Factors Affecting Bond Yields and
the Term Structure of Interest Rates
Zvi Wiener
Based on Chapter 5 in Fabozzi
Bond Markets, Analysis and Strategies
Fall-02
http://pluto.mscc.huji.ac.il/~mswiener/zvi.html
EMBAF
Base Interest Rate
Treasury
Libor
Prime
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Fabozzi Ch 5
slide 2
Term Structure of Interest Rates
rzero
Yield curve
Time to maturity
0
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3m
6m
1yr
3yr
Fabozzi Ch 5
5yr
10yr
30yr
slide 3
http://bond.yahoo.com/rates.html
http://www.ratecurve.com/yc2.html
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Fabozzi Ch 5
slide 4
Yield Curve = Term Structure of IR
r
Flat
Normal
Inverted
maturity
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Fabozzi Ch 5
slide 5
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slide 6
Types of Issuers
Governments
Agencies
Corporate
Municipals
Others …
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Fabozzi Ch 5
slide 7
Factors affecting Bond yields and TS
Base interest rate - benchmark interest rate
Risk Premium - spread
Expected liquidity
Market forces - Demand and supply
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Fabozzi Ch 5
slide 8
Taxability of interest
qualified municipal bonds are exempts from
federal taxes.
After tax yield = pretax yield (1- marginal tax rate)
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slide 9
Do not use yield curve to price bonds
Period
A
B
1-9
$6
$1
10
$106
$101
They can not be priced by discounting
cashflow with the same yield because of
different structure of CF.
Use spot rates (yield on zero-coupon
Treasuries) instead!
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Fabozzi Ch 5
slide 10
On-the-run Treasury issues
Off-the-run Treasury issues
Special securities
Lending
Repos and reverse repos
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slide 11
30 yr US Treasuries
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slide 12
10 yr US Treasuries
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slide 13
Duration and Term Structure of IR
Ct
~
rt t
P

C
e

t
t
t (1  rt )
t
Ct
( ~
rt  r ) t
P(r )  
  Ct e
t
t (1  rt  r )
t
1 dP(r )
D
P(0) dr r 0
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Fabozzi Ch 5
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Partial Duration
Ct
~
rt t
P(r1 , r2 ,, rT )  

C
e

t
t
t (1  rt )
t
1 dP
Drt  
P drt
Key rate duration
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Fabozzi Ch 5
slide 15
Determinants of the Yield Curve
Federal Reserve sets a target level for the fed
funds rate - the rate at which depository
institutions make uncollaterized overnight
loans to one another.
Long-term rates reflect expectations of future
rates and can be influenced by the outlook for
monetary policy.
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Fabozzi Ch 5
slide 16
Liquidity
Bid-offer spread 1-2 cents per $100 face
Corporate bonds for example 13 cents
High yield bonds 19 cents
on-the-run - recently issued in a particular
maturity class. With time became off-the-run.
Flight to Quality (fall 98) bid-ask 16-25
cents.
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Fabozzi Ch 5
slide 17
Term Structure of IR
If we knew the future IR:
0(Today) 8%
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1
10%
2
11%
3
11%
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slide 18
Term Structure of IR
If we knew the future IR:
0(Today) 8%
1
10%
2
11%
3
11%
$1,000
P
(1  0.08)(1  0.10)(1  0.11)(1  0.11)
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Fabozzi Ch 5
slide 19
r1 = 8%
r1 = 10%
r3 = 11%
r4 = 11%
Spot rate is the yield to maturity on zerocoupon bonds.
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slide 20
Future versus Spot Rates
r1 = 8%
r1 = 10%
r3 = 11%
r4 = 11%
y1= 8%
y2= 8.995%
y3= 9.66%
y4= 9.993%
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slide 21
Forward Rates
Suppose you will need a loan in two years
from now for one year.
How one can create such a loan today?
Go short a three-year zero coupon bond.
Go long a two-year zero coupon bond.
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slide 22
Suppose you will need a loan in two years
from now for one year.
How one can create such a loan today?
Go short a three-year zero coupon bond.
Go long a two-year zero coupon bond.
+1
0
0
-1.3187
-1
0
+1.188
0
0
1
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2
Fabozzi Ch 5
3
slide 23
Forward Rates
(1 + yn)n = (1 + yn-1)n-1(1 + fn)
(1 + yn)n
(1 + yn-1)n-1
+1
-1.3187
-1
0
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+1.188
1
2
Fabozzi Ch 5
3
slide 24
Forward Rates
(1 + yn)n = (1 + yn-1)n-1(1 + fn)
(1 + yn)n
(1 + yn-1)n-1
+1
-1.3187
-1
0
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+1.188
1
2
Fabozzi Ch 5
fn
3
slide 25
Forward Rates
In other words we can lock now interest rate
for a loan which will be taken in future.
To specify a forward interest rate one should
provide information about
today’s date
beginning date of the loan
end date of the loan
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slide 26
Forward Rates
Buy a two years bond
Buy a one year bond and then use the money
to buy another bond (the price can be fixed
today).
(1+r2)=(1+r1)(1+f12)
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slide 27
Forward Rates
(1+r3)=(1+r1)(1+f13)= (1+r1)(1+f12)(1+f13)
Term structure of instantaneous forward rates.
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slide 28
Forward Rates - Advanced
Let P(t,s) be the price at time t of a pure
discount bond maturing at time s > t. Then
the yield to maturity R(t,T) is the internal rate
of return at time t on a bond maturing at t+T.
P(t, t+T) = Exp[-R(t,T)*T]
Then
R(t,T) = - Log[P(t, t+T)]/T
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slide 29
Forward Rates - Advanced
The integral of the forward rates gives the
yield to maturity:
1
R (t , T ) 
T
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t T
 F (t , s)ds
t
Fabozzi Ch 5
slide 30
Forward Rates - Advanced
The integral of the forward rates gives the
yield to maturity:
1
R (t , T ) 
T
t T
 F (t , s)ds
t
or alternatively

