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**Chapter 12 Bond Prices and the Importance of Duration**

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**Outline Introduction Review of bond principles**

Bond pricing and returns Bond risk The meaning of bond diversification Choosing bonds Example: monthly retirement income

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Introduction 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

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**Review of Bond Principles**

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

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**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

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**Classification of Bonds**

Introduction Issuer Security Term

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**Introduction The bond indenture describes the details of a bond issue:**

Description of the loan Terms of repayment Collateral Protective covenants Default provisions

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Issuer 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

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

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**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

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**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

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**Secured Debt Municipalities issue: Revenue bonds Assessment 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

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**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

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**Term 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

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Terms of Repayment Interest only Sinking fund Balloon Income bonds

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**Interest Only Periodic payments are entirely interest**

The principal amount of the loan is repaid at maturity

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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

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Balloon 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

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**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

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**Bond Cash Flows Annuities Zero coupon bonds Variable rate bonds**

Consols

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Annuities 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

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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

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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

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**Consols 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.

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Definition 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

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**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

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**Commodity-Backed Bonds**

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

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**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.

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**Registered Bonds Registered bonds show the bondholder’s name**

Registered bondholders receive interest checks in the mail from the issuer

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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

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**Bond Pricing and Returns**

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

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**Bond Pricing and Returns (cont’d)**

The conversion feature The matter of accrued interest

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Introduction 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

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**Valuation equations Annuities Zero coupon bonds Variable rate bonds**

Consols

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Annuities For a semiannual bond:

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Annuities (cont’d) Separating interest and principal components:

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**Annuities (cont’d) 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?

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**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%.

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

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**Zero Coupon Bonds (cont’d)**

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)?

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**Zero Coupon Bonds (cont’d)**

Example (cont’d) Solution:

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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:

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Consols Consols are perpetuities:

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**Consols (cont’d) Example**

A consol is selling for $900 and pays $60 annually in perpetuity. What is this consol’s rate of return?

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Consols (cont’d) Example (cont’d) Solution:

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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

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**Realized Compound Yield**

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

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**Realized Compound Yield (cont’d)**

Example A bond has a yield to maturity of 9.00% and pays interest semiannually. What is this bond’s effective annual rate?

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**Realized Compound Yield (cont’d)**

Example (cont’d) Solution:

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**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

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**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

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**Current Yield (cont’d)**

Example A bond pays annual interest of $70 and has a current price of $870. What is this bond’s current yield?

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**Current Yield (cont’d)**

Example (cont’d) Solution: Current yield = $70/$870 = 8.17%

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**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

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**Information Used to Build A Yield Curve**

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**Theories of Interest Rate Structure**

Expectations theory Liquidity preference theory Inflation premium theory

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

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**Expectations Theory (cont’d)**

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?

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**Expectations Theory (cont’d)**

Example (cont’d) Solution:

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**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

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**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

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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

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**Spot Rates (cont’d) Spot Rate Curve Yield to Maturity Interest Rate**

Time Until the Cash Flow

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**Spot Rates (cont’d) 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.

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**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:

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**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

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**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

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**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

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**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

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**The Matter of Accrued Interest (cont’d)**

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?

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**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

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**Bond Risk Price risks Convenience risks**

Malkiel’s interest rate theories Duration as a measure of interest rate risk

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Price Risks Interest rate risk Default risk

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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

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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

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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

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**Convenience Risks Definition Call risk Reinvestment rate risk**

Marketability risk

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Definition 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

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**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

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**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

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**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

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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

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**Malkiel’s Interest Rate Theorems**

Definition Theorem 1 Theorem 2 Theorem 3 Theorem 4 Theorem 5

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Definition Malkiel’s interest rate theorems provide information about how bond prices change as interest rates change Any good portfolio manager knows Malkiel’s theorems

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**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

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Theorem 2 Bonds with longer maturities will fluctuate more if interest rates change Long-term bonds have more interest rate risk

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**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

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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

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Theorem 5 Capital gains from an interest rate decline exceed the capital loss from an equivalent interest rate increase

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**Duration as A Measure of Interest Rate Risk**

The concept of duration Calculating duration

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**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

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**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

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Calculating Duration The traditional duration calculation:

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**Calculating Duration (cont’d)**

The closed-end formula for duration:

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**Calculating Duration (cont’d)**

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?

