Bullets bonds
Let’s describe first a fixed rate bond without amortizing in a more general way :
Let’s note :




C the annual fixed rate, it’s a percentage
N the notional,
freq (=1, 2, 4…) the number of coupon per year,
R the redemption of capital,
then for a exactly n years bond, the cash flow at dates :
T1  1/freq, T2  2/freq, Ti  i/freq, … Tn freq  n (number of cash flows = freq × n) are :
C/freq×N if i < n
C/freq×N + R if i = n
Schedule Example with n= 10, freq = 1
-1-
A fixed rate bullet bond is a bond with constant coupon whose capital is redeemed at par :
R=N
In general : last cash flow = last coupon + redemption of capital, redemption of capital = notional
of the bond if bond redeemed at par).
To describe the schedule of a fixed rate bullet bond, one just need the annual fixed rate and the
frequency of coupons.
In the rest of the document we will consider only bond with a notional of 1 and redemption at par :
N=R=1
This is the market practice for probably 99% of fixed rate constant notional bond (no
amortizing).
Such bonds will be called bullet bond.
The definition given can also be used for more generalized bond, with exactly same formula and
very similar conclusions.
-2-
Yield for a series of cash
flows
 Notations

The current market date is T.

We have a series of positive cash flow F1 , F2 ,, Fn at increasing dates T1 , T2 ,,T n ,
with T1  T .

We will use the notation t1 , t 2 ,, t n for the year fractions corresponding to dates
T1 , T2 ,, Tn :
o

 T1  T 
 will be noted t1 , and so on…
 365 
For example, 
The year fractions are always calculated starting from current date T ; we will use the
same notation for any date T, obviously dt i  1 / 365  dT
 Definition of annual yield
The annual yield of a series of deterministic and positive cash flow F1 , F2 ,, Fn at dates
T1 , T2 ,,T n (also called internal rate of return) is defined by :
n

i 1
Fi
1  r 
ti
 Market _ PV
In other words, r is the unique discount rate that matches the discounted PV at a unique rate
and the market present value.
It can be shown easily that r is unique because all the cash-flow are positive.
r can be found in a few iterations by using a Newton-Raphson procedure (very efficient because
the PV function is convex).
We don’t explain how the market calculate the PV ! It’s a chicken and egg (or loop) story The
market practice is to use yields to quote bond, then a zero-coupon curve can be defined …
For swaps, again, the practice is of course to quote vanilla swaps and to derive a zero-coupon
curve.
-3-
Yield for a bullet bond
 Yield for annual bond
The yield (taux actuariel in French) of a bullet bond with one coupon by year and rate of coupon C
is then defined by :
n 1

i 1
C
1  r 
ti

1 C
1  r 
tn
 Market _ PV
Market_PV is by definition the market present value of the bond. Of course, one can say the yield
is derived from the market value, or the market value is derived from the yield.
It is well known and easy to demonstrate that if t1  1, t 2  2,, t n  n and C = r
n 1

i 1
C

1 C
1  C  1  C 
i
n
1
In other words, a (exactly) n years bond with coupon C and yield C is at par (price = notional).
 Yield for semi-annual, quarterly…bonds
For a bullet bond with more than a coupon per year, the market practice is to use yield to
maturity defined by /
n 1

