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Chapter 12 Bond Prices and the Importance of Duration

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Chapter 12Bond 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

- Identification of bonds
- Classification of bonds
- Terms of repayment
- Bond cash flows
- Convertible bonds
- Registration

- 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

- 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)

- 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

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

- Corporations:
- Debentures are signature loans backed by the good name of the company
- Subordinated debentures are paid off after original debentures

- 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

- Revenue bonds

- 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

- Mortgages

- 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

- Interest only
- Sinking fund
- Balloon
- Income bonds

- Periodic payments are entirely interest
- The principal amount of the loan is repaid at maturity

- 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

- The borrower can:

- 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 pay interest only if the firm earns it
- For example, an income bond may be issued to finance an income-producing project

- 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

- 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 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.

- 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 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 are convertible into a tangible asset
- For example, silver or gold

- Bearer bonds
- Registered bonds
- Book entry 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 show the bondholder’s name
- Registered bondholders receive interest checks in the mail from the issuer

- 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

- Introduction
- Valuation equations
- Yield to maturity
- Realized compound yield
- Current yield
- Term structure of interest rates
- Spot rates

- 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

- Annuities
- Zero coupon bonds
- Variable rate bonds
- Consols

- For a semiannual bond:

- Separating interest and principal components:

Example

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?

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%.

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

Example

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)?

Example (cont’d)

Solution:

- 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:

Example

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

What is this consol’s rate of return?

Example (cont’d)

Solution:

- 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

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

Example

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

What is this bond’s effective annual rate?

Example (cont’d)

Solution:

- 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

- 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

Example

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

What is this bond’s current yield?

Example (cont’d)

Solution:

Current yield = $70/$870 = 8.17%

- Yield curve
- Theories of interest rate structure

- 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

- Expectations theory
- Liquidity preference theory
- Inflation premium theory

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

Example

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?

Example (cont’d)

Solution:

- 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

- 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:
- 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 Rate Curve

Interest Rate

Yield to Maturity

Time Until the Cash Flow

Example

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.

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:

- 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 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

- 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

- 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

Example

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?

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

- Price risks
- Convenience risks
- Malkiel’s interest rate theories
- Duration as a measure of interest rate risk

- Interest rate risk
- Default 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 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

- 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

- 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

- 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

- 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

- Call protection

- 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 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

- 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

- 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

- Bonds with longer maturities will fluctuate more if interest rates change
- Long-term bonds have more interest rate risk

- 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

- 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

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

- The concept of duration
- Calculating 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

- Duration is a direct measure of interest rate risk:
- The higher the duration, the higher the interest rate risk

- The traditional duration calculation:

- The closed-end formula for duration:

Example

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?

Example (cont’d)

Solution: Using the closed-form formula for duration: