bond pres4

8/12/2019 Bond Pres4

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- 1 -

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 :

=1T 1/ freq, =2T 2/freq, =iT i/ freq, … =× freqnT 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

Bullets bonds



- 2 -

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.



- 3 -

Notations

• The current market date is T.

• We have a series of positive cash flow n F F F ,,, 21 L at increasing dates nT T T ,,, 21 L ,

with T T >1 .

• We will use the notation nt t t ,,, 21 L for the year fractions corresponding to dates

nT T T ,,, 21 L :

o For example,

−

365

1 T T will be noted 1t , and so on…

• The year fractions are always calculated starting from current date T ; we will use the

same notation for any date T , obviously dT dt i ×−= 365/1

Definition of annual yield

The annual yield of a series of deterministic and positive cash flow n F F F ,,, 21 L at dates

nT T T ,,, 21 L (also called internal rate of return) is defined by :

( ) PV Market

r

F in

it i

_ 11

=+

∑=

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.

Yield for a series of cash

flows



- 4 -

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 :

( ) ( ) PV Market

r

C

r

C ni t

n

it

_ 1

1

1

1

1

=+

++

+∑−

=

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 nt t t n === ,,2,1 21 L and C = r

( ) ( )1

1

1

1

1

1

=+

++

+∑−

= C

C

C

C n

n

ii

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 /

( ) ( ) PV Market freqr

freqC

freqr

freqC ni

t freq

n

it freq _

/1

/1

/1

/1

1 =+

+

++ ×

−

=×

∑

Yield for a bullet bond



- 5 -

If freq = 2, we have :

( ) ( )

PV Market

r

C

r

C ni t

n

i

t _

2/1

2/1

2/1

2/2

1

1

2=

+

++

+

×

−

=

×∑ ;

r is then called a semi-annual yield.

Then again, for freq = 2, if r = C,

( ) ( )1

2/1

2/1

2/1

2/1

1

=+

++

+∑−

= C

C

C

C n

n

ii

The relation between the semi-annual yield r and the annual yield r’ is :

( ) ( )22/1'1 r r +=+

For a quarterly bullet bond, in same way :

( ) ( )44/1'1 r r +=+

When comparing bond with different frequency, it’s better of course to compare yield using the

same convention, so in practice compare annual yield.

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 :

Dirty price, clean price,

accrued interest



- 6 -

Accrued Interest =( ) prevnext

prev

T T

T T freqC cc

−

−×= / ,

prevT being the coupon date for the previous coupon at date T and next T 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 :

( ) ( )r

C

r

C P

ni t

n

it

+

++

+=∑

−

= 1

1

1

1

1

,

( ) rdt P dt r P t

P ×−≈+−=∂∂ 1ln

For r = C, P = 1 at coupon dates , dt C t

P ×−=

∂

∂.In other words, around par, everyday we gain

365

1×C . This is the rational for accrued interest formula.

Remark :

For calculating the accrued interest or the year fractions nt t t ,,, 21 L

, the market practice is infact to use specific rules called “basis”.



- 7 -

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 nt t t ,,, 21 L .

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 n F F F ,,, 21 L at increasing dates nT T T ,,, 21 L will be defined

by :

( ) ( )

( )

( )

( )

1

,,1,0

,

,

,

,

,,,,

1

1

11

1

21

=

=>=

=×=

∑

∑

∑∑∑

=

=

==

=

n

i

i

n

i

i

ii

n

i

iii

n

in

i

i

in

w

ni

F T PV

F T PV w

t wt

F t PV

F t PV F F F T D

L

L

The duration is the iw weighted average of the cash-flows at maturities nt t t ,,, 21 L , and so :

• The duration of a zero-coupon is its maturity (the year fraction).

• The duration always have the following property :

( ) nn t F F F Dt ≤≤ L,, 211

We don’t explain here how to calculate the present value of each cash-flow, ( )i F T PV , . 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 n F F F ,,, 21 L and

their total present value ( )∑=

n

i

i F T PV 1

, .

