Markit iBoxx Spread
Analytics
July 2010
Markit iBoxx Spread Analytics
Contents
1
2
3
4
5
6
7
8
Benchmark Spread .............................................................................................................................................. 3
1.1
Methodology .................................................................................................................................................. 3
1.1.1 Annual Benchmark Spread ....................................................................................................................... 3
1.1.2 Semi-annual Benchmark Spread .............................................................................................................. 4
1.2
Index Benchmark Spread.............................................................................................................................. 4
1.2.1 Annual Index Benchmark Spread ............................................................................................................. 4
1.2.2 Semi-annual Index Benchmark Spread .................................................................................................... 4
Spread to Benchmark Curve............................................................................................................................... 5
2.1
Methodology .................................................................................................................................................. 5
2.1.1 Annual Spread to Benchmark Curve ........................................................................................................ 5
2.1.2 Semi-annual Benchmark Spread .............................................................................................................. 6
2.2
Index Spread to Benchmark Curve ............................................................................................................... 6
2.2.1 Annual Index Spread to Benchmark Curve .............................................................................................. 6
2.2.2 Semi-annual Index Spread to Benchmark Curve ..................................................................................... 6
Spread to Libor Curve ......................................................................................................................................... 7
3.1
Methodology .................................................................................................................................................. 7
3.1.1 Annual Spread to Libor Curve................................................................................................................... 7
3.1.2 Semi-annual Spread to Libor Curve.......................................................................................................... 7
3.2
Index Spread to Libor Curve ......................................................................................................................... 8
3.2.1 Annual Index Spread to Libor Curve......................................................................................................... 8
3.2.2 Semi-Annual Index Spread to Libor Curve ............................................................................................... 8
Z-spread .............................................................................................................................................................. 10
4.1
Methodology ................................................................................................................................................ 10
4.1.1 Benchmark Zero Coupon Curve ............................................................................................................. 10
4.1.2 Z-spread Calculation ............................................................................................................................... 11
4.2
Index Z-spread ............................................................................................................................................ 11
Z-spread over Libor Curve ................................................................................................................................ 12
5.1
Methodology ................................................................................................................................................ 12
5.1.1 Z-spread over Libor Curve Calculation ................................................................................................... 12
5.2
Index Z-spread over Libor ........................................................................................................................... 12
Option Adjusted Spread .................................................................................................................................... 13
6.1
Methodology ................................................................................................................................................ 13
6.1.1 Constructing Binomial Interest Rate Tree............................................................................................... 13
6.1.2 OAS Calculation...................................................................................................................................... 13
6.2
Index OAS ................................................................................................................................................... 14
Asset Swap Spread............................................................................................................................................ 15
7.1
Methodology ................................................................................................................................................ 15
7.1.1 Asset Swap Spread Calculation.............................................................................................................. 15
7.2
Index Asset Swap Spread ........................................................................................................................... 16
Further information............................................................................................................................................ 17
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Markit iBoxx Spread Analytics
1 Benchmark Spread
1.1 Methodology
The benchmark spread can be defined as a premium above the yield on a default-free bond necessary to
compensate for additional risk associated with holding the bond.
Markit calculates the benchmark spread as the difference between the yield of the bond and the benchmark bond.
Selection criteria for a benchmark bond are:
– Government bond is selected as an approximation of a ‘default-free bond’
– The difference between maturities of a bond and the benchmark bond is the smallest in comparison to other
alternatives
6%
5%
Benchmark Spread
4%
3%
2%
1%
Benchmark Yield
Bond Yield
0%
0.5
4.5
8.5
12.5
16.5
20.5
24.5
28.5
32.5
1.1.1 Annual Benchmark Spread
The annual benchmark spread of a bond
i at time t is:
0
BMSi,ta =  a
a
 yi ,t − yBM i ,t
where:
yia,t - Annualized yield of a bond i at time t
a
yBM
i ,t - Annualized yield of the benchmark of a bond i at time t
Note that for benchmark bonds
BMSi,a t = 0
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1.1.2 Semi-annual Benchmark Spread
The semi-annual benchmark spread of a bond
i at time t is:
0
BMSi,ts =  s
s
 yi ,t − yBM i ,t
where:
yis,t - Semi-annualized yield of a bond i at time t
s
yBM
i ,t - Semi -annualized yield of the benchmark of a bond i at time t
Note that for benchmark bonds
BMSi,s t = 0
1.2 Index Benchmark Spread
1.2.1 Annual Index Benchmark Spread
The annual index benchmark spread is calculated as follows:
a
i,t ⋅ MVi,t ⋅ D i,t )
∑ (BMSBond
n
a
BMSIndex,
t =
i=1
n
∑ (MVi,t ⋅ Di,t )
i=1
where:
–
a
BMSBond
i,t - Annual benchmark spread of bond
–
MVi, t - Market value of bond
–
i at time t
i at time t
Di, t - Effective duration of bond i at time t
1.2.2 Semi-annual Index Benchmark Spread
The semi-annual index benchmark spread is calculated as follows:
s
i,t ⋅ MVi,t ⋅ D i,t )
∑ (BMSBond
n
s
BMS Index,
t =
i=1
n
∑ (MVi,t ⋅ Di,t )
i=1
where:
–
BMSsBond i, t - Semi-annual benchmark spread of bond i at time t
–
MVi, t - Market value of bond
–
i at time t
Di, t - Effective duration of bond i at time t
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Markit iBoxx Spread Analytics
2 Spread to Benchmark Curve
2.1 Methodology
Spread to benchmark curve can be defined as a premium above the yield on a default-free bond necessary to
compensate for additional risk associated with holding the bond.
The default-free yield to maturity is found by a linear interpolation of two benchmark bonds with maturities being just
above and just below the time to maturity of a bond.
Selection criteria for benchmark bonds are:
–
Government bonds are selected as an approximation of a ‘default-free bond’
–
The difference between maturities of a bond and the benchmark bonds is the smallest in absolute terms in
comparison to other alternatives
6%
5%
Spread to Benchmark Curve
4%
3%
2%
1%
Benchmark Yield
Bond Yield
Interpolated Yield
0%
0.5
4.5
8.5
12.5
16.5
20.5
24.5
28.5
32.5
2.1.1 Annual Spread to Benchmark Curve
The annual spread to benchmark of a bond
i at time t is:
0
SBC i,ta =  a
a
 yi ,t − yInBM i ,t
where:
yia,t - Annualized yield of a bond i at time t
a
yInBM
i ,t - Annualized yield of the interpolated benchmark of a bond i at time t
Note that for benchmark bonds
SBC i,ta = 0
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Markit iBoxx Spread Analytics
2.1.2 Semi-annual Benchmark Spread
The semi-annual benchmark spread of bond
i at time t is:
0
SBC i,ts =  s
s
 yi ,t − yInBM i ,t
where:
yis,t - Semi-annualized yield of bond i at time t
s
yInBM
i ,t - Semi -annualized yield of the interpolated benchmark of a bond i at time t
Note that for benchmark bonds
SBC i,ts = 0
2.2 Index Spread to Benchmark Curve
2.2.1 Annual Index Spread to Benchmark Curve
The annual index spread to benchmark curve is calculated as follows:
a
i,t ⋅ MVi,t ⋅ D i,t )
∑ (SBC Bond
n
a
SBC Index,
t =
i=1
n
∑ (MVi,t ⋅ Di,t )
i=1
where:
–
a
SBC Bond
i,t - Annual spread to benchmark curve of bond
–
MVi, t - Market value of bond
–
i at time t
i at time t
Di, t - Effective duration of bond i at time t
2.2.2 Semi-annual Index Spread to Benchmark Curve
The semi-annual index benchmark spread is calculated as follows:
s
i,t ⋅ MVi,t ⋅ D i,t )
∑ (SBC Bond
n
s
SBC Index,
t =
i=1
n
∑ (MVi,t ⋅ Di,t )
i=1
where:
–
s
SBC Bond
i,t - Semi-annual spread to benchmark curve of bond
–
MVi, t - Market value of bond
–
i at time t
i at time t
Di, t - Effective duration of bond i at time t
–
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Markit iBoxx Spread Analytics
3 Spread to Libor Curve
3.1 Methodology
Spread to Libor curve can be defined as a premium above the yield on Markit SWAP curve, constructed from Libor
rates and ICAP swap rates, necessary to compensate for additional risk associated with holding the bond.
3.1.1 Annual Spread to Libor Curve
The spread to benchmark of a bond
i at time t is:
0
SLC ai ,t =  a
a
y i ,t − y SWAP t
where:
y ia,t i ,t - Annualized yield of a bond i at time t
a
y SWAP
t - Annualized value of Markit SWAP curve at time t
3.1.2 Semi-annual Spread to Libor Curve
The spread to benchmark of a bond
i at time t is:
0
SLC si ,t =  s
s
y i ,t − y SWAP t
where:
y is,t i ,t - Semi- annualized yield of a bond i at time t
s
y SWAP
t - Semi- annualized value of Markit SWAP curve at time t
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Markit iBoxx Spread Analytics
4.5%
4.0%
3.5%
Spread to Libor Curve
3.0%
2.5%
2.0%
1.5%
Markit SWAP Curve
Bond Yield
1.0%
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
5.5
6.0
6.5
7.0
7.5
8.0
8.5
9.0
9.5
3.2 Index Spread to Libor Curve
3.2.1 Annual Index Spread to Libor Curve
The index spread to Libor curve is calculated as follows:
∑ (SLC Bond i,t ⋅ MVi,t ⋅ Di,t )
n
a
SLC Index,
t
=
i=1
n
∑ (MVi,t ⋅ Di,t )
i=1
where:
–
a
SLC Bond
i,t - Annual spread to benchmark curve of bond
–
MVi, t - Market value of bond
–
i at time t
i at time t
Di, t - Effective duration of bond i at time t
3.2.2 Semi-Annual Index Spread to Libor Curve
The index spread to Libor curve is calculated as follows:
∑ (SLC Bond i,t ⋅ MVi,t ⋅ Di,t )
n
s
SLC Index,
t =
i=1
n
∑ (MVi,t ⋅ Di,t )
i =1
where:
–
s
SLC Bond
i,t - Semi-annual spread to benchmark curve of bond
i at time t
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Markit iBoxx Spread Analytics
–
–
i at time t
Di, t - Effective duration of bond i at time t
MVi, t - Market value of bond
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Markit iBoxx Spread Analytics
4 Z-spread
4.1 Methodology
The Z-spread is a measure of the spread the investor would realize over the entire benchmark zero coupon curve if
the bond is held to maturity.
4.1.1 Benchmark Zero Coupon Curve
Benchmark zero curve
zt ( L) also known as spot rate curve is calculated form dirty prices of defined benchmark
bonds.
The following equation is solved using the Nelder-Mead-Simplex Method

min = Pk ,t + Ak ,t −
k =1 

! m
∑
n
∑
CFi k, j
j =1
⋅
(
1 + zt (Lki , j
)
)
−Lki , j



2
zt ( L) - Function constructed by natural splines with defined knots
( Pk ,t + Ak ,t ) - Dirty price of defined benchmark bond
CFi ,kj - Cash flow of defined benchmark bond
In the figure below the estimated zero curve is compared to annual yields of the benchmark bonds
5.0%
4.5%
4.0%
3.5%
3.0%
2.5%
2.0%
1.5%
1.0%
0.5%
Spot Rate
0.0%
0
5
10
15
20
25
30
35
Annual Yield
40
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45
50
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Markit iBoxx Spread Analytics
4.1.2 Z-spread Calculation
The Z-spread is calculated as the spread that will make the present value of the cash flows of respective bond equal
to the market dirty price, when discounted at the benchmark spot rate plus the spread. The spread is found iteratively
using the Newton method.
In general, constant spread si , j over the spot curve zt (L ) for a bond at time t on an annual basis is calculated
iteratively by using the Newton method:
∑CFi, j ⋅ (1 + zt (Li , j ) + si , j )−L
n
Pi ,t + Ai ,t =
i, j
j =1
(Pk ,t + Ak ,t ) - Dirty price of a bond
CFi k, j - Cash flow of a bond
4.2 Index Z-spread
The index Z-spread is calculated as follows:
n
∑ (Z - spreadBond i,t ⋅ MVi,t ⋅ Di,t )
Z - spreadIndex, t =
i =1
 n

 MVi,t ⋅ D i,t 


 i =1

∑
where:
– MVi,t - Market value of a bond
–
i at time t
Di, t - Effective duration of a bond i at time t
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Markit iBoxx Spread Analytics
5 Z-spread over Libor Curve
5.1 Methodology
The Z-spread over Libor Curve is a measure of the spread that the investor would realize over the entire ICAP curve,
constructed from Libor rates and ICAP swap rates, if the bond is held to maturity.
5.1.1 Z-spread over Libor Curve Calculation
The Z-spread is calculated as the spread that will make the present value of the cash flows of respective bond equal
to the market dirty price, when discounted at the ICAP rate plus the spread. The spread is found iteratively using the
Newton method.
In general, Z-spread si , j over Markit SWAP curve zt (L ) for a bond at time t on an annual basis is calculated
iteratively using the Newton method:
Pi ,t + Ai ,t = ∑ CFi , j ⋅ (1 + zt ( Li , j ) + si , j )
n
− Li , j
j =1
( Pk ,t + Ak ,t ) - Dirty price of a bond
CFi ,kj - Cash flow of a bond
5.2 Index Z-spread over Libor
The index Z-spread is calculated as follows:
n
∑ (Z - spreadBond i,t ⋅ MVi,t ⋅ Di,t )
Z - spreadIndex, t =
i =1
 n

