Part 2 Interest Rates and Bond Valuation

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• Part 2Interest Rates and Bond Valuation
• Topics Covered
• Bonds and Bond Features
• Bond Valuation
• Price-Yield Relationship
• Term Structure of Interest Rate
• Default Risk and Bond Ratings

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2.1 Bond Features

• Bond - evidence of debt issued by a corporation
  or a governmental body. A bond represents a loan
  made by investors to the issuer. In return for
  his/her money, the investor receives a legaI
  claim on future cash flows of the borrower. The
  issuer promises to
• Make regular coupon payments every period until
  the bond matures, and
• Pay the face/par/maturity value of the bond when
  it matures.
• Default - since the abovementioned promises are
  contractual obligations, an issuer who fails to
  keep them is subject to legal action from the
  lenders (bondholders).

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2.2 Bond Rates and Yields

• Consider such a bond. It sells for 924.18, pays
  an annual coupon of 80, and it matures in 5
  years. It has a face value of 1000. What are its
  coupon rate, current yield, and yield to maturity
  (YTM)?
• 1. The coupon rate (or just coupon) is the
  annual dollar coupon as a percentage of the
  face value
• Coupon rate 80 /1000 8
• 2. The current yield is the annual coupon divided
  by the current market price of the bond
• Current yield 80 / 924.18 8.66

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2.3 Bond Rates and Yields (concluded)

• 3. The yield to maturity (or YTM) is the rate
  that makes the market price of the bond equal
  to the present value of its future cash flows.
  It is the unknown r in the equation
• 924.18 80 ? 1 - 1/(1 r)5/r
  1000/(1 r)5
• The way to find the YTM is used to be by
  trial and error,
• a. Try 8 80 ? 1 - 1/(1.08)5/.08
  1000/(1.08)5 1000
• b. Try 9 80 ? 1 - 1/(1.09)5/.09
  1000/(1.09)5 961.10
• c. Try 1080 ? (1 - 1/(1.10)5/.10
  1000/(1.10)5924.18
• So, the yield to maturity is 10.
• But this is so 1980s. Now, we use the IRR
  function on Excel.

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2.4 The Bond Pricing Equation

• Bond Value Present Value of the Coupons
• Present Value of the Face Value
• C ? 1 - 1/(1 r )t/r F /(1 r )t
• where C Coupon paid each period
• r Rate per period
• t Number of periods
• F Bonds face value

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2.5 Valuing a Bond An Example of Annual Coupon
Bond

• Barnhart, Inc. bonds have a 1000 face value. The
  promised annual coupon is 100. The bonds mature
  in 20 years. The markets required return on
  similar bonds is 10 (that is, YTM 10). What
  is the bonds value?
• 1 Calculate the present value of the face
  value
• 1000 ? 1/1.1020 1000 ? .14864
  148.64
• 2. Calculate the present value of the coupon
  payments
• 100 ? 1 - (1/1.1020)/.10 100 ? 8.5136
  851.36
• 3. The value of each bond 148.64 851.36
  1000

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2.6 Premium Bond

• One more example. Now you have the following
  information.
• Barnhart, Inc. bonds have a 1000 face value,
  mature in 20 years
• The promised annual coupon is 100
• The markets required return on similar bonds is
  8.
• 1. Calculate the present value of the face value
• 1000 ? 1/1.0820 1000 ? .21455
  214.55
• 2. Calculate the present value of the coupon
  payments
• 100 ? 1 - (1/1.0820)/.08 100 ? 9.8181
  981.81
• 3. The value of each bond 214.55 981.81
  1,196.36
• Why is this bond selling at a premium to par?

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2.7 Semi-annual coupon bond

• Joe Kernan Corporation has bonds on the market
  with 10.5 years to maturity, a yield-to-maturity
  of 8 percent, and a current price of 850. The
  bonds make semiannual payments. What must the
  coupon rate be on the bonds?
• Total number of coupon payments 10.5 ? 2 21
• Yield to maturity per period 8 / 2 4
• (Note YTM is a simple interest rate concept
  by industry convention.)
• Maturity value F 1000

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2.8 Semi-Annual Coupon Bond Valuation Example
(concluded)

• Substituting the values into the bond pricing
  equation
• Bond
• Value C/2 ? 1 - 1/(1 r )t / r F / (1
  r )t
• 850 C/2 ? 1 - 1/(1 .04)21 / .04
  1000/(1.04)21
• 850 C/2 ? 14.0291 438.83
• C/2 29.31
• So the annual coupon must be 29.31 ? 2
  58.62
• and the coupon rate is 58.62 / 1,000 .0586
  ? 5.86.

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2.9 Bond Price Sensitivity to YTM The
Price-Yield Relationship
Bond price
1,800
Coupon 10020 years to maturity1,000 face
value
1,600
Key Insight Bond prices and YTMs are inversely
related.
1,400
1,200
1,000
800
600
Yield to maturity, YTM
12
4
6
8
10
14
16
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2.10 Coupon Payment and Interest Rate Risk

• Bond J is a 4 coupon bond. Bond K is a 10
  coupon bond.
• Both bonds have 8 years to maturity, make
  semi-annual payments, and have a YTM of 9.
• If interest rates suddenly rise or fall by 2,
  then how does this affect the bonds prices?
• What does this exercise tell you about the
  interest rate risk of lower-coupon bonds?

