Transcript Term Structure of Interest Ratesx

Determinants of Interest Rates
• Higher Rates, More money!!!
Copyright © 2014 Diane Scott Docking
1
Learning Objectives
• Know what specific factors determine interest
rates.
• Know what “term structure of interest rates”
means.
• Understand how forward rates of interest can be
derived from the term structure of interest rates.
Copyright 2014 by Diane S. Docking
2
What affects interest rates
on bonds?
Copyright © 2014 Diane Scott Docking
3
The Determinants of Interest Rates
• Factors that may influence interest rates:
 Inflation
 Current economic activity
 Expectations of future growth
 Term to maturity
Copyright © 2014 Diane Scott Docking
4
Interest Rates on Different Maturity Bonds Move Together
Copyright 2014 by Diane S. Docking
http://www.federalreserve.gov/releases/h15/data.htm
5-5
Term to Maturity
• Maturity premium - is the additional return
required by investors on long-term bonds to
compensate investors for the greater price
fluctuations as market interest rates change.
6
Copyright © 2014 Diane
Scott Docking
The Determinants of Interest Rates (cont.)
• The Yield Curve and Discount Rates
 The relationship between the investment term and
the interest rate is called the term structure of
interest rates
 We can plot this relationship on a graph called the
yield curve
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7
Yield Curve
Yield
%
Time to Maturity
An upward-sloping yield curve indicates that Treasury
Securities with longer maturities offer higher annual yields
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8
3 Theories of Term Structure
1. Pure Expectations Theory (aka Unbiased
Expectations Theory)
2. Liquidity Premium Theory
3. Market Segmentation Theory
Copyright © 2014 Diane Scott Docking
9
Pure Expectations Theory
(aka Unbiased Expectations Theory)
Key Assumption:
Bonds of different
maturities are perfect
substitutes
In words:
Rates are a function of
investors’ expectations.
Copyright © 2014 Diane Scott Docking
10
Derivation – Pure Expectations Theory
• Investment strategies for a two-period horizon
 1. Buy $1 of one-year bond and when matures buy another one-year
bond
 2. Buy $1 of two-year bond and hold it
• The current spot rate on 1 year Treasury securities is 1R1%, while the
current spot rate on 2 year Treasury securities is 1R2%.
Year 1
time(beginning of year t) Rmaturity (yrs)
Option I:
1R1%,
Option II:
1R2%.
Year 2
2
f1 %
1R2%.
R = observable spot rate
f = expected forward rate
Copyright © 2014 Diane Scott Docking
11
Derivation – of Geometric Average for Predicting
Current L-T Rates under Pure Expectations Theory
•
Returns should be equal under both options
E  R I   E  R II 
E  R I   1 1 R1 1 2 f 1    1




E  R II   1 1 R 2  1 1 R 2   1  1 1 R 2   1
1 R  1
1
root
1
1 R 1
E 1 R 2 
1
1
f
2
   1  1 1 R 2 
1
  1  1
1/2
2
f

1
 1 R 2  1 1 R 1 1 2 f
1
R2
1

 1
  1
2
2
1
2
2
Cancel common terms, square
1
Copyright © 2014 Diane Scott Docking
Solve for current 2-year rate
12
Derivation – of Geometric Average for Predicting
Current L-T Rates under Pure Expectations Theory
(cont.)
In words: Interest rates on long bond (L-T rates) = geometric average of current and
expected short-term rates over life of long bond
In formula:
Predicting Current L-T rates


E  t R n    1 t R1 1 t 1 f 1 1 t  2 f 1  1 t  n 1 f 1 

where :
R  t he observed market rat e,
f  t he expected forward rat e,
t  t ime period for which t he rat e is applicable ,
n  maturit y of t he bond.
1
n
 1


Arithmetic average for predicting current L-T rates:
E  t Rn  
t
R1  t 1 f 1  t  2 f 1  ... t  n 1 f 1
n
Copyright © 2014 Diane Scott Docking
13
Example:
Predicting current long-term rates under Pure
Expectations Theory
The one-year interest rates over the next five years are
expected to be 5%, 6%, 7%, 8% and 9%. Based upon the
pure expectations theory, what should be today’s interest
rate on a 2-year bond?, 3-year bond? 4-year bond? 5-year
bond?
So Given:
1R1= 5%, 2f1= 6%, 3f1= 7%, 4f1= 8%, 5f1= 9%,
Find:
1R2, 1R3,
1R4,
1R5
Copyright © 2014 Diane Scott Docking
14
Example (cont.):
Predicting current long-term rates under Pure
Expectations Theory
Year 1
Option I:
1 R1 %
Option II:
1 R2 %
Option III:
1 R3 %
Option IV:
1 R4 %
Option V:
1 R5 %
Year 2
2
f1 %
Year 3
Year 4
Year 5
3
f1 %
4
f1 %
5
f1 %
3
f1 %
4
f1 %
5
f1 %
4
f1 %
5
f1 %
5
Copyright © 2014 Diane Scott Docking
f1 %
15
Solution:
Predicting current long-term rates under Pure
Expectations Theory
Year 1
Option I:
1R1%=5%
Option II:
1 R2 %
Year 2
2
Year 3
Year 4
Year 5
f1  6 %
(1+1R2)2= (1+1R1) (1+2f1)
1/2 -1
R
=
[(1+
R
)
(1+
f
)]
1 2
1 1
2 1
1

