ńņš. 6 |

1. The value of the investment acquired at t, calculated at this rate r(t).

2. The sum of:

ā” The value of the coupons falling due acquired at t, reinvested at the current rate

observed between 0 and t.

3

This is sometimes known as the ex post rate.

r(0) = r is of course the actuarial rate at issue.

4

118 Asset and Risk Management

ā” The discounted value in t of the ļ¬nancial ļ¬‚ows generated subsequent to t, calculated

using the market rate at the moment t.

Example

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 ļ¬rst two years and has now changed to 12 %. The above

deļ¬nition gives us the equation that deļ¬nes the actuarial rate of return for the speciļ¬c

moment t = 3.

10 10 115

98 Ā· (1 + r(3))3 = (10 Ā· 1.111 Ā· 1.12 + 10 Ā· 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 ļ¬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.

4.1.2.3 Accrued interest

When a bond is acquired between two coupon payment dates, the purchaser pays not

only the value of the bond for that speciļ¬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 + Īø (integer t and

0 ā¤ Īø <1) (Figure 4.1).

(d)

The dirty price P (d) of the bond is linked to the clean price P by the relation Pt+Īø =

Pt+Īø + Īø Ā· Ct+1 .

The accrued interest system affects the rate of return, as the purchaser, at the moment

of the transaction, pre-ļ¬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.

Īø

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

Figure 4.1 Accrued interest

Bonds 119

0.1005

0.1

Rate of return

0.0995

0.099

0.0985

0 1 2 3 4 5 6

t

Figure 4.2 Festoon effect

The actuarial rate of return is calculated in the classic way, and is deļ¬ned implicitly

by the relation:

T

Ck (1 + r)ā’k+t+Īø + R(1 + r)ā’T +t+Īø

Pt+Īø + Īø Ā· Ct+1 =

k=t+1

Example

Let us consider a bond, issued and repayable at par, with a price at a constant 100, which

presents annual coupons at 10 for ļ¬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 ļ¬nancial ļ¬‚ows generated (residual coupons and ļ¬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:

T

Ck (1 + rm )ā’k+t + R(1 + rm )ā’T +t

Pt =

k=t+1

At a given moment t + Īø (integer t and 0 ā¤ Īø < 1), we will, taking the accrued interest

into account, have:

T

Ck (1 + rm )ā’k+t+Īø + R(1 + rm )ā’T +t+Īø

Pt+Īø = ā’Īø Ā· Ct+1 +

k=t+1

4.2 BONDS AND FINANCIAL RISK

4.2.1 Sources of risk

A bond is a ļ¬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.

5

This arises from the law of supply and demand.

120 Asset and Risk Management

4.2.1.1 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 quantiļ¬ed. Note, however, that bond market

professionals attach considerable importance to ratings allocated to issuers through certain

agencies such as Moodyā™s and Standard & Poorā™s, with the aim of determining the quality

of the issuers and therefore ā˜estimatingā™ 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 inļ¬‚uence on another

nonquantiļ¬able risk, the liquidity of the security in question.

4.2.1.2 Rate ļ¬‚uctuation risk

Another type of risk associated with bonds, this one quantiļ¬able, is the risk linked to

ļ¬‚uctuations in interest rates.

For certain speciļ¬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 ā˜classicā™

bond. Two aspects of this risk, which in one sense are contradictory, may be taken into

consideration.

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.

Example

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 ļ¬rst two years and at 12 % afterwards)

gives, in t = 4, a purchase value of6 A4 = 100 Ā· 1.1 Ā· 1.122 + 100 Ā· 1.122 + 100 Ā· 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.

6

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 deļ¬ned by the relation P Ā· (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

4.2.2.1 Deļ¬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 deļ¬ned

as follows:

T

t Ā· Ct (1 + r)ā’t + T Ā· R(1 + r)ā’T

t=1

D= T

Ct (1 + r)ā’t + R(1 + r)ā’T

t=1

The concept of modiļ¬ed duration is also entered, as is the sensitivity coefļ¬cient, deļ¬ned by:

D

Dm =

1+r

Note

In addition to this ā˜simpleā™ concept of duration, a more developed concept of ā˜stochastic

durationā™ will be issued in Section 4.6.

