Transcript Presentation

PROPERTIES OF YIELD CURVES
AND FORWARD CURVES
FOR AFFINE TERM STRUCTURE MODELS
Gennady Medvedev
Belarusian State University
Minsk, Belarus
The term structure of interest rates plays the
key role in pricing of bonds. In this paper an attempt
of detailed description of every possible shapes of
term structure is made for the class of affine interest
rate models because for these models the solutions
in closed form are attainable.
As basis the general model (GM) with
arbitrary lower boundary for interest rate is taken.
The results for well-known models the CIR model
and the Vasiček model are obtained as particular
cases.
It is found that there is four modes for yield
curve shapes. The empirical evidences are presents
that are based on the parameter estimates for 13
different models of real time series of yield interest
rates.
The Short Interest Rate r(t) Model
The affine models of the term structure are
occurred if the short interest rate r(t) follows the
stochastic process described by a stochastic
differential equation: for r(0) +   0,
dr(t)  (r(t) + )dt +
 r (t )   dW(t),
(1)
where parameters , , , and  are constants, and
W(t) is a standard Wiener process. It is supposed that
the values of parameters , ,  and  are such that a
stationary solution of equation (1) exists. Then this
equation may be rewritten in the more convenient
form: for r(0)  х
dr(t)  k(  r(t))dt +
2kD
r (t )  x
dW(t). (2)
 x
The relations between parameters of equations
(1) and (2) are set by compare: k  –   0,
–


  
 0, D 

0,
х

–
 .
2


2
(3)
If the no arbitrage condition is valid then such
interest rate models generate the affine yield class of
term structures.
Parameters of equation (2) have specific practical
interpretation:
 – stationary expectation of short interest rate r(t);
D – stationary variance of r(t);
х – lower boundary for the process r(t): r(t)  х;
this boundary is inaccessible if ( – x)2  D;
k – parameter that determines correlation
coefficient of process (1) in the form
()  E[(r(t) – )(r(t ) – )]/D  exp– k||.
Example of short rate process r(t) when
Expectation  = 6%, Variance D = 16,
Lower boundary x = 1%, Parameter k = 0,05.
10
Rate (%)
8
6
4
2
0
0
Short Rate
10
20
Expectation
30 Time 40
Lower Boundary
The Price of Zero Coupon Bond
Within the affine class of term structures of
interest rate at current time point t, when r(t)  r, the
price P(t, r, Т) of discount bond with maturity date Т
and unit payment is determined by formula
P(t, r, Т) = exp{A()  rВ()},   Т  t.
(5)
The term structure functions A() and В() are
1
 

В() =  
V  ,
 e 1

(8)
(  x ) 2
A   x [ B ()  ] 
[v  ln(1  vB( ))] , (9)
D
where for short designation it is assumed
2
2kD 
4kD

       2    k  
,
 
 x  x

2
1
2kD 
1
2kD 
v    k  
,
V



k




.
2
 x
2
 x
Here the market price of risk (r) is determined as
(r)    2kD
rx
,   0.
 x
Yield Rates and Forward Rates
The yield to maturity y(t, T) of zero coupon
bond in framework of the affine term structure has a
form
y(t, T)  
ln P(t , r , T )
rB( )  A()
 y() 
.
T t

The (instantaneous) forward rate
f(t, T)  
 ln P(t , r , T )
dB() dA()
 f()  r

,
T
d
d
that connected with yield to maturity rather simple
relations
1 T
1
y(t, T) 
 f (t , s )ds  y()   f ( s ) ds
0
T t t
and conversely
f(t, Т) 
 [(T  t ) y (t , T )]
dy ()
 f()  y ( )  
.
T
d
Yield Curves and Forward Curves
The explicit forms of functions y() and f() are
у()  x  ( r  x)
B() k (  x)  ln(1  vB( )) 

1 
,

V
v


f()  r + [k(  x)  (V  v)(r  x)]B() 
– vV(r  x)[В()] 2.
The function B() plays the key role in
determination of the functions A(), y(), and f().
1
 