E  1 R 2    1 . 05 1 . 06  2   1  5 . 499 %


Copyright © 2014 Diane Scott Docking
16
Solution:
Predicting current long-term rates under Pure
Expectations Theory
Year 1
Option I:
1R1%=5%
Option III:
1 R3 %
Year 2
2
f1  6 %
Year 3
3
Year 4
Year 5
f1  7 %
(1+1R3)3= (1+1R1) (1+2f1) (1+3f1)
1/3 -1
R
=
[(1+
R
)
(1+
f
)
(1+
f
)]
1 3
1 1
2 1
3 1
1

E  1 R 3    1 . 05 1 . 06 1 . 07  3   1  5 . 997 %


Copyright © 2014 Diane Scott Docking
17
Solution: Predicting current long-term rates under
Pure Expectations Theory – Geometric Average
1

E  1 R 2    1 . 05 1 . 06  2   1  __________ _


1

E  1 R 3    1 . 05 1 . 06 1 . 07  3   1  _________


1

E  1 R 4    1 . 05 1 . 06 1 . 07 1 . 08  4   1  ________


1

E  1 R 5    1 . 05 1 . 06 1 . 07 1 . 08 1 . 09  5   1  _______


Copyright © 2014 Diane Scott Docking
18
Solution: Predicting current long-term rates under
Pure Expectations Theory – Arithmetic Average
E 1 R2  
E 1 R3  
56
 ________
2
567
 ________
3
E 1 R4  
5678
E 1 R5  
56789
Copyright © 2014 Diane Scott Docking
 _________
4
 _________
5
19
Derivation – Pure Expectations Theory:
of Geometric Average for Predicting Future S-T Rates
• Investment strategies for a two-period horizon
 1. Buy $1 of one-year bond and when matures buy another one-year
bond
 2. Buy $1 of two-year bond and hold it
• The current spot rate on 1 year Treasury securities is 1R1%, while the
current spot rate on 2 year Treasury securities is 1R2%.
Year 1
time(beginning of year t) Rmaturity (yrs)
Option I:
1R1%,
Option II:
1R2%.
Year 2
2
f1 %
1R2%.
R = observable spot rate
f = expected forward rate
Copyright 2014 by Diane S. Docking
20
Derivation – of Geometric Average for Predicting
Future S-T Rates under Pure Expectations Theory
•
Returns should be equal under both options
E  R I   E  R II 
E  R I   1 1 R1 1 2 f 1    1




E  R II   1 1 R 2  1 1 R 2   1  1 1 R 2   1
1 R  1
1 R 1
1
1
2
1
1
f1 
2
f
   1  1 1 R 2 
1
2
f
1
  1  1
1 1 R 2 2
1
1 1 R1 
1
1
R2
 1
  1
2
2
2
Cancel common terms
Solve for “Implied” 1-year forward 1-year rate
(i.e. 1-yr “Implied” forward rate at beginning of
year 2)
Copyright © 2014 Diane Scott Docking
21
Derivation – of Geometric Average for Predicting
Future S-T Rates under Pure Expectations Theory
(cont.)
In words: Future S-T interest rates are a function of current S-T and L-T rates
In formula:
Predicting Future S-T rates
E  t  n 1
 1 t R n  n 
f1   
1
n 1 
 1 t R n 1 

