Bond Valuation Econ 181 Corporate Finance

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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.

 Floating-Rate Bonds

» Pay a variable coupon, reset periodically to a reference rate.

 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.

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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 Bonds

B = Market price of the Bond of bond F = Face value

R = Annual percentage rate

m = compounding period (annual  m = 1, semiannual  m = 2,…) i = Effective periodic interest rate; i=R/m

T = 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:



^Fⁱ



^N

B   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 Rate Bond 1 Bond 2 Bond 3 1-Year 2-Year 10-Year

9.0% $917.43 $841.68 $422.41

10.0% $909.09 $826.45 $385.54

11.0% $900.90 $811.62 $352.18

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Bond Prices and Interest Rates

 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 Sensitivity

Zero 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 .

0 01  B  DV

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Computing DV01: An Example

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

 Method 3:

1

1 1.10

* 1 10 . 0

$

* 0001

. 10 0

. 1

000 , 1 0001 $

.

0  _   _

 

  T _T T _T

R B

Interest Rate Bond 1 Bond 2 Bond 3 1-Year 2-Year 10-Year 9.990% $909.1736 $826.5966 $385.8940 9.995% $909.1322 $826.5214 $385.7186 10.000% $909.0909 $826.4463 $385.5433 10.005% $909.0496 $826.3712 $385.3681 Method 1 $0.082652 $0.150283 $0.350669 Method 2 $0.082645 $0.150263 $0.350494 Method 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 Bonds

Example 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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Valuing Coupon Bonds

Example 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

0 1 2 3 4

Buy 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 $0.00 Sell 1-Year Zero $907.03 $0.00 -$1,000.00 $0.00 $0.00 Sell 1.5-Year Zero $863.84 $0.00 $0.00 -$1,000.00 $0.00 Sell 2-Year Zero $822.70 $0.00 $0.00 $0.00 -$1,000.00 Portfolio $3,545.95 -$1,000.00 -$1,000.00 -$1,000.00 -$1,000.00

Net 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 Bonds The General Formula

   

   

^_

 





 



 





 







N

N i

F i i

c

Zero Annuity

B

1 1

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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:



^yield ^m



^N



^yield^F ^m



^N

m yield B c

/ 1

1 1

/  

 





 



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Yield to Maturity

A 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

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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

   

^_

 





 



 





 

 _N _N

i F i i

B c

1 1

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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 % Change 7.995% $1,080.06 $0.21019 0.0195% $1,134.57 $0.36585 0.0323%

8.000% $1,079.85 $1,134.20

8.005% $1,079.64 -$0.21013 -0.0195% $1,133.84 -$0.36569 -0.0322%

DV01 $0.42032 $0.73154

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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

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 F PV TN

B cN PV TN

B c T PV

Duration ( 2) ( ) ( )

2 1)

(

1       

 



 



 



 

 

 



 B R

B B

R R

Duration B 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 CHANGE

3-Month 08/16/2007 4.72 / 4.84 0.01 / .010 13:41

6-Month 11/15/2007 4.78 / 4.98 0.01 / .015 13:41

Notes/Bonds

COUPON MATURITY CURRENT PRICE/YIELD TIME

DATE PRICE/YIELD CHANGE

2-Year 4.500 04/30/2009 99-121⁄4 / 4.84 -0-02 / .035 14:08 3-Year 4.500 05/15/2010 99-081⁄2 / 4.77 -0-031⁄2 / .040 14:06 5-Year 4.500 04/30/2012 98-281⁄2 / 4.75 -0-06 / .043 14:07 10-Year 4.500 05/15/2017 97-15 / 4.82 -0-091⁄2 / .038 14:07 30-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 r

₁

, r

₂

, r

₃

,…

» 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 1

^st

2005.

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Term Structure, September 12

^th

,

2006

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Term Structure, May 20

^th

,

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.