i 1
C / freq
1  r / freq 
freqt i

1  C / freq
1  r / freq 
-4-
freqt n
 Market _ PV
If freq = 2, we have :
n 1

i 1
C/2
1  r / 2
2t i

1 C / 2
1  r / 2
2t n
 Market _ PV ;
r is then called a semi-annual yield.
Then again, for freq = 2, if r = C,
n 1

i 1
C/2
1  C / 2 
i

1 C / 2
1  C / 2 
n
1
The relation between the semi-annual yield r and the annual yield r’ is :
1  r '  1  r / 22
For a quarterly bullet bond, in same way :
1  r '  1  r / 44
When comparing bond with different frequency, it’s better of course to compare yield using the
same convention, so in practice compare annual yield.
Dirty price, clean price,
accrued interest
It is easy to see that, if we use the same yield to calculate the present value of a bond at two
consecutive dates, there is a jump in the present value when the coupon falls. This is why the
present value is called the dirty price. The jump is of course the coupon (C for annual bond or C/2
for semi-annual bond)
In order to have “continuity” (i.e no jump) from one day to the following , the yield being
unchanged, the market practice is to use “clean price” to quote bonds.
The clean price (“prix pied de coupon” in French) is the dirty price (“prix plein coupon” in
French) minus the accrued interest :
Clean price = Dirty price - cc
Accrued interest (“coupon couru” in French) at date t, is defined by :
-5-
Accrued Interest = cc  C / freq 
T  T 
T
next
prev
 T prev 
,
T prev being the coupon date for the previous coupon at date T and Tnext the coupon date for the
next coupon at date T.
Roughly speaking, the accrued interest is the “pro rata” of coupon at date T.
It’s easy to check that if :
n 1
P
i 1
C
1  r 
ti

1 C
1  r 
tn
,
P
  P ln 1  r dt   P  rdt
t
For r = C, P = 1 at coupon dates,
C
P
 C  dt .In other words, around par, everyday we gain
t
1
. This is the rational for accrued interest formula.
365
Remark :
For calculating the accrued interest or the year fractions t1 , t 2 ,, t n , the market practice is in
fact to use specific rules called “basis”.
-6-
For each market, i.e corporate bonds, USD, EUR, GBP, JPY…government bonds, there is a
specific basis for the exact calculation of accrued interest (30/360, ACT/ACT,30/360E,
ACT/365…)and another specific basis for the calculation of the year fractions t1 , t 2 ,, t n .
The purpose of this document is not to enter into these kind of details, so we assume ACT/365
basis through all the document. It doesn’t change any conclusion or interpretation. Using the exact
rules change very marginally the calculation (duration, risk…) and is only necessary when
working in dealing room.
The duration at date t of a series of deterministic and positive deterministic
cash flows F1 , F2 ,, Fn at increasing dates T1 , T2 ,,T n will be defined
Duration, sensitivity for a
series
of
cash-flows
accrued interest
by :
PV t , Fi 
n
DT , F1 , F2 , Fn   
i 1
wi 
PV T , Fi 
n
 PV T , Fi 
n
 PV t , Fi 
n
 t i   wi t i ,
i 1
i 1
 0, i  1,, n
i 1
n
 wi  1
i 1
The duration is the wi weighted average of the cash-flows at maturities t1 , t 2 ,, t n , and so :


The duration of a zero-coupon is its maturity (the year fraction).
The duration always have the following property :
t1  DF1 , F2 , Fn   t n
We don’t explain here how to calculate the present value of each cash-flow, PV T , Fi  . It
will be of course calculated using discount factors derived from a zero-coupon curve, but here they
are supposed to be given.
Let’s assume now that we use the annual yield, r, associated to the cash-flows F1 , F2 ,, Fn and
n
their total present value
 PV T , F  .
i
i 1
n
n
 PV T , F    1  r 
i 1
i
i 1
1
ti
Fi
Then it is easy to check that the duration can also be defined by :
-7-
Duration 

PV F1 , F2 ,, Fn , r 
r
 1  r 
PV F1 , F2 ,, Fn , r 
We will also define the sensitivity by :
Sensitivity 

PV F1 , F2 ,, Fn , r 
r
.
PV F1 , F2 ,, Fn , r 
Remarks :

The values of the duration and sensitivity are not very different for usual values of r.

If we multiply the cash-flows F1 , F2 ,, Fn by a constant  , the yield, duration, and
sensitivities are obviously unchanged.

If we go back to the general definition of duration and use discount factors instead of
using r :
n
DT , F1 , F2 , Fn   
i 1
BT , Ti Fi
n
 BT ,Ti Fi
,
i 1
tThe numerical value of the duration will of course be different.
Risk, sensitivity, duration
for a bond
 Risk, Duration and Sensitivity for annual bonds
We now work with bullet bonds of notional 1.
Let’s note :
n 1
P
i 1
C
1  r 
ti

1 C
1  r 
tn
,
the market present value of this bond, r being the annual yield associated to P.
The risk with respect to r will be defined as :
-8-
Risk  
P
r
The convexity with respect to the annual yield r will be defined as :
2P
convexity  2
r
The duration by :
Duration 

P
r  1  r   risk  1  r 
P
P
The sensitivity by :
Sensitivity 

P
r  duration / 1  r 
P
P is the dirty price. It is easy to see there is not jump for the risk at the coupon date, r being constant.
To avoid jump for duration and sensitivity at the coupon date, it is necessary to use a slightly different
definition, using P-cc, the clean price :
P
r  1  r  for the duration
P  cc

