. 6
( 16)


which we will call4 r(t), is the rate for which there is equality between:

1. The value of the investment acquired at t, calculated at this rate r(t).
2. The sum of:
” The value of the coupons falling due acquired at t, reinvested at the current rate
observed between 0 and t.
This is sometimes known as the ex post rate.
r(0) = r is of course the actuarial rate at issue.
118 Asset and Risk Management

” The discounted value in t of the ¬nancial ¬‚ows generated subsequent to t, calculated
using the market rate at the moment t.

Let us take the same example as above. Suppose that we are at the moment in time
immediately subsequent to payment of the third coupon (t = 3) and the market rate has
remained at 11.1 % for the ¬rst two years and has now changed to 12 %. The above
de¬nition gives us the equation that de¬nes the actuarial rate of return for the speci¬c
moment t = 3.

10 10 115
98 · (1 + r(3))3 = (10 · 1.111 · 1.12 + 10 · 1.12 + 10) + + +
1.12 1.122 1.123
= 35.33 + 98.76
= 134.09

This gives r(3) = 11.02 %.
It will of course be evident that if the rate of interest remains constant (and equal to
11.1) for the ¬rst three years, the above calculation would have led to r(3) = 11.1 %, this
being consistent with the principle of equivalence.
This example clearly shows the phenomenon of bond risk linked to changes in interest
rates. This phenomenon will be studied in greater detail in Section 4.2.1. Accrued interest
When a bond is acquired between two coupon payment dates, the purchaser pays not
only the value of the bond for that speci¬c moment but also the portion of the coupon
to come, calculated in proportion to the period that has passed since payment of the last
coupon. The seller, in fact, has the right to partial interest relating to the period from
the last coupon payment to the moment of the deal. This principle is called the accrued
interest system and the price effectively paid is the dirty price, as opposed to the clean
price, which represents merely the rate for the bond at the time of the deal.
Let us consider a bond of maturity T and non integer moment t + θ (integer t and
0 ¤ θ <1) (Figure 4.1).
The dirty price P (d) of the bond is linked to the clean price P by the relation Pt+θ =
Pt+θ + θ · Ct+1 .
The accrued interest system affects the rate of return, as the purchaser, at the moment
of the transaction, pre-¬nances part of the next coupon. A slight reduction in the actuarial
rate therefore occurs, and will be smaller the closer the moment of the deal is (before or
after) to a coupon payment date. For this reason, it is known as the festoon effect.

0 1 ... t (t + 1) ... T

Figure 4.1 Accrued interest
Bonds 119


Rate of return


0 1 2 3 4 5 6

Figure 4.2 Festoon effect

The actuarial rate of return is calculated in the classic way, and is de¬ned implicitly
by the relation:
Ck (1 + r)’k+t+θ + R(1 + r)’T +t+θ
Pt+θ + θ · Ct+1 =

Let us consider a bond, issued and repayable at par, with a price at a constant 100, which
presents annual coupons at 10 for ¬ve years and clearly has an actuarial rate at issue of
10 %. This will produce the graph shown in Figure 4.2.

4.1.3 Valuing a bond
The value of a bond at any given moment (that is, its price) is the discounted value of
the ¬nancial ¬‚ows generated (residual coupons and ¬nal repayment), the discounting rate
being the market rate5 for bonds of the same nature and the same residual time (we will
term it rm ).
On the day after the payment of the t th coupon, the valuation is made using the equation:
Ck (1 + rm )’k+t + R(1 + rm )’T +t
Pt =

At a given moment t + θ (integer t and 0 ¤ θ < 1), we will, taking the accrued interest
into account, have:
Ck (1 + rm )’k+t+θ + R(1 + rm )’T +t+θ
Pt+θ = ’θ · Ct+1 +

4.2.1 Sources of risk
A bond is a ¬nancial asset generally considered to be low risk in comparison to equities
or derivatives with high leverage effect. This does not mean, however, that holding bonds
does not pose certain risks.

This arises from the law of supply and demand.
120 Asset and Risk Management Default risk
The default risk, which relates to cases in which the issuer fails to honour his undertaking,
can be described as late payment of coupons, bankruptcy with consequent nonpayment
of coupons, and nonrepayment at maturity date.
This default risk cannot, of course, be quanti¬ed. Note, however, that bond market
professionals attach considerable importance to ratings allocated to issuers through certain
agencies such as Moody™s and Standard & Poor™s, with the aim of determining the quality
of the issuers and therefore ˜estimating™ the probability of default. This means that for
the same maturity date, a coupon for an issuer rating of AAA (the highest) will be less
than the coupon offered by a B-rated issuer; the concept intuitively recognised here is
that of market price of risk. In addition, the rating will have a direct in¬‚uence on another
nonquanti¬able risk, the liquidity of the security in question. Rate ¬‚uctuation risk
Another type of risk associated with bonds, this one quanti¬able, is the risk linked to
¬‚uctuations in interest rates.
For certain speci¬c bonds (known as call-associated bonds), the issuer has the right
to repay the bond before its maturity date. If the repayment is made just after a fall in
interest rates, it will not be possible for the investor to reinvest the total repaid at a rate
equivalent to that of the security, and the investor therefore suffers a loss of income.
However, as we stated in the introduction (Section 4.1.1), we are not looking at this
type of product here. We will therefore limit ourselves to analysing the risk of a ˜classic™
bond. Two aspects of this risk, which in one sense are contradictory, may be taken into
First, there is the risk of reinvestment. In the event of a change in market rates, the
coupons (and sometimes the repayment value itself) will be reinvested at a different rate.
In this instance, an increase (decrease) in the interest rate will be favourable (unfavourable)
to the investor.
Then, there is the risk of realisation if the bond is sold before its maturity date. The
sale price is determined by the discounted value of the term coupons (at the rate in force
on the market at the time) and by the repayment value. In this case, an increase (decrease)
in the interest rate will of course be unfavourable (favourable) to the investor.

