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Chapter 12 Bond Prices and the Importance of Duration. We cannot gamble with anything so sacred as money. - William McKinley. Outline. Introduction Review of bond principles Bond pricing and returns Bond risk. Introduction.

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Chapter 12 Bond Prices and the Importance of Duration

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Chapter 12 bond prices and the importance of duration

Chapter 12Bond Prices and the Importance of Duration

Chapter 12 bond prices and the importance of duration

We cannot gamble with anything

so sacred as money.

- William McKinley



  • Introduction

  • Review of bond principles

  • Bond pricing and returns

  • Bond risk



  • The investment characteristics of bonds range completely across the risk/return spectrum

  • As part of a portfolio, bonds provide both stability and income

    • Capital appreciation is not usually a motive for acquiring bonds

Review of bond principles

Review of Bond Principles

  • Identification of bonds

  • Classification of bonds

  • Terms of repayment

  • Bond cash flows

  • Convertible bonds

  • Registration

Identification of bonds

Identification of Bonds

  • A bond is identified by:

    • The issuer

    • The coupon

    • The maturity

  • For example, five IBM “eights of 10” means $5,000 par IBM bonds with an 8% coupon rate and maturing in 2010

Classification of bonds

Classification of Bonds

  • Introduction

  • Issuer

  • Security

  • Term



  • The bond indenture describes the details of a bond issue:

    • Description of the loan

    • Terms of repayment

    • Collateral

    • Protective covenants

    • Default provisions



  • Bonds can be classified by the nature of the organizations initially selling them:

    • Corporation

    • Federal, state, and local governments

    • Government agencies

    • Foreign corporations or governments



  • Definition

  • Unsecured debt

  • Secured debt



  • The security of a bond refers to what backs the bond (what collateral reduces the risk of the loan)

Unsecured debt

Unsecured Debt

  • Governments:

    • Full faith and credit issues (general obligation issues) is government debt without specific assets pledged against it

      • E.g., U.S. Treasury bills, notes, and bonds

Unsecured debt cont d

Unsecured Debt (cont’d)

  • Corporations:

    • Debentures are signature loans backed by the good name of the company

    • Subordinated debentures are paid off after original debentures

Secured debt

Secured Debt

  • Municipalities issue:

    • Revenue bonds

      • Interest and principal are repaid from revenue generated by the project financed by the bond

    • Assessment bonds

      • Benefit a specific group of people, who pay an assessment to help pay principal and interest

Secured debt cont d

Secured Debt (cont’d)

  • Corporations issue:

    • Mortgages

      • Well-known securities that use land and buildings as collateral

    • Collateral trust bonds

      • Backed by other securities

    • Equipment trust certificates

      • Backed by physical assets

Chapter 12 bond prices and the importance of duration


  • The term is the original life of the debt security

    • Short-term securities have a term of one year or less

    • Intermediate-term securities have terms ranging from one year to ten years

    • Long-term securities have terms longer than ten years

Terms of repayment

Terms of Repayment

  • Interest only

  • Sinking fund

  • Balloon

  • Income bonds

Interest only

Interest Only

  • Periodic payments are entirely interest

  • The principal amount of the loan is repaid at maturity

Sinking fund

Sinking Fund

  • A sinking fund requires the establishment of a cash reserve for the ultimate repayment of the bond principal

    • The borrower can:

      • Set aside a potion of the principal amount of the debt each year

      • Call a certain number of bonds each year



  • Balloon loans partially amortize the debt with each payment but repay the bulk of the principal at the end of the life of the debt

  • Most balloon loans are not marketable

Income bonds

Income Bonds

  • Income bonds pay interest only if the firm earns it

  • For example, an income bond may be issued to finance an income-producing project

Bond cash flows

Bond Cash Flows

  • Annuities

  • Zero coupon bonds

  • Variable rate bonds

  • Consols



  • An annuity promises a fixed amount on a regular periodic schedule for a finite length of time

  • Most bonds are annuities plus an ultimate repayment of principal

Zero coupon bonds

Zero Coupon Bonds

  • A zero coupon bond has a specific maturity date when it returns the bond principal

  • A zero coupon bond pays no periodic income

    • The only cash inflow is the par value at maturity

Variable rate bonds

Variable Rate Bonds

  • Variable rate bonds allow the rate to fluctuate in accordance with a market index

  • For example, U.S. Series EE savings bonds



  • Consols pay a level rate of interest perpetually:

    • The bond never matures

    • The income stream lasts forever

  • Consols are not very prevalent in the U.S.

