# Review of basic Bonds, Interest rates and yield curve

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```					    Markov Chains and Processes                                                          41

Review of basic Bonds, Interest rates and yield curve mathematics
Charles S. Tap iero, Bar Ilan University

A bond is an obligation to pay at a given future date (the maturity date), T , a certain
amount of money. For simplicity, let this amount be \$1. The price of such a bond (that
we are willing to pay for) at a given time t is denoted by b(t , T ) and expresses the time
preference (value) of money in a g iven market. One of the essential topics of study in
finance consists in studying the properties of such a function, for it underlies the
dynamic process of interest rates formation (Boy le 1992). At this time, we shall merely
define what bonds, yield, spot and forward rates are. Let b(t, T ) be the price at time t
of a \$1 payment at maturity ( T  t periods hence). So me obvious properties of this
function are :
         b(t , t )  1

     Lim b(t, T )  0
 (T  t ) 
b(t, s )  b(t, s' ) if s'  s
Equivalently, the time preference for money can be represented by a « yield » wh ich
expresses the bond discount rate. This is also called the term structure interest rate. We
let y(t , T ) be the yield of a bond whose price is b(t, T ) . The relat ionship between the
two is clearly,
( T t )
 b(t , T )  1  y(t , T )
                                        for discrete time discounting

b(t , T ) = exp- y(t , T )(T - t ) for continuous time discounting

Considering the continuous time case, the yield of a bond at time t whose maturity at
time T , is given by
ln(b(t , T ))
y (t , T )  
(T  t )
Pure discount bonds such as the above are one of the « building blocks » of finance
and can be used to evaluate a variety of financial instruments. For examp le, if a default
free bond pays a periodic payment of \$c (also called the coupon payment) as well as a
terminal pay ment of F at time t (also called the bond face value at maturity), then the
price of such a coupon paying bond is :
                   T
bc (t , T )  c  b(t , k  t )  Fb(t , T )     for a discrete stream
                k  t 1
                T
b (t , T )  c b(t ,   t )d  Fb(t , T ) for a continuous stream
  c             
                t
Or, in terms of the yield, for a discrete and continuous stream, we have :

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Markov Chains and Processes                                                                                               42

                  T                            k t
1                                 1
bc (t , T )  c                                   F


k  t 1  1  y (t , k )                1  y(t , T ) T t
T
                          y ( t ,  t )(  t )
 bc (t , T )  c  e                              d  Fe  y ( t ,T )( T  t )
                     t
In the same manner, forward rates represent a relationship between time to maturity
and interest rates. A forward rate is also agreed on at time t but for payments starting
to take effect at the future time t1 and for a certain amount of t ime t2  t1 . Such rates
will be denoted by f (t , t1 , t 2 ) . A relat ionship between forward rates and spot rates (or
alternatively the yield) exists but it hinges on an arbitrage argument. Roughly, this
argument states that two equivalent investments (fro m all points of view) have
necessarily the same returns. Explicitly, say that at time t we invest \$1 for a given
amount of time t2  t at the available spot rate (yield). The price of such an investment
is then :
(t2 t )
b1  1  y(t , t 2 )                      in d iscrete time
b1  exp y(t, t2 )(t2  t ) in continuous time
Alternatively, we could invest this \$1 for a certain amount of time t1  t , t1  t 2 at
which time the moneys available will be reinvested at a forward rate for the remain ing
time interval t2  t1 . The price of such an investment will then be
b2  1  y(t, t1 )     1  f (t, t1, t2 )(t  t ) in discrete time
 ( t1  t )                       2     1

b2  exp y (t , t1 )(t1  t ) exp  f (t , t1 , t 2 )(t 2  t1 ) in continuous time

Arbitrage means that if these investments are equivalent (i.e. have the same risk),
then their value must be the same, or b1  b2 and therefore in discrete and continuous
time we have,
t t
b1  1  y(t, t2 ) 2
t t
= b2  1  y(t , t1 ) 1             1  f (t , t1 , t 2 )t2 t1 (discrete)
b1  exp y(t , t 2 )(t 2  t )
b2  exp y (t , t1 )(t1  t ) exp  f (t , t1 , t 2 )(t 2  t1 ) (continuous)
t2  t1     1  y(t , t 2 )t2 t1
1  f (t , t1 , t 2 )                                             in discrete time
1  y(t , t1 )t1 t
y(t , t 2 )(t 2  t )  y(t , t1 )(t1  t )
f (t , t1 , t 2 )                                      in continuous time
t 2  t1
This is a general equation which can be specialized to specific periods. In particular, we
let f k denote the k periods forward rate expressing the rate agreed on at time t for k

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Markov Chains and Processes                                                                           43

periods hence (and for all subsequent periods of time). In this case, it is simp le to verify
that :

1 fk      
1  y(t , t  k  1)k 1               (discrete time)
1  y(t , t  k )k
as well as :
T t 1
1  y(t , T )    T t
    (1  f k ) (discrete time)
k 0
which provides a relationship between the spot and the forward interest rates. In this
sense, in a perfect market where arbitrage is possible, the spot rate (yield) contains all
the information regarding the forward market interest rate and vice versa. Simila rly, in
continuous                      time,                     note                       that :
b(t , t1 )  e  y (t ,t1 )(t1 t ) , t1  t and
b(t , t 2 ) e  y (t ,t2 )(t2 t )   or
b f (t , t1 , t 2 )  e  f (t ,t1 ,t2 )(t2 t ) =               =
b(t , t1 ) e  y (t ,t1 )(t1 t )
1        b( t , t 1 ) 
f ( t , t1 , t 2 )             ln               
t 2  t 1  b( t , t 2 ) 

1  b( t , t1   ) 
f (t , t1 )  f (t , t1 , t1 )  Lim f (t , t1 , t1   )  Lim ln                  
 0                        0   b(t , t1 ) 
d                       1        d
=         ln b(t , t1 )                  b( t , t 1 )
dt1                   b(t , t1 ) dt1

If we set t1  t  t , then
1            d
f (t , t  t )                                    b(t , t  t )
b(t , t  t ) d (t  t )

and therefore by integration of the differential equation we obtain :
 t  t            

b(t , t  t )  exp  f (t , t   )d 
                    or
 t                 

 t  t              
exp  y (t , t  t )  exp   f (t , t   )d  or

 t                   


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Markov Chains and Processes                                                         44

t  t
1                         
 y(t , t  t )  t   f (t , t   )d 
    t                

which establishes a direct relationship between the yield and the forward rate in
continuous time. In some cases, a concept of duration is used in finance for interest rates
futures hedging. Durat ion of a bond measures the average time the holder of the bond
has to wait before receiving cash payments. A zero coupon bond maturing at time T
has, of course, a duration of T (since only one payment is received at T ). However, if
the bond provides payments  i at times ti , i  1,... n and if the price and yield are B
and y , then :
n
 yt i
n                                                  i ti e
B    i e  yti wh ile the duration is :        D   i 1
n
 yt i
i 1                                                 ie
i 1
B                ln( B)
with D          or D               . Thus, a small duration will mean that the bond is
By                 y
insensitive to the change in interest rates.

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