F (t , s)   log P(t , s)
s
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slide 31
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slide 32
FRA Forward Rate Agreement
A contract entered at t=0, where the parties (a
lender and a borrower) agree to let a certain
interest rate R*, act on a prespecified principal,
K, over some future time period [S,T].
Assuming continuous compounding we have
at time S: -K
at time T: KeR*(T-S)
Calculate the FRA rate R* which makes PV=0
hint: it is equal to forward rate
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Fabozzi Ch 5
slide 33
The Expectations Hypothesis
Suggested by Lutz.
Forward interest rates is the expected future
spot rate.
Cox-Ingersoll-Ross have investigated this
hypothesis and find that it is not consistent
with an economic equilibrium.
However it gives often a right direction for
expectations.
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Fabozzi Ch 5
slide 34
Liquidity Preference
Hicks (1939) suggested that lenders demand a
premium for locking up their money for long
period of time.
This implies that the term structure will be
always upward sloping.
The theory ignores the borrowing side of the
market.
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slide 35
Market Segmentation and
Preferred Habitat Theories
Modigliani and Sutch
The market is segmented, investors absolutely
prefer one maturity over another.
This means that there is no connection
between interest rates for different maturities.
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Fabozzi Ch 5
slide 36
Modern Theories
Equilibrium Theories: CIR, BP
Non-equilibrium Theories: Dothan, Vasicek,
Ho-Lee, Hull-White, HJM
Most of them are based on a Brownian
Motion as a source of market uncertainty.
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Fabozzi Ch 5
slide 37
Brownian Motion
B
Time
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slide 38
Brownian Motion
Starts at the origin
Is continuous
Is normally distributed at each time
Increments are independent
Markovian property
Technical conditions
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Fabozzi Ch 5
slide 39
Home Assignment
Chapter 5
Ch. 5: Questions 2, 3, 10, 13.
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Fabozzi Ch 5
slide 40
Measuring the Term Structure
There are too many data plus some noise.
The easiest way to measure the TS is with
liquid zero coupon bonds.
We obtain a series of points.
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Fabozzi Ch 5
slide 41
Measuring the Term Structure
rzero
Time to maturity
0
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3m
6m
1yr
3yr
Fabozzi Ch 5
5yr
10yr
30yr
slide 42
First Order Spline
rzero
Time to maturity
0
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3m
6m
1yr
3yr
Fabozzi Ch 5
5yr
10yr
30yr
slide 43
Second Order Spline
rzero
Time to maturity
0
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3m
6m
1yr
3yr
Fabozzi Ch 5
5yr
10yr
30yr
slide 44
Measuring the Term Structure
There are too many data plus some noise.
The easiest way to measure the TS is with
liquid zero coupon bonds.
We obtain a series of points.
One can connect them with a spline.
First order is good for pricing simple bonds.
For swaps one need a very high precision.
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Fabozzi Ch 5
slide 45