 MVi,t ⋅ D i,t 


 i =1

∑
where:
– Z - spread Bond i,t - Z-spread of a bond
–
–
i at time t
MVi, t - Market value of a bond i at time t
Di, t - Effective duration of a bond i at time t
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Markit iBoxx Spread Analytics
6 Option Adjusted Spread
6.1 Methodology
Similar to Z-spread, OAS is the spread over the benchmark zero coupon curve realized if the bond is held until
maturity. The major difference is the interest rate volatility assumption used in OAS.
Due to the fact that interest rate changes can affect the cash flows of the security with embedded option the following
relationship cab be highlighted:
Z-spread = OAS + Option cost
6.1.1 Constructing Binomial Interest Rate Tree
Two components are required to determine the interest rate tree:
–
Benchmark Zero Coupon Curve
–
Empirical Volatility
Empirical volatility is annualized daily standard deviation of percentage change in daily yields from their mean.
The number of observations equals to the number of trading days in one year period minus 5% of observations from
each tail of the distribution of percentage change in daily yields.
The daily standard deviation is annualized by multiplying it by the square root of 252 trading days in a year.
6.1.2 OAS Calculation
The OAS is calculated as the spread that will make the present value of the cash flows of the respective bond equal
to the market dirty price when discounted at the benchmark spot rate plus the OAS spread.
Given spot rates and empirical volatility the interest rate tree is derived using the iterative search method.
9.568%
5.383%
2.717%
0.989%
0.094%
3.553%
1.830%
0.742%
0.147%
1.319%
0.622%
0.203%
0.490%
0.212%
0.182%
Interest rate tree of 2.5-year bond with semi-annual coupon payments.
After the binomial interest rate tree is derived, the theoretical price of the bond is found by conventional backward
induction of future cash flows.
The OAS is found iteratively by using the Newton method, such that adding the spread to every node of the interest
rate tree would make the present value of the cash-flows equal to the market dirty price of the bond.
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6.2 Index OAS
n
∑ (OASBond i,t ⋅ MVi,t ⋅ Di,t )
OASIndex, t =
i=1




n

i=1

∑ MVi,t ⋅ Di,t 
where:
– OAS Bond i,t - OAS of a bond
–
–
i at time t
MVi, t - Market value of a bond i at time t
Di, t - Effective duration of a bond i at time t
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Markit iBoxx Spread Analytics
7 Asset Swap Spread
7.1 Methodology
Markit SWAP curve, constructed from Libor rates and ICAP swap rates, is central to the process of asset swap
spreads calculation. As soon as the curve is defined the present value of fixed and floating payoffs is calculated, and
the asset swap spread is determined.
The curve is interpolated to account for fixed and floating payoffs dates.
7.1.1 Asset Swap Spread Calculation
To evaluate asset swap spread one needs to distinguish between fixed and floating payments.
Given the frequency of fixed rate payments, the floating rate payment frequency is determined as follows:
1
2
3
4
Fixed rate paid yearly = Floating rate paid semiannually
Fixed rate paid semiannually = Floating rate paid quarterly
Fixed rate paid quarterly = Floating rate paid monthly
else: Fixed frequency = Floating frequency
Floating Rate Payments
Fixed Rate Payments
DFnFixed =
1
Fixed
(SWAPn + 1)Ln
/360
where:
1
(SWAPn + 1)
LFloating / 360
n
where:
Markit SWAP curve rate at the next
coupon payment day
Number of days between next coupon
payment day and calculation day (given
bond’s day count convention)
SWAPn
LFixed
n
Fixed Rate Payments
PVFixed =
DFnFloating =
T
∑
C t ⋅ DFtFixed + Principal T ⋅ DFtFixed
t =1
where:
C t - Coupon payment
SWAPn
LFloating
n
Markit SWAP curve rate at the next
coupon payment day
Number of days between next coupon
payment day and calculation day
(ACT/360 day count convention)
Floating Rate Payments
PVFloating =
T
L
∑ 360t ⋅ DFtFloating
t =1
where:
L t - Number of days between floating rate payments
Given the present value of fixed and floating rate payments, the asset swap spread is calculated as follows:
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ASW =
PVFixed − DP
PVFloating
where:
DP - Bond’s market price (dirty price)
7.2 Index Asset Swap Spread
n
∑ (ASWBond i,t ⋅ MVi,t ⋅ Di,t )
ASWIndex, t =
i=1
n
∑ (MVi,t ⋅ Di,t )
i=1
where:
– ASWBond i,t - Asset swap spread of a bond
i at time t
–
MVi, t - Market value of a bond i at time t
–
Di, t - Effective duration of a bond i at time t
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Markit iBoxx Spread Analytics
8 Further information
For contractual or content issues please refer to
Markit Indices Limited
Goetheplatz 5
60313 Frankfurt am Main
Germany
Tel
Fax
+49 (0) 69 299 868 100
+49 (0) 69 299 868 149
E-mail
internet:
iBoxx@Markit.com
indices.markit.com
For technical issues please refer to
E-mail
iBoxx@Markit.com
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