Bond Valuation

1
Bond Valuation

• Econ 181
• Corporate Finance
• Wadia Haddaji
• Department of Economics
• Duke University
• RK-CH

2
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

3
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

4
Repayment Schemes

• Pure Discount or Zero-Coupon Bonds
• Pay no coupons prior to maturity.
• Pay the bonds face value at maturity.
• Coupon Bonds
• Pay a stated coupon at periodic intervals prior
  to maturity.
• Pay the bonds face value at maturity.
• Floating-Rate Bonds
• Pay a variable coupon, reset periodically to a
  reference rate.
• Pay the bonds 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.

5
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

6
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

7
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.

8
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.

9
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

10
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.

11
Bond Ratings
12
The US Bond Market
Amount (bil.). Source U.S. Federal Reserve
(Table L.4, September/2006)
13
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 iR/m
• T Maturity (in years)
• N Number of compounding periods N Tm
• 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

14
Valuing Zero Coupon BondsAn Example

• Value a 5 year, U.S. Treasury strip with face
  value of 1,000. The APR is R7.5 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?

15
Interest Rate SensitivityZero Coupon Bonds

• Consider the following 1, 2 and 10-year
  zero-coupon bonds, all with
• face value of F1,000
• APR of R10, 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

16
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.

17
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
  0.010.0001
• 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
• DV01B(R-0.005) -B(R0.005).
• Method 3 Use calculus

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Computing DV01 An Example

• Reconsider the 1, 2 and 10- year bonds discussed
  before
• Method 3

19
DV01 A Graphical Approach

• DV01 estimates the change in the Price-Interest
  rate curve using a linear approximation.
• ?higher slope implies greater sensitivity

20
Valuing Coupon BondsExample 1 Amortization Bonds

• Consider Amortization Bond
• T2
• m2
• C2,000 ?c C/m 2,000/2 1,000
• R10 ? 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

21
Valuing Coupon BondsExample 1 Amortization Bonds

• Compare with a portfolio of zero coupon bonds

22
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?

23
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

24
Valuation of Coupon BondsExample 2 Straight
Bonds

• 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

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Valuing Coupon BondsThe General Formula

• 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
  cF

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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

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Yield to MaturityA Graphical Interpretation

• 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 B1,000 we
  obviously find R10.

28
Interest Rate SensitivityCoupon 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?

29
Implication of Interest Rate Changes on Coupon
Bond Prices

• Recall the general formula
• 
  
• What is the price of the bond if the APR is 8
  compounded semiannually?
• Similarly
• If R12 B 827.95
• If R 9 B1,000.00

30
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.

31
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?

32
Bond Prices and Interest Rates
33
Bond Yields and Prices

• 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?

34
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

35
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 bonds price for a 1
  percentage change in the interest rate.

36
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 / (1yield)

37
A Few Bond Markets Statistics U.S. Treasuries,
May 20th 2007.
Bills MATURITY DISCOUNT/YIELD DISCOUNT/YIELD
TIME DATE CHANGE 3-Month 08/16/2007 4.72
/ 4.84 0.01 / .010 1341 6-Month 11/15/2007 4.78
/ 4.98 0.01 / .015 1341 Notes/Bonds
COUPON MATURITY CURRENT PRICE/YIELD TIME
DATE PRICE/YIELD CHANGE 2-Year 4.500 04/30/20
09 99-121/4 / 4.84 -0-02 / .035 1408 3-Year 4.500
05/15/2010 99-081/2 / 4.77 -0-031/2 /
.040 1406 5-Year 4.500 04/30/2012 98-281/2 /
4.75 -0-06 / .043 1407 10-Year 4.500 05/15/2017 9
7-15 / 4.82 -0-091/2 / .038 1407 30-Year 4.750 02
/15/2037 96-17 / 4.97 -0-17 / .035 1407
38
Spot Rates

• A spot rate is a rate agreed upon today, for a
  loan that is to be made today
• r15 indicates that the current rate for a
  one-year loan is 5.
• r26 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 RatesAn Example
1
2
3
40
Term Structure, July 1st 2005.
41
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

46
The US Bond Market Amount (bil.). Source U.S.
Federal Reserve (June/2005)

• For comparison, the total market capitalization
  of NYSE firms is approximately 12.6 trillion as
  of April/2005.

47
A Few Bond Markets Statistics
U.S. Treasuries, October 12th 2006.
48
Valuing Coupon BondsExample 1 Amortization Bonds

• Consider Amortization Bond
• T2
• m2
• C2,000 ?c C/m 2,000/2 1,000
• R10 ? i R/m 10/2 5
• Compare with a portfolio of zero coupon bonds

49
Duration

• The logical way to measure sensitivity of the
  bond price to changes in interest rates is to
  take the derivative of the price B with respect
  to effective rate i
• We adjust the derivative by dividing by minus the
  bond price and the number of periods per year m,
  and multiply by one plus the effective rate.
• The measure obtained is often called Macaulay
  Duration.

50
Duration (cont.)

• If we replace n/m with Tn -- which will be the
  time (in years) until the nth cash flow, the
  formula is
• Duration is a weighted average term to maturity
  where the cash flows are in terms of their
  present value. We can rewrite the above equation
  in a simpler format

51
Zero Coupon Valuation Examples

• Find the current market price of a STRIP that
  matures in 5 years and has a face value of 100.
  Assume the APR is 7.5 with annual compounding.
• What if the compounding is quarterly? Continuous?

52
Personal Finance ExamplesZero Coupons

• One and half years from today, you want to buy a
  new car that requires a down payment of 5,000.
  How much do you have to save today assuming Bank
  A is offering a nominal interest rate of 3.2
  with quarterly compounding?
• Bank B is advertising a special rate 3.5 with
  semiannual compounding. Should you switch your
  account to bank B?

53
Personal Finance ExampleCoupon Bonds

• You are buying a new home and you can only afford
  monthly payments of 2000. The current APR is
  7.9. How large of a loan can you afford if the
  term is 20 years? 30 years?

54
YTM Examples

• Example
• Find the yield to maturity on a self-amortizing
  bond that matures in 1 year, makes semiannual
  payments of 1000 and is selling for 1,876.
• Mathematically
• Beyond 5 compound periods, there is no analytic
  solution for coupon bonds (i.e. must rely on
  numerical analysis)

55
Limitations of Yield to Maturity

• Yield to maturity is simply the internal rate of
  return with a different name. As such, this
  measure experiences the same problems when used
  for comparing investments.
• Example (from Brealey Myers)
• Consider 2 bonds, both with 1000 face value,
  annual coupon payments and maturing in 5 years.
• Bond A 5 coupon 8.78 yield market price
  852.11
• Bond B 10 coupon 8.62 yield market price
  1,054.29
• Is bond A a better buy than bond B? That is, has
  the market erred in pricing these two bonds at
  different yields?

56
Limitations of Yield to Maturity (Cont.)

• The discrepancy lies with the timing of cash
  flows.
• Increasing interest rate
• Decreasing interest rate

57
Duration Hedging ExampleGrinblatt Titman

• I own a 1,000, 10 year zero coupon bond with 8
  yield compounded semiannually. How can I
  perfectly hedge this position with a short
  position in a 5-year zero with yield to maturity
  of 8 (semiannual compounding)

58
Yield Curve

• A sample yield curve

59
Real Rates and Nominal Rates

• Money is of value because of its ability to
  purchase goods.
• Measure price changes by inflation defined as the
  percent change in the Consumer Price Index (CPI).
• Adjust the return for our investment by the rate
  of inflation. The adjusted rate is called the
  real rate.
• Often see the real interest rate written as