chp.19 term structure of interest rates
DESCRIPTION
Chp.19 Term Structure of Interest Rates. 报告人：陈焕华. 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. - PowerPoint PPT PresentationTRANSCRIPT
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报告人：陈焕华 指导老师：郑振龙 教授
厦门大学金融系
Chp.19 Term Structure of Interest Rates
2012 年 12 月 19日
报告人：陈焕华
>> 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
)( ,)(
jtttj
t mEP
>> 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:• Log price:• Log yield:• Log holding period return:• Instantaneous return:• Forward rate:• Instantaneous forward rate:
)()1(1
)(1
Nt
Nt
Nt pphpr
)(NtP
)()( ln Nt
Nt Pp
Npy Nt
Nt /)()(
dtNtNP
PdPtNdP
),(1),(
)1()()1( Nt
Nt
NNt ppf
PtNP
PtNf
),(1),(
>> Some proof(1)
Log yield: the yield is just a convenient way to quote the price
– Or
( ) ( )
( )( )
exp(- )
-
N Nt t
NN tt
P Ny
pyN
>> Remark
• Holding Period Returns
>>
Chen, Huanhua Dept. of Finance, XMU 8
>> Some proof(2)• Instantaneous return:
• 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.
dtNtNP
PPtNdP
tNPtNPtNPtNPtNP
tNPtNPtNPhpr
),(1),(),(
),(),(),(),(),(
),(),(
lim
lim
0
0
>> Some proof(3)• Forward rate:• Consider a zero cost investment
strategy:– Buy one N-period zero ;– sell N+1 period zero.– The cost is zero.– The payoff is 1 at time N, and
at time N+1;
)(NtP
)1()( / Nt
Nt PP
)1()( / Nt
Nt PP
>> Some proof(3)
According to no arbitrage condition,
)1()()1()1()()1(
)1()()1(
,/
,0/*1
Nt
Nt
NNt
Nt
Nt
NNt
Nt
Nt
NNt
ppfPPF
PPF
>> Some proof(4)
• Instantaneous forward rate:
NtNp
NtNP
PtNPtNPtNPtNf
),(),(1
),(),(),(),(
>> Some extensions
1
0
1)1()(
)1()21()12()1(
)1()1()1()1()()(
)(
...
...
)(N
j
jjt
PNt
ttNN
tNN
t
ttNt
Nt
Nt
Nt
FeP
yfff
pppppp
Nt
Forward rates have the lovely property that you can always express a bond price as its discounted present value using forward rates,
>> Some extensions
>> Some extensions
)()1()1(
2
)1(
),,(),(),(
),(1),(1),(1),(),(
Nt
Nt
NNt NyyNf
tNyNtNyNtNf
tNfN
tNyNN
tNpNN
tNpNtNy
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
>> EH (expectations hypothesis)
>> Log(Net) return: consistent
• EH1:
• EH2:
• EH3:• When risk premium equals zero, this is
PEH.
( ) (1) (1) (1)1 1
(1) ( 1)1
1 ( ... )( )
1 ( ( 1) )( )
Nt t t t t N
Nt t t
y E y y y riskpremiumN
E y N y riskpremiumN
1 (1)( )( )N Nt t t Nf E y riskpremium
( ) (1)1( ) ( )N
t t tE hpr y riskpremium
>> Proof of consistence(1)
• By EH(1) and suppose risk premium is zero,
• By EH(3),
)1(1
)()1(
)1(1
)1()(
)1(
))1((/1
Ntt
Ntt
Nttt
Nt
yENNyy
yNyNEy
)1(1
)()()1(1
)(1
)1( )1()()(
N
ttNt
Nt
Ntt
Nttt yENNyppEhprEy
>> Proof of consistence(2)
• By EH(2),
)(
)()1()2()1()1()0(12110
)1(1
)1(1
)1(12110
11
1
)(...)()(...
)...(...
),(
Nt
Nt
Nt
Nttttt
NNttt
NttttNN
ttt
NttNN
t
Nyp
ppppppfff
yyyEfff
yEf
>>Level (Gross) Return: Self-contradiction • EH(1):
• EH(3)：
(N) (1) (N)t t t t+1
(1) (N) (N)t t t t+1
exp(Ny )= E exp(y +(N -1)y ),
exp(y )= exp(Ny ) / E (exp(N -1)y )
)))1exp((/)(exp(
)))1((exp(
)/()exp(
)1(1
)(
)1(1
)(
)()1(1
)1(
Nt
Ntt
Nt
Ntt
Nt
Nttt
yNNyE
yNNyE
PPEy
>> 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 22
>>
Chen, Huanhua Dept. of Finance, XMU 23
>>
Chen, Huanhua Dept. of Finance, XMU 24
>> 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 co-movement 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),
• Based on EH(1),
• Remark: not from discount factor and may not be arbitrage.
1)1()1(
1 )( ttt yy
)(2
1))((2/1)(2/1
)1(
)1()1()1(1
)1()2(
t
ttttttt
y
yyEyyEy
)(111 )1()(
t
NNt y
Ny
>>implications
• If the short rate is below its mean,
• 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.
• The average slope is zero.• But we can not produce humps or other
interesting yield curve.
0)(
Ny Nt
)(,)( )(1 Ntt yEyE
>>Implications(2)
• All bond yields move together.
1)()(
1
1)2(
1)1()1(
1)2(1
1)1()1(
1
111)(
21)(
])([2
1)(2
1)(
t
NNt
Nt
tt
tttt
ttt
Nyy
y
yyy
yy
>> 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•
mE ln
11 )(lnln ttt mm
>> An example
• Consumption-based power utility model with normal errors:
111
11
)(
)(
ttttt
t
tt
ccccCCem
>> Bond prices and yields
)(ln2/1
)(ln21
1
lnln)2()2(
ln)1()1(
tt
t
mmttt
mttt
eEpy
eEpy
212
21
32123
3
212
2
)1())(ln(lnln
)(lnln
)(lnln
ttttt
ttttt
tttt
mmm
mm
mm
>> Bond prices and yields(2)
2)1(
2
21
)(ln2/1lnln
2/1)(ln
2/1)(ln
2/1ln)ln(ln 12
11
tt
t
ttmmEm
t
my
m
mEeeE tttt
222
)2(
4)1(1)(ln
2
tt my
>> Bond prices and yields(3)
2ln)1( 2/1)(ln)( meEyE
2
2
0
1
0)1()1()(
22
)1()1(2
)2(
)1()1(1
2)1()1()1(
2
)())((
)1()1(
4)1(1))((
2
))((2/1))((
NyEy
Ny
yEyy
yEyyEyy
j
k
kN
jt
NNt
tt
tttt
>> 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.
>>
报告人：陈焕华 指导老师：郑振龙 教授
厦门大学金融系
Thank you for listening and
Comments are welcome.
2012 年 12 月 19日