P
r for the sensitivity.
P  cc

 Risk, Duration and Sensitivity for non annual bonds
Let’s assume now we have a semi-annual bullet bond.
We remind that the semi-annual yield is defined by :
n 1
P
i 1
C/2
1  r / 2 

2t i
1 C / 2
1  r / 2 
The risk with respect the semi-annual yield, r, is :
Risk  
The duration by :
-9-
P
r
2tn
Duration 

P
r  1  r / 2
P
The sensitivity by :
P
Sensitivity  r  duration / 1  r / 2
P
2
The risk with respect the annual yield, defined by 1  r '  1  r / 2  is :


P
/(1  r / 2)
r
This formula also gives the way of calculating the sensitivity with respect the annual yield.
It’s very easy to check that whatever the frequency of coupon on the bond, the risk, convexity,
sensitivity depends on the frequency used for the yield, not the duration.
More generally, for a bullet bond of frequency freq, if r is the yield consistent with the frequency of
the bond :
n 1
P
i 1
C / freq
1  r / freq 

freqt i

1  C / freq
1  r / freq 
freqt n
P
freq1
/ 1  r / freq 
r
will be the risk with respect the annual yield of the bond.
Convexity
1  1 / freq 
1
 risk 
2 freq 2
1  r / freq 
1  r / freq 2 freq1
will be the convexity with respect the annual yield of the bond.
To give a more concrete idea of the different indicators, we give their values for various annual bullet
bonds with increasing maturities and increasing yield, in two ways :
First, the coupon is at 4%
Second, the coupon is equal to the yield
- 10 -
 Coupon = 4%
Risk
maturity
1Y
2Y
3Y
4Y
5Y
6Y
7Y
8Y
9Y
10Y
12Y
15Y
20Y
30Y
40Y
50Y
yield
1%
1.02
2.06
3.12
4.19
5.28
6.39
7.52
8.66
9.82
11.00
13.39
17.08
23.50
37.12
51.54
66.48
2%
1.00
2.00
3.00
3.99
4.99
5.98
6.97
7.96
8.94
9.92
11.87
14.77
19.51
28.56
36.95
44.64
3%
0.98
1.94
2.88
3.81
4.71
5.60
6.47
7.32
8.15
8.97
10.55
12.80
16.25
22.13
26.85
30.62
4%
0.96
1.89
2.78
3.63
4.45
5.24
6.00
6.73
7.44
8.11
9.39
11.12
13.59
17.29
19.79
21.48
5%
0.94
1.83
2.67
3.46
4.21
4.91
5.58
6.20
6.79
7.35
8.36
9.68
11.41
13.62
14.81
15.44
6%
0.93
1.78
2.57
3.31
3.98
4.61
5.19
5.72
6.21
6.66
7.46
8.44
9.61
10.82
11.25
11.36
7%
0.91
1.73
2.48
3.16
3.77
4.33
4.83
5.28
5.68
6.05
6.67
7.38
8.12
8.67
8.69
8.57
8%
0.89
1.69
2.39
3.02
3.57
4.06
4.49
4.87
5.21
5.50
5.97
6.47
6.90
7.01
6.81
6.61
9%
0.88
1.64
2.31
2.88
3.39
3.82
4.19
4.51
4.78
5.01
5.36
5.68
5.88
5.72
5.43
5.21
10%
0.86
1.60
2.22
2.76
3.21
3.59
3.91
4.17
4.39
4.56
4.81
5.00
5.03
4.71
4.39
4.20
2%
1.00
1.96
2.89
3.78
4.65
5.48
6.29
7.08
7.84
8.58
10.00
11.99
14.99
20.12
24.36
27.96
3%
1.00
1.96
2.89
3.78
4.64
5.47
6.27
7.04
7.79
8.51
9.88
11.78
14.57
19.06
22.46
25.08
4%
1.00
1.96
2.89
3.78
4.63
5.45
6.24
7.00
7.73
8.44
9.76
11.56
14.13
