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					    Bond Markets

  Investments Chapter 7
QFE Section 1.1—1.2 & 20.1
            Bond Markets
• Payments: Redemption value, M, paid at
  maturity, n, and Coupons, Ct, paid at
  specified dates, t, until t = n.
• Ct a form of interest and typically
  expressed as a percentage of M.
• Typically longer term to maturity compared
  to money markets (1<n<30yrs)
• Possiblilty of Capital gain/loss (trade at
  discount/premium).
                  Bond Markets             2
• Issued by government (state/ county/
  municipality) or (large) corporations.
• Domestic currency issued bonds assumed
  risk-free (No exchange risk + right to print
  money).
• Assumed risk-premium, rp, above safe
  rate, rf, commensurate in size to preceived
  risk level of income stream (Security of Ct
  + time-to-maturity related interest rate
  risk).
                   Bond Markets              3
    Prices & Rates of Return?
• From the redemtion value, M, the size and
  number of coupon payments, C, and time-
  to-maturity, n, markets determine a price,
  P, given other instruments.
• P should provide a rate of return
  commensurate with the return on similar
  assets.
• Many ways to calculate a return,
  depending on needs.
• Conventionally involves compounding.
                  Bond Markets             4
        Pure Discount Bonds
• Recall,
                             FVn                      M
            DPV                            P
                                                  1  rs 
                                  n                   n  n
                          (1  rs )     n



• Thus, the price today of money to be received
  in n periods time should be commensurate with
  the discount rate:
                                                       1
                    1     M P      n  M 
                                                n
               rs                rs     1
                             P             P
• rs(n) is the price of n-period money (annual)
• Spot Rate for n-period money: rsn = f(P, M, n)
                            Bond Markets                           5
 Example: Zeroes & Spot Rates
• P = 62,321.30, M = 100,000, n = 6years

• Spot rate?

                           1
        6     100, 000  6
  rs                      1  0.082  8.2% p.a.
                62,321.3 


                          Bond Markets                6
     Coupon Paying Bonds?
• Stream of coupon payments Ct, which are
  ‘known’ at issue.
• Government bonds’ coupons are generally
  fixed. May actually be indexed or variable
  [in corporate bond case].
• Redeemable at €M at some specified time
  in the future [perpetuities aside].
• Prices quoted clean. Prices paid involve
  accrued interest, i.e. dirty price.
                  Bond Markets             7
                Current Yield
• A.k.a. running/flat/interest yield:
                      C Annual
                       PClean
• Quick summary of simple interest, i.e. annual
  income relative to expenditure
• Caveats:
   – No capital gain
   – Time to Maturity & Face Value?
   – Interest on coupons?
                        Bond Markets          8
         Yield to Maturity (YTM)
• YTM can be calculated ex ante:
YTM = y = f(P,M,C,n)
       C       C                  C             C       M
P 
 (n)
                      ...          n 1
                                                     
    (1  y) (1  y)
           1        2
                              (1  y)        (1  y) (1  y) n
                                                    n



• YTM the discount rate (rate of return) that sets
  the price of a n-period bond, P(n), equal to the
  PDV of its income stream.
• YTM is the internal rate of return of the bond’s
  cash-flow.

                          Bond Markets                       9
    YTM: Assumptions & Caveats
• Assumes bond held to maturity.
• Assumes reinvest C at rate y. Why?
• The rate of return is constant at y.
• If several payments per period (year) the rate is
  grossed-up in simple annual terms. [See
  examples on semi-annual coupons.]
• Inverse relationship among P and y.
• P = f(y), and that function is convex in y (non-
  linear).
• If y = C => P = M. Why?
                      Bond Markets               10
       Aside: Pricing an Annuity
• Recall, the price of a perpetuity:

                  P  0
                                  C
                    Perpetuity            y
• Thus, an annuity in n periods time should cost:

                                C  1 
                P  n
                                            
                                y  1  y  
                  Perpetuity                n
                                             
• The difference should be the price of a T-period annuity.

                           Bond Markets                   11
   Breaking-Up a Coupon Paying
              Bond

           C      1        M
        P  1          
           y  1  y   1  y 
                       n           n
                        
• For simplification, we can view the coupon
  paying bond as having a one-off lump-sum
  payment at redemption, M, and a series of
  periodic payments, the coupons, which can be
  priced as an annuity.
                   Bond Markets             12
        Example: YTM of Semi-Annual
           Coupon Paying Bond
 • P = 900 C = 10% of M = 1,000, 3 years to
   maturity semi-annual. y?
900 
  50              50               50               50               50               50              1000
        1
                        2
                                         3
                                                          4
                                                                           5
                                                                                            6
                                                                                                             6
   y             y              y              y              y              y              y
 1            1             1             1             1             1             1 
 2             2              2              2              2              2              2



 • y = 0.142 = 14.2% p.a.

