift in interest rates P+ = the estimated price after a upward shift in interest rates P = the current price S = the assumed shift in the term structure

Copyright © 2000 by Harcourt, Inc. All rights reserved Effective Duration Effective duration greater than maturity Negative effective duration

Copyright © 2000 by Harcourt, Inc. All rights reserved Empirical Duration Actual percent change for an asset in response to a change in yield during a specified time period

Copyright © 2000 by Harcourt, Inc. All rights reserved End of Chapter 16 –The Analysis and Valuation of Bonds

Copyright © 2000 by Harcourt, Inc. All rights reserved Future topics Chapter 19 Why do industry analysis? Competition and expected industry returns Estimating an industry earnings multiplier

Copyright © 2000 by Harcourt, Inc. All rights reserved