FIN 472
Fixed-Income Securities
Yield-Curve Games
Professor Robert B.H. Hauswald
Kogod School of Business, AU
Yield Curves and Interest-Rate Risk
• Introduction to the yield curve
– revisit yields and their calculation
– the term structure of interest rates
– spot rates
• Playing the yield curve
– creating and managing interest-rate exposure
– yield-curve hypotheses
1/18/2016
Yield-Curve Games
© Robert B.H. Hauswald
2
Yield to Maturity
• A bond’s yield is a measure of its life-time
return given its current price
– only true life-time return under certain
assumptions about how the future will unfold
• Different forms of yield besides YTM:
– current yield (garbage) = current YTM
– yield to call, put, conversion, etc.
– yield to worst
31/18/2016
Yield-Curve Games
© Robert B.H. Hauswald
The Term Structure of Interest Rates
• The term structure of interest rates, or spot curve, or
yield curve, at a certain time t defines the relation
between the level of interest rates and their time to
maturity T
• The term spread is the difference between long term
interest rates (e.g. 10 year rate) and the short term
interest rates (e.g. 3 month interest rate)
• The term spread depends on many variables: expected
future inflation, expected growth of the economy, agents
attitude towards risk, etc.
• The term structure varies over time, and may take
different shapes
1/18/2016
Yield-Curve Games
© Robert B.H. Hauswald
4
Yield Curve Function
• What is a yield curve?
• A key function of the yield curve is to serve
as a benchmark for pricing bonds
– determine yields in all other sectors of the debt
market
– corporates, agencies, mortgages, bank loans, etc.
51/18/2016
Yield-Curve Games
© Robert B.H. Hauswald
Yield Curve Changes
• Four “classic” yield-curve shapes
–
–
–
–
Upward sloping
Downward sloping
Flat
Humped
• Four “classic” yield-curve shifts
– Parallel
– A flattening or steepening of the yield curve (i.e. a
change in slope)
– A change in the curvature of the yield curve
61/18/2016
Yield-Curve Games
© Robert B.H. Hauswald
Yield Curve Shifts
• An upward shift of the YC is typically
accompanied by
– a flattening of the yield curve
– a decrease in its curvature
• A downward shift of the YC is typically
accompanied by
– a steepening of the yield curve
– an increase in its curvature
71/18/2016
Yield-Curve Games
© Robert B.H. Hauswald
Yield Curve Facts
• The yield curve changes shape and slope.
• The yield curve changes level.
• The yield curve is typically upward sloping.
• Short rates are more volatile than long rates.
81/18/2016
Yield-Curve Games
© Robert B.H. Hauswald
Spot Rates and Strips
• A spot rate is the rate used to discount a single
expected future cash flow.
• Strip rates are created from Separate Trading of
Interest and Principal.
• The most recently issued securities are used to
create a theoretical spot curve
– these are called the on-the-run or actives Treasury
securities
• In practice the yield on the on-the-run Treasury is
adjusted such that the bond is at par
– this is the par yield.
91/18/2016
Yield-Curve Games
© Robert B.H. Hauswald
Yield Curve
• Most fixed income securities were priced at a
spread relative to the Treasury yield curve.
– If the yield to maturity on the 10-year Treasury
bond was 7%, then a 10-year Baa corporate bond
would be priced to yield 7% plus the Baa credit
spread.
The problem with this approach is that it
ignores differences in duration and convexity
that may be priced. Can you see why?
1/18/2016
Yield-Curve Games
© Robert B.H. Hauswald
10
On-the-run & Off-the-run
1/18/2016
11
Yield-Curve Games
© Robert B.H. Hauswald
Par Curve
• The par rate is the discount rate at which the
bond’s price equals its par value
M=
C
C
C
M
+
+K+
+
(1 + y )1 (1 + y ) 2
(1 + y ) n (1 + y ) n
• The reason for this adjustment is that the
observed price and yield may reflect cheap
repo financing available from an issue if it
is “on special”.
12
1/18/2016
Yield-Curve Games
© Robert B.H. Hauswald
Building the On-the-Run Curve
• Yields for missing maturities are interpolated using linear approximation.
• Yield (n) = Yield (lower) + (Yield (upper) - Yield (lower)) x (n-lower)/(upper-lower)
• What is the interpolated yield for a 4-year Treasury?
Yield (4) = Yield (3) + (Yield (5) - Yield (3)) x (4-3)/(5-3)
Yield (4) = 4.6437% + (4.6172% - 4.6437%) x (4-3)/(5-3)
Yield (4) = 4.6305%
1/18/2016
13
Yield-Curve Games
© Robert B.H. Hauswald
Theoretical Spot Rates
• Definition: set of interest or discount rates that
should be used to value default-free cash flows.
– construct default-free theoretical spot rates from the
observed Treasury yield curve or par curve.
– yield curve analysis starts with the set of yields on the
most recently issue (i.e. on the run) UST yields.
• The U.S. Treasury routinely issues seven
securities – the 3- and 6-month T-bills, the 2-, 3-,
5-, and 10-year notes, and the 30-year bond
– read about market conventions and auction procedures
1/18/2016
14
Yield-Curve Games
© Robert B.H. Hauswald
Yield Curve
• Consider the following yield-curve data for
on-the-run Treasuries of various maturities:
Notice the
range of
coupons.
These bonds
have very
different cash
flow patterns.
1/18/2016
Coupon
8.50%
7.38%
9.00%
8.88%
6.75%
7.75%
6.25%
5.63%
6.50%
7.50%
Yield-Curve Games
Term (yrs)
1/2
1
1 1/2
2
2 1/2
3
3 1/2
4
4 1/2
5
Yield
5.10%
5.49%
5.63%
5.81%
5.86%
5.93%
6.03%
6.09%
6.10%
6.16%
Price
$101.66
$101.81
$104.78
$105.72
$102.03
$104.94
$100.69
$98.38
$101.56
$105.69
© Robert B.H. Hauswald
15
A Better Approach
• The problem: differing cash flow patterns
among on-the-run Treasuries
– but realize that each coupon bond is really a
package of single payment bonds.
• For example, a 2-year 10% coupon bond is
really a package of five single payment bonds:
– four for the semi-annual coupon payments and
– one for the repayment of the corpus.
1/18/2016
Yield-Curve Games
© Robert B.H. Hauswald
16
Zeroes
• A single payment bond is called a “zero.”
• A coupon bond can be thought of as a
package of zeroes,
– one for each of the coupon payments and
– one for the corpus.
• Any coupon bond could be “stripped” or
“unbundled” into its constituent zeroes.
– US Treasury STRIPS are unbundled coupon
bonds.
1/18/2016
Yield-Curve Games
© Robert B.H. Hauswald
17
Spot Yields
• A “spot yield” is the current yield to maturity
on a zero coupon bond.
– For example, the 1-year spot yield is the yield to
maturity on a 1-year zero.
• The price ($1 face value) of an n-year zero is
related to the n-year spot rate by the formula:
P =
0 n
1/18/2016
Yield-Curve Games
1
 in 
1+

2
2n
© Robert B.H. Hauswald
18
From Spot Yield to Price
• Yields imply prices:
– if the 3 1/2 year spot yield is 6.05%, then the
price (per $1 face value) of the 3 1/2 year zero is:
P =
0 3.5
1/18/2016
1
1+ .0605
(
)
2*3.5
2
Yield-Curve Games
=
1
7 = .811
(1.03025)
© Robert B.H. Hauswald
19
From Price to Spot Yield
• Prices imply yields
– we can express the n-year spot yield as a
function of the price of an n-year zero:
1

 1  2n 
 −1
in = 2 

 0 Pn 


1/18/2016
Yield-Curve Games
© Robert B.H. Hauswald
20
From Price to Spot Yield:
Example
• For example, if a 4 year zero is priced at
$.79 per dollar of face value,
– then the 4-year spot rate is:
1
1

 1  2*4    1  8 
 − 1 = 2
i4 = 2 
− 1 = 2(1.03059 −1) = 6.12%

 .79 
 0 P4 


 
1/18/2016
Yield-Curve Games
© Robert B.H. Hauswald
21
Spot Yields and Discount Factors
Term (yrs)
1/2
1
1 1/2
2
2 1/2
3
3 1/2
4
4 1/2
5
1/18/2016
Spot Yield
5.10%
5.49%
5.64%
5.82%
5.88%
5.95%
6.05%
6.12%
6.12%
6.19%
Price of zero
$0.98
$0.95
$0.92
$0.89
$0.87
$0.84
$0.81
$0.79
$0.76
$0.74
Yield-Curve Games
P =
0 2.5
1
 .0588 
1+

2 
5
= .87
1


 1 9
i4.5 = 2
−1 = 6.12%
 .76 


© Robert B.H. Hauswald
22
Price of a Coupon Bond
– In principle, the price of an n-year coupon bond
ought to be equal to the total value of all its
constituent zeroes:
c
2n
1
1
2
2 +
P=∑
s +
2n = ∑
s
2n
is   in 
y  y
s=1 
s=1 
1+
1+
1+ 2  1+ 2 
 2  2
2

2n
c
Priced using yield to
maturity Yield-Curve Games
1/18/2016
Priced using spot
yields
© Robert B.H. Hauswald
23
Pricing a Coupon Bond
n
1/2
1
1 1/2
2
2 1/2
3
3 1/2
4
4 1/2
5
5-year 7.5% coupon bond
Spot Yield
Price of zero
5.10%
$0.98
5.49%
$0.95
5.64%
$0.92
5.82%
$0.89
5.88%
$0.87
5.95%
$0.84
6.05%
$0.81
6.12%
$0.79
6.12%
$0.76
6.19%
$0.74
Cash flow
$0.0375
$0.0375
$0.0375
$0.0375
$0.0375
$0.0375
$0.0375
$0.0375
$0.0375
$1.0375
Value
$0.0366
$0.0355
$0.0345
$0.0334
$0.0324
$0.0315
$0.0304
$0.0295
$0.0286
$0.7647
$1.0571
This bond actually traded at a price of $1.0569 or a yield to
maturity of 6.16%
1/18/2016
Yield-Curve Games
© Robert B.H. Hauswald
24
Term Structure
• The term structure of interest rates is the pattern of
spot rates over the range of maturities.
– A flat term structure means that spot yields are equal at
all maturities.
– A normal term structure slopes upward
– An inverted term structure slopes downward
• Modern pricing practice is to regard any bond as a
package of zeros and price the package using
spreads relative to the term structure.
1/18/2016
Yield-Curve Games
© Robert B.H. Hauswald
25
Finding Yield Curves
• We can derive the theoretical term structure
from the yield curve using a procedure
known as “bootstrapping.”
• Here’s yield curve information
Coupon
8.50%
7.38%
9.00%
8.88%
6.75%
7.75%
6.25%
5.63%
6.50%
7.50%
1/18/2016
Term (yrs)
1/2
1
1 1/2
2
2 1/2
3
3 1/2
4
4 1/2
5
Yield-Curve Games
Yield
5.10%
5.49%
5.63%
5.81%
5.86%
5.93%
6.03%
6.09%
6.10%
6.16%
Price
$101.66
$101.81
$104.78
$105.72
$102.03
$104.94
$100.69
$98.38
$101.56
$105.69
© Robert B.H. Hauswald
26
The Shortest Zero…
• The first bond has 1/2 year to run
• It is a single payment bond that will pay
$1.0425 per dollar of face value at maturity.
• Its price is $1.0166 per dollar of face value.
• Therefore the 1/2 year spot rate is
 1.0425 1 
i1/ 2 = 2
−1 = 2(.0255) = 5.10%
 1.0166  
1/18/2016
Yield-Curve Games
© Robert B.H. Hauswald
27
From Spot Rate to Discount Factor
• Given the 1/2 year spot rate, we can
determine the price of the 1/2 year zero:
P =
0 1/ 2
1/18/2016
1
 i1/ 2 
1+

2
2 (1/ 2)
Yield-Curve Games
=
1
= .9751
 .0510 
1+

2 
© Robert B.H. Hauswald
28
The Recursion Principle: Pricing
• For each dollar of face value, the 1-year
bond will pay $.03675 in 6 months and
$1.03675 in one year.
• It’s price ($1.0181 per dollar of face value)
should equal
$1.0181 =
1/18/2016
$0.03675
1
 i1/ 2 
1+

2 
Yield-Curve Games
+
$1.03675
 i1 
1+
 2
2
© Robert B.H. Hauswald
29
Setting up the Recursion
• But since the 6-months spot rate is 5.10%,
$1.0181 =
$0.03675
1
 .0510 
1+

2 
+
$1.03675
 i1 
1+
 2
2
Which we can solve for the 1-year spot rate as
1/ 2


1.03675

i1 = 2 
− 1 = 5.49%

  1.0181 − .03584 



1/18/2016
Yield-Curve Games
© Robert B.H. Hauswald
30
Alternatively…
• Start with
$1.0181 =
$0.03675 $1.03675
1 +
2
 i1/ 2 
 i1 
1+
1+

 2
2 
$1.0181 = $0.03675*0 P1/ 2 + $1.03675*0 P1
= $0.3675(.9751) + $1.03675*0 P1
= $0.03596 + $1.03675*0 P1
as
solve for 0P1 and then i1
1/18/2016
Yield-Curve Games
© Robert B.H. Hauswald
31
Bootstrapping
• Continue this process to find the theoretical
term structure of (spot) interest rates
Coupon
8.50%
7.38%
9.00%
8.88%
6.75%
7.75%
6.25%
5.63%
6.50%
7.50%
1/18/2016
Maturity
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
Yield
5.10%
5.49%
5.63%
5.81%
5.86%
5.93%
6.03%
6.09%
6.10%
6.16%
Yield-Curve Games
Price
$101.66
$101.81
$104.78
$105.72
$102.03
$104.94
$100.69
$98.38
$101.56
$105.69
© Robert B.H. Hauswald
Zero Price
$97.51
$94.73
$92.00
$89.16
$86.51
$83.88
$81.16
$78.58
$76.23
$73.71
Spot Yield
5.10%
5.49%
5.64%
5.82%
5.88%
5.95%
6.05%
6.12%
6.12%
6.19%
32
Valuation of an 8% 10-Year
Treasury Using a Spot Curve
Illustration of Spot/YTM
Years
0.50
1.00
1.50
2.00
2.50
3.00
3.50
4.00
4.50
5.00
5.50
6.00
6.50
7.00
7.50
8.00
8.50
9.00
9.50
10.00
Cash Flow
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
104
Spot Rate
6.05
6.15
6.21
6.26
6.29
6.37
6.38
6.40
6.41
6.48
6.49
6.53
6.63
6.78
6.79
6.81
6.84
6.93
7.05
7.20
Present Value
3.8826
3.7649
3.6494
3.5361
3.4263
3.3141
3.2107
3.1090
3.0113
2.9079
2.8151
2.7203
2.6178
2.5082
2.4242
2.3410
2.2583
2.1666
2.0711
51.2670
107.0018
7.4
7.2
7
6.8
6.6
Yield
Period
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
Spot Rate
6.4
Yield to Maturity
6.2
6
5.8
5.6
5.4
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
Time Period
The yield to maturity on this bond is 7.014%
1/18/2016
33
Yield-Curve Games
© Robert B.H. Hauswald
Riding the Yield Curve •
for Fun and Profit
Assuming a positively
sloped yield curve,
– purchase a security with
a maturity longer than
your expected holding
period.
• Rationale: You will
make money because
Yield
1.
.07
.04
1
1/18/2016
2
Yield-Curve Games
longer maturities pay
higher rates,
2. when you sell it later
the security will have
a shorter maturity,
hence lower rates,
hence a capital gain.
maturity (years)
© Robert B.H. Hauswald
34
Example
• Investment horizon is 1 year
– current 1-year rate is 4%, 2-year rate is 7%.
• If you buy 1-year security make 4%
• If you“ride,” price per $1of face of 2-year security is 0.8734. If
sell in one year when 1-year rate is 4%, get 0.9615
• Spot the fallacy?
Profit =
.9615 − .8734
= .1009 = 10.1%
.8734
2
Price * (1.07) = 1.00
Price =
1.00
(1.07)
1/18/2016
2
= .8734
Yield-Curve Games
Price* (1.04)= 1.00
1.00
Price =
= .9615
(1.04)
© Robert B.H. Hauswald
35
Market Efficiency?
• Will this work in an “efficient market”?
– what can you sell the security at next year?
• The market seems to expect the rate on 1Y paper
to be 10%: implies a per $1 price of 0.9090
Profit =
.9090 − .8734
= .040 = 4%
.8734
• Riding the YC works as long as the 1Y rate next
year is less than the market forecast
• if higher you will lose money: the market forecast is a
“breakeven” rate.
1/18/2016
Yield-Curve Games
© Robert B.H. Hauswald
36
Three Yield-Curve Facts
1. The downward sloping (inverted) yield
curve is unusual but not rare.
2. Interest rates of bonds of all maturities
move together (are positively correlated)
3. The downward sloping (inverted) yield
curve tends to occur when interest rates in
general are high.
1/18/2016
Yield-Curve Games
© Robert B.H. Hauswald
37
Three (Yield) Spread Measures
• Nominal Spread: an issue’s yield to maturity
minus the yield to maturity of a Treasury
security of similar maturity.
• Static Spread (Z-Spread): spread over the spot
rates in a Treasury term structure.
– The same spread is added to all risk-free spot rates.
• Option Adjusted Spread (OAS): used when a
bond has embedded options
– Z-spread - OAS = Option cost in percentage terms
1/18/2016
38
Yield-Curve Games
© Robert B.H. Hauswald
Yield Analysis - YAS
1/18/2016
39
Yield-Curve Games
© Robert B.H. Hauswald
Summary
• Properly pricing bonds: use spot rates
– discount rates on zero bonds
• Extraction of spot rates from coupon bonds
– recursive construction of term structure of
interest rates: spot curve
• Yield curve games: investment strategies
• Stylized facts and possible explanations
– sets the stage for yield-curve modellilng
1/18/2016
Yield-Curve Games
© Robert B.H. Hauswald
40
Appendix
• Yield curve and portfolio strategies
– laddered portfolios
– barbell portfolios
• Yield curve hypotheses: three explanations
– expectations
– segmentation
– preferred habitat
• Finding spot rates
– more on stripping
1/18/2016
Yield-Curve Games
© Robert B.H. Hauswald
41
Laddered Portfolio
• In a laddered strategy, the fixed-income dollars
are distributed throughout the yield curve
• A laddered strategy eliminates the need to
estimate interest rate changes
• For example, a $1 million portfolio invested in
bond maturities from 1 to 25 years
– yields what in terms of portfolio weights?
1/18/2016
Yield-Curve Games
© Robert B.H. Hauswald
42
Par Value Held ($ in Thousands)
Laddered Portfolio Allocation
50000
45000
40000
35000
30000
25000
20000
15000
10000
5000
0
1
3
5
7
9
11
13
15
17
19
21
23
25
Years Until Maturity
1/18/2016
Yield-Curve Games
© Robert B.H. Hauswald
43
Barbell Portfolio
• The barbell strategy differs from the
laddered strategy in that less amount is
invested in the middle maturities
• For example, a $1 million portfolio invests
$70,000 par value in bonds with maturities
of 1 to 5 and 21 to 25 years, and $20,000
par value in bonds with maturities of 6 to 20
years
1/18/2016
Yield-Curve Games
© Robert B.H. Hauswald
44
Par Value Held ($ in Thousands)
Barbell Portfolio Allocation
50000
45000
40000
35000
30000
25000
20000
15000
10000
