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Bond Valuation

Econ 181

Corporate Finance

Wadia Haddaji

Department of Economics

Duke University

RK-CH

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Bond Valuation An Overview

Introduction to bonds and bond markets» What are they? Some examples

Zero coupon bonds» Valuation» Interest rate sensitivity

Coupon bonds» Valuation» Interest rate sensitivity

The term structure of interest rates

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What is a Bond?

A bond is a security that obligates the issuer to make specified interest and principal payments to the holder on specified dates.

» Coupon rate

» Face value (or par)

» Maturity (or term) Bonds are also called fixed income securities. Bonds differ in several respects:

» Repayment type

» Issuer

» Maturity

» Security

» Priority in case of default

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Repayment Schemes

Pure Discount or Zero-Coupon Bonds» Pay no coupons prior to maturity.» Pay the bond’s face value at maturity.

Coupon Bonds» Pay a stated coupon at periodic intervals prior to maturity.» Pay the bond’s face value at maturity.

Floating-Rate Bonds» Pay a variable coupon, reset periodically to a reference rate. » Pay the bond’s face value at maturity.

Perpetual Bonds (Consols)» No maturity date.» Pay a stated coupon at periodic intervals.

Annuity or Self-Amortizing Bonds » Pay a regular fixed amount each payment period.» Principal repaid over time rather than at maturity.

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Types of Bonds: Issuers

Bonds Issuer

Government Bonds US Treasury, Government Agencies

Mortgage-Backed Securities Government agencies (GNMA etc)

Municipal Bonds State and local government

Corporate Bonds Corporations

Asset-Back Securities Corporations

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U.S. Government Bonds

Treasury Bills» No coupons (zero coupon security)

» Face value paid at maturity

» Maturities up to one year

Treasury Notes» Coupons paid semiannually

» Face value paid at maturity

» Maturities from 2-10 years

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U.S. Government Bonds (Cont.)

Treasury Bonds» Coupons paid semiannually» Face value paid at maturity» Maturities over 10 years» The 30-year bond is called the long bond.

Treasury Strips» Zero-coupon bond» Created by “stripping” the coupons and principal from Treasury

bonds and notes. No default risk. Considered to be risk free. Exempt from state and local taxes. Sold regularly through a network of primary dealers. Traded regularly in the over-the-counter market.

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Agency and Municipal Bonds

Agency bonds: mortgage-backed bonds» Bonds issued by U.S. Government agencies that are backed by a

pool of home mortgages.» Self-amortizing bonds. (mostly monthly payments)» Maturities up to 30 years.» Prepayment risk.

Municipal bonds» Maturities from one month to 40 years.» Usually exempt from federal, state, and local taxes.» Generally two types:

– Revenue bonds– General Obligation bonds

» Riskier than U.S. Government bonds.

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Corporate Bonds

Bonds issued by corporations» Bonds vs. Debentures

» Fixed-rate versus floating-rate bonds.

» Investment-grade vs. Below investment-grade bonds.

» Additional features:– call provisions

– convertible bonds

– puttable bonds

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Seniority of Corporate Bonds

In case of default, different classes of bonds have different claim priority on the assets of a corporation.

Secured Bonds (Asset-Backed)» Secured by real property.

» Ownership of the property reverts to the bondholders upon default.

Debentures» Same priority as general creditors.

» Have priority over stockholders, but subordinate to secured debt.

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Bond Ratings

Moody’s S&P Quality of Issue

Aaa AAA Highest quality. Very small risk of default.

Aa AA High quality. Small risk of default.

A A High-Medium quality. Strong attributes, but potentially vulnerable.

Baa BBB Medium quality. Currently adequate, but potentially unreliable.

Ba BB Some speculative element. Long-run prospects questionable.

B B Able to pay currently, but at risk of default in the future.

Caa CCC Poor quality. Clear danger of default.

Ca CC High speculative quality. May be in default.

C C Lowest rated. Poor prospects of repayment.

D - In default.