( )( )

∑∑== +

=n

i

it

n

i

i F r

F T PV i

11 1

1,

Then it is easy to check that the duration can also be defined by :

Duration, sensitivity for a

series of cash-flows



- 8 -

( )

( ) ( )r

r F F F PV

r

r F F F PV

Durationn

n

+×∂

∂−

= 1,,,,

,,,,

21

21

L

L

We will also define the sensitivity by :

( )

( )r F F F PV

r

r F F F PV

ySensitivit n

n

,,,,

,,,,

21

21

L

L

∂

∂

−= .

Remarks :

• The values of the duration and sensitivity are not very different for usual values of r .

• If we multiply the cash-flows n F F F ,,, 21 L by a constant λ , 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

in

i

ii

iin

F T T B

F T T B F F F T D

1

1

21

,

,,,, L ,

tThe numerical value of the duration will of course be different.

Risk, Duration and Sensitivity for annual bonds

We now work with bullet bonds of notional 1.

Let’s note :

( ) ( )r

C

r

C P

ni t

n

it

+

++

+=∑

−

= 1

1

1

1

1

,

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 :

Risk, sensitivity, duration

for a bond



- 9 -

r

P Risk

∂

∂−=

The convexity with respect to the annual yield r will be defined as :

2

2

r

P convexity

∂

∂=

The duration by :

( ) ( )

P

r risk r

P

r

P

Duration +×

=+×∂

∂−

=1

1

The sensitivity by :

( )r duration P

r

P

ySensitivit +=∂

∂−

= 1/

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 :

( )r cc P

r

P

+×−∂

∂−

1 for the duration

cc P

r

P

−∂∂−

for the sensitivity.

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 :

( ) ( )2/1

2/1

2/1

2/2

1

12

r

C

r

C P

ni t

n

it

+

++

+=∑

−

=

The risk with respect the semi-annual yield, r , is :

r

P Risk

∂

∂−=

The duration by :



- 10 -

( )2/1 r P

r

P

Duration +×∂

∂−

=

The sensitivity by :

( )2/1/ r duration P

r

P

ySensitivit +=∂

∂−

=

The risk with respect the annual yield, defined by ( ) ( )22/1'1 r r +=+ is :

)2/1/( r r

P +

∂

∂−

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 :

( ) ( ) freqr

freqC

freqr

freqC P

ni t freq

n

it freq

/1

/1

/1

/1

1 +

++

+=

×

−

=×∑

( ) 1/1/

−+

∂

∂−

freq freqr

r

P

will be the risk with respect the annual yield of the bond.

( )( )

( ) 1222/1

/11

/1

1−−

+

−×+

+×

freq freq freqr

freqrisk

freqr Convexity

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



- 11 -

Coupon = 4%

Risk

yield

maturity 1% 2% 3% 4% 5% 6% 7% 8% 9% 10%

1Y 1.02 1.00 0.98 0.96 0.94 0.93 0.91 0.89 0.88 0.86

2Y 2.06 2.00 1.94 1.89 1.83 1.78 1.73 1.69 1.64 1.60

3Y 3.12 3.00 2.88 2.78 2.67 2.57 2.48 2.39 2.31 2.22

4Y 4.19 3.99 3.81 3.63 3.46 3.31 3.16 3.02 2.88 2.76

5Y 5.28 4.99 4.71 4.45 4.21 3.98 3.77 3.57 3.39 3.21

6Y 6.39 5.98 5.60 5.24 4.91 4.61 4.33 4.06 3.82 3.59

7Y 7.52 6.97 6.47 6.00 5.58 5.19 4.83 4.49 4.19 3.91

8Y 8.66 7.96 7.32 6.73 6.20 5.72 5.28 4.87 4.51 4.17

9Y 9.82 8.94 8.15 7.44 6.79 6.21 5.68 5.21 4.78 4.39

10Y 11.00 9.92 8.97 8.11 7.35 6.66 6.05 5.50 5.01 4.56

12Y 13.39 11.87 10.55 9.39 8.36 7.46 6.67 5.97 5.36 4.81

15Y 17.08 14.77 12.80 11.12 9.68 8.44 7.38 6.47 5.68 5.00

20Y 23.50 19.51 16.25 13.59 11.41 9.61 8.12 6.90 5.88 5.03

30Y 37.12 28.56 22.13 17.29 13.62 10.82 8.67 7.01 5.72 4.7140Y 51.54 36.95 26.85 19.79 14.81 11.25 8.69 6.81 5.43 4.39

50Y 66.48 44.64 30.62 21.48 15.44 11.36 8.57 6.61 5.21 4.20

Duration

1% 2% 3% 4% 5% 6% 7% 8% 9% 10%

1Y 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00

2Y 1.96 1.96 1.96 1.96 1.96 1.96 1.96 1.96 1.96 1.96

3Y 2.89 2.89 2.89 2.89 2.88 2.88 2.88 2.88 2.88 2.88

4Y 3.79 3.78 3.78 3.78 3.77 3.77 3.76 3.76 3.75 3.75

5Y 4.66 4.65 4.64 4.63 4.62 4.61 4.60 4.59 4.58 4.57

6Y 5.50 5.48 5.47 5.45 5.43 5.42 5.40 5.38 5.36 5.357Y 6.32 6.29 6.27 6.24 6.22 6.19 6.16 6.13 6.10 6.07

8Y 7.12 7.08 7.04 7.00 6.96 6.92 6.88 6.84 6.79 6.75

9Y 7.89 7.84 7.79 7.73 7.68 7.62 7.56 7.50 7.44 7.37

10Y 8.65 8.58 8.51 8.44 8.36 8.28 8.20 8.12 8.03 7.95

12Y 10.11 10.00 9.88 9.76 9.64 9.51 9.37 9.24 9.10 8.95

15Y 12.19 11.99 11.78 11.56 11.34 11.11 10.87 10.62 10.37 10.12

20Y 15.40 14.99 14.57 14.13 13.68 13.22 12.74 12.26 11.78 11.30

30Y 21.13 20.12 19.06 17.98 16.90 15.82 14.78 13.77 12.82 11.92

40Y 26.22 24.36 22.46 20.58 18.77 17.06 15.49 14.07 12.81 11.70

50Y 30.86 27.96 25.08 22.34 19.83 17.59 15.64 13.98 12.58 11.40