Let us consider a bond with nominal value of 1000, without issue or repayment premium
(N V = P = R = 1000), with a period of eight years and a coupon constant of 100. In
this case, the actuarial rate at issue r is equal to the nominal return rate, that is, 10 %. Let
us suppose also that at the end of the second year, the market rate changes from 10 %
to 12 %.
What will happen if the investor wants to resell the bond after four years? To analyse
the situation, we will determine the actuarial rate of return r(4) at moment 4.
Reinvesting the coupons (at 10 % during the ¬rst two years and at 12 % afterwards)
gives, in t = 4, a purchase value of6 A4 = 100 · 1.1 · 1.122 + 100 · 1.122 + 100 · 1.12 +
100 = 475.424.
While without a rate change, we would have obtained A4 = 464.1, which clearly shows that a rise in rates is favourable.
Bonds 121

In addition, the resale value at t = 4 is given by the discounted value (12 %) of the
term coupons and the repayment:7

100 100 100 1100
B4 = + + + = 939.253
1.12 1.122 1.123 1.124

The ex post return rate is de¬ned by the relation P · (l + r(4))4 = 1414.677, which
ultimately leads to r(4) = 9.06 %. This rate is lower than the one found by replacing A4
with B4 by 1464.1, as we would be using 10 %, the actuarial rate of the bond at its issue.
By applying the same reasoning to an investor who resells his bond after seven years, we
obtain for a reinvestment value A7 = 1005.376 and for a realisation value B7 = 982.143
(instead of 948.717 and 1000 respectively if the rate had not changed). The ex post rate
of return is then given by r(7) = 10.31 (instead of 10 %). This time, the combination of
the two effects is reversed and favours the investor.

4.2.2 Duration De¬nition
The observations made in the above example raise the question of identifying at which
the increase/decrease in the interest rate is unfavourable/favourable to the investor and at
which point it becomes favourable/unfavourable to the investor.
The answer to this question is found in the concept of duration.8 Let us consider a
bond at the moment of its issue9 with actuarial rate r. Its duration, termed D, is de¬ned
as follows:
t · Ct (1 + r)’t + T · R(1 + r)’T
D= T
Ct (1 + r)’t + R(1 + r)’T

The concept of modi¬ed duration is also entered, as is the sensitivity coef¬cient, de¬ned by:
Dm =

In addition to this ˜simple™ concept of duration, a more developed concept of ˜stochastic
duration™ will be issued in Section 4.6. Interpretations
The concept of duration can be interpreted in several different ways; the two best known
While with an initial rate of 10 %, we would have obtained B4 = 1000. This time, the rise in interest rates clearly does

not favour the investor.
This concept was introduced in Macauley F., Some Theoretical Problems Suggested by the Movement of Interest Rates,
Bond Yields and Stock Prices in the United States since 1856, National Bureau of Economic Research, 1938, pp. 44“53.
This concept can easily be generalised at any time, provided it is assumed that the rate r stays constant.
122 Asset and Risk Management

Duration can ¬rst of all be considered as the average life span of the bond, the weighting
coef¬cients for this average being proportional to the discounted values for the rate r of
the ¬nancial ¬‚ows (coupons and repayment) at the corresponding times. In fact, given
that the denominator is none other than the price P on issue, we can write:
Ct (1 + r)’t R(1 + r)’T
D= t +T

C1 (1 + r)’1 CT ’1 (1 + r)’(T ’1) (CT + R)(1 + r)’T
=1· + · · · + (T ’ 1) · +T ·
Duration can also be interpreted in terms of sensitivity of prices to interest rates. When
two economic variables x and y are linked by a relation y = y(x), one way of describing
the way in which y depends on x is to use the derivative y (x).
The relation that de¬nes the actuarial rate, P = T Ct (1 + r)’t + R(1 + r)’T ,
de¬nes a function P (r). The derivative of this function can be written
t · Ct (1 + r)’t’1 ’ T · R(1 + r)’T ’1
P (r) = ’
t · Ct (1 + r)’t + T · R(1 + r)’T
= ’(1 + r)

= ’(1 + r)’1 D · P (r)

This shows how the duration works in the measurement of changes in P according to r.
This relation can also be written:
P (r)
D = ’(1 + r)
P (r)
P (r)
Dm = ’
P (r)

By taking a small variation in the r rate into account instead of the in¬nitesimal vari-
ation, and noting that P = P (r + r) ’ P (r), the previous relation is written as:

D = ’(1 + r) r
(1 + r)
= µ(P /1 + r)

r is the same as (1 + r).
because the variation
Bonds 123

We are looking here at the classic economic concept of elasticity. Generally speaking,
the elasticity µ(y/x) of y with respect to x,

µ(y/x) =

is interpreted as follows: if x increases by 1 %, y will vary (increase or decrease according
to the sign of y) by |µ(y/x)| %.
Duration is therefore shown as elasticity (with minus sign) of the price of the bond
with respect to (1 + r), and a relative increase of 1 % in (1 + r) will lead to a relative
decrease in P of D %.


Let us take the data used above (a bond issued and repaid at par, 1000 in value, eight years
in maturity and coupon constant at 100; r = 10 %). It is easily shown that D = 5.868.
If the relative increase in (1 + r) is 1 % (that is, if (1 + r) changes from 1.1 to 1.1
—1.01 = 1.111, or if r changes from 10 % to 11.1 %), it is easy to see from the direct
calculation that the price changes from P (10 %) = 1000 to:

100 · 1.111’t + 1000 · 1.111’8 = 943.59
P (11.1 %) =

This total therefore decreases by 5.64 %, the difference with respect to duration arising
simply because we have not used an in¬nitesimal increase. Duration of speci¬c bonds
The duration of a zero-coupon bond is equal to its maturity. In fact, it is given by:

T · R(1 + r)’T
D= =T
R(1 + r)’T

In all other cases, duration is strictly less than maturity; we are looking in fact at a
weighted average of the numbers 1, 2, . . . , T .
A perpetual bond is a security that goes on issuing coupons ad in¬nitum (and issues
no repayment). It is supposed here that the coupons are for the same total C.
Here, the relation that de¬nes the actual rates on return takes the form10