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The US Bond Market

Debt Instrument 2006 Q2

Treasury securities 4759.6

Municipal securities 2305.7

Corporate and foreign bonds 8705.3

Consumer Credit 2327.4

Mortgages 12757.7

Corporate equities 18684.5

Amount ($bil.). Source: U.S. Federal Reserve (Table L.4, September/2006)

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Bond Valuation: Zero Coupon BondsB = Market price of the Bond of bondF = Face valueR = Annual percentage ratem = compounding period (annual m = 1, semiannual m = 2,…)i = Effective periodic interest rate; i=R/mT = Maturity (in years)N = Number of compounding periods; N = T*m

Two cash flows to purchaser of bond:» -B at time 0» F at time T

What is the price of a bond? Use present value formula:

Ni

FB

1

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Valuing Zero Coupon Bonds:An Example

Value a 5 year, U.S. Treasury strip with face value of $1,000. The APR is R=7.5% with annual compounding? What about quarterly compounding?

What is the APR on a U.S. Treasury strip that pays $1,000 in exactly 7 years and is currently selling for $591.11 under annual compounding? Semi-annual compounding?

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Interest Rate Sensitivity:Zero Coupon Bonds

Consider the following 1, 2 and 10-year zero-coupon bonds, all with

» face value of F=$1,000

» APR of R=10%, compounded annually.

We obtain the following table for increases and decreases of the interest rate by 1%:

Bond prices move up if interest rates drop, decrease if interest rates rise

Interest Rate Bond 1 Bond 2 Bond 31-Year 2-Year 10-Year

9.0% $917.43 $841.68 $422.4110.0% $909.09 $826.45 $385.5411.0% $900.90 $811.62 $352.18

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Bond Prices and Interest Rates

$0

$200

$400

$600

$800

$1,000

$1,200

0.0% 5.0% 10.0% 15.0% 20.0% 25.0%

1-Year

2-Year

10-Year

Bond prices are inversely related to IR

Longer term bonds are more sensitive to IR changes than short term bonds

The lower the IR, the more sensitive the price.

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Measuring Interest Rate SensitivityZero Coupon Bonds

We would like to measure the interest rate sensitivity of a bond or a portfolio of bonds.» How much do bond prices change if interest rates change by a small

amount?» Why is this important?

Use “Dollar value of a one basis point decrease” (DV01):» Basis point (bp): 1/100 of one percentage point =0.01%=0.0001» Calculate DV01:

– Method 1: Difference of moving one basis point down:DV01= B(R-0.01%)-B(R).

– Method 2: Difference of moving 1/2bp down minus 1/2pb up:DV01=B(R-0.005%) -B(R+0.005%).

– Method 3: Use calculus: 0001.001

R

BDV

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Computing DV01: An Example

Reconsider the 1, 2 and 10- year bonds discussed before:

Method 3:

11 10.1

1*10.0$*0001.0

10.1

000,1$0001.0

TT

TTR

B

Interest Rate Bond 1 Bond 2 Bond 31-Year 2-Year 10-Year

9.990% $909.1736 $826.5966 $385.89409.995% $909.1322 $826.5214 $385.7186

10.000% $909.0909 $826.4463 $385.543310.005% $909.0496 $826.3712 $385.3681

Method 1 $0.082652 $0.150283 $0.350669Method 2 $0.082645 $0.150263 $0.350494Method 3 $0.082645 $0.150263 $0.350494

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DV01: A Graphical Approach

DV01 estimates the change in the Price-Interest rate curve using a linear approximation.

higher slope implies greater sensitivity

10-Year

$0.00

$200.00

$400.00

$600.00

$800.00

$1,000.00

$1,200.00

Interest Rate

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Valuing Coupon BondsExample 1: Amortization Bonds

Consider Amortization Bond» T=2

» m=2

» C=$2,000 c = C/m = $2,000/2 = $1,000

» R=10% i = R/m = 10%/2 = 5%

How can we value this security?» Brute force discounting

» Similar to another security we already know how to value?

» Replication

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0 1 2 3 4Buy Coupon Bond -$3,545.95 $1,000.00 $1,000.00 $1,000.00 $1,000.00

Buy 6-Month Zero -$952.38Buy 1-Year Zero -$907.03Buy 1.5-Year Zero -$863.84Buy 2-Year Zero -$822.70Portfolio -$3,545.95

Valuing Coupon BondsExample 1: Amortization Bonds

Compare with a portfolio of zero coupon bonds:

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A First Look at Arbitrage

Reconsider amortization bond; suppose bond trades at $3,500 (as opposed to computed price of $3,545.95)» Can we make a profit without any risk?

– What is the strategy?

– What is the profit?