Sensitivity

1% 2% 3% 4% 5% 6% 7% 8% 9% 10%

1Y 0.99 0.98 0.97 0.96 0.95 0.94 0.93 0.93 0.92 0.91

2Y 1.94 1.92 1.90 1.89 1.87 1.85 1.83 1.81 1.80 1.78

3Y 2.86 2.83 2.80 2.78 2.75 2.72 2.69 2.67 2.64 2.61

4Y 3.75 3.71 3.67 3.63 3.59 3.55 3.51 3.48 3.44 3.41

5Y 4.61 4.56 4.50 4.45 4.40 4.35 4.30 4.25 4.20 4.15

6Y 5.45 5.38 5.31 5.24 5.18 5.11 5.05 4.98 4.92 4.86

7Y 6.26 6.17 6.09 6.00 5.92 5.84 5.76 5.68 5.60 5.52

8Y 7.05 6.94 6.84 6.73 6.63 6.53 6.43 6.33 6.23 6.13

9Y 7.81 7.69 7.56 7.44 7.31 7.19 7.06 6.94 6.82 6.70

10Y 8.56 8.41 8.26 8.11 7.96 7.81 7.66 7.52 7.37 7.22

12Y 10.01 9.80 9.59 9.39 9.18 8.97 8.76 8.55 8.34 8.14

15Y 12.06 11.75 11.44 11.12 10.80 10.48 10.16 9.84 9.52 9.20

20Y 15.24 14.70 14.15 13.59 13.03 12.47 11.91 11.36 10.81 10.28

30Y 20.92 19.72 18.51 17.29 16.09 14.93 13.81 12.75 11.76 10.84

40Y 25.96 23.88 21.81 19.79 17.88 16.10 14.48 13.03 11.75 10.63

50Y 30.55 27.41 24.35 21.48 18.88 16.59 14.62 12.94 11.54 10.36



- 12 -

Convexity

1% 2% 3% 4% 5% 6% 7% 8% 9% 10%

1Y 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02

2Y 0.06 0.06 0.06 0.05 0.05 0.05 0.05 0.05 0.04 0.04

3Y 0.12 0.12 0.11 0.11 0.10 0.10 0.09 0.09 0.08 0.08

4Y 0.20 0.19 0.18 0.17 0.16 0.15 0.14 0.14 0.13 0.12

5Y 0.31 0.29 0.27 0.25 0.23 0.22 0.21 0.19 0.18 0.17

6Y 0.43 0.40 0.37 0.34 0.32 0.29 0.27 0.25 0.24 0.22

7Y 0.57 0.53 0.48 0.44 0.41 0.38 0.35 0.32 0.29 0.278Y 0.74 0.67 0.61 0.56 0.51 0.46 0.42 0.38 0.35 0.32

9Y 0.93 0.83 0.75 0.68 0.61 0.55 0.50 0.45 0.41 0.37

10Y 1.14 1.01 0.90 0.81 0.72 0.65 0.58 0.52 0.47 0.42

12Y 1.62 1.42 1.24 1.09 0.96 0.84 0.74 0.66 0.58 0.51

15Y 2.50 2.13 1.82 1.55 1.33 1.14 0.98 0.85 0.73 0.63

20Y 4.41 3.60 2.94 2.41 1.98 1.63 1.35 1.12 0.93 0.77

30Y 9.87 7.38 5.55 4.20 3.19 2.44 1.88 1.46 1.14 0.89

40Y 17.48 12.05 8.38 5.89 4.18 3.00 2.18 1.60 1.19 0.90

50Y 27.16 17.32 11.20 7.36 4.92 3.34 2.32 1.64 1.18 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%.



- 13 -

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 λ .



