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					    Introduction to Bond Valuation
 Valuing a bond is very similar to valuing an
  annuity.
 As before, we care about the size, timing,
  and risk of the cash flows.
 The basic idea in bond valuation is
  discounting a stream of level cash flows with
  return of principal at the end of the bond’s
  life.
 Most of what is new is terminology.


                                                  1
                   What is a Bond
 A bond is a loan, typically made by investors
  to a corporation or government.
 The indenture spells out the terms of the
  loan:
      Coupon
      Maturity

      Seniority


   A corporation can deduct the interest
    payments on bonds (dividends paid on stock
    are not deductible).
                                                  2
                US Treasury Securities
   The US Treasury is the largest security issuer in the
    world, with £5.2 trillion in debt in 1996.
   It issues three basic kinds of securities:
        Bills (maturities less than one year at issuance)
        Notes
        Bonds (up to 30 year maturity)
   There are other hybrid securities such as STRIPS
    and inflation-indexed bonds.
   Treasury securities are important “benchmark”
    instruments (e.g., “riskless” interest rate).


                                                             3
                Corporate Bonds
 Generally have £1000 par value, semiannual
  coupons.
 Default risk is rated by agencies such as
  Moody’s and S&P.
 Other features may include:
     call provision
     convertibility

     floating rate

     option features




                                               4
                Junk Bonds


 Junk bonds are high-yield instruments with a
  significant probability of default.
 Made popular by Michael Milken of Drexel
  Burnham Lambert in the 1980’s.
 Often used in “leveraged” transactions such
  as LBOs, mergers, acquisitions.



                                                 5
       Characteristics of Bond Prices
 The cash flows on a bond are constant (“fixed
  income”).
 A bond’s market price changes in response to
  the market interest rate.
    When market rates increase, the fixed payments
     from the bond are worth less so the price falls.
    If rates decrease, the fixed payments are now

     worth more.

 [A bond’s price also changes in response to changes in the risk
 of the cash flows, but we are not quite ready for that discussion.]
                                                                   6
             Basics of Bond Valuation
   The bond pricing equation consists of two
    components
        PV of Coupons
        PV of Face Value
   The price of a bond (these PVs) depends on:
      Discount Rate (r)

      Number of Periods (N)


      Size of Cash Flows (C and P N)




                                                  7
                 Yield to Maturity

   The yield to maturity is an important number in bond
    valuation.
   It is the rate which equates the market price of the
    bond with the value of the discounted cash flows.
   That is, YTM is the r such that the bond equation
    holds.
   Finding the YTM requires a financial calculator, a
    goal-seeking solver, or trial and error.



                                                           8
             Example - Annual Coupon
   £1000 10 year bond paying a 10% annual coupon

        What is the value when the interest rate is 10%?




        If r = 11%?



        If r = 9%?


                                                            9
         Example - Semiannual Coupon
   Now the coupon is split semiannually

        At 10%


        At 11%


        At 9%



                                           10
          YTM and the Coupon Rate
   Relationship between YTM and Coupon Rate
      YTM = Coupon  bond is selling at par (P0 = PN).
      YTM > Coupon  bond is at a discount (P0 < PN).

      YTM < Coupon  bond is at a premium (P0 > PN).




   Why does the YTM differ from the coupon?
      The coupon is set when the bond is issued.
      The YTM is the market’s required interest rate. It

       may change as economic fundamentals shift.


                                                            11
Remembering the YTM-Coupon Relationship

   Zero Coupon Bonds
      Pays no coupon so interest comes in the form of a
       discount from the repayment (P0 < PN).
      Since Coupon = 0, YTM must be greater than

       Coupon.
      Putting these pieces together gives the answer.


   Capital Gains
      If the YTM is greater than the Coupon, the extra
       return must be coming from somewhere.
      The extra return comes from capital gains (P0 < PN).



                                                           12
             Example - Solving for YTM
   Consider a £1000 5 year bond with a 8%
    coupon
        What is the YTM if it is selling for £1000?

        If it is selling for £900?

        If it is priced at £1100?




                                                       13
                       Duration

   As we have seen, bonds have value from two
    sources: coupons and return of principal.
   Intuitively, bonds with high coupon rates or short
    maturities will return value more quickly than those
    with low coupons or long maturities.
   At the extreme is a zero coupon bond, which returns
    all value at maturity.

   Duration is a measure of how quickly the (present)
    value of a bond is returned.
                                                           14
                            Duration
   To calculate duration:
        Find the present value of each cash flow individually
        Sum these to get the present value of all cash flows (price)
        Calculate the proportion of the total value from each
         individual cash flow
        Multiply each proportion by the corresponding number of
         periods and sum
   The answer will give a measure of the average life of
    the bond in a present value sense.
   A bonds with a low duration gets most of its value
    from cash flows occurring early.

                                                                        15
              Bond Theorems
 Price and interest rates move inversely.
 A decrease in interest rates raises bond
  prices by more than a corresponding
  increase in rates lowers price.
 Price volatility is inversely related to coupon.
 Price volatility is directly related to maturity.
 Price volatility increases at a diminishing rate
  as maturity increases.