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A First Look at Arbitrage

Reconsider amortization bond; suppose bond trades at $3,500 (as opposed to computed price of $3,545.95)

» Can make risk less profit

– Buy low: buy amortization bond

– Sell high: Sell portfolio of zero coupon bonds

– riskless profit of $45.95

– no riskless profit if price is correct

0 1 2 3 4Buy Coupon Bond -$3,500.00 $1,000.00 $1,000.00 $1,000.00 $1,000.00

Sell 6-Month Zero $952.38 -$1,000.00 $0.00 $0.00 $0.00Sell 1-Year Zero $907.03 $0.00 -$1,000.00 $0.00 $0.00Sell 1.5-Year Zero $863.84 $0.00 $0.00 -$1,000.00 $0.00Sell 2-Year Zero $822.70 $0.00 $0.00 $0.00 -$1,000.00Portfolio $3,545.95 -$1,000.00 -$1,000.00 -$1,000.00 -$1,000.00Net Cash Flow $45.95 $0.00 $0.00 $0.00 $0.00

Time Period

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What is the market price of a U.S. Treasury bond that has a coupon rate of 9%, a face value of $1,000 and matures exactly 10 years from today if the interest rate is 10% compounded semiannually?

0 6 12 18 24 ... 120 Months

45 45 45 45 1045

Valuation of Coupon Bonds:Example 2: Straight Bonds

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What is the market price of a bond that has an annual coupon C, face value F and matures exactly T years from today if the required rate of return is R, with m-periodic compounding?

» Coupon payment is: c = C/m

» Effective periodic interest rate is: i = R/m

» number of periods N = Tm

0 1 2 3 4 ... … N

c c c c … … c+F

Valuing Coupon BondsThe General Formula

NN i

F

ii

c

ZeroAnnuityB

11

11

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The Concept of a “Yield to Maturity”

So far we have valued bonds by using a given interest rate, then discounted all payments to the bond.

Prices are usually given from trade prices» need to infer interest rate that has been used

Definition: The yield to maturity is that interest rate that equates the present discounted value of all future payments to bondholders to the market price:

Algebraic:

NN myield

F

myieldmyield

cB

/1/1

11

/

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Yield to MaturityA Graphical Interpretation

$0.00

$500.00

$1,000.00

$1,500.00

$2,000.00

$2,500.00

0% 2% 4% 6% 8% 10%

12%

14%

16%

18%

20%

22%

24%

Consider a U.S. Treasury bond that has a coupon rate of 10%, a face value of $1,000 and matures exactly 10 years from now. » Market price of $1,500, implies a yield of 3.91% (semi-annual

compounding); for B=$1,000 we obviously find R=10%.

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Interest Rate Sensitivity:Coupon Bonds

Coupon bonds can be represented as portfolios of zero-coupon bonds» Implication for price sensitivity

Consider purchasing the US Treasury bond discussed earlier (10 year, 9% coupon, $1,000 face)» Suppose immediately thereafter interest rates fall to 8%,

compounded semiannually.

» Suppose immediately thereafter interest rate rises to 12% compounded semiannually.

» Suppose the interest rate equals 9%, compounded semiannually.

What are the pricing implications of these scenarios?

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Recall the general formula:

What is the price of the bond if the APR is 8% compounded

semiannually?

Similarly:

If R=12%: B=$ 827.95

If R= 9%: B=$1,000.00

Implication of Interest Rate Changes on Coupon Bond Prices

NN i

F

ii

cB

11

11

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Relationship Between Coupon Bond Prices and Interest Rates

Bond prices are inversely related to interest rates (or yields).

A bond sells at par only if its interest rate equals the coupon rate

A bond sells at a premium if its coupon rate is above the interest rate.

A bond sells at a discount if its coupon rate is below the interest rate.

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DV01 and Coupon Bonds

Consider two bonds with 10% annual coupons with maturities of 5 years and 10 years.

The APR is 8% What are the responses to a .01% (1bp) interest rate change?

Does the sensitivity of a coupon bond always increase with the term to maturity?

Yield 5-Year Bond $ Change % Change 10-Year Bond $ Change % Change7.995% $1,080.06 $0.21019 0.0195% $1,134.57 $0.36585 0.0323%8.000% $1,079.85 $1,134.208.005% $1,079.64 -$0.21013 -0.0195% $1,133.84 -$0.36569 -0.0322%DV01 $0.42032 $0.73154

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$0.00

$500.00

$1,000.00

$1,500.00

$2,000.00

$2,500.00

0% 2% 4% 6% 8%

10%

12%

14%

16%

18%

20%

22%

24%

Interest Rate (R)

Pri

ce (

P)

5-Year Bond

10-Year Bond

Bond Prices and Interest Rates

Longer term bonds are more sensitive to changes in interest rates than shorter term bonds, in general.

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Consider the following two bonds:» Both have a maturity of 5 years» Both have yield of 8%» First has 6% coupon, other has 10% coupon, compounded

annually. Then, what are the price sensitivities of these bonds, measured by

DV01 as for zero coupon bonds?

Why do we get different answers for two bonds with the same yield and same maturity?

Bond Yields and Prices

Yield 6%-Bond $ Change % change 10%-Bond $ Change % change

7.995% $920.33 $0.1891 $1,080.06 $0.2102

8.000% $920.15 $1,079.85

8.005% $919.96 ($0.1891) $1,079.64 ($0.2101)

0.0411% 0.0389%

DV01 $0.3782 $0.4203

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Maturity and Price Risk

Zero coupon bonds have well-defined relationship between maturity and interest rate sensitivity:

Coupon bonds can have different sensitivities for the same maturity» DV01 now depends on maturity and coupon

Need concept of “average maturity” of coupon bond:» Duration

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Duration

Duration is a weighted average term to maturity where the weights are relative size of the contemporaneous cash flow.

Duration is a unitless number that quantifies the percentage change in a bond’s price for a 1 percentage change in the interest rate.

B

FPVN

TB

NcPV

NT

B

cPVT

B

cPVTDuration )()()2(

2

)1(

1

R

RB

B

B

R

R

BDuration

1

1

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Duration (cont.)

The duration of a bond is less than its time to maturity (except for zero coupon bonds).

The duration of the bond decreases the greater the coupon rate. This is because more weight (present value weight) is being given to the coupon payments.

As market interest rate increases, the duration of the bond decreases. This is a direct result of discounting. Discounting at a higher rate means lower weight on payments in the far future. Hence, the weighting of the cash flows will be more heavily placed on the early cash flows -- decreasing the duration.

Modified Duration = Duration / (1+yield)

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Bills MATURITY DISCOUNT/YIELD DISCOUNT/YIELD TIME

DATE CHANGE3-Month 08/16/2007 4.72 / 4.84 0.01 / .010 13:416-Month 11/15/2007 4.78 / 4.98 0.01 / .015 13:41

Notes/Bonds COUPON MATURITY CURRENT PRICE/YIELD TIME

DATE PRICE/YIELD CHANGE2-Year 4.500 04/30/2009 99-121⁄4 / 4.84 -0-02 / .035 14:083-Year 4.500 05/15/2010 99-081⁄2 / 4.77 -0-031⁄2 / .040 14:065-Year 4.500 04/30/2012 98-281⁄2 / 4.75 -0-06 / .043 14:0710-Year 4.500 05/15/2017 97-15 / 4.82 -0-091⁄2 / .038 14:0730-Year 4.750 02/15/2037 96-17+ / 4.97 -0-17 / .035 14:07

A Few Bond Markets Statistics U.S. Treasuries, May 20th 2007.

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Spot Rates

A spot rate is a rate agreed upon today, for a loan that is to be made today» r1=5% indicates that the current rate for a one-year loan is 5%.

» r2=6% indicates that the current rate for a two-year loan is 6%.

» Etc. The term structure of interest rates is the series of spot rates

r1, r2, r3,…

» We can build using STRIPS or coupon bond yields.

» Explanations of the term structure.

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The Term Structure of Interest Rates

An Example

1 2 3

Yield

Maturity

5.00

5.75

6.00

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Term Structure, July 1st 2005.

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Term Structure, September 12th, 2006

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Term Structure, May 20th, 2007

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Term Structure of Interest Rates

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Summary

Bonds can be valued by discounting their future cash flows

Bond prices change inversely with yield Price response of bond to interest rates depends on term

to maturity.» Works well for zero-coupon bond, but not for coupon bonds

Measure interest rate sensitivity using ‘DV01’ and duration.

The term structure implies terms for future borrowing:» Forward rates» Compare with expected future spot rates