(1 + r)’t =
P =C

By using the concept of the geometric series.
124 Asset and Risk Management

This allows the duration of a perpetual bond to be calculated:

P (r) 1+r
= ’(1 + r) r =
D = ’(1 + r)
P (r) r
r Duration and characteristics of a bond
Duration is a function of the parameters rn (nominal rate), r (actuarial rate) and T (matu-
rity). Here we show the way in which it depends on these three parameters, without strict
supporting calculations.11
Duration is a decreasing function of rn , the two parameters r and T being ¬xed. In the
same way, it is a decreasing function of r, the two parameters r and T being ¬xed.
The dependency of D on T (r and rn being ¬xed) is more complex. We have already
indicated that the duration of a zero-coupon bond is equal to its maturity and that of a
perpetual obligation is equal to (1 + r)/r. For a bond repayable at par and with constant
coupons, it can be shown that:
• If it is issued at par or above par, D increases concavely along with T , with:
lim D(T )
< r
T ’∞

depending on whether the bond is issued at par or above par.
• If it is issued below par, D shows an increase according to T until a certain moment
passes, and then decreases. Immunisation of bonds
The most important aspect of duration is that the reinvestment risk and the sale risk will
balance if the bond is sold at moment D. Interest rate changes will not have any more
in¬‚uence on the actuarial ex post rate of return. The bond is said to be immunised at
horizon D against the interest-rate risk.
This result is not in fact accurate unless the interest-rate changes are of low magnitude
and occur just after the security is purchased.
Let us therefore suppose that a small change r in the market rate (that is, the actuarial
rate on issue) occurs just after the security is issued; this rate changes from r to r + r.
Let us now go to a time t + θ (integer t and 0 ¤ θ < 1) in the life of the bond. We
have to prove that if the security is immunised at the horizon t + θ (that is, if the actuarial
rate of return r(t + θ ) is equal to the rate r at issue), t + θ is equal to the duration D of
the bond.
The reinvestment and sale values at the moment t + θ are given respectively by:
At+θ = Ck (1 + r +
r)’k+t+θ + R(1 + r + r)’T +t+θ
Bt+θ = Ck (1 + r +

The reasoning is not very complicated, but is quite demanding.
Bonds 125

Their sum is therefore equal to:
r)’k + R(1 + r + r)’T
At+θ + Bt+θ = (1 + r + r) Ck (1 + r +

Let us express the powers shown in this expression using a Taylor development to the
¬rst degree:

r)m = (1 + r)m + m(1 + r)m’1 r + O(( r)2 )
(1 + r +

This allows the following to be written:

At+θ + Bt+θ
= [(1 + r)t+θ + (t + θ )(1 + r)t+θ’1 r + O(( r)2 )]
Ck [(1 + r)’k ’ k(1 + r)’k’1 r + O(( r)2 )]

+R[(1 + r)’T ’ T (1 + r)’T ’1 r + O(( r)2 )]

t +θ
= (1 + r)t+θ 1 + r + O(( r)2 )
Ck (1 + r)’k + R(1 + r)’T
k · Ck (1 + r)’k’1 + T · R(1 + r)’T ’1
’ r + O(( r)2 )

t +θ D·P
= (1 + r)t+θ 1 + r + O(( r)2 ) · P ’ r + O(( r)2 )
1+r 1+r
t +θ ’D
= (1 + r)t+θ P · 1 + r + O(( r)2 )

By de¬nition of the actuarial rate at moment t, we also have

At+θ + Bt+θ = P · (1 + r(t + θ ))t+θ

t +θ ’D
is zero, that is, if t + θ = D, which we
This is only possible if the coef¬cient
wished to prove.

Let us take the data used above (a bond issued and repaid at par, value 1000, maturity
eight years and coupon constant at 100, r = 10 % and D = 5.868), and assume that
126 Asset and Risk Management

immediately after its issue the rate of interest changes from 10 % to 9 % and remains at
that level subsequently.
The reinvestment and sale values at moment D will be given respectively by:

AD = 100 · (1.09D’1 + 1.09D’2 + 1.09D’3 + 1.09D’4 + 1.09D’5 ) = 644.98
100 100 1100
BD = + + = 1104.99
1.096’D 1.097’D 1.098’D

The actuarial rate of return at moment D will be given by:

1000 · (l + r(D))D = 1104.99 + 644.98

We therefore ¬nd that r(D) = 10.005 %, which clearly shows that the two sources of risk
vanish at a horizon that is equal to the duration.12 Approximation of a bond price
Applying the Taylor formula to the expression P (r + r) gives

P (r + r) = P (r) + P (r) · r + O(( r)2 )
D · P (r)
= P (r) ’ r + O(( r)2 )

This allows an approximate formula to be written for evaluating P (r + r):

P (r + r) ≈ P (r) 1 ’ r
≈ P (r)[1 ’ Dm · r]

Following the previous example (a bond issued and repaid at par, value 1000, maturity
eight years and coupon constant at 1000, r = 10 % and D = 5.868), we have for this
modi¬ed duration:
Dm = = 5.335

Let us verify the interpretation of this parameter for an increase of 0.8 % in r. The new
price (exact development) is then given by

1.108’t + 1000 · 1.108’8 = 958.536
P (0.108) = 100 ·

The very slight difference between r(D) and the actuarial rate of return at issue (r = 10 %) arises because the result

that we have given (demonstrated using the Taylor formula) is only valid for an in¬nitesimal variation in the rate of interest,
while in our example the variation is 1 %.
Bonds 127

While the above approximation gives:

P (0.108) ≈ 1000 · (1 ’ 5.335 · 0.008) = 957.321

The relation P (r + r) ’ P (r) ≈ ’Dm P (r) · r used with r = 0.0001 (this one-
hundredth per cent is termed the basis point), gives the price variation for a bond as
triggered by an increase in the actuarial rate. This variation is known as the value of one
basis point and termed the VBP.