- 14 -

Coupon = yield

Risk

yieldmaturity 1% 2% 3% 4% 5% 6% 7% 8% 9% 10%

1Y 0.99 0.98 0.97 0.96 0.95 0.94 0.93 0.93 0.92 0.91

2Y 1.97 1.94 1.91 1.89 1.86 1.83 1.81 1.78 1.76 1.74

3Y 2.94 2.88 2.83 2.78 2.72 2.67 2.62 2.58 2.53 2.49

4Y 3.90 3.81 3.72 3.63 3.55 3.47 3.39 3.31 3.24 3.17

5Y 4.85 4.71 4.58 4.45 4.33 4.21 4.10 3.99 3.89 3.79

6Y 5.80 5.60 5.42 5.24 5.08 4.92 4.77 4.62 4.49 4.36

7Y 6.73 6.47 6.23 6.00 5.79 5.58 5.39 5.21 5.03 4.87

8Y 7.65 7.33 7.02 6.73 6.46 6.21 5.97 5.75 5.53 5.33

9Y 8.57 8.16 7.79 7.44 7.11 6.80 6.52 6.25 6.00 5.76

10Y 9.47 8.98 8.53 8.11 7.72 7.36 7.02 6.71 6.42 6.14

12Y 11.26 10.58 9.95 9.39 8.86 8.38 7.94 7.54 7.16 6.81

15Y 13.87 12.85 11.94 11.12 10.38 9.71 9.11 8.56 8.06 7.61

20Y 18.05 16.35 14.88 13.59 12.46 11.47 10.59 9.82 9.13 8.51

30Y 25.81 22.40 19.60 17.29 15.37 13.76 12.41 11.26 10.27 9.43

40Y 32.83 27.36 23.11 19.79 17.16 15.05 13.33 11.92 10.76 9.78

50Y 39.20 31.42 25.73 21.48 18.26 15.76 13.80 12.23 10.96 9.91

Duration

1% 2% 3% 4% 5% 6% 7% 8% 9% 10%

1Y 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00

2Y 1.99 1.98 1.97 1.96 1.95 1.94 1.93 1.93 1.92 1.91

3Y 2.97 2.94 2.91 2.89 2.86 2.83 2.81 2.78 2.76 2.74

4Y 3.94 3.88 3.83 3.78 3.72 3.67 3.62 3.58 3.53 3.49

5Y 4.90 4.81 4.72 4.63 4.55 4.47 4.39 4.31 4.24 4.17

6Y 5.85 5.71 5.58 5.45 5.33 5.21 5.10 4.99 4.89 4.79

7Y 6.80 6.60 6.42 6.24 6.08 5.92 5.77 5.62 5.49 5.36

8Y 7.73 7.47 7.23 7.00 6.79 6.58 6.39 6.21 6.03 5.87

9Y 8.65 8.33 8.02 7.73 7.46 7.21 6.97 6.75 6.53 6.3310Y 9.57 9.16 8.79 8.44 8.11 7.80 7.52 7.25 7.00 6.76

12Y 11.37 10.79 10.25 9.76 9.31 8.89 8.50 8.14 7.81 7.50

15Y 14.00 13.11 12.30 11.56 10.90 10.29 9.75 9.24 8.79 8.37

20Y 18.23 16.68 15.32 14.13 13.09 12.16 11.34 10.60 9.95 9.36

30Y 26.07 22.84 20.19 17.98 16.14 14.59 13.28 12.16 11.20 10.37

40Y 33.16 27.90 23.81 20.58 18.02 15.95 14.26 12.88 11.73 10.76

50Y 39.59 32.05 26.50 22.34 19.17 16.71 14.77 13.21 11.95 10.91

Sensitivity

1% 2% 3% 4% 5% 6% 7% 8% 9% 10%

1Y 0.99 0.98 0.97 0.96 0.95 0.94 0.93 0.93 0.92 0.912Y 1.97 1.94 1.91 1.89 1.86 1.83 1.81 1.78 1.76 1.74

3Y 2.94 2.88 2.83 2.78 2.72 2.67 2.62 2.58 2.53 2.49

4Y 3.90 3.81 3.72 3.63 3.55 3.47 3.39 3.31 3.24 3.17

5Y 4.85 4.71 4.58 4.45 4.33 4.21 4.10 3.99 3.89 3.79

6Y 5.80 5.60 5.42 5.24 5.08 4.92 4.77 4.62 4.49 4.36

7Y 6.73 6.47 6.23 6.00 5.79 5.58 5.39 5.21 5.03 4.87

8Y 7.65 7.33 7.02 6.73 6.46 6.21 5.97 5.75 5.53 5.33

9Y 8.57 8.16 7.79 7.44 7.11 6.80 6.52 6.25 6.00 5.76

10Y 9.47 8.98 8.53 8.11 7.72 7.36 7.02 6.71 6.42 6.14

12Y 11.26 10.58 9.95 9.39 8.86 8.38 7.94 7.54 7.16 6.81

15Y 13.87 12.85 11.94 11.12 10.38 9.71 9.11 8.56 8.06 7.61

20Y 18.05 16.35 14.88 13.59 12.46 11.47 10.59 9.82 9.13 8.51

30Y 25.81 22.40 19.60 17.29 15.37 13.76 12.41 11.26 10.27 9.43

40Y 32.83 27.36 23.11 19.79 17.16 15.05 13.33 11.92 10.76 9.78

50Y 39.20 31.42 25.73 21.48 18.26 15.76 13.80 12.23 10.96 9.91



- 15 -

Convexity

1% 2% 3% 4% 5% 6% 7% 8% 9% 10%

1Y 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02

2Y 0.06 0.06 0.06 0.05 0.05 0.05 0.05 0.05 0.05 0.05

3Y 0.12 0.11 0.11 0.11 0.10 0.10 0.10 0.09 0.09 0.09

4Y 0.19 0.18 0.18 0.17 0.16 0.16 0.15 0.15 0.14 0.14

5Y 0.29 0.27 0.26 0.25 0.24 0.23 0.22 0.21 0.20 0.19

6Y 0.40 0.38 0.36 0.34 0.32 0.31 0.29 0.28 0.27 0.26

7Y 0.53 0.50 0.47 0.44 0.42 0.40 0.38 0.36 0.34 0.32

8Y 0.67 0.63 0.59 0.56 0.52 0.49 0.46 0.44 0.41 0.39

9Y 0.84 0.78 0.73 0.68 0.63 0.59 0.55 0.52 0.49 0.46

10Y 1.02 0.94 0.87 0.81 0.75 0.70 0.65 0.61 0.56 0.5312Y 1.42 1.30 1.19 1.09 1.00 0.92 0.85 0.78 0.72 0.67