В() =  
V  ,
 e 1

(8)
The function B() is monotonically increasing and
such that
B(0)  0  B()  B()  V  1, 0    .
B()   
B() 
1
(k  2vV )  2 + О( 3) for small ;
2
1  
 2 e  O(e 2 ) for large .
V V
The functions y() and f() will be named further
the yield curve and the forward curve.
The plots of functions B()
for some real time series
B ()
5
4
3
2
1
0
0
5
10
15
Parameter estimation by
Gibbons & Ramaswamy (1993), III
Sun (1992)
Gibbons & Ramaswamy (1993), I
Ait-Sahalia (1996)
Duffie & Singleton (1997), I
Chen & Scott (1993)
CKLS (1992)

20
B()
0,830
0,864
0,965
1,121
1,838
2,500
4,275
y (), f ()
Yield Curves y() and Forward Curves f()
for parameters:
 0,06; D  0,02; x  0,01; k  0,03;
r  0,03 or 0,07.
0,07
0,06
0,05
0,04
0,03
0
10
20
y (), f ()
y
30

40
10

100
f
0,07
0,06
0,05
0,04
0,03
0,01
0,1
1
y
f
Property 1. The yield curve y() and the forward
curve f() take the same values on limiting times to
maturity   0 and   :
f(0)  y(0)  r,
f()  f*(x)  y()  y*(x) 
k
 k
  1   x.
V
 V
(20)
(21)
Because 0  k/V  1 then
x  f()  f*(x)  y()  y*(x)  .
From this it follows in particular that as   
limiting values of yield curve and forward curve
always less of the stationary expectation  of short
interest rate r(t).
Note also that from the definitions of the yield to
maturity and the forward rate following limiting
relations follow as T  t
y(t, T)  y(t, t) = r(t), f (t, T)  f (t, t) = r(t)
that are equivalent the equations (20).
Property 2. If the time to maturity  is small then
the yield curve y() and the forward curve f() can be
presented in the forms
y()  r + ½ [(  r)(k + 2vV)  2kD] + O(2) 
 r + ½ [k(  x)  (V  v)(r  x)] + O(2), (22)
f()  r + [(  r)(k + 2vV)  2kD] + O(2) 
 r + [k(  x)  (V  v)(r  x)] + O(2).
(23)
These formulae indicate that for   0 (the term
structure model is risk neutral)
 if r   then the yield curve y() and the
forward curve f() have a negative slope (decrease)
in neighborhood of value   0;
 if r   then these curves have a positive
slope (increase) in neighborhood of value   0.
Moreover the forward curve f() varies doubly
rather than the yield curve y().
Property 3. The limiting value y*(x) of the yield
curve y() (and f*(x) of the forward curve f() too) as
 that is determined by equation (21) is a
monotonic increasing function of boundary x and on
interval [] takes values
y*() =   (1 + 2k)D/k  y*(х)  y*() = .
Thus if k  (1 + 2k)D then
the limiting value y*(x) is positive for every x  .
In the case when k  (1 + 2k)D
the limiting values of curves f*(x)  y*(x)  0 for
k  kD  (kD) 2  kD
x  x*   
.
D  k  2kD
Property 4. The necessary conditions in order to
the yield curve y() and the forward curve f() take
only nonnegative values for 0    , that is
f()  y()  0
(26)
and for r  0
df ()
 0,
d   0
dy( )
 0,
d   0
(27)
are held for every x  if k  (1 + 2k)D and for
  x  x* if k  (1 + 2k)D.
More about function B()
As already it was said the affine term structure
function B() have the key role for determination the
yield curves and the forward curves and them
properties. It follows from (8) that B() is monotonic
increasing on interval 0, . Note also that from (8)
we have that the inverse function B() is found in
form
(B)  [ln(1 + vB)  ln(1  VB)]/ .
(28)
In future it is convenient for analysis to consider
the forward curve f() and the yield curve y() as the
composite functions that depend on time to maturity
 only by the affine term structure function B(), that
is y()  Y(B()) and f()  F(B()). First, it is
convenient because the possible values of function
B() are situated into a finite interval, therefore the
properties of functions Y(B) and F(B) are visually
displayed by graphical charts on finite interval.
Second, the function B() may be regarded as a
duration measure because [P/r]/P  B().
It is obtained from the equation (28) that
Y(B) 
k (  x )
(r  x) B  k (  x) ln(1  vB) vV
 x