Convertible bonds

Convertible Bonds

  • Definition

  • Security-backed bonds

  • Commodity-backed bonds



  • A convertible bond gives the bondholder the right to exchange them for another security or for some physical asset

  • Once conversion occurs, the holder cannot elect to reconvert and regain the original debt security

Security backed bonds

Security-Backed Bonds

  • Security-backed convertible bonds are convertible into other securities

    • Typically common stock of the company that issued the bonds

    • Occasionally preferred stock of the issuing firm, common stock of another firm, or shares in a subsidiary company

Commodity backed bonds

Commodity-Backed Bonds

  • Commodity-backed bonds are convertible into a tangible asset

  • For example, silver or gold



  • Bearer bonds

  • Registered bonds

  • Book entry bonds

Bearer bonds

Bearer Bonds

  • Bearer bonds:

    • Do not have the name of the bondholder printed on them

    • Belong to whoever legally holds them

    • Are also called coupon bonds

      • The bond contains coupons that must be clipped

    • Are no longer issued in the U.S.

Registered bonds

Registered Bonds

  • Registered bonds show the bondholder’s name

  • Registered bondholders receive interest checks in the mail from the issuer

Book entry bonds

Book Entry Bonds

  • The U.S. Treasury and some corporation issue bonds in book entry form only

    • Holders do not take actual delivery of the bond

    • Potential holders can:

      • Open an account through the Treasury Direct System at a Federal Reserve Bank

      • Purchase a bond through a broker

Bond pricing and returns

Bond Pricing and Returns

  • Introduction

  • Valuation equations

  • Yield to maturity

  • Realized compound yield

  • Current yield

  • Term structure of interest rates

  • Spot rates

Bond pricing and returns cont d

Bond Pricing and Returns (cont’d)

  • The conversion feature

  • The matter of accrued interest



  • The current price of a bond is the market’s estimation of what the expected cash flows are worth in today’s dollars

  • There is a relationship between:

    • The current bond price

    • The bond’s promised future cash flows

    • The riskiness of the cash flows

Valuation equations

Valuation equations

  • Annuities

  • Zero coupon bonds

  • Variable rate bonds

  • Consols



  • For a semiannual bond:

Annuities cont d

Annuities (cont’d)

  • Separating interest and principal components:

Annuities cont d1

Annuities (cont’d)


A bond currently sells for $870, pays $70 per year (Paid semiannually), and has a par value of $1,000. The bond has a term to maturity of ten years.

What is the yield to maturity?

Annuities cont d2

Annuities (cont’d)

Example (cont’d)

Solution: Using a financial calculator and the following input provides the solution:

N = 20

PV = $870

PMT = $35

FV = $1,000

CPT I = 4.50

This bond’s yield to maturity is 4.50% x 2 = 9.00%.

Zero coupon bonds1

Zero Coupon Bonds

  • For a zero-coupon bond (annual and semiannual compounding):

Zero coupon bonds cont d

Zero Coupon Bonds (cont’d)


A zero coupon bond has a par value of $1,000 and currently sells for $400. The term to maturity is twenty years.

What is the yield to maturity (assume semiannual compounding)?

Zero coupon bonds cont d1

Zero Coupon Bonds (cont’d)

Example (cont’d)


Variable rate bonds1

Variable Rate Bonds

  • The valuation equation must allow for variable cash flows

  • You cannot determine the precise present value of the cash flows because they are unknown:



  • Consols are perpetuities:

Consols cont d

Consols (cont’d)


A consol is selling for $900 and pays $60 annually in perpetuity.

What is this consol’s rate of return?

Consols cont d1

Consols (cont’d)

Example (cont’d)


Yield to maturity

Yield to Maturity

  • Yield to maturity captures the total return from an investment

    • Includes income

    • Includes capital gains/losses

  • The yield to maturity is equivalent to the internal rate of return in corporate finance

Realized compound yield

Realized Compound Yield

  • The effective annual yield is useful to compare bonds to investments generating income on a different time schedule

Realized compound yield cont d

Realized Compound Yield (cont’d)


A bond has a yield to maturity of 9.00% and pays interest semiannually.

What is this bond’s effective annual rate?

Realized compound yield cont d1

Realized Compound Yield (cont’d)

Example (cont’d)


Current yield

Current Yield

  • The current yield:

    • Measures only the return associated with the interest payments

    • Does not include the anticipated capital gain or loss resulting from the difference between par value and the purchase price

Current yield cont d

Current Yield (cont’d)

  • For a discount bond, the yield to maturity is greater than the current yield

  • For a premium bond, the yield to maturity is less than the current yield

Current yield cont d1

Current Yield (cont’d)


A bond pays annual interest of $70 and has a current price of $870.

What is this bond’s current yield?

Current yield cont d2

Current Yield (cont’d)

Example (cont’d)


Current yield = $70/$870 = 8.17%

Term structure of interest rates

Term Structure of Interest Rates

  • Yield curve

  • Theories of interest rate structure

Yield curve

Yield Curve

  • The yield curve:

    • Is a graphical representation of the term structure of interest rates

    • Relates years until maturity to the yield to maturity

    • Is typically upward sloping and gets flatter for longer terms to maturity

Information used to build a yield curve

Information Used to Build A Yield Curve

Theories of interest rate structure

Theories of Interest Rate Structure

  • Expectations theory

  • Liquidity preference theory

  • Inflation premium theory

Expectations theory

Expectations Theory

  • According to the expectations theory of interest rates, investment opportunities with different time horizons should yield the same return:

Expectations theory cont d

Expectations Theory (cont’d)


An investor can purchase a two-year CD at a rate of 5 percent. Alternatively, the investor can purchase two consecutive one-year CDs. The current rate on a one-year CD is 4.75 percent.

According to the expectations theory, what is the expected one-year CD rate one year from now?

Expectations theory cont d1

Expectations Theory (cont’d)

Example (cont’d)


Liquidity preference theory

Liquidity Preference Theory

  • Proponents of the liquidity preference theory believe that, in general:

    • Investors prefer to invest short term rather than long term

    • Borrowers must entice lenders to lengthen their investment horizon by paying a premium for long-term money (the liquidity premium)

  • Under this theory, forward rates are higher than the expected interest rate in a year

Inflation premium theory

Inflation Premium Theory

  • The inflation premium theory states that risk comes from the uncertainty associated with future inflation rates

  • Investors who commit funds for long periods are bearing more purchasing power risk than short-term investors

    • More inflation risk means longer-term investment will carry a higher yield

Spot rates

Spot Rates

  • Spot rates:

    • Are the yields to maturity of a zero coupon security

    • Are used by the market to value bonds

      • The yield to maturity is calculated only after learning the bond price

      • The yield to maturity is an average of the various spot rates over a security’s life

Spot rates cont d

Spot Rates (cont’d)

Spot Rate Curve

Interest Rate

Yield to Maturity

Time Until the Cash Flow

Spot rates cont d1

Spot Rates (cont’d)


A six-month T-bill currently has a yield of 3.00%. A one-year T-note with a 4.20% coupon sells for 102.

Use bootstrapping to find the spot rate six months from now.

Spot rates cont d2

Spot Rates (cont’d)

Example (cont’d)

Solution: Use the T-bill rate as the spot rate for the first six months in the valuation equation for the T-note:

The conversion feature

The Conversion Feature

  • Convertible bonds give their owners the right to exchange the bonds for a pre-specified amount or shares of stock

  • The conversion ratio measures the number of shares the bondholder receives when the bond is converted

    • The par value divided by the conversion ratio is the conversion price

    • The current stock price multiplied by the conversion ratio is the conversion value

The conversion feature cont d

The Conversion Feature (cont’d)

  • The market price of a bond can never be less than its conversion value

  • The difference between the bond price and the conversion value is the premium over conversion value

    • Reflects the potential for future increases in the common stock price

  • Mandatory convertibles convert automatically into common stock after three or four years

The matter of accrued interest

The Matter of Accrued Interest

  • Bondholders earn interest each calendar day they hold a bond

  • Firms mail interest payment checks only twice a year

  • Accrued interest refers to interest that has accumulated since the last interest payment date but which has not yet been paid

The matter of accrued interest cont d

The Matter of Accrued Interest (cont’d)

  • At the end of a payment period, the issuer sends one check for the entire interest to the current bondholder

    • The bond buyer pays the accrued interest to the seller

    • The bond sells receives accrued interest from the bond buyer

The matter of accrued interest cont d1

The Matter of Accrued Interest (cont’d)


A bond with an 8% coupon rate pays interest on June 1 and December 1. The bond currently sells for $920.

What is the total purchase price, including accrued interest, that the buyer of the bond must pay if he purchases the bond on August 10?

