>>
Chp.19 Term Structure of
Interest Rates
报告人：陈焕华
报告人：陈焕华
指导老师：郑振龙
厦门大学金融系
教授
2012年12月19日
>> Main Contents
• Some Basic Definitions;
• Yield Curve and Expectation Hypothesis;
• Term Structure Models-A Discrete Time
Introduction;
• Continuous Time Term Structure Models;
• Three Linear Term Structure Models;
• Some Comments
>> Remark
Pt ( j )  Et (mt ,t  j )
>> Definition and Notation
• Bonds:
– Zero-Coupon Bonds(the simplest instrument):
– Coupon Bonds : portfolio of zero coupon bonds.
– Bonds with Default Risk: such as corporate
bonds.
• In this chapter, we only study the bonds without default
risk. And since coupon bonds can be regarded as
portfolio of zero-coupon bonds, the main research is
done to zero coupon bonds.
>> Zero Coupon Bonds
• Price: Pt ( N )
• Log price: pt( N )  ln Pt ( N )
• Log yield: yt( N )   pt( N ) / N
• Log holding period return:
• Instantaneous return:
• Forward rate:
hprt (N1 )  pt(N11)  pt( N )
dP ( N , t )
1 P ( N , t )

dt
dP
P
N
f t ( N  N 1)  pt( N )  pt( N 1)
• Instantaneous forward rate:
f (N , t)  
1 P ( N , t )
P
P
>>
Some proof(1)
Log yield: the yield is just a convenient way to quote the price
Pt ( N )  exp(- Nyt( N ) )
(N )
t
y
– Or
(N )
t
p
N
>> Remark
• Holding Period Returns
>>
Chen, Huanhua Dept. of Finance, XMU
7
>> Some proof(2)
• Instantaneous return:
P ( N  , t   )  P ( N , t )
P( N , t )
 0
P ( N  , t   )  P ( N , t   )  P ( N , t   )  P ( N , t )
 lim
P( N , t )
 0
dP( N , t ) 1 P( N , t )


dt
P
P N
hpr  lim
• Remark: hpr is the time value, dP/P is the total value, the
second item in right equation is the term value. Total value
equals the time value plus the term value.
>> Some proof(3)
• Forward rate:
• Consider a zero cost investment strategy:
–
–
–
–
Buy one N-period zero Pt ( N ) ;
sell Pt ( N ) / Pt ( N 1) N+1 period zero.
The cost is zero.
The payoff is 1 at time N, and  Pt ( N ) / Pt ( N 1) at
time N+1;
>> Some proof(3)
According to no arbitrage condition,
1* Ft ( N  N 1)  Pt ( N ) / Pt ( N 1)  0,
Ft
( N  N 1)
 Pt
(N)
/ Pt
( N 1)
, ft
( N  N 1)
p
(N)
t
( N 1)
t
p
>> Some proof(4)
• Instantaneous forward rate:
P ( N , t )  P ( N  , t )
1 P( N , t )
p( N , t )
f (N , t) 


P ( N  , t )
P N
N
>> Some extensions
Forward rates have the lovely property that you
can always express a bond price as its discounted
present value using forward rates,
pt( N )  pt( N )  pt( N 1)  pt( N 1)  ...  pt(1)  pt(1)
  f t ( N 1 N )  f t ( N  2 N 1)  ...  f t (12 )  yt(1)
Pt ( N )  e Pt
(N)
N 1
 ( Ft ( j  j 1) ) 1
j 0
>> Some extensions
>> Some extensions
Since yield is related to price, we can relate forward
rates to the yield curve directly. Differentiating the
definition of yield y(N , t ) = −p(N , t)/N
y ( N , t ) p( N , t ) 1 p( N , t )
1
1


