Download forward rate

International Fixed Income
Topic IA: Fixed Income BasicsValuation
January 2000
Readings
• Overview of Forward Rate Analysis
(Ilmanen, Salomon Brothers)
Outline
I. Bond pricing basics
A. The discount function
B. Valuing fixed cash flows
C. Yields
D. Forward rates
E. Term structure theories
Discount Factors
The most basic debt instrument is a zero-coupon
bond. Because of the time value of money, a dollar
today is worth more than a dollar received in the
future, so the price of a zero is always less than its
face value. Let
dt
denote the t-period discount factor, i.e., the price
today of an asset that pays $1 in t periods.
Discount Factors: An Example
US Strips Market
6-month 97.30
1-year
94.76
1.5-year 92.22
2-year
89.72
5-year
75.41
The Discount Function
• The Discount Function gives the discount
factor or unit zero price as a function of
maturity.
• Because of the time value of money, longer
zeroes have lower prices. Therefore, the
discount function is always downwardsloping.
The Discount Function: An Example
1
0.9
0.8
0.7
0.6
d t 0.5
0.4
0.3
0.2
0.1
0
0
10
20
Years to Maturity
US Strips Market
30
40
Spot Rates of Interest
Spot rate: the t-year spot rate is the semiannually
compounded rate of return implied by the market
price of the t-year zero.
Discount factors and spot rates of interest are closely
related:
1
dt 
(1  rt / 2) 2t
 1
rt  2 
d

 t
1/ 2t





 1


Spot Rates of Interest: An Example
US Strips Market

  1  5.54%
1/ 2
1
r1  2 0.9476   1  5.45%
1 / 10
1
r5  2 0.7370   1  5.73%
r0.5  2 
1/1
1
0.9730
The Term Structure of Interest Rates
The relation between spot rates and time to
maturity is called the term structure of interest
rates (also known as the spot, yield, or zero
curve).
The Spot Curve
7
6.5
6
Spot Cve
5.5
5
0
5
10
US Strips Market
15
20
25
30
Yield Curve Shapes
Upward
Humped
Downward
Flat
Spot Curve (1/9/98 - 1/7/00)
7
6
5
4
3mo 6mo 1yr 2yr 3yr 5yr 7yr 10yr 15yr 20yr 30yr
1/9/1998
1/11/1999
1/7/2000
Outline
I. Bond pricing basics
A. The discount function
B. Valuing fixed cash flows
C. Yields
D. Forward rates
E. Term structure theories
A Coupon Bond as a Portfolio of Zeroes
Example: $10,000 par of a 1.5 year, 8.5% T-bond
makes the following payments:
6
m
o
n
t
h
s
1
y
e
a
r
1
1
/
2
y
e
a
r
s
$
4
2
5
$
4
2
5
$
1
0
4
2
5
This is the same as a porfolio of three zeroes:
• $425 par of a 6-mth zero
•$425 par of a 1-yr zero
•$10,425 par of a 1.5 yr zero
The Principle of No Arbitrage or
The Law of One Price
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Valuation by Replication
The previous example shows that we can
construct a coupon bond from zeros.
Therefore, given a set of discount factors, we
can value a coupon bond by valuing the
portfolio of zeros that replicates its cash
flows.
Valuing A 1.5-Year, 8.5% T-Note
V  $425  d.5  $425  d1  $10425  d1.5
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t
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0
An Arbitrage Opportunity
• What if the 1.5-year 8.5% coupon bond were
worth only 104% of par value?
• You could buy, say $1 million par of the bond for
$1,040,000 and sell the cash flows off individually
as zeroes for total proceeds of $1,043,000, making
$3000 of riskless profit.
• Similarly, if the bond were worth more than
104.3% of par, you could buy the portfolio of
zeroes, reconstitute them, and sell the bond for
riskless profit.
The General Approach
Suppose we have an asset whose cash flows
are risk-free.
Then, by no arbitrage, the market value of the
asset must be:
V
CFt1

CFt2
1   1  
rt1
2
rt2
2
2

CFt N
1  
rt
N
N
2
 (d t1  CF1 )  (d t2  CF2 )    (d t N  CFN )
Application: Synthetic Bonds
• A synthetic bond is a portfolio of securities
that is constructed to produce a specific
pattern of cash flows
• Uses
– exploit mispricing (conversion arbitrage)
– hedge a series of future cash flows
– price complex securities by replication
Outline
I. Bond pricing basics
A. The discount function
B. Valuing fixed cash flows
C. Yields
D. Forward rates
E. Term structure theories
Definition
• Definition: the yield-to-maturity of a bond
is defined as the discount rate that makes
the market price of the bond equal to the
discounted value of its future cash flows.
• The yield-to-maturity is often called the
internal rate of return, or the redemption
yield.
Mathematical Definition
Mathematically, the yield-to-maturity is the
interest rate (y) that solves the equation:
P
M (c / 2) M (c / 2)
M (c / 2)  M