17.98
20.58
22.34
5%
1.00
1.96
2.88
3.77
4.62
5.43
6.22
6.96
7.68
8.36
9.64
11.34
13.68
16.90
18.77
19.83
6%
1.00
1.96
2.88
3.77
4.61
5.42
6.19
6.92
7.62
8.28
9.51
11.11
13.22
15.82
17.06
17.59
7%
1.00
1.96
2.88
3.76
4.60
5.40
6.16
6.88
7.56
8.20
9.37
10.87
12.74
14.78
15.49
15.64
8%
1.00
1.96
2.88
3.76
4.59
5.38
6.13
6.84
7.50
8.12
9.24
10.62
12.26
13.77
14.07
13.98
9%
1.00
1.96
2.88
3.75
4.58
5.36
6.10
6.79
7.44
8.03
9.10
10.37
11.78
12.82
12.81
12.58
10%
1.00
1.96
2.88
3.75
4.57
5.35
6.07
6.75
7.37
7.95
8.95
10.12
11.30
11.92
11.70
11.40
2%
0.98
1.92
2.83
3.71
4.56
5.38
6.17
6.94
7.69
8.41
9.80
11.75
14.70
19.72
23.88
27.41
3%
0.97
1.90
2.80
3.67
4.50
5.31
6.09
6.84
7.56
8.26
9.59
11.44
14.15
18.51
21.81
24.35
4%
0.96
1.89
2.78
3.63
4.45
5.24
6.00
6.73
7.44
8.11
9.39
11.12
13.59
17.29
19.79
21.48
5%
0.95
1.87
2.75
3.59
4.40
5.18
5.92
6.63
7.31
7.96
9.18
10.80
13.03
16.09
17.88
18.88
6%
0.94
1.85
2.72
3.55
4.35
5.11
5.84
6.53
7.19
7.81
8.97
10.48
12.47
14.93
16.10
16.59
7%
0.93
1.83
2.69
3.51
4.30
5.05
5.76
6.43
7.06
7.66
8.76
10.16
11.91
13.81
14.48
14.62
8%
0.93
1.81
2.67
3.48
4.25
4.98
5.68
6.33
6.94
7.52
8.55
9.84
11.36
12.75
13.03
12.94
9%
0.92
1.80
2.64
3.44
4.20
4.92
5.60
6.23
6.82
7.37
8.34
9.52
10.81
11.76
11.75
11.54
10%
0.91
1.78
2.61
3.41
4.15
4.86
5.52
6.13
6.70
7.22
8.14
9.20
10.28
10.84
10.63
10.36
Duration
1Y
2Y
3Y
4Y
5Y
6Y
7Y
8Y
9Y
10Y
12Y
15Y
20Y
30Y
40Y
50Y
1%
1.00
1.96
2.89
3.79
4.66
5.50
6.32
7.12
7.89
8.65
10.11
12.19
15.40
21.13
26.22
30.86
Sensitivity
1Y
2Y
3Y
4Y
5Y
6Y
7Y
8Y
9Y
10Y
12Y
15Y
20Y
30Y
40Y
50Y
1%
0.99
1.94
2.86
3.75
4.61
5.45
6.26
7.05
7.81
8.56
10.01
12.06
15.24
20.92
25.96
30.55
- 11 -
Convexity
1Y
2Y
3Y
4Y
5Y
6Y
7Y
8Y
9Y
10Y
12Y
15Y
20Y
30Y
40Y
50Y
1%
0.02
0.06
0.12
0.20
0.31
0.43
0.57
0.74
0.93
1.14
1.62
2.50
4.41
9.87
17.48
27.16
2%
0.02
0.06
0.12
0.19
0.29
0.40
0.53
0.67
0.83
1.01
1.42
2.13
3.60
7.38
12.05
17.32
3%
0.02
0.06
0.11
0.18
0.27
0.37
0.48
0.61
0.75
0.90
1.24
1.82
2.94
5.55
8.38
11.20
4%
0.02
0.05
0.11
0.17
0.25
0.34
0.44
0.56
0.68
0.81
1.09
1.55
2.41
4.20
5.89
7.36
5%
0.02
0.05
0.10
0.16
0.23
0.32
0.41
0.51
0.61
0.72
0.96
1.33
1.98
3.19
4.18
4.92
6%
0.02
0.05
0.10
0.15
0.22
0.29
0.38
0.46
0.55
0.65
0.84
1.14
1.63
2.44
3.00
3.34
7%
0.02
0.05
0.09
0.14
0.21
0.27
0.35
0.42
0.50
0.58
0.74
0.98
1.35
1.88
2.18
2.32
8%
0.02
0.05
0.09
0.14
0.19
0.25
0.32
0.38
0.45
0.52
0.66
0.85
1.12
1.46
1.60
1.64
9%
0.02
0.04
0.08
0.13
0.18
0.24
0.29
0.35
0.41
0.47
0.58
0.73
0.93
1.14
1.19
1.18
10%
0.02
0.04
0.08
0.12
0.17
0.22
0.27
0.32
0.37
0.42
0.51
0.63
0.77
0.89
0.90
0.87
Remarks :
For any given coupon, risk, duration, sensitivity and convexity are :