The matter of accrued interest cont d2

The Matter of Accrued Interest (cont’d)

Example (cont’d)

Solution: The accrued interest for 71 days is:

$80/365 x 71 = $15.56

Therefore, the total purchase price is:

$920 + $15.56 = $935.56

Bond risk

Bond Risk

  • Price risks

  • Convenience risks

  • Malkiel’s interest rate theories

  • Duration as a measure of interest rate risk

Price risks

Price Risks

  • Interest rate risk

  • Default risk

Interest rate risk

Interest Rate Risk

  • Interest rate risk is the chance of loss because of changing interest rates

  • The relationship between bond prices and interest rates is inverse

    • If market interest rates rise, the market price of bonds will fall

Default risk

Default Risk

  • Default risk measures the likelihood that a firm will be unable to pay the principal and interest on a bond

  • Standard & Poor’s Corporation and Moody’s Investor Service are two leading advisory services monitoring default risk

Default risk cont d

Default Risk (cont’d)

  • Investment grade bonds are bonds rated BBB or above

  • Junk bonds are rated below BBB

  • The lower the grade of a bond, the higher its yield to maturity

Convenience risks

Convenience Risks

  • Definition

  • Call risk

  • Reinvestment rate risk

  • Marketability risk



  • Convenience risk refers to added demands on management time because of:

    • Bond calls

    • The need to reinvest coupon payments

    • The difficulty in trading a bond at a reasonable price because of low marketability

Call risk

Call Risk

  • If a company calls its bonds, it retires its debt early

  • Call risk refers to the inconvenience of bondholders associated with a company retiring a bond early

    • Bonds are usually called when interest rates are low

Call risk cont d

Call Risk (cont’d)

  • Many bond issues have:

    • Call protection

      • A period of time after the issuance of a bond when the issuer cannot call it

    • A call premium if the issuer calls the bond

      • Typically begins with an amount equal to one year’s interest and then gradually declining to zero as the bond approaches maturity

Reinvestment rate risk

Reinvestment Rate Risk

  • Reinvestment rate risk refers to the uncertainty surrounding the rate at which coupon proceeds can be invested

  • The higher the coupon rate on a bond, the higher its reinvestment rate risk

Marketability risk

Marketability Risk

  • Marketability risk refers to the difficulty of trading a bond:

    • Most bonds do not trade in an active secondary market

    • The majority of bond buyers hold bonds until maturity

  • Low marketability bonds usually carry a wider bid-ask spread

Malkiel s interest rate theorems

Malkiel’s Interest Rate Theorems

  • Definition

  • Theorem 1

  • Theorem 2

  • Theorem 3

  • Theorem 4

  • Theorem 5



  • Malkiel’s interest rate theorems provide information about how bond prices change as interest rates change

  • Any good portfolio manager knows Malkiel’s theorems

Theorem 1

Theorem 1

  • Bond prices move inversely with yields:

    • If interest rates rise, the price of an existing bond declines

    • If interest rates decline, the price of an existing bond increases

Theorem 2

Theorem 2

  • Bonds with longer maturities will fluctuate more if interest rates change

  • Long-term bonds have more interest rate risk

Theorem 3

Theorem 3

  • Higher coupon bonds have less interest rate risk

  • Money in hand is a sure thing while the present value of an anticipated future receipt is risky

Theorem 4

Theorem 4

  • When comparing two bonds, the relative importance of Theorem 2 diminishes as the maturities of the two bonds increase

  • A given time difference in maturities is more important with shorter-term bonds

Theorem 5

Theorem 5

  • Capital gains from an interest rate decline exceed the capital loss from an equivalent interest rate increase

Duration as a measure of interest rate risk

Duration as A Measure of Interest Rate Risk

  • The concept of duration

  • Calculating duration

The concept of duration

The Concept of Duration

  • For a noncallable security:

    • Duration is the weighted average number of years necessary to recover the initial cost of the bond

    • Where the weights reflect the time value of money

The concept of duration cont d

The Concept of Duration (cont’d)

  • Duration is a direct measure of interest rate risk:

    • The higher the duration, the higher the interest rate risk

Calculating duration

Calculating Duration

  • The traditional duration calculation:

Calculating duration cont d

Calculating Duration (cont’d)

  • The closed-end formula for duration:

Calculating duration cont d1

Calculating Duration (cont’d)


Consider a bond that pays $100 annual interest and has a remaining life of 15 years. The bond currently sells for $985 and has a yield to maturity of 10.20%.

What is this bond’s duration?

Calculating duration cont d2

Calculating Duration (cont’d)

Example (cont’d)

Solution: Using the closed-form formula for duration:

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