                                              Bond Markets                                             13
 Example: Price when YTM given?
• 20-year, C = 10% of M = 1,000, YTM = 0.11 p.a.
  semi-annual
• P?
         50       50                  50            1000
  P                     ...               
      0.11   0.11  2
                                  0.11 
                                           40
                                                 0.11 
                                                           40

      1     1                1            1 
          2  
                     
                   2            
                                         
                                       2            2 
                                                         


             50          1           1000      
• or     P       1                            802.31
            0.055  1  0.055   1  0.055 
                               40              40
                                               
                          Bond Markets                      14
     (One Period) Holding Period
           Return (HPR)
• Ex post measure of return
                        n 1            n
              n     Pt 1  Pt                 Ct
           H t 1                     n
                                 Pt




                        Bond Markets                   15
 Realised Compound Yield (RCY)
• A.k.a. Total return/effective holding period
  return.
• Ex post measure of return.
• Assumes the interest earned on each coupon is
  known, plus resale price.
• The RCY of a bond held for n-periods:
                                         1
                        TV           TV  n
    1  rRCY 
                  n
                           rRCY       1
                         P           P 
                          Bond Markets          16
                 Example: RCY
• 5 year bond with C = 10% pf M = 1,000, trading
  at par.
• RCY assuing 2 year horizon, interest rate r = 8%
  and a YTM after 2 years of 9%?
• TV of Coupons: 100  100(1.08)  208
  3     100    100      100     1000
P2                                    1025.31
          1.09 1.09  1.09  1.09 
                      2       3        3



                              1233.31
          1  rRCY 
                        2
                                      rRCY  0.1105
                               1000
                              Bond Markets              17
            Pricing a Bond
• A coupon paying bond must be priced
  such that each its payments is discounted
  by the pertinent spot rate.
• Deviations from this policy will result in
  arbitrage opportunities from coupon
  stripping.
• Hence, if arbitrage opportunities exist
  traders will exploit these, thus exerting
  pressure on prices. This behaviour will
  eliminate the arbitrage opportunities.
                   Bond Markets            18
                Bond Pricing
             C          C                CM
       P                       ... 
          1  r1  1  r2  2
                                        1  rn 
                                                  n



• Each coupon represents a single payoff at a
  certain time in the future.
• Each payment can thus be treated as
  comparable to a zero of equal maturity.
• If provided with spot rates you should be able to
  find a price for a bond.
                        Bond Markets                  19
         Bond Pricing: Spots
• Bond A: coupon 8¾% of FV = 100 annual, 2
  years to maturity
• Bond B: coupon 12% of FV = 100 annual, 2
  years to maturity
• Spots: r1 = 0.05 r2 = 0.06
               8.75 108.75
          PA             2
                                105.12
               1.05 1.06
                12   112
          PB           2
                              111.11
               1.05 1.06
                     Bond Markets            20
  Calculating Spot (Bootstrapping)
• Riskless deep discount securities only
  have short maturities.
   Spot rates of longer maturities have to
  be imputed.
• Take the spot rates you have, say up to a
  year, then calculate the spot rate for the
  next period (e.g. six months, year) using
  comparable (riskless) instruments, such as
  coupon paying government bonds of that
  maturity.
                   Bond Markets           21
       Example: Bootstrapping
• Spot rates (annual return) given for first six
  months, r1 = 8%, and year, r2 = 8.3%.
• Calculate the 18-month spot rate given an 18-
  month coupon paying bond with C = 8.5% of M
  = 100 semi-annual.
              4.25            4.25            104.25
   99.45            1
                                     2
                                                      3
              r1            r2           r3 
             1            1           1  2 
              2               2               
                                          104.25
   99.45  4.0865  3.9180                        3
                                     1     r3
                                         2
                                          
   r3  0.0893                  Bond Markets               22
                Coupon Stripping
• C = 12.5% of FV = 100 = P, semi-annual, 20 yrs
• YTM?
               6.25       6.25              106.25
          P                    2
                                    ...         20
              1.0625 1.0625                1.0625
          P  5.88  5.54  ...  31.61  100
•   Spots: r6months = 0.08 and r12months = 0.083
•   PV(C1) = 6.25/1.04 = 6.0096
•   PV(C2) = 6.25/(1.0415)2 = 5.7618
•   Profits?
                             Bond Markets              23
         Equilibrium Price
• Spot 1-year r1 = 0.1
• Spot 2-year r2 = 0.11
• Consider 2-year coupon bond, C= 9% of M
  = 1000 & P = 966.4866
• Stripping coupons:
• PV(C1)= 90/1.1 = 81.8182
• PV(C1)= 1090/1.112 = 884.668
• What is your guess as to the YTM?
                 Bond Markets          24
          Accrued Interest
• Cum-dividend: clean + accrued
• Ex-dividend: clean - rebate




                  Bond Markets    25
 Example: Accrued Interest (Cum
           Dividend)
• 31.03.1993 a 9% T-Bill 2012 quoted at
  £106(3/16) for settlement 1.04.1993.
• Last coupon on 6.02.1993
• Accrued Interest?
• 22 days in February + 31 in March + 1
  April.
• N = 54  9(54/365) = 1.3315

                  Bond Markets            26
     Example: Accrued Interest
            (Rebate)
• 31.03.1993 a 9% Treasury 2004 quoted at
  £111(5/32)xd
• Next coupon date is 25.04.1993 (i.e. 24
  days)
• Rebate? 9(24/365) = 0.592
• Dirty Price? £111(5/32) – 0.592 = 110.56



                  Bond Markets           27
          Convertible Bonds
• A bond that can be converted to a
  specified number of shares from a certain
  date on.
• Allows for a lower initial cost of capital,
  since the option to convert provides the
  holder with upside potential.



                    Bond Markets                28
           Call Provisions
• Bonds are described as callable if they
  can be redeemed from a certain date on at
  (above) a specified strike price.
• The bond will tend not to trade above the
  strike price.
• Implies that if interest rates fall the
  company can refinance.


                  Bond Markets            29

				
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