decreasing function of the yield for a given maturity
increasing function of the maturity for a given yield
For a given bond, i.e maturity and coupon being given (and also frequency, basis …) the risk is very
often interpreted as follows :
The risk is the number of bp in price for 1 bp1 in yield.
For example, we see that for a 10years bond with coupon 4%, the price decreases of 8.11bp when
yield moves of 1bp.
Somebody who holds 10000 bonds of notional 1 euro, will loose 8 euros if rates increases by 1bp.
If we forget the convexity, the same person will loose approximately 811 euros when rates increases
by 100bp.
The convexity can be interpreted as follows :
Holding bonds, we gain more when rates decrease by 100bp than we loose when rates increases by
100bp.
As the maturity increases, the convexity increases (strongly).
So, as there is no free lunch in the financial markets, it means the yield curve moves cannot be only
parallel shift. Otherwise, a strategy such as :


Being long 30 year bonds,
Short 10 years bonds so as total risk is zero
would gain whatever the yield curve move !
To hedge a portfolio, the risk is the relevant indicator, not the duration.
1
1bp = 1 basis point = 0.01%.
- 12 -
But the duration is a good indicator of the average maturity of a portfolio, as it’s unchanged if you
multiply your portfolio by a constant  .
- 13 -
 Coupon = yield
Risk
maturity
1Y
2Y
3Y
4Y
5Y
6Y
7Y
8Y
9Y
10Y
12Y
15Y
20Y
30Y
40Y
50Y
yield
1%
0.99
1.97
2.94
3.90
4.85
5.80
6.73
7.65
8.57
9.47
11.26
13.87
18.05
25.81
32.83
39.20
2%
0.98
1.94
2.88
3.81
4.71
5.60
6.47
7.33
8.16
8.98
10.58
12.85
16.35
22.40
27.36
31.42
3%
0.97
1.91
2.83
3.72
4.58
5.42
6.23
7.02
7.79
8.53
9.95
11.94
14.88
19.60
23.11
25.73
4%
0.96
1.89
2.78
3.63
4.45
5.24
6.00
6.73
7.44
8.11
9.39
11.12
13.59
17.29
19.79
21.48
5%
0.95
1.86
2.72
3.55
4.33
5.08
5.79
6.46
7.11
7.72
8.86
10.38
12.46
15.37
17.16
18.26
6%
0.94
1.83
2.67
3.47
4.21
4.92
5.58
6.21
6.80
7.36
8.38
9.71
11.47
13.76
15.05
15.76
7%
0.93
1.81
2.62
3.39
4.10
4.77
5.39
5.97
6.52
7.02
7.94
9.11
10.59
12.41
13.33
13.80
8%
0.93
1.78
2.58
3.31
3.99
4.62
5.21
5.75
6.25
6.71
7.54
8.56
9.82
11.26
11.92
12.23
9%
0.92
1.76
2.53
3.24
3.89
4.49
5.03
5.53
6.00
6.42
7.16
8.06
9.13
10.27
10.76
10.96
10%
0.91
1.74
2.49
3.17
3.79
4.36
4.87
5.33
5.76
6.14
6.81
7.61
8.51
9.43
9.78
9.91
2%
1.00
1.98
2.94
3.88
4.81
5.71
6.60
7.47
8.33
9.16
10.79
13.11
16.68
22.84
27.90
32.05
3%
1.00
1.97
2.91
3.83
4.72
5.58
6.42
7.23
8.02
8.79
10.25
12.30
15.32
20.19
23.81
26.50
4%
1.00
1.96
2.89
3.78
4.63
5.45
6.24
7.00
7.73
8.44
9.76
11.56
14.13
17.98
20.58
22.34
5%
1.00
1.95
2.86
3.72
4.55
5.33
6.08
6.79
7.46
8.11
9.31
10.90
13.09
16.14
18.02
19.17
6%
1.00
1.94
2.83
3.67
4.47
5.21
5.92
6.58
7.21
7.80
8.89
10.29
12.16
14.59
15.95
16.71
7%
1.00
1.93
2.81
3.62
4.39
5.10
5.77
6.39
6.97
7.52
8.50
9.75
11.34
13.28
14.26
14.77
8%
1.00
1.93
2.78
3.58
4.31
4.99
5.62
6.21
6.75
7.25
8.14
9.24
10.60
12.16
12.88
13.21
9%
1.00
1.92
2.76
3.53
4.24
4.89
5.49