,
V
ln(1  vB)  ln(1  VB)
F(B)  r + [k(  x)  (V  v)(r  x)] B  vV(r  x) В 2.
(30)
y (), f ()
Yield Curves Y(B) and Forward Curves F(B)
for parameters:
 0,06; D  0,02; x  0,01; k  0,03;
r  0,03 or 0,07.
0,07
0,06
0,05
0,04
0,03
0
10
20
y
30

40
f
Y (B ), F (B )
0,07
0,06
0,05
0,04
0,03
0
0,2
Y
Y
0,4
F
F
B
0,6
Property 5. The forward curve F(B) is a concave
function.
If (positive) parameter r in (30) meets
inequalities
k
rx
k


(31)
V v  x V v
then forward curve F(B) on the interval 0  В  V  1
has maximum in point
B* 
1   x

 V  v .
k
2vV  r  x

(32)
In this case the maximum value of the forward curve
F(B) is determined by formula
[ k (  x)  (V  v)(r  x)]2
F(В*)  r +
.
4vV (r  x)
If the parameter r meets the inequality
rx
k

then the forward curve F(B) strongly
 x V v
increase on the interval 0  В  V  1.
If the parameter r meets the inequality
rx
k

then the forward curve F(B) strongly
 x V v
decrease on the interval 0  В  V  1.
Property 6. If the value of short rate r meets the
inequality
rx k  v
 ln1  
 x v  V 
then the yield curve Y(B) is a concave function on
interval 0  В  V  1.
If the value of short rate r meets the inequality
rx
k

  x v V
then the yield curve Y(B) is a convex function on
interval 0  В  V  1.
If the value of short rate r meets the
inequalities
k
rx k  v

 ln1  
v V   x v  V 
then the yield curve Y(B) has a point of inflexion Bi
on interval 0  В  V  1. In this case the yield curve
Y(B) is a concave function on the interval 0  В  Bi
and a convex function on the interval Bi  В  V  1.
The shapes of the yield curve Y(B) and the forward
curve F(B) versus value the interest rate r
(r – a value of short rate on pricing date)
THE FULFILMENT OF INEQUALITIES
(36)
(35)
(34)
(33)
concave,
concave,
concave,
F(В)
has maximum
increases
decreases
at point В*
has
concave
convex,
a point of
maximum
increases
inflexion, at point В0, decreases
Y(В)
increases
В0  В*
(mode D)
(mode C)
(mode B) (mode A)
there is
intersection Y(В) 
Y(В)  F(В)
at point В0,  F(В)
В0  В*
rx
k
k
k  v rx