4.2.3 Convexity De¬nition
The duration (or modi¬ed duration) allows the variations in price to be understood in
terms of interest rates, but only in a linear way. This linear variation is clearly shown by
the ¬rst-degree relation:

P (r + r) ≈ P (r) · 1 ’ D

It is, however, easy to show that the price curve is not in fact linear (see Figure 4.3). In
order to take this curve into consideration, a new parameter is reduced.
In this way, the convexity of a bond, termed C, is de¬ned by the relation:
t (t + 1)Ct (1 + r)’t + T (T + 1)R(1 + r)’T
C= T
Ct (1 + r)’t + R(1 + r)’T
(1 + r)2 ·

Let us take once again the data used above (a bond issued and repaid at par, value 1000,
maturity eight years and coupon constant at 1000, r = 10 %).

100 · [1 · 2 · 1.1’1 + 2 · 3 · 1.1’2 + · · · + 8 · 9 · 1.1’8 ] + 1000 · 8 · 9 · 1.1’8
C= = 38.843
1.12 · {100 · [1.1’1 + 1.1’2 + · · · + 1.1’8 ] + 1000 · 1.1’8 }



Figure 4.3 Price of a bond
128 Asset and Risk Management Interpretation
The mathematical concept that allows this curve to be taken into consideration is the
second derivative of the function. We therefore have:
t (t + 1) · Ct (1 + r)’t’2 + T (T + 1) · R(1 + r)’T ’2
P (r) =
t (t + 1) · Ct (1 + r)’t + T (T + 1) · R(1 + r)’T
= (1 + r)

= C · P (r)

P (r)
This allows us to write: C = .
P (r) Approximating a bond price
We can now apply the Taylor formula to the expression P (r + r) up to order 2:

P (r)
P (r + r) = P (r) + P (r) · r+ ( r)2 + O ( r)3
D · P (r) C · P (r)
= P (r) ’ r+ ( r)2 + O ( r)3
1+r 2

This gives us the approximation formula

P (r + r) ≈ P (r)[1 ’ Dm · r + 1 C · ( r)2 ]

Let us continue with the previous example (a bond issued and repaid at par, value
1000, maturity eight years and coupon constant at 100; r = 10 %, D = 5.868 and C =
38.843. Assuming once again an increase of 0.8 % in r, we have the following price

P (0.108) ≈ 1000 · [1 ’ 5.335 · 0.008 + 1 38.843 · 0.0082 ] = 958.564

This is a much more accurate approximation than the one that uses only duration, as
duration only will give a value of 957.321 when the precise value is 958.536.

As the second-degree term C( r)2 /2 of the approximation formula is always pos-
itive, it therefore appears that when one has to choose between two bonds with the
same return (actuarial rate) and duration, it will be preferable to choose the one with
Bonds 129

the greater convexity regardless of the direction of the potential variation in the rate
of return.

4.3.1 Yield curves
The actuarial rate at the issue of a bond, as de¬ned in Section 4.1.2 is obviously a
particular characteristic to the security in question. The rate will vary from one bond to
another, depending mainly on the quality of the issuer (assessed using the ratings issued
by public rating companies) and the maturity of the security.
The ¬rst factor is of course very dif¬cult to model, and we will not be taking account
of it, assuming throughout this section 4.3 that we are dealing with a public issuer who
does not carry any risk of default. As for the second factor, it can be assumed that for
equal levels of maturity, the rate is the same for all securities in accordance with the
law of supply and demand. In reality, the coupon policies of the various issuers introduce
additional differences; in the following paragraphs, therefore, we will only be dealing with
zero-coupon bonds whose rate now depends only on their maturities. This simpli¬cation
is justi¬ed by the fact that a classic bond is a simple ˜superimposition™ of zero-coupon
securities, which will be valuated by discounting of the various ¬nancial ¬‚ows (coupons
and repayment) at the corresponding rate.14
We are only dealing with deterministic structures for interest rates; random cases are
dealt with in Section 4.5.
If we describe P (s) as the issue price of a zero-coupon bond with maturity s and R(s)
as the rate observed on the market at moment 0 for this type of security, called the spot
rate, these two values are clearly linked by the relation P (s) = (1 + R(s))’s .
The value R(s), for all the values of s > 0, constitutes the term interest-rate structure
at moment 0 and the graph for this function is termed the yield curve.
The most natural direction of the yield curve is of course upwards; the investor should
gain more if he invests over a longer period. This, however, is not always the case;
in practice we frequently see ¬‚at curves (constant R(s) value) as well as increasing
curves, as well as inverted curves (decreasing R(s) value) and humped curves (see
Figure 4.4).

R(s) R(s) R(s) R(s)

s s s s

Figure 4.4 Interest rate curves
A detailed presentation of these concepts can be found in Bisi` re C., La Structure par Terme des Taux d™int´ rˆ t, Presses
e ee
Universitaires de France, 1997.
This justi¬es the title of this present section, which mentions ˜interest rates™ and not bonds.
130 Asset and Risk Management

4.3.2 Static interest rate structure
The static models examine the structure of interest rates at a ¬xed moment, which we
will term 0, and deal with a zero-coupon bond that gives rise to a repayment of 1, which
is not a restriction.
In this and the next paragraph, we will detail the model for the discrete case and then
generalise it for the continuous case. These are the continuous aspects that will be used
in Section 4.5 for the stochastic dynamic models. Discrete model
The price at 0 for a bond of maturity level s is termed15 P0 (s) and the associated spot
rate is represented by R0 (s). We therefore have: P0 (s) = (1 + R0 (s))’s .
The spot interest rate at R0 (s) in fact combines all the information on interest rates
relative to period [0,1], [1, 2] . . ., [s ’ 1, s]. We will give the symbol r(t) and the term
term interest rate or short-term interest rate to the aspects relative to the period [t ’ 1;
t]. We therefore have: (1 + R0 (s))s = (1 + r(1)). (1 + r(2)). . . .. (1 + r(s)).
Reciprocally, it is easy to express the terms according to the spot-rate terms:
 r(1) = R0 (1)
(1 + R0 (s))s
 1 + r(s) = s = 2, 3, . . .
(1 + R0 (s ’ 1))s’1
In the same way, we have:
P0 (s ’ 1)
r(s) = ’1 (s > 0)
P0 (s)
To sum up, we can easily move from any one of the following three structures to
another; the price structure {R0 (s) : s = 1, 2, . . .} and the term interest structure {r(s) :
s = 1, 2, . . .}.

