Bond Valuation

Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall. 1

CHAPTER 18

Valuation of Debt Contracts and Their Price Volatility Characteristics

Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall.

Learning Objectives• The cash flow characteristics of a bond• How the price of a bond is determined• Why the yield to maturity is used as a

measure of a bond’s return• The importance of the reinvestment rate

in realizing the yield to maturity• Why the price of a bond changes

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Learning Objectives (continued)

• That the price/yield curve of an option-free bond is convex

• That the two characteristics of a bond that affect its price volatility are its coupon and its maturity

• What duration is and how it is calculated

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Learning Objectives (continued)

• The limitations of duration as a measure of price volatility of a bond when interest rates change

• What the convexity of a bond is and how it is related to bond price volatility

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Features of Debt Contracts• Bullet maturity means that the entire

principal is due at the maturity date• When the principal is paid variously

throughout the life of the loan, then the last remaining payment at maturity is the balloon payment

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Features of Debt Contracts (continued)

• A special type of debt contract is a bond, and the amount paid at maturity is called par value, maturity value, or face value

• A debt contract’s coupon is the periodic interest payment made to owners during the life of the contract

• The coupon is coupon rate or the rate of interest

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Features of Debt Contracts (continued)

• When both the principal and interest are paid at maturity, the debt is call zero-coupon instruments

• Typically, but not universally, for bonds issued in the United States the coupon payments are made every 6 months

• The price of most debt contracts are quoted as percentages of par value

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Basic Valuation Principles• The cash flow from a bond consists of

each period coupon payments and the repayment of the principal

• The price of a debt instrument must equal the sum of the stream of discounted payments the debtor is required to make until maturity. The discounted value of the debt is

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Basic Valuation Principles (continued)

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Return from a Bond: Yield to Maturity Measure• yield to maturity• A measure that will permit a comparison of

the rate of return of instruments having different cash flows and different maturities is needed

• It is defined as that interest rate which makes the present value of the cash flows equal to the market value (price) of the instrument

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Return from a Bond: Yield to Maturity Measure (continued)• yield to maturity

• where P is the par value, C the coupon payment, M the maturity value, n the number of periods, and y is the yield to maturity

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Return from a Bond: Yield to Maturity Measure (continued)• The ratio of the par value to the maturity

value P/M is the par value relation, usually expressed as a percentage. If it is equal to one, the bond sells "at par": If this is larger than one, it sells at a "premium"; and if less than one, it sells at a "discount"

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Price of an Option-Free Bond• The yield to maturity calculation for a

bond that pays interest semiannually results in the calculation of a semiannual interest rate

• The resulting yield to maturity is said to be calculated on a bond-equivalent or coupon-equivalent yield basis

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Price of an Option-Free Bond (continued)

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Price of an Option-Free Bond (continued)

• The yield to maturity computed using this• convention—doubling the semiannual

yield—is called the bond equivalent yield

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Reasons for Changes in Bond Prices• The price of a bond can change over time

for any one of the following reasons– A change in the level of interest rates in the

economy– A change in the price of a bond selling at a

price other than par as it moves towards maturity without any change in the required yield

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Reasons for Changes in Bond Prices (continued)

– For a non-Treasury security, a change in the required yield due to a change in the yield spread between non-Treasury and Treasury securities

– A change in the perceived credit quality of the issuer

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What Determines the Premium-Par Yield• When a bond is selling at par, its yield to

maturity is equal to its coupon rate• The coupon rate of an n-period bond

selling at par is called the n-period par yield

• This occurs when the following equation holds true

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What Determines the Premium-Par Yield (continued)

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Reinvestment of Cash Flow and Yield• An investor will only realize the yield to

maturity that is calculated at the time of purchase if – all the coupon payments can be reinvested

at the promised yield to maturity– the bond is held to maturity

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Bond Price Volatility• Review of Price/Yield Relationship• If the price/yield relationship for any option-free

bond is graphed, its shape is slightly convex• a fundamental characteristic of an option-free

bond is that its price changes in the opposite direction from the change in yield. This behavior follows from the fact that the price of a bond is equal to the present value of its cash flow

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Bond Price Volatility (continued)• Price Volatility Properties• The absolute dollar price change and the

absolute percentage price change are not the same for an equal increase and decrease in the yield, except for very small changes

• In general, the dollar price increase and the percentage price increase when the yield declines are greater than the dollar price decrease and percentage price decrease when the yield increases

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Bond Price Volatility (continued)

• Characteristics of a Bond that Affect Price Volatility

• Two characteristics of a bond that are the primary determinants of its price volatility– Coupon– Term to maturity

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• For a given term to maturity and initial market yield, percentage price volatility is greater the lower the coupon rate

• For a given coupon rate and initial yield, the longer the term to maturity, the greater the price volatility, in terms of percentage price change

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• Measure of Price Volatility: Duration• The sensitivity of prices to rate changes

can be estimated using a pricing model for the security and small changes in market rates. The following formula is useful for measuring duration

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Bond Price Volatility (continued)• Measure of Price Volatility: Duration

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Bond Price Volatility (continued)• Measure of Price Volatility: Duration• The previous formula is an approximation

of the Macaulay duration• This measure of duration can be interpreted

as the number of years for the returns (as reinvested) to equal the initial payout. The higher the number of years, the greater is the duration or price sensitivity

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• Measure of Price Volatility: Duration• Using duration to approximate the

percentage price change The relationship between duration and the approximate price change is as follows: Approximate Percentage Price Change = –Duration (Δy) 100

• Interpretation of duration: Duration can be interpreted as the approximate percentage price change for a 100 basis point change in yields

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• Measure of Price Volatility: Duration– Dollar duration: If duration measures the

percentage price change, then dollar duration of a bond measures the dollar price change

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• Macaulay duration is a weighted average term-to-maturity of the components of a bond's cash flows, in which the time of receipt of each payment is weighted by the present value, of that component

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Bond Price Volatility (continued)• D in equation (18.13) is called Macaulay

duration

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• Modified Macaulay Duration• Modified duration = Macaulay duration/(1 + y)

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• Convexity• Duration is in fact a first approximation for a

small change in yield. The approximation can be improved by using a second approximation. This approximation is the bond’s convexity. The convexity measure of a security is the approximate change in the price that is not explained by the duration

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• A simple formula for estimating convexity is

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• Modified convexity and effective convexity

• If "Con" is calculated assuming that expected cash flows change with rates it is known as effective convexity or option-adjusted convexity

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• Modified convexity and effective convexity

• All option-free bonds exhibit positive convexity

• Bonds which contain options might exhibit negative convexity at certain points along their price/yield function

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Summary• The cash flow from a bond consists of

periodic coupon payments (semiannual payments in the United States) and the repayment of the principal

• A bond’s price changes over time for several reasons

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Summary (continued)

• Two characteristics of a bond that affect its price volatility and therefore its interest rate risk exposure are maturity and coupon rate

• A measure of price volatility that relates coupon and maturity is duration. Duration is interpreted as the approximate percentage price change of a bond for a 100 basis point change in interest rates

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Summary (continued)

• The best way to think about duration is as a measure of price sensitivity rather than some weighted time measure

• Convexity is another measure of price volatility to be used in conjunction with duration to improve the estimate for price volatility for large changes in interest rates.

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Bond Valuation

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