15Y 2.15 1.92 1.73 1.55 1.40 1.27 1.15 1.05 0.95 0.87

20Y 3.63 3.16 2.75 2.41 2.11 1.86 1.65 1.46 1.30 1.16

30Y 7.54 6.16 5.07 4.20 3.50 2.95 2.49 2.12 1.82 1.57

40Y 12.47 9.60 7.47 5.89 4.70 3.79 3.10 2.55 2.13 1.80

50Y 18.19 13.21 9.77 7.36 5.64 4.40 3.49 2.81 2.30 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.

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 1021 ,,, T T T L the coupon payment dates dates on the fixed leg

We note'

20

'

2

'

1 ,,, T T T L

the coupon payment dates on the float leg

Of course the 1021 ,,, T T T L are included in the'

20

'

2

'

1 ,,, T T T L .

Link between swap and

bonds



- 16 -

We know that the EURIBOR6M paid at date'

iT is the EURIBOR6M fixed at 2'

1 −−iT business days

note (the fixing dates) and calculated on period ( ) M T T ii 6, ''

1 +− .

We first give the schedule of the two legs :

Float leg schedule

• Pay dates are'

20

'

2

'

1 ,,, T T T L , the payment dates

• Fix dates are the fixing dates : 2'

1 −−iT business days is the fixing dates for date'

iT

• (Lib Start, LibEnd) = ) M T T ii 6, '1

'1 +−− for date

'

iT is the period of calculation of the

EURIBOR6M2

.

• Pay Cvge is the coverage of payment :( )

360

'

1

'

−− ii T T =

'

iδ

• Lib coverage is the coverage for the calculation of the EURIBOR6M :( )

360

6'

1

'

1 −− −+ ii T M T =

''

iδ

• DF Pay Dates = Discount Factor at Payment dates = ( )',0 iT B

2 Lib is for Libor as for most currencies the trading place is London



- 17 -

• Zc Pay dates is the associated zero-coupon rate.

• FRA'

iT =( )

( )

−

+−

− 16,0

,01'

1

'

1

'' M T B

T B

i

i

iδ is the forward EURIBOR6M for period

M T T ii 6, ''

1 +− , paid at date'

iT and fixed at 2'

1 −−iT business days.

• The flow at date'

iT is FRA''

iiT δ × (otherwise column 3 × column 9 = column 10 in

previous schedule).

Looking at the schedule, we see that M T i 6'

1 +− is not always equal to '

iT , so '

iδ is not always to

''

iδ .

The present value of the float leg is :

( ) ( ) ( )''20

1

',0 ii

i

i T FRAT B floatleg PV δ ∑=

= = 0.3255



- 19 -

( ) ( ) ( )( ) ( ) ( )∑=

− −=−=20

1

'

20

'

0

''

1 ,0,0,0,0 _ i

ii T BT BT BT Bleg float PV

Of course, quants don’t take into account the 2 business rules, they do as if fixing date for date'

iT is

'

1−iT .

They do as if .0'

0 =T

Of course 10

'

20 T T = , 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 nT :

( )nT B ,01− .

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 equalto 1).



- 20 -

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 :

( )( ) ( ) ( )10

'

1

'

1

'

1

'

0 ,0,0,61 T BT BT T M Euribor −+ δ .

To get this formula, just write that the leg is the sum of a floating leg starting at date'

1T , whose

valuation is ( ) ( )10

'

1 ,0,0 T BT B − , and the present value of the known fixing, whose valuation is

( ) ( )'

1

'

1

'

1

'

0 ,0,6 T BT T M Euribor δ .

This formula is of course valuable if'

1

'

0 0 T T ≤≤ . “0” being the current market date…

To come to the point of our demonstration after all these perhaps painful preliminaries, let’s now startfrom 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 ( ) ( )( )1010 ,0,01 T BT B +− , 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 a10Y bond.



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 ( 5.0−≈ for our example as the float leg is payer).

bond pres4

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