6.03
6.53
7.00
7.81
8.79
9.95
11.20
11.73
11.95
10%
1.00
1.91
2.74
3.49
4.17
4.79
5.36
5.87
6.33
6.76
7.50
8.37
9.36
10.37
10.76
10.91
2%
0.98
1.94
2.88
3.81
4.71
5.60
6.47
7.33
8.16
8.98
10.58
12.85
16.35
22.40
27.36
31.42
3%
0.97
1.91
2.83
3.72
4.58
5.42
6.23
7.02
7.79
8.53
9.95
11.94
14.88
19.60
23.11
25.73
4%
0.96
1.89
2.78
3.63
4.45
5.24
6.00
6.73
7.44
8.11
9.39
11.12
13.59
17.29
19.79
21.48
5%
0.95
1.86
2.72
3.55
4.33
5.08
5.79
6.46
7.11
7.72
8.86
10.38
12.46
15.37
17.16
18.26
6%
0.94
1.83
2.67
3.47
4.21
4.92
5.58
6.21
6.80
7.36
8.38
9.71
11.47
13.76
15.05
15.76
7%
0.93
1.81
2.62
3.39
4.10
4.77
5.39
5.97
6.52
7.02
7.94
9.11
10.59
12.41
13.33
13.80
8%
0.93
1.78
2.58
3.31
3.99
4.62
5.21
5.75
6.25
6.71
7.54
8.56
9.82
11.26
11.92
12.23
9%
0.92
1.76
2.53
3.24
3.89
4.49
5.03
5.53
6.00
6.42
7.16
8.06
9.13
10.27
10.76
10.96
10%
0.91
1.74
2.49
3.17
3.79
4.36
4.87
5.33
5.76
6.14
6.81
7.61
8.51
9.43
9.78
9.91
Duration
1Y
2Y
3Y
4Y
5Y
6Y
7Y
8Y
9Y
10Y
12Y
15Y
20Y
30Y
40Y
50Y
1%
1.00
1.99
2.97
3.94
4.90
5.85
6.80
7.73
8.65
9.57
11.37
14.00
18.23
26.07
33.16
39.59
Sensitivity
1Y
2Y
3Y
4Y
5Y
6Y
7Y
8Y
9Y
10Y
12Y
15Y
20Y
30Y
40Y
50Y
1%
0.99
1.97
2.94
3.90
4.85
5.80
6.73
7.65
8.57
9.47
11.26
13.87
18.05
25.81
32.83
39.20
- 14 -
Convexity
1Y
2Y
3Y
4Y
5Y
6Y
7Y
8Y
9Y
10Y
12Y
15Y
20Y
30Y
40Y
50Y
1%
0.02
0.06
0.12
0.19
0.29
0.40
0.53
0.67
0.84
1.02
1.42
2.15
3.63
7.54
12.47
18.19
2%
0.02
0.06
0.11
0.18
0.27
0.38
0.50
0.63
0.78
0.94
1.30
1.92
3.16
6.16
9.60
13.21
3%
0.02
0.06
0.11
0.18
0.26
0.36
0.47
0.59
0.73
0.87
1.19
1.73
2.75
5.07
7.47
9.77
4%
0.02
0.05
0.11
0.17
0.25
0.34
0.44
0.56
0.68
0.81
1.09
1.55
2.41
4.20
5.89
7.36
5%
0.02
0.05
0.10
0.16
0.24
0.32
0.42
0.52
0.63
0.75
1.00
1.40
2.11
3.50
4.70
5.64
6%
0.02
0.05
0.10
0.16
0.23
0.31
0.40
0.49
0.59
0.70
0.92
1.27
1.86
2.95
3.79
4.40
7%
0.02
0.05
0.10
0.15
0.22
0.29
0.38
0.46
0.55
0.65
0.85
1.15
1.65
2.49
3.10
3.49
8%
0.02
0.05
0.09
0.15
0.21
0.28
0.36
0.44
0.52
0.61
0.78
1.05
1.46
2.12
2.55
2.81
9%
0.02
0.05
0.09
0.14
0.20
0.27
0.34
0.41
0.49
0.56
0.72
0.95
1.30
1.82
2.13
2.30
10%
0.02
0.05
0.09
0.14
0.19
0.26
0.32
0.39
0.46
0.53
0.67
0.87
1.16
1.57
1.80
1.91
Remarks :
It’s very easy to check that as now we work with bonds at par, the sensitivity is equal to the risk (and
so the risk is very close to the duration)
All the numbers above are very similar when moving to semi annual bonds.
Link between swap and
bonds
We will now remind why bonds and swaps are in fact very similar financial products.
In fact, in terms of risk, we can say that a 10 years swap and a 10 years bond are exactly the same
product (if we forget the use of coverage for the details of the schedule of a swap).
We will use as example a 10Y swap vanilla swap to be more concrete, the demonstration and so the
conclusion will be the same for any maturity.
We suppose the reader familiar with vanilla swap definition and pricing.
 Notations
The swap is a 10Y swap, receiver fixed rate, payer EURIBOR 6M
Market date 25/10/06