Let us consider a spot rate structure de¬ned for maturity dates 1“6 shown in Table 4.1.
This (increasing) structure is shown in Figure 4.5.
From this, it is easy to deduce prices and term rates: for example:

P0 (5) = 1.075’5 = 0.6966
r(5) = 1.0755 /1.0734 ’ 1 = 0.0830

This generally gives data shown in Table 4.2.

Table 4.1 Spot-rate structure

s 1 2 3 4 5 6

R0 (s) 6.0 % 6.6 % 7.0 % 7.3 % 7.5 % 7.6 %

Of course, P0 (0) = 1.
Bonds 131

0 1 2 3 4 5 6 7
Maturity dates

Figure 4.5 Spot-rate structure

Table 4.2 Price and rate structures at 0

s R0 (s) P0 (s) r(s)

0 1.0000
1 0.060 0.9434 0.0600
2 0.066 0.8800 0.0720
3 0.070 0.8163 0.0780
4 0.073 0.7544 0.0821
5 0.075 0.6966 0.0830
6 0.076 0.6444 0.0810 Continuous model
If the time set is [0; +∞], we retain the same de¬nitions and notations for the price
structures and spot rates: {P0 (s): s > 0] and {R0 (s) : s > 0}. This last will be an instant
rate; after a period s, a total 1 will become, at this rate : es·R0 (s) .
We will also note, before taking limits, R0 d (s) being the spot rate for the discrete
model (even applied to a non integer period). It is therefore linked to the spot rate for the
continuous model by the relation R0 (s) = ln(1 + R0 d (s)).
With regard to the term rate, we are provisionally introducing the notation r(t1 , t2 )
to represent the interest rate relative to the period [t1 ;t2 ] and we de¬ne the instant term
interest rate by:
r(t) = lim r(t, u) du
s’t+ s ’ t t

We can readily obtain, as above:

s+ s
1 + R0 (s + s)
(1 + r(s, s + s)) = s
1 + R0 (s)

Thanks to the Taylor formula, this is written:

s) + O(( s)2 )].(1 + R0 (s))s = (1 + R0 (s +
d d
s))s+ s
[1 + s.r(s, s +
132 Asset and Risk Management

This relation can be rewritten as:
s+ s s
d d
1 + R0 (s + s) ’ 1 + R0 (s)
r(s, s + s) · 1 + R0 (s) + O( s) =

After taking the limit, this becomes:

(1 + R0 (s))s s
d d
r(s) = = ln 1 + R0 (s) = s · ln 1 + R0 (s) = [s · R0 (s)]
1+ R0 (s)

This relation, which expresses the spot rate according to the instant term rate, can easily
be inverted by integrating:
R0 (s) = r(u) du

It can also be expressed easily by saying that the spot rate for the period [0; s] is the
average of the instant term rate for the same period. The price is of course linked to the
two rates by the relations:
’ r(u) du
’s·R0 (s)
P0 (s) = e =e 0

For a ¬‚at rate structure (that is, R0 (s) independent of s), it is easy to see, by developing
the relation [s · R0 (s)] = r0 (s), that

R0 (s) = r(s) = r for every s

and that the price structure is given, P0 (s) = e’rs .

4.3.3 Dynamic interest rate structure
The dynamic models examine the structure of the interest rates at any given moment t.
They always deal with zero-coupon bonds, issued at 0 and giving rise to a repayment
of 1.
They may allow the distortions in the rate curve to be taken into account; in fact, we
will be studying the link that exists between price and rate structures for the various
observation periods. Discrete model
The price at the moment t for the bond issued at 0 and maturing at s is termed16 Pt (s).
The term Rt (s) is given to the spot rate relative to the interval] t; s]. Finally, the term
rate relative to the period] t ’ 1; t] is termed r(t).

It is of course supposed that 0 < t < s.
Bonds 133

Following reasoning similar in every way to that used for the static models, we will
readily obtain the relations

Pt (s) = (1 + Rt (s))’(s’t)
(1 + Rt (s))s’t = (1 + r(t + 1)) · (1 + r(t + 2)) · . . . · (1 + r(s))

This will invert readily to
 r(t + 1) = Rt (t + 1)
(1 + Rt (s))s
 1 + r(s) = s = t + 2, t + 3, . . .
(1 + Rt (s ’ 1))s’1

We also have, between the structure of the prices and that of the interest rates:

Pt (s ’ 1)
r(s) = ’1 (s > t)
Pt (s)

The link between the price structures at different observation times is expressed by the
following relation:

Pt (s) = [(1 + r(t + 1)) · (1 + r(t + 2)) · . . . · (1 + r(s))]’1
(1 + r(t)) · (1 + r(t + 1)) · (1 + r(t + 2)) · . . . · (1 + r(s))
(1 + r(t))
(1 + Rt’1 (s))’(s’t+1)
(1 + Rt’1 (t))’1
Pt’1 (s)
Pt’1 (t)

This result can easily be generalised, whatever u may be, placed between t and s (t ¤
u ¤ s), we have:
Pu (s)
Pt (s) =
Pu (t)

From this relation it is possible to deduce a link, which, however, has a rather ungainly
expression, between the spot-rate structures at the various times.

Let us take once again the spot interest-rate structure used in the previous paragraph: 6 %,
6.6 %, 7 %, 7.3 %, 7.5 % and 7.6 % for the respective payment dates at 1, 2, 3, 4, 5 and
6 years. Let us see what happens to the structure after two years. We can ¬nd easily:

P0 (5) 0.69656
P2 (5) = = = 0.7915
P0 (2) 0.88001
R2 (5) = P2 (5) 5’2 ’ 1 = 0.7915’1/3 ’ 1 = 0.0810
134 Asset and Risk Management
Table 4.3 Price and rate structures at 2

s P2 (s) R2 (s)

2 1.0000
3 0.9276 0.0780
4 0.8573 0.0800
5 0.7915 0.0810
6 0.7322 0.0810

and more generally as shown in Table 4.3.
Note that we have:
P0 (4) P2 (4)
r(5) = ’1= ’ 1 = 0.0830
P0 (5) P2 (5) Continuous model
The prices Pt (s) and the spot rates Rt (s) are de¬ned as for the static models, but with
an observation at moment t instead of 0. The instant term rates r(t) are de¬ned in the
same way.
It can easily be seen that the relations that link the two are:

r(s) = [(s ’ t) · Rt (s)] s ∀t
Rt (s) = r(u) du
s ’t t

Meanwhile, the relations that link rates to prices are given by:
’ r(u) du
’(s’t)·Rt (s)
Pt (s) = e =e t

4.3.4 Deterministic model and stochastic model
The relations mentioned above have been established in a deterministic context. Among
other things, the short instant rate and the term rate have been assimilated. More generally
(stochastic model), the following distinction should be made.

1. The instant term rate, de¬ned by: r(t) = lim Rt (s).
2. The instant term or forward rate, de¬ned as follows: if ft (s1 , s2 ) represents the rate of
interest seen since time t for a bond issued at s1 and with maturity at s2 , the forward
rate (in s seen from t, with t < s) is: ft (s) = lim ft (s, u).

In a general model, this forward rate must be used to ¬nd the price and spot-rate
’ ft (u) du
Pt (s) = e t

Rt (s) = ft (u) du
s ’t t
Bonds 135

It can easily be seen that these two rates (instant term and forward) are linked by the
relation r(t) = ft (t).
It can be demonstrated that in the deterministic case, ft (s) is independent of t and the
two rates can therefore be identi¬ed: ft (s) = r(s). It is therefore only in this context that
we have:
’ r(u) du
Pt (s) = e t

Rt (s) = r(u) du
s’t t
Pu (s)
Pt (s) =
Pu (t)

4.4.1 Passive strategy: immunisation
The aim of passive management is to neutralise the portfolio risk caused by ¬‚uctuations
in interest rates. Duration and convexity of portfolio
Let us consider a bond portfolio consisting at moment 0 of N securities (j = 1, . . . , N ),
each characterised by:

• a maturity (residual life) Tj ;
• coupons yet to come Cj , t (t = 1, . . . , Tj );
• a repayment value Rj ;
• an actuarial rate on issue rj ;
• a price Pj .

The highest of the maturity values Tj will be termed T , and Fj,t the ¬nancial ¬‚ow
generated by the security j at the moment t:
 Cj if t < Tj
= CTj + Rj if t = Tj

if t > Tj

The duration of the j th security is given by

Tj T
’t ’Tj
t · Fj,t (1 + rj )’t
t · Cj,t (1 + rj ) + Tj · Rj (1 + rj )
t=1 t=1
Dj = =
’t ’Tj
Cj,t (1 + rj ) + Rj (1 + rj )
136 Asset and Risk Management

Finally, let us suppose that the j th security is present within the portfolio in the number
nj . The discounted ¬nancial ¬‚ow generated by the portfolio at moment t totals:

nj Fj,t (1 + rj )’t
j =1

Its price totals: N=1 nj Pj .
The duration of the portfolio can therefore be written as:

nj Fj,t (1 + rj )’t

t=1 j =1
DP = N
nk Pk
t · Fj,t (1 + rj )’t
= N
j =1 t=1
nk Pk
t · Fj,t (1 + rj )’t
nj Pj t=1
= ·
j =1
nk Pk
= Xj Dj
j =1

nj Pj
Where: Xj = represents the proportion of the j th security within the portfolio,
k=1 nk Pk
expressed in terms of capitalisation.
The same reasoning will reveal the convexity of the portfolio:

CP = Xj Cj
j =1 Immunising a portfolio
A portfolio is said to be immunised at horizon H if its value at that date is at least the
value that it would have had if interest rates had remained constant during the period
[0; H ]. By applying the result arrived at in Section 4.2.2 for a bond in the portfolio, we
obtain the same result: a bond portfolio is immunised at a horizon that corresponds to
its duration.
Bonds 137

Of course, whenever the interest rate changes, the residual duration varies suddenly. A
careful bond portfolio manager wishing to immunise his portfolio for a horizon H that
he has ¬xed must therefore:

• Put together a portfolio with duration H .
• After each (signi¬cant) interest rate change, alter the composition of the portfolio by
making sales and purchases (that is, alter the proportions of Xj ) so that the residual
duration can be ˜pursued™.

Of course these alterations to the portfolio composition will incur transaction charges,
which should be taken into consideration and balanced against the bene¬ts supplied by
the immunising strategy.

It was stated in Section 4.2.3 that of two bonds that present the same return (actuarial
rate) and duration, the one with the higher convexity will be of greater interest. This result
remains valid for a portfolio, and the manager must therefore take it into consideration
whenever revising his portfolio.

