# Bond Markets

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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
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?
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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.

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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.
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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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