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2.11 Coupon Payment and Interest Rate Risk
(continued)

• Prices if market rates(YTM)s remain at 9
• Bond J
• PV 20 ? 1 - 1/(1.045)16/.045
  1000/(1.045)16
• 719.05
• Bond K
• PV 50 ? 1 - 1/(1.045)16/.045
  1000/(1.045)16
• 1056.17

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2.12 Coupon Payment and Interest Rate Risk
(continued)

• Prices if market rates fall by 2 to 7
• Bond J
• PV 20 ? 1 - 1/(1.035)16/.035
  1,000/(1.035)16
• 818.59
• Bond K
• PV 50 ? 1 - 1/(1.035)16/.035
  1,000/(1.035)16
• 1181.41

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2.13 Coupon Payment and Interest Rate Risk
(concluded)

• Percentage Changes in Bond Prices
• Bond Prices and Market Rates
• 7 9 11
• _________________________
  ________
• Bond J 818.59 719.15 633.82
• chg. (13.83) (11.87)
• Bond K 1181.41 1056.17 947.69
• chg. (11.86) (10.27)
• __________________________
  _______
• All else equal, the price of the lower-coupon
  bond changes more (in percentage terms) than the
  price of the higher-coupon bond when market rates
  change.

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2.14 Interest Rate Risk and Time to Maturity
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2.15 Summary of Bond Valuation

• I. Finding the value of a bond
• Bond value C 1 - 1/(1 r )t/r F/(1 r)t
• where C Coupon paid each period
• r Rate per period
• t Number of periods
• F Bonds face value
• Finding the yield on a bond
• Given a bond value, coupon, time to maturity,
  and face value, it is possible to find the
  implicit discount rate, or yield to maturity, by
  using the IRR function on Excel. Remember that
  increasing the rate decreases the bond value.

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2.16 Bond Pricing Theorems

• The following statements about bond pricing are
  always true.
• 1. Given two bonds identical but for maturity,
  the price of the longer-term bond will change
  more (in percentage terms) than that of the
  shorter-term bond, for a given change in market
  interest rates.
• 2. Given two bonds identical but for coupon rate,
  the price of the lower coupon bond will change
  more (in percentage terms) than that of the
  higher coupon bond, for a given change in market
  interest rates.

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2.17 Term Structure of Interest Rates

• The term structure of interest rates, also called
  the yield curve, is the relationship between
  interest rates on loans of different maturities.
• Short term interest rates can be different from
  long term interest rates.
• Long term rates typically are higher than short
  term rates, but this is not always true.
• Therefore, we should compute the PV for cash flow
  at t 1 with the short term interest rate r1,
  and for cash flow occurred at t2 with the long
  term interest rate r2, etc.

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2.18 The Term Structure of Interest Rates
(Graph)
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2.19 Terminology Spot Rate vs Forward Rate

• The yield to maturity on zero-coupon bonds that
  prevails today is called the spot interest rate.
• The forward interest rate is the rate fixed today
  on a loan to be made at some future date.
• Forward contracts are common in commodities and
  foreign exchanges markets, where both parties
  (buyers and sellers) lock in a price for the
  transaction of goods in the future.
• For example, farmers and food processing firms
  may sign forward contracts to mitigate price risk.

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2.20 Spot Rates and Forward Rates

• The following two approaches should yield the
  same return if there is no arbitrage.
• The no-arbitrage principle requires one good,
  one price.
• (1) buy a 2-yr bond today
• (2) buy a 1-yr bond and then buy a 1-yr bond in
  one years time.
• Because two approaches achieve the same purpose,
  then,

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2.21 Forward Rates from Observed Long-Term Rates
fn one-year forward rate for period n rn
yield for a zero-coupon bond with a maturity of n
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2.22 Yield To Maturity and the Term Structure

• Coupon bonds are priced using the term structure
  information.
• An example What is the YTM of a two year 6
  coupon bond, given r1 4, r2 6?
• Answer YTM IRR(-1001.0885, 60, 1060) 5.941
• Somehow, the YTM is the average of the two spot
  rates in the example.

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2.23 Term Structure Theory and Applications

• What Determines the Shape of the Term Structure?
• 1 - Unbiased Expectations Theory (We just went
  through the example)
• 2 - Liquidity Premium Theory (Volatility of
  long-term bonds creates extra risk for investors)
  
• Term Structure Capital Budgeting
• Cash Flows should be discounted using Term
  Structure info
• If possible, you should use the spot rates as the
  basis to discount the cash flows of your project
• But the CF from the project is not as safe as the
  Treasury zero coupon bonds, thus, we also need to
  consider the risk of the project

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2.24 The Default Risk Structure of Interest
Rates

• Government bonds issued by some well developed
  countries are considered (almost) default-risk
  free. However, bonds issued by emerging countries
  and corporations carry significant default risk.
• Based on a bonds default risk, we need to derive
  a new set of discount rates to calculate risky
  bonds.
• Default means that the firm cannot pay the
  interest and/or principal on time
• Bond ratings by SP and Moodys give us the guide
  to adjust for the risk premium required. For
  example, check www.bondsonline.com.

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2.25 Default Risk Premium and Bond Rating

• Agencies such as Moodys and Standard and Poor
  (SP) issue ratings for bonds. The highest rating
  is Aaa (Moodys) or AAA(SP). For bonds rated
  above Baa(Moodys) or BBB(SP), they are
  considered investment grade.
• For those rated below Baa or BBB, they are
  considered speculative grade, or junk bonds.
• Investing in lower rated bonds demand higher
  coupons and YTMs to compensate for the default
  risk involved.

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2.26 Factors Affecting Bond Yields A Summary

• Key Issue
• What factors affect observed bond yields?
• The real rate of interest
• Expected future inflation
• Interest rate risk
• Default risk premium
• Taxability premium