Frequency 12M, basis 30/360 on the fixed leg
Frequency 6M, basis ACT/360 on the float leg
The notional of the swap is 1€
We note T1 , T2 ,, T10 the coupon payment dates dates on the fixed leg
We note T1' , T2' ,, T20' the coupon payment dates on the float leg
- 15 -
Of course the T1 , T2 ,, T10 are included in the T1' , T2' ,, T20' .
We know that the EURIBOR6M paid at date Ti ' is the EURIBOR6M fixed at Ti ' 1  2 business days


note (the fixing dates) and calculated on period Ti ' 1 , Ti '  6M .
We first give the schedule of the two legs :
 Float leg schedule

Pay dates are T1' , T2' ,, T20' , the payment dates

Fix dates are the fixing dates : Ti ' 1  2 business days is the fixing dates for date Ti '

(Lib Start, LibEnd) = Ti'1 , Ti'1  6M
EURIBOR6M2.

Pay Cvge is the coverage of payment :


 for date
T
i
'
Ti ' is the period of calculation of the

 Ti ' 1
=  i'
360
Lib coverage is the coverage for the calculation of the EURIBOR6M :
 i''
2
Lib is for Libor as for most currencies the trading place is London
- 16 -
T
'
i 1

 6M  Ti ' 1
=
360



DF Pay Dates = Discount Factor at Payment dates = B 0, Ti '

Zc Pay dates is the associated zero-coupon rate.

FRA Ti '

The flow at date Ti ' is FRA Ti '   i' (otherwise column 3  column 9 = column 10 in
previous schedule).



1  B 0, Ti ' 1

 1 is the forward EURIBOR6M for period
'' 
'
 i  B 0, Ti 1  6M

'
'
'
Ti1 ,Ti  6M  , paid at date Ti and fixed at Ti' 1  2 business days.
 

=

 
Looking at the schedule, we see that Ti ' 1  6 M is not always equal to Ti ' , so  i' is not always to
 i'' .
The present value of the float leg is :
20


 
PV  floatleg    B 0, Ti '  i' FRA Ti ' = 0.3255
i 1
- 17 -
 Fixed leg schedule
Pay. Cvge
Pay. Dates
Fix. Dates
DF Pay Dates
flow
1.0055556
29/10/07
25/10/06
0.96294926
4.0004%
0.9944444
27/10/08
25/10/07
0.927323186
3.9758%
1
27/10/09
23/10/08
0.893046999
3.9980%
1
27/10/10
23/10/09
0.859704415
3.9980%
1
27/10/11
25/10/10
0.827219006
3.9980%
1.0055556
29/10/12
25/10/11
0.795085426
4.0202%
0.9972222
28/10/13
25/10/12
0.763922449
3.9869%
0.9972222
27/10/14
24/10/13
0.733555613
3.9869%
1
27/10/15
23/10/14
0.703542439
3.9980%
1
27/10/16
23/10/15
0.674342197
3.9980%
The definitions are exactly the same, of course the fixed leg is much simple to describe.