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**Calculating Duration (cont’d)**

Example (cont’d) Solution: Using the closed-form formula for duration:

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**Bond Selection - Introduction**

In most respects selecting the fixed-income components of a portfolio is easier than selecting equity securities There are ways to make mistakes with bond selection

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**The Meaning of Bond Diversification**

Introduction Default risk Dealing with the yield curve Bond betas

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**Introduction It is important to diversify a bond portfolio**

Diversification of a bond portfolio is different from diversification of an equity portfolio Two types of risk are important: Default risk Interest rate risk

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Default Risk Default risk refers to the likelihood that a firm will be unable to repay the principal and interest of a loan as agreed in the bond indenture Equivalent to credit risk for consumers Rating agencies such as S&P and Moody’s function as credit bureaus for credit issuers

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**Default Risk (cont’d) To diversify default risk:**

Purchase bonds from a number of different issuers Do not purchase various bond issues from a single issuer E.g., Enron had 20 bond issues when it went bankrupt

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**Dealing With the Yield Curve**

The yield curve is typically upward sloping The longer a fixed-income security has until maturity, the higher the return it will have to compensate investors The longer the average duration of a fund, the higher its expected return and the higher its interest rate risk

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**Dealing With the Yield Curve (cont’d)**

The client and portfolio manager need to determine the appropriate level of interest rate risk of a portfolio

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**Bond Betas The concept of bond betas:**

States that the market prices a bond according to its level of risk relative to the market average Has never become fully accepted Measures systematic risk, while default risk and interest rate risk are more important

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**Choosing Bonds Client psychology and bonds selling at a premium**

Call risk Constraints

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**Client Psychology and Bonds Selling at A Premium**

Premium bonds held to maturity are expected to pay higher coupon rates than the market rate of interest Premium bond held to maturity will decline in value toward par value as the bond moves towards its maturity date

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**Client Psychology & Bonds Selling at A Premium (cont’d)**

Clients may not want to buy something they know will decline in value There is nothing wrong with buying bonds selling at a premium

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**Call Risk If a bond is called:**

The funds must be reinvested The fund manager runs the risk of having to make adjustments to many portfolios all at one time There is no reason to exclude callable bonds categorically from a portfolio Avoid making extensive use of a single callable bond issue

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**Constraints Specifying return Specifying grade**

Specifying average maturity Periodic income Maturity timing Socially responsible investing

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**Specifying Return To increase the expected return on a bond portfolio:**

Choose bonds with lower ratings Choose bonds with longer maturities Or both

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Specifying Grade A legal list specifies securities that are eligible investments E.g., investment grade only Portfolio managers take the added risk of noninvestment grade bonds only if the yield pickup is substantial

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**Specifying Grade (cont’d)**

Conservative organizations will accept only U.S. government or AAA-rated corporate bonds A fund may be limited to no more than a certain percentage of non-AAA bonds

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**Specifying Average Maturity**

Average maturity is a common bond portfolio constraint The motivation is concern about rising interest rates Specifying average duration would be an alternative approach

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Periodic Income Some funds have periodic income needs that allow little or not flexibility Clients will want to receive interest checks frequently The portfolio manager should carefully select the bonds in the portfolio

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**Maturity Timing Maturity timing generates income as needed**

Sometimes a manager needs to construct a bond portfolio that matches a particular investment horizon E.g., assemble securities to fund a specific set of payment obligations over the next ten years Assemble a portfolio that generates income and principal repayments to satisfy the income needs

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**Socially Responsible Investing**

Some clients will ask that certain types of companies not be included in the portfolio Examples are nuclear power, military hardware, “vice” products

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**Example: Monthly Retirement Income**

The problem Unspecified constraints Using S&P’s Bond Guide Solving the problem

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**The Problem A client has: Primary objective: growth of income**

Secondary objective: income $1,100,000 to invest Inviolable income needs of $4,000 per month

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**The Problem (cont’d) You decide:**

To invest the funds between common stocks and debt securities To invest in ten common stock in the equity portion (see next slide) You incur $1,500 in brokerage commissions

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**The Problem (cont’d) Stock Value Qrtl Div. Payment Month 3,000 AAC**

$51,000 $380 Jan./April/July/Oct. 1,000 BBL 50,000 370 2,000 XXQ 49,000 400 Feb./May/Aug./Nov. 5,000 XZ 52,000 270 March/June/Sept./Dec. 7,000 MCDL 53,000 -- 1,000 ME 2,000 LN 51,000 500 4,000 STU 47,000 260 3,000 LLZ 290 6,000 MZN 43,000 170 Total $494,000 $3,010

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**The Problem (cont’d) Characteristics of the fund:**

Quarterly dividends total $3,001 ($12,004 annually) The dividend yield on the equity portfolio is 2.44% Total annual income required is $48,000 or 4.36% of fund Bonds need to have a current yield of at least 6.28%

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**Unspecified Constraints**

The task is meeting the minimum required expected return with the least possible risk You don’t want to choose CC-rated bonds You don’t want the longest maturity bonds you can find

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**Using S&P’s Bond Guide Figure 11-4 is an excerpt from the Bond Guide:**

Indicates interest payment dates, coupon rates, and issuer Provides S&P ratings Provides current price, current yield

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**Using S&P’s Bond Guide (cont’d)**

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**Solving the Problem Setup**

Dealing with accrued interest and commissions Choosing the bonds Overspending What about convertible bonds?