                                                  16
       Illustration of Bond Theorems
   A decrease in interest rates raises bond prices by
    more than a corresponding increase in rates lowers
    price. This is known as convexity.




                                                         17
        Illustration of Bond Theorems
   Price volatility is inversely related to coupon.




                                                       18
        Illustration of Bond Theorems
   Price volatility is directly related to maturity.
   Price volatility increases at a diminishing rate as
    maturity increases.




                                                          19
        Illustration of Bond Theorems
   Price volatility is directly related to maturity.
   Price volatility increases at a diminishing rate as
    maturity increases.




                                                          20
            Risk and Bond Valuation

   So far we have ignored risk in valuing the bonds.
   We will now discuss qualitatively the types of risk a
    bondholder faces.
   Quantification of the price impact due to risk is still
    coming.
   In all cases, adding risk to a security increases
    the required, or expected, return.
   This implies that an increase in the risk of a bond will
    lower its current price.


                                                           21
      Types of Risk Bondholders Face
   Interest Rate Risk
     The risk of a bond changing in value when interest rates
      change. This affects all bonds regardless of credit quality,
      but is more severe for longer maturity bonds.
   Reinvestment Risk
     The risk that investors will be unable to reinvest the coupon
      payments at the coupon rate. This is more important for
      high coupon bonds.
   Default (Credit) Risk
     The risk that the firm will go bankrupt and not make all
      payments to bondholders.
   Other Risks: Inflation, Call, Liquidity

                                                                     22
           Inflation and Interest Rates

   Inflation is the increase in the nominal (or cash) cost of goods
    and services over time.
   Put differently, it is the decrease in purchasing power over time.
   In the end, we are generally concerned with consumption in
    finance (and in life). The amount of dollars you have is really
    much less important than their purchasing power.
   Nominal rates are the rates observed in the market and quoted
    in contracts.
   Real rates are actually very illusive since measuring inflation
    accurately is difficult.



                                                                     23
    Term Structure of Interest Rates


 A graph of interest rates on securities of
  various maturities.
 Generally constructed using riskless zero
  coupon bonds (i.e., Treasuries).
 Serves as a measure of the Time Value of
  Money.
 Generally upward sloping, but can also be
  downward sloping, inverted, or humped.

                                               24
%




                                                 40%




                                                 20%




                                                  0%




                                                 -20%




                                                 -40%
                                                        4%   6%   8%        10%        12%   14%            16%

                                                                       Interest Rate


                                                                                                                  18
        Illustration of Bond Theorems
   Price volatility is directly related to maturity.
   Price volatility increases at a diminishing rate as
    maturity increases.
                                                       90%
                                                                                                        30 yr, 10%
                                                                                                        20 yr, 10%
                                                       80%                                              10 yr, 10%
              Price Volatility (|% Change from par|)




                                                       70%



                                                       60%



                                                       50%



                                                       40%



                                                       30%



                                                       20%



                                                       10%



                                                       0%
                                                             4%   6%   8%        10%        12%   14%                16%

                                                                            Interest Rate




                                                                                                                           19
        Illustration of Bond Theorems
   Price volatility is directly related to maturity.
   Price volatility increases at a diminishing rate as
    maturity increases.

                                       200%
                                                                     Illustration of Bond Theorems
                                                  5% Interest Rate
                                                  10% Interest Rate
                                       180%
                                                  15% Interest Rate

                                       160%
             Percentage Price Change




                                       140%


                                       120%


                                       100%


                                       80%


                                       60%


                                       40%


                                       20%


                                        0%
                                              0              5             10          15          20   25   30

                                                                                Years to Maturiy



                                                                                                                  20
            Risk and Bond Valuation

   So far we have ignored risk in valuing the bonds.
   We will now discuss qualitatively the types of risk a
    bondholder faces.
   Quantification of the price impact due to risk is still
    coming.
   In all cases, adding risk to a security increases
    the required, or expected, return.
   This implies that an increase in the risk of a bond will
    lower its current price.


                                                           21
      Types of Risk Bondholders Face
   Interest Rate Risk
     The risk of a bond changing in value when interest rates
      change. This affects all bonds regardless of credit quality,
      but is more severe for longer maturity bonds.
   Reinvestment Risk
     The risk that investors will be unable to reinvest the coupon
      payments at the coupon rate. This is more important for
      high coupon bonds.
   Default (Credit) Risk
     The risk that the firm will go bankrupt and not make all
      payments to bondholders.
   Other Risks: Inflation, Call, Liquidity

                                                                     22
           Inflation and Interest Rates

   Inflation is the increase in the nominal (or cash) cost of goods
    and services over time.
   Put differently, it is the decrease in purchasing power over time.
   In the end, we are generally concerned with consumption in
    finance (and in life). The amount of dollars you have is really
    much less important than their purchasing power.
   Nominal rates are the rates observed in the market and quoted
    in contracts.
   Real rates are actually very illusive since measuring inflation
    accurately is difficult.



                                                                     23
    Term Structure of Interest Rates


 A graph of interest rates on securities of
  various maturities.
 Generally constructed using riskless zero
  coupon bonds (i.e., Treasuries).
 Serves as a measure of the Time Value of
  Money.
 Generally upward sloping, but can also be
  downward sloping, inverted, or humped.

                                               24

				
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