where :
R  t he observed market rat e,
f  t he expected forward rat e,
t  t ime period for which t he rat e is applicable ,
n  maturit y of t he bond.
Copyright © 2014 Diane Scott Docking
22
Example:
Forecasting Future Short-term Interest Rates under
the Pure Expectations Theory
Current rates on bonds are as follows:
1-year bond = 5%
4-year bond = 6.5%
2-year bond = 5.5%
5-year bond = 7%
3-year bond = 6%
What is the expected future rate on a 1-year bond at the beginning
of year 2 (i.e., 1 year from today)? At the beginning of year 3
(i.e., 2 years from today)? At the beginning of year 4? At the
beginning of year 5?
So Given: 1R1= 5%, 1R2 = 5.5%, 1R3 = 6%, 1R4 = 6.5%, 1R5 = 7%
Find: 2f1, 3f1, 4f1, 5f1
Copyright © 2014 Diane Scott Docking
23
Solution:
Forecasting Future Short-term Interest Rates under the
Pure Expectations Theory
Year 1
Option I:
Option II:
1R1=5%
1R2
Year 2
2
Year 3
Year 4
Year 5
f1 %
5.5%
(1+1R2)2= (1+1R1) (1+2f1)
2 /(1+ R )] -1
f
=
[(1+
R
)
2 1
1 2
1 1
E2
 1 . 055 2 
f1   
 1  6 . 002 %
1 
 1 . 05  
Copyright © 2014 Diane Scott Docking
24
Solution (cont.):
Forecasting Future Short-term Interest Rates under the
Pure Expectations Theory
Year 1
Option II:
1R2=5.5%
Option III:
1R3=6%
Year 2
Year 3
3
Year 4
Year 5
f1 %
(1+1R3)3= (1+1R2)2 (1+3f1)
3
2
3f1 = [(1+1R3) /(1+1R2) ] -1
E 3
 1 . 06 3 
f1   
 1  7 . 0071 %
2 
 1 . 055  
Copyright © 2014 Diane Scott Docking
25
Solution: Forecasting future Short-term Interest
Rates under the Pure Expectations Theory
E2
E3
 1 . 055 2
f1   
1
 1 . 05 
 1 . 06 
f1   
2
 1 . 055 
Copyright © 2014 Diane Scott Docking
3

  1  __________


  1  __________ ___

26
Solution: Forecasting future Short-term Interest
Rates under the Pure Expectations Theory
E4
 1 . 065 4 
f1   
 1  ________
3 
 1 . 06  
E5
 1 . 07 5
f1   
4
 1 . 065 
Copyright © 2014 Diane Scott Docking

  1  _______

27
3 Theories of Term Structure
1. Pure Expectations Theory (aka Unbiased
Expectations Theory)
2. Liquidity Premium Theory
Copyright © 2014 Diane Scott Docking
28
Liquidity Premium Theory
• Key Assumption:
• Implication:
Bonds of different maturities
are substitutes, but are not
perfect substitutes
Modifies Pure Expectations
Theory with features of Market
Segmentation Theory
• Investors prefer short-term rather than long-term bonds. This implies that
investors must be paid positive liquidity premium, ℓnt, to hold long term
bonds.
Copyright © 2014 Diane Scott Docking
29
Derivation – Liquidity Premium Theory
• Investment strategies for a two-period horizon
 1. Buy $1 of one-year bond and when matures buy another one-year
bond
 2. Buy $1 of two-year bond and hold it
Long-term interest rates are geometric averages of current and expected
future short-term interest rates plus liquidity risk premiums that increase
with maturity
Year 1
Option I:
Option II:
1R1%,
1R2%.
Year 2
2
f1 %   2 %
1R2%.
Copyright © 2014 Diane Scott Docking
30
Derivation – of Geometric Average for Predicting
Current L-T Rates under Liquidity Premium Theory
•
Returns should be equal under both options
E  R I   E  R II 
E  R I   1 1 R1 1 2 f 1   2    1