4.4.2 Active strategy
The aim of active management is to obtain a return higher than that produced by immu-
nisation, that is, higher than the actuarial return rate on issue.
In the case of increasing rates (the commonest case), when the rate curve remains
unchanged over time, the technique is to purchase securities with a higher maturity than
the investment horizon and to sell them before their maturity date.17

Let us take once again the rate structure shown in the previous section (Table 4.4).
Let us suppose that the investor ¬xes a two-year horizon. If he simply purchases a
security with maturity in two years, he will simply obtain an annual return of 6.6 %. In
addition, the return over two years can be calculated by

1 ’ 0.8800
= 0.1364 and 1.1364 = 1.066

Table 4.4 Price and rate structures

S R0 (s) P0 (s) r(s)

0 1.0000
1 0.060 0.9434 0.06000
2 0.066 0.8800 0.07203
3 0.070 0.8163 0.07805
4 0.073 0.7544 0.08205
5 0.075 0.6966 0.08304
6 0.076 0.6444 0.08101

If the rate curve is ¬‚at and remains ¬‚at, the strategy presented will produce the same return as the purchase of a security
with a maturity equivalent to the investment horizon.
138 Asset and Risk Management

If he purchases a security with maturity in ¬ve years (at a price of 0.6966) and sells it
on after two years (at the three-year security price if the rate curves remain unchanged,
that is 0.8163), he will realise a total return of
0.8163 ’ 0.6966
= 1.1719

This will give 1.1719 = 1.0825, that is, an annual return of 8.25 %, which is of con-
siderably greater interest than the return (6.6 %) obtained with the two-year security.
Note that we have an interpretation of the term rate here, as the total return for the period
[3; 5], effectively used, is given by (1 + r(4)) · (1 + r(5)) = 1.0821 · 1.0830 = 1.1719.
The interest rate obtained using this technique assumes that the rate curve remains
unchanged over time. If, however the curve, and more speci¬cally the spot rate used
to calculate the resale price, ¬‚uctuates, the investor will be exposed to the interest-rate
¬‚uctuation risk. This ¬‚uctuation will be favourable (unfavourable) to him if the rate in
question falls (rises).
In this case, the investor will have to choose between a safe return and a higher but
potentially more risky return.

With the same information, if after the purchase of a security with maturity in ¬ve years
the spot rate for the three-year security shifts from 7.6 % to 8 %, the price of that security
will fall from 0.8163 to 0.7938 and the return over the two years will be
0.7938 ’ 0.6966
= 1.1396

1.1396 = 1.0675, which corresponds to an annual return of 6.75 %.
We therefore have

The models presented here are actually generalisations of the deterministic interest-rate
structures. The aim is to produce relations that govern changes in price Pt (s) and spot
rates Rt (s). There are two main categories of these models: distortion models and arbi-
trage models.
The distortion models examine the changes in the price Pt (s) when the interest-rate
structure is subject to distortion. A simple model is that of Ho and Lee,18 in which
the distortion of the rate curve shows in two possible movements in each period; it
is therefore a binomial discrete type of model. A more developed model is the Heath,
Jarrow and Morton model,19 which has a discrete and a continuous version and in which
the distortions to the rate curve are more complex.
The arbitrage models involve the compilation, and where possible the resolution, of an
equation with partial derivatives for the price Pt (s, v1 , v2 , . . .) considered as a function
of t, v1 , v2 , . . . (s ¬xed), using:
Ho T. and Lee S., Term structure movement and pricing interest rate contingent claims, Journal of Finance, Vol. 41,
No. 5., 1986, pp. 1011“29.
Heath D., Jarrow R. and Morton A., Bond Pricing and the Term Structure of Interest Rates: a New Methodology, Cornell
University, 1987. Heath D., Jarrow R. and Morton A., Bond pricing and the term structure of interest rates: discrete time
approximation, Journal of Financial and Quantitative Analysis, Vol. 25, 1990, pp. 419“40.
Bonds 139

• the absence of arbitrage opportunity;
• hypotheses relating to stochastic processes that govern the evolutions in the state vari-
ables v1 , v2 etc.

The commonest of the models with just one state variable are the Merton model,20 the
Vasicek model21 and the Cox, Ingersoll and Ross model;22 all these use the instant term
rate r(t) as the state variable. The models with two state variables include:

• The Brennan and Schwarz model,23 which uses the instant term rate r and the long
rate l as variables.
• The Nelson and Schaefer model24 and the Schaefer and Schwartz model,25 for which
the state variables are the long rate l and the spread s = l ’ r.
• The Richard model,26 which uses the instant term rate and the rate of in¬‚ation.
• The Ramaswamy and Sundaresan model,27 which takes the instant market price of risk
linked to the risk of default alongside the instant term rate.

In this section we will be dealing with only the simplest of arbitrage models: after a
general introduction to the principle of these models (Section 4.5.1), we will examine in
succession the Vasicek model (Section 4.5.2) and the Cox, Ingersoll and Ross model28
(Section 4.5.3). Finally, in Section 4.5.4, we will deal with the concept of ˜stochas-
tic duration™.

4.5.1 Arbitrage models with one state variable General principle
It is once again stated (see Section 4.3) that the stochastic processes of interest to us
here are:

• The price Pt (s) in t of a zero-coupon bond (unit repayment value) maturing at the
moment s (with t < s). The spot rate Rt (s), linked to the price by the relation

Pt (s) = e’(s’t)Rt (s)

Merton R., Theory of rational option pricing, Bell Journal of Economics and Management Science, Vol. 4, No. 1, 1973,
pp. 141“83.
Vasicek O., An equilibrium characterisation of the term structure, Journal of Financial Economics, Vol. 5, No. 2, 1977,
pp. 177“88.
Cox K., Ingersoll J. and Ross J., A theory of the term structure of interest rates, Econometrica, Vol. 53, No. 2, 1985,
pp. 385“406.
Brennan M. and Schwartz E., A continuous time approach to the pricing of bonds, Journal of Banking and Finance,
Vol. 3, No. 2, 1979, pp. 133“55.
Nelson J. and Schaefer S., The dynamics of the term structure and alternative portfolio immunization strategies, in
Bierwag D., Kayfman G. and Toevs A., Innovations in Bond Portfolio Management: Duration Analysis and Immunization, JAI
Press, 1983.
Schaefer S. and Schwartz E., A two-factor model of the term structure: an approximate analytical solution, Journal of
Financial and Quantitative Analysis, Vol. 19, No. 4, 1984, pp. 413“24.
Richard S., An arbitrage model of the term structure of interest rates, Journal of Financial Economics, Vol. 6, No. 1,
1978, pp. 33“57.
Ramaswamy K. and Sundaresan M., The valuation of ¬‚oating-rate instruments: theory and evidence, Journal of Financial
Economics, Vol. 17, No. 2, 1986, pp. 251“72.
The attached CD-ROM contains a series of Excel ¬les that show simulations of these stochastic processes and its rate
curves for the various models, combined together in the ˜Ch4™ ¬le.
140 Asset and Risk Management