(33);
ln1   

(34);
 x V v
v  V   x V v
k
rx k  v

 ln1   (35);
v V   x v  V 
rx
k

(36).
  x v V
0,04
0,08
0,07
0,06
0,03
0,05
0,04
0,03
0,02
0,02
0,2
0
0,6
0,4
B
0,6
0,4
B
0,8
c
F
Y
c
F
Y
0,2
0
0,8
Figure 1a. F – the forward curve F(B),
Y – the yield curve Y(B) of mode A,
c – the straight-line segment c(B).
Figure 1b. F – the forward curve F(B),
Y – the yield curve Y(B) of mode B,
c – the straight-line segment c(B).
0,03
0,03
0,02
0,02
0,01
0
0,01
0,2
0
Y
0,6
0,4
F
B
0,8
c
Figure 1c. F – the forward curve F(B),
Y – the yield curve Y(B) of mode C,
c – the straight-line segment c(B).
0,2
0
Y
0,6
0,4
B
F
0,8
c
Figure 1d. F – the forward curve F(B),
Y – the yield curve Y(B) of mode D,
c – the straight-line segment c(B).
Parameter values:   0,06; D  0,002; x  0,05; k  0,03;
r 0,07 (mode A), 0,03 (mode B), 0,014 (mode C), 0,005 (mode D).
Probabilities of Modes for Yield Curves
To classify the shape mode of the yield curve
Y(B) it is convenient to use the (random) variable
(r  x)/(  x). The yield curve Y(B) has the
mode D if   k/(v  V), the yield curve Y(B) has the
mode C if k/(v  V)    kln(1  v/V)/v, the curve
Y(B) has the mode B if kln(1  v/V)/v    k/(V  v),
the yield curve Y(B) has the mode A if k/(V  v)  .
The probabilities of these random events are
presented in the four last columns.
Probabilities of modes
D
C
B
A
CKLS (1992)
0,453 0,025 0,080 0,442
Sun (1992)
0,535 0,003 0,009 0,453
Gibbons & Ramaswamy (1993), I 0,422 0,000 0,000 0,578
Gibbons & Ramaswamy (1993), II 0,500 0,000 0,000 0,500
Gibbons & Ramaswamy (1993), III 0,378 0,000 0,000 0,622
Chen & Scott (1993)
0,583 0,021 0,073 0,323
Pearson & Sun (1994)
0,511 0,004 0,011 0,474
Ait-Sahalia (1996)
0,458 0,007 0,022 0,513
Duffie & Singleton (1997), I
0,036 0,001 0,003 0,960
Duffie & Singleton (1997), II
0,001 0,001 0,084 0,914
Bali (1999)
0,201 0,062 0,292 0,445
Ait-Sahalia (1999)
0,282 0,045 0,329 0,344
Ilieva (2001)
0,490 0,006 0,019 0,485
DATA SOURCE
From Table one could say that the modes B and C
for real models are unlikely occurred. This property
is similar to property of the Vasiček Model where
these modes are absent.
0,09
0,08
0,07
0,06
0,0
0,5
1,0
1,5
2,0
2,5
3,0
3,5 B () 4,0
4,5
Figure 2. The yield curves Y(B) (solid line) and the forward curves F(B) (dashed line) for the
CKLS (1992) parameters of models for different values of spot rate r  0,065; 0,073; 0,077;
0,085; and barrier x  0 (CIR model case). The limiting point of curves is shown by circle.
The markers on axis B mean terms to maturity   0,5; 1; 2; 3; 5; 7; 10; 20; 30 years.
0,10
0,09
0,08
0,07
0,06
0,0
0,5
1,0
1,5
2,0
2,5
3,0
3,5
4,0 B () 4,5
Figure 3. The yield curves Y(B) (solid line) and the forward curves F(B) (dashed line) for the
CKLS (1992) parameters of models for different values of spot rate r  0,061; 0,071; 0,077;
0,09; and barrier x  – 1000 (quasi Vasiček model case). The limiting point of curves is
shown by circle. The markers on axis B mean terms to maturity
  0,5; 1; 2; 3; 5; 7; 10; 20; 30 years.
0,085
0,075
0,065
0,055
0,045
0,035
0
5
10
15
20
B ()
25
Figure 4. The yield curves Y(B) (solid line) and the forward curves F(B) (dashed line) for the
Bali (1999) parameters of models for different values of spot rate r  0,0385; 0,044; 0,055;
0,08; and barrier x  0 (CIR model case). The limiting point of curves is shown by circle.
The markers on axis B mean terms to maturity   0,5; 1; 2; 3; 5; 7; 10; 20; 30 years.
0,09
0,07
0,05
0,03
0,01
0
5
10
15
20
25
30
B () 35
Figure 5. The yield curves Y(B) (solid line) and the forward curves F(B) (dashed line) for the
Bali (1999) parameters of models for different values of spot rate r  0,014; 0,025; 0,045;
0,08; and barrier x  – 1000 (quasi Vasiček model case). The limiting point of curves is
shown by circle. The markers on axis B mean terms to maturity
  0,5; 1; 2; 3; 5; 7; 10; 20; 30 years.
0,080
0,075
0,070
0,065
0,060
-0,05
-0,03
-0,01
0,01
0,03
y(0) = f(0)
y(2)
f(2)
f(10)
y(20)
f(20)
x
0,05
y(10)
Figure 6. The yield curves (solid line) and the forward curves (dashed line) for the CKLS
(1992) parameters of models for the spot rate r  0,06 as functions of parameter x.
The designations y() and f() are pointing to the curves for
different terms to maturity   0; 2; 10; 20 years.
0,07
0,06
0,05
0,04
0
2
y(0)
4
f(0)
6
y(0,03)
8
f(0,03)
10
y(0,05)

12
f(0,05)
Figure 7. The yield curves (solid line) and the forward curves (dashed line) for the CKLS
(1992) parameters of models as functions of parameter  for the spot rate r  0,06 and the
term to maturity   2. The designations y(x) and f(x) are pointing to the curves for
different barriers x  0; 0,03; 0,05.