Pay cvge are the payment coverage :  i is the payment coverage at date Ti , calculated as the
year fraction between date Ti 1 , and Ti in basis 30/360.

10
LVL fixed _ leg     i B0, Ti  = 8.1411 is the Level associated to the schedule of this
i 1
10Y swap fixe leg.
The swap rate is by definition equal to
PV  floatleg 
= 3.9980%.
LVL fixed _ leg 
Knowing the value of the swap rate, we can calculate each cash-flow at date Ti :  i  3.9980% .
 Simplifications
We start with the float leg :
When reading a quantitative book or working paper, you will find much simple way of describe a float
leg.
Quants always do the following assumptions :
Ti ' 1  6M  Ti '   i'   i'' .
In the float leg schedule shown above, it means the columns Lib End and Pay Dates are identical,
same thing for columns Pay Cvge and Lib Coverage.
Then we get :
- 18 -
20

 
 
 
PV  float _ leg    B 0, Ti ' 1  B 0, Ti '  B 0, T0'  B 0, T20'

i 1
Of course, quants don’t take into account the 2 business rules, they do as if fixing date for date Ti ' is
Ti ' 1 .
They do as if T0'  0.
Of course T20'  T10 , it can be checked on the two above schedules.
So they get the well known formula for valuing a float leg at a fixing date (but just before the fixing
time !) of maturity Tn :
1 B0, Tn  .
We see then a very important property :
The value of the float leg then doesn’t depend of the index of the float leg : EURIBOR3M, EURIBOR
6M, EURIBOR 12M…
In or example, doing all these simplifications; we get for the floatleg :
0.3257
and then a swap rate of : 4.0001%.
As it is well known the approximation is very good.
When use these approximation ?
Not to value a swap portfolio : then you have to take into account the basis swap between
EURIBOR3M and EURIBOR6M, for example.
It means you have one forward curve by index, one unique discount curve. As you will use different
curves to calculate the forwards and to calculate the discount factors, there is no value to make such
simplifications.
In addition, the p & l impact on the valuation of thousands swaps with notional from 10M€ to more
1000M€ is not so small.
If you work on a model for a very exotic interest product (for which the margins fortunately much
more important than for vanilla swaps), there is value to make these approximations, at least when you
write the documentation of your model !
In the rest of the document we will do theses simplifications.
On the fixed leg, in order to get closer to a bond, we will assume the coverage to be identical (so equal
to 1).
- 19 -
We get now a swap rate of 4.0004%; instead of 4.0001% after the simplifications of the float leg. The
level is now 8.1407.
Overall, we just mean the details of convention (basis, business days, 2 days rule for the fixing vs the
payment…) are of course important for exact valuation, not for understand the properties of a vanilla
swap in terms of risk, exposure to the change in interest curves.
We also remind that when the fixing is known, the valuation of the flow leg becomes :
1  Euribor 6M T ,T  B0,T   B0,T  .
'
0
'
1
'
1
'
1
10
To get this formula, just write that the leg is the sum of a floating leg starting at date T1' , whose

  
Euribor 6M T , T  B0, T  .
valuation is B 0, T1'  B 0, T10 , and the present value of the known fixing, whose valuation is
'
0
'
1
'
1
'
1
This formula is of course valuable if T0'  0  T1' . “0” being the current market date…
To come to the point of our demonstration after all these perhaps painful preliminaries, let’s now start
from the schedules of the two legs of our 10Y swap :
Fix is now equal to 4.0004%.
Let’s assume now we add a cash-flow of 1 on each leg. We don’t change anything to the total value of
the swap of course, whatever the curve.
The value of the float leg is now 1 1  B0, T10   B0, T10  , the fixed leg is now a 10Y bond of
coupon 4.0004% !
For a trader holding a 10Y bond, financing the position at EURIBOR6M, the analysis is the same. In
other words, the float leg of a 10year vanilla receiver swap is the financing leg of a long position on a
10Y bond.
- 20 -
When we add the notional on the two legs of the swaps, at each fixing dates, (just before the fixing
time, 11am !), we just move all the risk of the swap on the fixed leg.
When the fixing is known, there is a slight risk on the float leg, which is initially around the index of
the float leg (  0.5 for our example as the float leg is payer).
- 21 -