• The instant term rate, which we will refer hereafter as rt 29 or r if there is no risk of
confusion, and which is the instant rate at moment t, being written as

rt = lim Rt (s) = lim ft (u) du
s’t+ s ’ t
s’t+ t

It is this instant term rate that will be the state variable. The price and spot rate will be
written as Pt (s, r) and Rt (s, r) and will be considered as functions of the variables t and
r alone, the maturity date s being ¬xed. In addition, it is assumed that these expressions
are random via the intermediary of rt only.
It is assumed here that the changes in the state variable rt are governed by the general
stochastic differential equation30 drt = a(t, rt ) dt + b(t, rt ) dwt , where the coef¬cients a
and b respectively represent the average instant return of the instant term rate and the
volatility of that rate, and wt is the standard Brownian motion.
Applying the Itˆ formula to the function Pt (s, rt ) leads to the following, with simpli-
¬ed notations:

dPt (s, rt ) = (Pt + Pr a + 1 Prr b2 ) · dt + Pr b · dwt

= Pt (s, rt ) · µt (s, rt ) · dt ’ Pt (s, rt ) · σt (s, rt ) · dwt

Here, we have:
Pt + Pr a + 1 Prr b2

 µt = 2
Pr b

σ = ’

(Note that σt > 0 as Pr < 0). The expression µt (s, rt ) is generally termed the average
instant return of the bond.
Let us now consider two ¬xed maturity dates s1 and s2 (> t) and apply an arbitrage
reasoning by putting together, at the moment t, a portfolio consisting of:

• The issue of a bond with maturity date s1 .
• The purchase of X bonds with maturity date s2 .

The X is chosen so that the portfolio does not contain any random components; the term
involving dwt therefore has to disappear.
The value of this portfolio at moment t is given by Vt = ’Pt (s1 ) + XPt (s2 ), and the
hypothesis of absence of opportunity for arbitrage allows us to express that the average
return on this portfolio over the interval [t; t + dt] is given by the instant term rate rt :

= rt · dt + 0 · dwt

Instead of r(t) as in Section 4.3, for ease of notation.
See Appendix 2.
Bonds 141

By differentiating the value of the portfolio, we have:

dVt = ’Pt (s1 )(µt (s1 ) dt ’ σt (s1 ) dwt ) + X · Pt (s2 )(µt (s2 ) dt ’ σt (s2 ) dwt )
= [’Pt (s1 )µt (s1 ) + XPt (s2 )µt (s2 )] · dt + [Pt (s1 )σt (s1 ) ’ XPt (s2 )σt (s2 )] · dwt

The arbitrage logic will therefore lead us to
 ’Pt (s1 )µt (s1 ) + XPt (s2 )µt (s2 ) = r

 t
’P (s ) + XP (s )
t t
1 2
 Pt (s1 )σt (s1 ) ’ XPt (s2 )σt (s2 )

 =0
’Pt (s1 ) + XPt (s2 )

In other words:

XPt (s2 ) · (µt (s2 ) ’ rt ) = Pt (s1 ) · (µt (s1 ) ’ rt )
XPt (s2 ) · σt (s2 ) = Pt (s1 ) · σt (s1 )

We can eliminate X, for example by dividing the two equations member by member,
which gives:
µt (s1 ) ’ rt µt (s2 ) ’ rt
σt (s1 ) σt (s2 )

µt (s) ’ rt
This shows that the expression »t (rt ) = is independent of s; this expression
σt (s)
is known as the market price of the risk.
By replacing µt and σt with their value in the preceding relation, we arrive at

Pt + (a + »b)Pr + P ’ rP = 0
2 rr
What we are looking at here is the partial derivatives equation of the second order, which
together with the initial condition Ps (s, rt ) = l, de¬nes the price process. This equation
must be resolved for each speci¬cation of a(t, rt ), b(t, rt ) and »t (rt ). The Merton model31
Because of its historical interest,32 we are showing the simplest model, the Merton model.
This model assumes that the instant term rate follows a random walk model: drt =
± · dt + σ · dwt with ± and σ being constant and the market price of risk being zero
(» = 0).
The partial derivatives equation for the prices takes the form:

Pt + ±Pr + P ’ rP = 0.
2 rr
Merton R., Theory of rational option pricing, Bell Journal of Economics and Management Science, Vol. 4, No. 1, 1973,
pp. 141“83.
This is in fact the ¬rst model based on representation of changes in the spot rate using a stochastic differential equation.
142 Asset and Risk Management

It is easy to verify that the solution to this equation (with the initial condition) is given by

± σ2
Pt (s, rt ) = exp ’(s ’ t)rt ’ (s ’ t) + (s ’ t)3
2 6

The average instant return rate is given by

Pt + ±Pr + P
2 rr = rt · P = r
µt (s, rt ) = t
which shows that in this case, the average return is independent of the maturity date.
The spot rate totals:

Rt (s, rt ) = ’ ln Pt (s, rt )
± σ2
= rt + (s ’ t) ’ (s ’ t)2
2 6
This expression shows that the spot rate is close to the instant term rate in the short term,
which is logical, but also (because of the third term) that it will invariably ¬nish as a
negative for distant maturity dates; this is much less logical.

If one generalises the Merton model where the market price of risk » is a strictly positive
constant, we arrive at an average return µt that grows with the maturity date, but the
inconvenience of the Rt spot rate remains.
The Merton model, which is unrealistic, has now been replaced by models that are


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