2
2T
y
y
y


1
1  
1  
2
2
2


c is the annual coupon rate
M is the face value of the bond
P is the market price of the bond
Example
• Consider the aforementioned 8.5% 1.5-year
T-bond. What’s its yield?
4.25
4.25
104.25
104.3 


2
3
y 
y 
y
1
2 1  2  1  2 
Solving for the yield, we get 5.47%
Valuation Formulas
We have expressions for the value of a
portfolio of fixed cash flows in terms of
– discount factors (by no arbitrage)
– discount rates (by the definition of the discount
rates)
– yield (by the definition of yield).
Valuation Formulas continued...
N
V   Ct j  d t j
j 1
N
  Ct j 
j 1
1
rt j
(1 2
)
2t j
N
  Ct j 
j 1
1
(1 2y )
2t j
Interpretation
Compare the formula with discount rates and yield
in our example:
0.0425
0.0425
1.0425
1.0430 


1
2
3
(1  .0554 / 2) (1  .0545 / 2) (1  .0547 / 2)
0.0425
0.0425
1.0425
1.0430 


1
2
3
(1  0.0547 / 2) (1  0.0547 / 2) (1  0.0547 / 2)
Interpretation...
Yield to maturity is just a complex, nonlinear
“average” of spot rates of interest.
– Because most of the bond’s cash flow arrives at
maturity (the principal), the T-year spot rate
gets the most weight in the yield-to-maturity
calculation.
– High coupon bonds pay a larger percentage of
their face value as coupons than low coupon
bonds; thus, their yields-to-maturity give more
weight to earlier spot rates.
Interpretation...
• The yield of a portfolio of fixed cash flows
depends on the size and timing of those
cash flows.
• The yield is more heavily influenced by
cash flows that are
– larger in size (in present value terms)
– later in time (for a given present value)
Conclusions about Yield
• Yields are not necessarily a good measure
of value:
– When the term structure is not flat, bonds with
different cash flows should generally have
different yields in the absence of arbitrage.
– This is true even if the bonds have the same
maturity.
Upward Sloping Yield Curve
Yield
Zero curve
Low coupon
High coupon
Maturity
Downward Sloping Yield Curve
Yield
High coupon
Low coupon
Zero curve
Maturity
Outline
I. Bond pricing basics
A. The discount function
B. Valuing fixed cash flows
C. Yields
D. Forward rates
E. Term structure theories
Forward Contracts
• Forward contract: a binding agreement to
buy or sell a fixed quantity of an asset at an
agreed upon price (called the forward price)
at a specific time in the future.
• The terms of the contract are set today.
• No money exchanges hands today. Delivery
and settlement occur on the future date.
Forward Loan
• One way to think of a forward purchase of a
zero is as a forward loan.
• You contract today to lend $F at date t and
be repaid $1 at date T.
T
$
1
F
t
0
t
T
Intuition
This is equivalent to
• Going long a T-period zero
• Financed by going short a t-period zero
No money exchanges hands today (as true of
all forward contracts) since your short position
in the t-year zero pays for your current purchase
of the T-year zero.
Forward Rates
• Basic concept: a forward contract on a zero-coupon bond
is a binding agreement to make or take a loan on a future
date. The loan principal is the face value of the zero, so
the forward price determines the rate of interest on the
loan. This rate is called the forward rate of interest.
• Definition: the forward rate of interest for the period t to
t+T is the rate of interest available today for a loan which
goes from date t (the expiration date of the forward
contract) to date t+T (the maturity date of the loan).
Forward Rate
• People try to summarize the terms of the
forward loan by quoting the forward rate.
• The annualized, semi-annually compounded
forward rate f is defined by:
1
Ft 
T
2 (T  t )
(1  f t / 2)
T
Forward Rate as Discount rates:
PROOF
1
(
1

f/
2
) 
T
F
t
d
t

d
T
T2
(
T

t
)
t
(
1

r
/
2
)