E  R II   1 1 R 2  1 1 R 2   1  1 1 R 2   1
1 R  1
1
1
1 R 1
1
E 1 R 2 
1


  1  1 R   1

2
f 1   2   1  1 1 R 2   1
2
2
f 1  2

2
1/2
2
1
 1 R 2  1 1 R 1 1 2 f 1   2 
2

1
2
Cancel common terms, square root
1
Copyright © 2014 Diane Scott Docking
Solve for current 2-year rate
31
Derivation – of Geometric Average for Predicting Current L-T
Rates under Liquidity Premium Theory (cont.)
• Long-term interest rates are geometric averages of current and
expected future short-term interest rates plus liquidity risk
premiums that increase with maturity
Geometric average for predicting current L-T rates:
E  t R n   1 t R1 1 t 1 f 1   t 1 ... 1 t  n 1 f 1   t  n 1 
1
n
1
ℓt = liquidity premium for period t
ℓ2 < ℓ3 < …< ℓn
Copyright © 2014 Diane Scott Docking
32
Example:
Predicting current long-term rates under
Liquidity Premium Theory
The one-year interest rates over the next four years are expected to be 5%,
6%, 7%, 8% and 9%.
Investors' preferences for holding short-term bonds, so liquidity premiums
for one- to five-year bonds are 0%, 0.25%, 0.5%, 0.75%, and 1%.
Based upon the liquidity premium theory, what should be today’s interest
rate on a 2-year bond?, 3-year bond? 4-year bond?, 5-year bond?
So Given:
1R1= 5%, 2f1= 6%, 3f1= 7%, 4f1= 8%, 5f1= 9%,
ℓ1= 0%, ℓ2= 0.25%, ℓ3= 0.5%, ℓ4= 0.75%, ℓ5= 1%,
Find:
1R2, 1R3,
1R4,
1R5
Copyright © 2014 Diane Scott Docking
33
Example (cont.):
Predicting current long-term rates under Liquidity
Premium Theory
Year 1
Option I:
1 R1 %
Option II:
1 R2 %
Option III:
1 R3 %
Option IV:
1 R4 %
Option V:
1 R5 %
Year 2
2
f1 %   2 %
Year 3
Year 4
Year 5
3
f1 %   3 %
4
f1 %   4 %
5
f1 %   5 %
3
f1 %   3 %
4
f1 %   4 %
5
f1 %   5 %
4
f1 %   4 %
5
f1 %   5 %
5
f1 %   5 %
Copyright © 2014 Diane Scott Docking
34
Solution:
Predicting current long-term rates under Liquidity
Premium Theory
Year 1
Option I:
1 R1 %
Option II:
1 R2 %
Year 2
2
Year 3
Year 4
Year 5
f1 %   2 %
(1+1R2)2= (1+1R1) (1+2f1+ℓ2)
1/2 -1
R
=
[(1+
R
)
(1+
f
+ℓ
)]
1 2
1 1
2 1
2
1
R 2  1 1 R1 1 2 f 1   2 
1
 1 . 05 1  . 06  . 0025
 1 . 05 1 . 0625

1
2
2
1

1
2
1
 1  5 . 6232 %
Copyright © 2014 Diane Scott Docking
35
Solution: Predicting current long-term rates
under Liquidity Premium Theory
R 2  1 1 R1 1 2 f 1   2 
1
1
2
 1 . 05 1  . 06  . 0025
 1 . 05 1 . 0625
1

1
2
1

1
2
1
 1  _________
R 3  1 1 R1 1 2 f 1   2 1 3 f 1   3 
 1 . 05 1 . 0625
1  . 07  . 005 
 1 . 05 1 . 0625
1 . 075 
Copyright © 2014 Diane Scott Docking
1
3
1
1
3
3
1
1
 1  _________
36
Solution: Predicting current long-term rates
under Liquidity Premium Theory
1
1
R 4  1 1 R1 1 2 f 1   2 1 3 f 1   3 1 4 f 1   4 
 1 . 05 1 . 0625
1 . 075 1  . 08  . 0075 
 1 . 05 1 . 0625
1 . 075 1 . 0875 
1
4
1
4
1
4
1
1
 1  ________
R 5  1 1 R1 1 2 f 1   2 1 3 f 1   3 1 4 f 1   4 1 5 f 1   5 
 1 . 05 1 . 0625
1 . 075 1  . 0875 1  . 09  . 01 
 1 . 05 1 . 0625
1 . 075 1 . 0875 1 . 10 
Copyright © 2014 Diane Scott Docking
1
5
1
5
1
5
1
1
 1  ________
37
Relationship Between the Liquidity
Premium and Pure Expectations
Theory
Copyright © 2014 Diane Scott Docking
38
Relationship Between the Liquidity Premium and Pure
Expectations Theory
Our Example
Copyright © 2014 Diane Scott Docking
39
Derivation – Liquidity Premium Theory
Predicting Future S-T Rates
• Investment strategies for a two-period horizon
 1. Buy $1 of one-year bond and when matures buy another one-year
bond
 2. Buy $1 of two-year bond and hold it
Assumption: Long-term interest rates already include liquidity premium
Year 1
Option I:
Option II:
1R1%
- ℓ1%
1R2%
- ℓ2%
Year 2
2
1 R2 %
f1 %
- ℓ2%
Copyright © 2014 Diane Scott Docking
40
Derivation – of Geometric Average for Predicting
Future S-T Rates under Liquidity Premium Theory
•
•
Returns should be equal under both options
Assume current long-term rates already include the liquidity premium.
E  R I   E  R II 
E  R I   1 1 R1   1 1 2 f 1    1




E  R II   1 1 R 2   2  1 1 R 2   2   1  1 1 R 2   2   1
1 R
1
1 R
1
1
1
  1  1 2 f
  1 1 2 f
1
   1  1 1 R 2   2 
1
  1  1 R
1
1 1 R 2   2 
1 1 R1   1 1
2
2
f1 
1