5000
0
1
3
5
7
9
11
13
15
17
19
21
23
25
Years Until Maturity
1/18/2016
Yield-Curve Games
© Robert B.H. Hauswald
45
Yield Curve Hypotheses
• Several explanations (“hypotheses” not
“theories”) for the stylized yield curve behavior
• Key difference in assumptions:
– How close of substitutes are bonds of different
maturities?
• Assess each hypothesis – how well does it
predict the three observed phenomena?
1/18/2016
Yield-Curve Games
© Robert B.H. Hauswald
46
1. Expectations Hypothesis
• The interest rate on a long-term bond is equal
to the average of short-term rates expected to
occur over the lifetime of the long-term bond.
– sounds familiar?
• Key assumption: Bonds of different maturities
are perfect substitutes
– means what?
– are they perfect substitutes?
1/18/2016
Yield-Curve Games
© Robert B.H. Hauswald
47
Assessment:
Expectations Hypothesis
• The downward sloping (inverted) yield curve is unusual
but not rare.
– fails to predict this occurrence: predicts 50-50 probability of
upward versus downward slope
• Interest rates of bonds of all maturities move together (are
positively correlated)
– accurately predicts this occurrence: bonds are (partial) substitutes
• YCs tend to slope downward (inverted) when interest rates
in general are high.
– accurately predicts this occurrence: when are interest rates
expected to decrease?
1/18/2016
Yield-Curve Games
© Robert B.H. Hauswald
48
2. Market Segmentation Hypothesis
• Each investor has his/her preferred maturity
sector and stays within it.
– consequently, yield are determined by independent
demand and supply conditions within each sector.
• Key Assumption: Bonds of different maturities
are not substitutes at all.
1/18/2016
Yield-Curve Games
© Robert B.H. Hauswald
49
Market Segmentation
Hypothesis: Predictions
• Upward sloping yield curve is the usual
state of affairs
– more demand for short-term bonds than longterm bonds.
• Downward sloping yield curve
– contractionary Federal Reserve policy,
– increases short-term rates without affecting
long-term rates.
1/18/2016
Yield-Curve Games
© Robert B.H. Hauswald
50
Assessment: Market
Segmentation Hypothesis
• The downward sloping (inverted) yield curve is unusual but not
rare
– accurately predicts this occurrence: highly contractionary monetary policy
• Interest rates of bonds of all maturities move together (are
positively correlated)
– fails to predict this occurrence: independently determined are uncorrelated
• The downward sloping (inverted) yield curve tends to occur when
interest rates in general are high.
– accurately predicts occurrence: high expected inflation = contractionary
monetary policy
1/18/2016
Yield-Curve Games
© Robert B.H. Hauswald
51
3. Preferred Habitat Hypothesis
• The interest rate on a long-term bond equals
the average of short-term rates expected to
occur over the lifetime of the long-term bond
– plus a risk premium due to higher market risk in
the long-term bond
• Key assumption: bonds of different maturities
are close but not perfect substitutes
• Major non-price difference at long end
– longer term bonds have higher market risk
1/18/2016
Yield-Curve Games
© Robert B.H. Hauswald
52
Assessment:
Preferred Habitat Hypothesis