(
1

r
/
2
)
2
T
T
2
t
t
Forward Rate Diagram
T2
()
T

t
(
12

f
/
)
t
2
t
(
1

r
/
2
)
t
0
0
t
T
(
12

r
/
)
T
2
T
2
T
(
12

r
/
)
T2
()
T

t
T
(
1

f
/
2
)

t
2
t
(
1

r
/
2
)
t
Forward Rate Example
What is the semi-annual compounded forward
for a six-month loan starting in six months?
(1  f
(1  f
 f
2 (1 0.5 )
(1  r1 / 2)

20.5
(1  r0.5 / 2)
1
0. 5
/ 2)
1
0. 5
(1  .0545 / 2)
/ 2) 
(1  .0554 / 2)
1
0 .5
 5.36%
21
2
Spot Rates as Averages of Forward Rates
d

d

F

F





F
1
1
.
5
t
t
0
.
5
0
.
5
1
t

0
.
5

(
1

r
/
2
)

(
1

r
/
2
)

(
1

f
/
2
)





(
1

f
/
2
)
2
t
1t
t
0
.
5
0
.
5t

0
.
5
The discount rate is the geometric average of all
the forward rates. In terms of our example, the
6-month spot rate is 5.54%, and the forward rate
is 5.36%. The average is equal to the one-year
rate of 5.45%.
Interpretation
• The forward rate is the marginal rate for
extending the length of the loan.
– In the above example, the marginal rate from
investing in 1-year strips instead of 6-month
strips is 5.36%, which is below the current
5.54% 6-month rate.
The Forward Curve
• Generally, when someone uses the term “forward
curve”, they are referring to the set of 1-period
implied forward rates today as a function of time.
• The T-year forward curve is composed of 2(T-1)
forward rates: the rate from year .5 to 1, the rate
from year 1 to 1.5, ..., the rate from year T-.5 to T.
The Forward Curve
8
7.5
7
Y 6.5
i
e
6
l
d 5.5
Spot
Forward
5
4.5
4
0
5
10
15
Time to Maturity
US Spot and Forward Curve
20
25
30
Practical Consideration
• One problem with the forward curve estimated via
the strip rates is that it is too jagged.
– errors in observable prices (bid/ask spread)
– only finite number of bonds
– liquidity of bonds
• Alternative method in practice is to use spline
estimation, which takes a limited number of bonds
and fits a smooth curve to estimate the discount
function.
The Forward Curve
8
7.5
7
Y 6.5
i
e 6
l
d 5.5
Spot
Forward
5
4.5
4
0
5
10
15
Time to Maturity
US Spot and Forward Curve
20
25
30
Outline
I. Bond pricing basics
A. The discount function
B. Valuing fixed cash flows
C. Yields
D. Forward rates
E. Term structure theories
The Expectations Theory
Expectations Theory: implied forward rates
are unbiased forecasts of future spot rates.
Note that nominal spot rates are a function of
inflation and real interest rates.
Implications for the Yield Curve
Under the expectations theory
– an upward sloping yield curve signals that
interest rates are expected to rise in the future
– a flat yield curve signals that they are expected
to stay the same
– a downward sloping yield curve signals that
interest rates are expected to fall
Implied Spot Curve Curve for 1/16/2001
7.5
7
6.5
1/16/2001
6
5.5
3mo 6mo 1yr 2yr 3yr 5yr 7yr 10yr 20yr
Implied Spot Curve Curve for 1/11/2000
Based on Forward Curve 1/11/1999
7
6
Actual Spot
Implied Spot
5
4
3mo 6mo 1yr 2yr 3yr 5yr 7yr 10yr 20yr
Empirical Evidence
• Historically, the yield curve has been
upward sloping on average.
• If the expectations theory is true, then the
market is forecasting an increase in interest
rates most of the time. This is not
consistent with rational expectations.
The Liquidity Preference Hypothesis
Liquidity preference hypothesis: For
liquidity reasons, investors may have to sell a
bond before it reaches maturity. Thus, they
face less price risk (interest rate risk) if they
invest in short-term instruments.
Risk Premiums
Under the liquidity preference theory, the
expected future spot rate and the forward rate
differ by a liquidity premium
f 1,2  E [r1,2 ]  L1,2
Implications for the Yield Curve
Under the liquidity preference theory
– an upward sloping yield curve is ambiguous
– a flat yield curve signals that interest rates are
expected to fall
– a downward sloping yield curve signals that
interest rates are expected to fall substantially
Empirical Evidence
• Average holding-period returns for bonds
increase with maturity. This supports the
hypothesis that investors have liquidity
preferences.
• But premiums appear to move substantially
over time.