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**Setup You have two constraints:**

Include only bonds rated BBB or higher Keep the average maturities below fifteen years Set up a worksheet that enables you to pick bonds to generate exactly $4,000 per month (see next slide)

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**Setup (cont’d) Security Price Jan. Feb. March April May June 3,000 AAC**

$51,000 $380 1,000 BBL 50,000 370 2,000 XXQ 49,000 $400 5,000 XZ 52,000 $270 7,000 MCDL 53,000 1,000 ME 2,000 LN 51,000 500 4,000 STU 47,000 260 3,000 LLZ 290 6,000 MZN 43,000 170 Equities $494,000 $1,420 $1,060 $530

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**Dealing With Accrued Interest and Commissions**

Bond prices are typically quoted on a net basis (already include commissions) Calculate accrued interest using the mid-term heuristic Assume every bond’s accrued interest is half of one interest check

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**Choosing the Bonds The following slide shows one possible solution:**

Stock cost: $494,000 Bond cost: $557,130 Accrued interest: $9,350 Stock commissions: $1,500 Do you think this solution could be improved?

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**Bonds Security Price Jan. Feb. March April May June**

$80,000 Empire 71/2s02 $86,400 $3,000 $80,000 Energen 8s07 82,900 $3,200 $100,000 Enhance 61/4s03 105,500 $3,370 $80,000 Enron 65/8s03 84,500 $2,650 $90,000 Enron 6.7s06 97,200 $3,010 $100,000 Englehard 6.95s28 100,630 $3,470 Bonds subtotal $557,130 Total income $4,420 $4,260 $3,900 $4,070 $4,000

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Overspending The total of all costs associated with the portfolio should not exceed the amount given to you by the client to invest The money the client gives you establishes another constraint

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**What About Convertible Bonds?**

Convertible bonds can be included in a portfolio Useful for a growth of income objective People buy convertible bonds in hopes of price appreciation Useful if you otherwise meet your income constraints

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**Immunization Strategies**

A portfolio of bonds is said to be immunized (from interest rate risk) if its payoff at some future date is independent of the future levels of interest rates. Immunization is closely related to the concept of duration.

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Immunization consists of matching the duration of the portfolio’s assets and liabilities (obligations). Suppose a firm has a future obligation Q. The prevailing interest rate is r, and the liability is N periods away. The present value of this liability is denoted by V0=Q/(1+r)N.

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Now suppose that the firm is currently hedging this liability with a bond whose value VB = V0 and whose coupon payments are denoted by P1,…,PM. We thus have:

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**Suppose now that interest rates change from r to r+Dr**

Suppose now that interest rates change from r to r+Dr. The new values of the future obligation and of the bond are:

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**Rearranging terms and recalling that V0=VB yields the following expression:**

The left-hand side represents the duration of the bond, while the right-hand side represents the duration of the obligation (Since the obligation consisted of only one payment, the duration is its maturity).

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**In conclusion, in order for a portfolio to be immunized, you need to have:**

DURATIONASSETS = DURATIONLIABILITIES Caveat: this works only if the interest rates of various maturities all change in the same manner, i.e. if the yield curve shifts upward or downward in a parallel shift.

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Immunization Example You need to immunize an obligation whose present value V0 is $1,000. The payment is to be made 10 years from now, and the current interest rate is 6%. The payment is thus the future value of 1,000 at 6%, therefore it is: 1,000(1.06)10 = $1,790.85 The Excel spreadsheet on the next slide shows three bonds that you have at your disposition to immunize the liability.

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**Values 10 years later, assuming interest rates do not change**

(The goal of getting $1, is still met)

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**Values 10 years later, assuming interest rates change to 5% right after we buy the bonds**

(The goal of getting $1, is not met by Bond 1 anymore)

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Observations If interest rates go down to 5%, Bond 1 does not meet the requirement anymore. Bond 3, on the other hand, exceeds the payment that must be made in year 10. The ability of Bond 2 to meet the obligation is barely affected. Why? Because its duration is 10 years, exactly matching the duration of the liability. Pick Bond 2.

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**We can compute and plot the bonds’ terminal values in year 10**

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