• The downward sloping (inverted) yield curve is
unusual but not rare.
– accurately predicts occurrence: investors must expect
interest rates to decrease a lot in the future
• Interest rates of bonds of all maturities are positively
correlated
– accurately predicts this occurrence: close substitutes
• The downward sloping (inverted) yield curve tends
to occur when interest rates in general are high
– accurately predicts occurrence: when are interest rates
expected to decrease a lot?
1/18/2016
Yield-Curve Games
© Robert B.H. Hauswald
53
Finding Spot Rates
• Suppose you observe the following data for
three Treasury securities with annual
compounding and exact maturities:
Maturity
1 year
2 years
3 years
Yield to Maturity
3.62%
3.80%
4.00%
Coupon
0%
3.80%
4.00%
Price (% of par)
96.6463
100
100
• What do you notice?
• What is your starting point?
54
1/18/2016
Yield-Curve Games
© Robert B.H. Hauswald
Stripped Cash Flows
Maturity
1 year
2 years
3 years
Yield to Maturity
3.62%
3.80%
4.00%
Coupon
0%
3.80%
4.00%
Price (% of par)
96.6463
100
100
• The cash-flow timeline from the stripped
cash flows is as follows:
0
1
1-year bond
-96.6463
+100
2-year bond
-100.000
+3.8
3-year bond
-100.000
+4
1/18/2016
55
Yield-Curve Games
2
3
+103.8
+4
+104
© Robert B.H. Hauswald
Spot-Rate Facts
• Spot rates are used to discount a single cash
flow to be received at some specific future date
– bonds all have the same issuer: all cash flows
received at t=1 discounted at the same rate.
• The 1-year zero coupon bond has only one
cash flow,
– use its YTM as to discount the other t=1 cash
flows, i.e. use 3.62% as the one-year spot rate Z1
• “Bootstrapping” a theoretical spot rate curve
1/18/2016
56
Yield-Curve Games
© Robert B.H. Hauswald
Two-Year Spot Rate
• Use the one-year spot rate to discount the
cash flows for the 2-year bond as follows:
1.
2.
3.
4.
5.
6.
100.000 = 3.8/(1 + Z1)1 + 103.8/(1 + Z2)2
100.000 = 3.8/(1.0362)1 + 103.8/(1 + Z2)2
100.000 – 3.66725 = 103.8/(1 + Z2)2
96.33275 = 103.8/(1 + Z2)2
1.077515 = (1 + Z2)2
1.038034 = 1 + Z2
• Z2 = 3.8034%
1/18/2016
57
Yield-Curve Games
© Robert B.H. Hauswald
Three-Year Spot Rate
• Use the one- and two-year spot rates to find
the 3-year spot rate as follows:
1. 100.000 = 4/(1 + Z1)1 + 4/(1 + Z2)2 + 104/(1
+ Z3)3
2. 100.000 = 4/(1.0362)1 + 4/(1.038034)2 +
104/(1 + Z3)3
3. 1 1.040105 = (1 + Z3)3
• Z3 = 4.0105%
1/18/2016
58
Yield-Curve Games
© Robert B.H. Hauswald
Stripping…
•
•
•
•
A coupon-bearing bond can be purchased and have the coupons
stripped.
If the stripped security, discounted at spot rates, is worth more than the
coupon security, then dealers can engage in arbitrage to drive the
prices back into equilibrium (by the coupon security and sell the
stripped cash flows).
This process of stripping and reconstitution assures that the price of a
Treasury issue will not depart materially from its arbitrage-free value.
Empirical evidence suggests that non-U.S. government issues have
also moved toward their arbitrage-free value as stripping and
reconstitution of cash flows has been allowed.
1/18/2016
59
Yield-Curve Games
© Robert B.H. Hauswald
Strip Curve
1/18/2016
60
Yield-Curve Games
© Robert B.H. Hauswald