 1
 2  1
2
2
2
2
Cancel common terms, square root
Solve for “Implied” 1-year forward 1-year rate
(i.e. 1-yr “Implied” forward rate at beginning of
year 2)
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41
Formula for Predicting Future S-T Rates under
Liquidity Premium Theory
Liquidity Premium Theory: tRn- ℓnt = same as pure expectations theory; replace
tRn by tRn- ℓnt in above equation to get adjusted forward-rate forecast
E  t  n 1
 1 t R n   nt n

f1   
1
n 1 
 1 t R n 1   nt 1 

where :
R  t he observed market rat e,
f  t he expected forward rat e,
t  t ime period for which t he rat e is applicable ,
n  maturit y of t he bond.
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Example:
Forecasting future Short-term Interest Rates under
the Liquidity Premium Theory
Current rates on bonds are as follows:
1-year bond = 5%
4-year bond = 6.5%
2-year bond = 5.5%
5-year bond = 7%
3-year bond = 6%
Investors' preferences for holding short-term bonds, so liquidity premiums for
one- to five-year bonds are 0%, 0.25%, 0.5%, 0.75%, and 1%
What is the expected future rate on a 1-year bond at the beginning of year 2 (i.e.,
1 year from today)? At the beginning of year 3 (i.e., 2 years from today)? At
the beginning of year 4? At the beginning of year 5?
So Given: 1R1= 5%, 1R2 = 5.5%, 1R3 = 6%, 1R4= 6.5%, 1R5 = 7%
ℓ1= 0%, ℓ2= 0.25%, ℓ3= 0.5%, ℓ4= 0.75%, ℓ5= 1%,
Find:
2f1,
3f1,
4f1,
5f1
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43
Solution:
Forecasting future Short-term Interest Rates under the
Liquidity Premium Theory
Year 1
Year 2
Option I:
1R1- ℓ1=5%-0%
Option II:
1R2- ℓ2=5.5%-0.25%
2
Year 3
Year 4
Year 5
f1 %
(1+1R2-ℓ2)2= (1+1R1-ℓ1) (1+2f1)
2
2f1 = [(1+1R2-ℓ2) /(1+1R1-ℓ1)] -1
E2
 1  0 . 055  0 . 0025
f1   
1


1

0
.
05

0

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2 
 1

 1 . 0525  2 
 
  1  5 . 5006 %
1
 1 . 05  
44
Solution:
Forecasting future Short-term Interest Rates under the
Liquidity Premium Theory
Year 1
Year 2
Option II:
1R2- ℓ2=5.5%-0.25%
Option III:
1R3- ℓ3=6%-0.5%
(1+1R3-ℓ3)3= (1+1R2-ℓ2)2 (1+3f1)
3
2
3f1 = [(1+1R3-ℓ3) /(1+1R2-ℓ2) ] -1
Year 3
3
E3
Year 4
Year 5
f1 %
 1  0 . 06  0 . 005 3 
f1   
1
2 
 1  0 . 055  0 . 0025  
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 1 . 055 3 
 
 1  6 . 0018 %
2 
 1 . 0525  
45
Solution: Forecasting future Short-term Interest
Rates under the Liquidity Premium Theory
E2
E3
 1  0 . 055  0 . 0025
f1   
1
1 . 05 


2

  1  ________

 1  0 . 06  0 . 005 3
f1   
2
 1  0 . 055  0 . 0025 
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
  1  ________

46
Solution: Forecasting future Short-term Interest
Rates under the Liquidity Premium Theory
E4
 1  0 . 065  0 . 0075 4
f1   
3
 1  0 . 06  0 . 005 
E 5

  1  _______

 1  0 . 07  0 . 01 5
f1   
4
 1  0 . 065  0 . 0075 
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
  1  _______

47
3 Theories of Term Structure
1. Pure Expectations Theory (aka Unbiased
Expectations Theory)
2. Liquidity Premium Theory
3. Market Segmentation Theory
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Market Segmentation Theory
• Key Assumption:
Bonds of different maturities are
not substitutes at all
• Implication:
Markets are completely segmented;
interest rate at each maturity are
determined separately
•
•
Short- and long-term markets are independent of one another and yields are determined
by supply and demand in each maturity market sector.
Assumes borrowers and lenders prefer certain maturities and cannot be induced to
substitute between maturities by small yield changes
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Yield Curves
http://fixedincome.fidelity.com/fi/FIHistoricalYield?refpr=obrfi14
http://stockcharts.com/charts/YieldCurve.html
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