  y( N , t )  f ( N , t )
2
N
N
N N
N
N
y ( N , t )
f (N , t)  N
 y ( N , t ),
N
f t ( N  N 1)  ( N  1) yt( N 1)  Nyt( N )
>> EH (expectations hypothesis)
>> Log(Net) return: consistent
• EH1:
• EH2:
• EH3:
1
Et ( yt(1)  yt(1)1  ...  yt(1) N 1 )( riskpremium)
N
1
 Et ( yt(1)  ( N  1) yt(N11) )( riskpremium)
N
yt( N ) 
ft N  N 1  Et ( yt(1) N )( riskpremium)
Et (hprt (N1 ) )  yt(1) ( riskpremium)
• When risk premium equals zero, this is PEH.
>> Proof of consistence(1)
• By EH(1) and suppose risk premium is zero,
(N)
t
y
( N 1)
t 1
 1 / NEt ( y  ( N  1) y
(1)
t
)
yt(1)  Nyt( N )  ( N  1) Et yt(N11)
• By EH(3),
( N 1)
t 1
y  Et (hpr )  Et ( p
(1)
t
(N)
t 1
 p )  Ny
(N)
t
(N )
t
( N 1)
t t 1
 ( N  1) E y
>> Proof of consistence(2)
• By EH(2),
f t N 1 N  Et ( yt1 N 1 ),
f t 01  f t12  ...  f t N 1 N  Et ( yt(1)  yt(1)1  ...  yt(1)N 1 )
ft
01
 ft
12
 ...  f t
  p  Ny
N
t
(N)
t
N 1 N
( N 1)
t
 ( p  p )  ( p  p )  ...  ( p
(0)
t
(1)
t
(1)
t
( 2)
t
p )
(N)
t
>>Level (Gross) Return: Self-contradiction
• EH(1):
(N)
exp(N yt(N) )= Et exp(yt(1) +(N - 1)yt+1
),
(1)
t
(N)
t
exp(y )= exp(Ny
(N)
t+1
) / Et (exp(N - 1)y
• EH(3)：
exp( yt(1) )  Et ( Pt (N1 1) / Pt ( N ) )
 Et (exp( Nyt( N )  ( N  1) yt(N11) ))
 Et (exp( Nyt( N ) ) / exp(( N  1) yt(N11) ))
)
>> Discrete Time Model
• Term Structure Models:
– specify the evolution of short rate and potentially
other state variables.
– The prices of bonds of various maturities at any
given time as a function of short rate and other
state variables.
• A way of generating term structure model:
write down the process for discount factor, and
prices of bonds as conditional mean of the
discount factor. This can guarantee the
absence of arbitrage.
>> Properties of the Term Structure
Properties of the Term Structure
Chen, Huanhua Dept. of Finance, XMU
21
>>
Chen, Huanhua Dept. of Finance, XMU
22
>>
Chen, Huanhua Dept. of Finance, XMU
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>> Other term structure model
• Model yields statistically.
– Run regressions;
– Factor analysis.
• Trouble: reach a conclusion that admits the
arbitrage opportunity, which will not be used
for derivative pricing.
• Example: Level factor will result in the comovement of all yields. This means the long
term forward rate must never fall.
>> a model based on EH
• Suppose the one period yield follows AR(1),
yt(1)1     ( yt(1)   )   t 1
• Based on EH(1),
yt( 2)  1 / 2Et ( yt(1)  yt(1)1 )  1 / 2Et ( yt(1)     ( yt(1)   ))
1   (1)
 
( yt   )
2
(N)
t
y
1 1   N (1)
 
( yt   )
N 1 
• Remark: not from discount factor and may not be
arbitrage.
>>implications
• If the short rate is below its mean,
yt( N )
0
N
• Long term bond yields are moving upward. yield
curve is sloping upward.
• If the short rate is above its mean, we get
inverted yield curve.
E ( yt1 )   , E ( yt( N ) )  
• The average slope is zero.
• But we can not produce humps or other
interesting yield curve.
>>Implications(2)
• All bond yields move together.
yt(1)1     ( yt(1)   )   t 1
1   (1)
1 
y  
( yt 1   ) 
[  ( yt(1)   )   t 1 ]
2
2
1 
( 2)
  ( yt   ) 
 t 1
2
N
1
1


yt(N1)     ( yt( N )   ) 
 t 1
N 1 
( 2)
t 1
>> Implication(3)
• AR(1) may result in negative interest rate.
>> Direction for generalization
• More complex driving process than AR(1),
such as hump-shape conditionally expected
short rate and multiple state variables. The
short rate should be positive in all states.
• Add some market price of risk to get average
yield curve not to be flat.
• Term structure literature: specify a short rate
process and the risk premium, and find the
price of long term bonds.
>>The Simplest Discrete Time Model
• Log of the discount factor follows AR(1) with
normal shocks.
• Log rather than level so that the discount factor
is positive to avoid arbitrage.
• Log discount factor is slightly negative.
• Unconditional mean E ln m  
•
ln mt 1     (ln mt   )   t 1
>> An example
• Consumption-based power utility model with
normal errors:
Ct 1 
mt 1  e (
)
Ct

ct 1  ct   (ct  ct 1 )   t 1
>> Bond prices and yields
y   p   ln Et (e
(1)
t
( 2)
t
y
(1)
t
 1 / 2 p
( 2)
t
ln mt 1
)
  ln Et (e
ln mt 1  ln mt 2
)
ln mt  2     (ln mt   )   t 1   t  2
2
ln mt 3     (ln mt   )    t 1   t  2   t 3
3
2
ln mt 1    ln mt  2    (    )(ln mt   )  (1   ) t 1   t  2
2
>> Bond prices and yields(2)
ln Et e
ln mt 1
 ln( e
Et ln mt 1 1/ 2 2 (ln mt 1 )
  (ln mt   )    1 / 2
2

y     (ln mt   )  1 / 2
(1)
t
)  Et ln mt 1  1 / 2
2

2
2



1

(
1


)
( 2)
2
yt   
(ln mt   ) 

2
4
2
>> Bond prices and yields(3)
E ( y )   ln E (e
(1)
ln m
)    1 / 2 
2
yt(1)     ( yt(1)  E ( y (1) ))  1 / 2 2   ( yt(11)  E ( y (1) ))   t
yt( 2 )
2
   2 (1)
1

(
1


)
 
( yt  E ( y (1) )) 
 2
2
4
N 1
yt( N )
 (1   N ) (1)
 
( yt  E ( y (1) )) 
N (1   )
j
k 2
(

 )
j 0 k 0
2N
 2
>> Remark
• It is not a very realistic term structure model.
• The real yield curve is slightly upward. this model
gets the slightly downward yield curve if the noise
term piles up.
• This model can only produces smoothly upward or
downward yield curve.
• No conditional heteroskedasticicy.
• All yields move together, one factor and perfectly
conditionally correlated.
• Possible solution: more complex discount factor
process.
>>
Thank you for listening
and
Comments are welcome.
报告人：陈焕华
指导老师：郑振龙
厦门大学金融系
教授
2012年12月19日