4.2.2.2 Interpretations

The concept of duration can be interpreted in several different ways; the two best known

follow.

While with an initial rate of 10 %, we would have obtained B4 = 1000. This time, the rise in interest rates clearly does

7

not favour the investor.

8

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. 44ā“53.

This concept can easily be generalised at any time, provided it is assumed that the rate r stays constant.

9

122 Asset and Risk Management

Duration can ļ¬rst of all be considered as the average life span of the bond, the weighting

coefļ¬cients for this average being proportional to the discounted values for the rate r of

the ļ¬nancial ļ¬‚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:

T

Ct (1 + r)ā’t R(1 + r)ā’T

D= t +T

P P

t=1

C1 (1 + r)ā’1 CT ā’1 (1 + r)ā’(T ā’1) (CT + R)(1 + r)ā’T

=1Ā· + Ā· Ā· Ā· + (T ā’ 1) Ā· +T Ā·

P P P

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 deļ¬nes the actuarial rate, P = T Ct (1 + r)ā’t + R(1 + r)ā’T ,

t=1

deļ¬nes a function P (r). The derivative of this function can be written

T

t Ā· Ct (1 + r)ā’tā’1 ā’ T Ā· R(1 + r)ā’T ā’1

P (r) = ā’

t=1

T

ā’1

t Ā· Ct (1 + r)ā’t + T Ā· R(1 + r)ā’T

= ā’(1 + r)

t=1

= ā’(1 + r)ā’1 D Ā· 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 = ā’(1 + r)

P (r)

or:

P (r)

Dm = ā’

P (r)

By taking a small variation in the r rate into account instead of the inļ¬nitesimal vari-

ation, and noting that P = P (r + r) ā’ P (r), the previous relation is written as:

P

D = ā’(1 + r) r

P

P

P

=ā’

(1 + r)

1+r

= Īµ(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 Īµ(y/x) of y with respect to x,

y

y

Īµ(y/x) =

x

x

is interpreted as follows: if x increases by 1 %, y will vary (increase or decrease according

to the sign of y) by |Īµ(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 %.

Example

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

Ć—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:

8

100 Ā· 1.111ā’t + 1000 Ā· 1.111ā’8 = 943.59

P (11.1 %) =

t=1

This total therefore decreases by 5.64 %, the difference with respect to duration arising

simply because we have not used an inļ¬nitesimal increase.

4.2.2.3 Duration of speciļ¬c bonds

The duration of a zero-coupon bond is equal to its maturity. In fact, it is given by:

T Ā· R(1 + r)ā’T

D= =T

R(1 + r)ā’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 inļ¬nitum (and issues

no repayment). It is supposed here that the coupons are for the same total C.

Here, the relation that deļ¬nes the actual rates on return takes the form10

ā

C

(1 + r)ā’t =

P =C

r

t=1

10

By using the concept of the geometric series.

124 Asset and Risk Management

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

C

ā’

P (r) 1+r

= ā’(1 + r) r =

2

D = ā’(1 + r)

C

P (r) r

r

4.2.2.4 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 ļ¬xed. In the

same way, it is a decreasing function of r, the two parameters r and T being ļ¬xed.

The dependency of D on T (r and rn being ļ¬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:

ā¢ If it is issued at par or above par, D increases concavely along with T , with:

1+r

=

lim D(T )

< r

T ā’ā

depending on whether the bond is issued at par or above par.

ā¢ If it is issued below par, D shows an increase according to T until a certain moment

passes, and then decreases.

4.2.2.5 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

inļ¬‚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 + Īø (integer t and 0 ā¤ Īø < 1) in the life of the bond. We

have to prove that if the security is immunised at the horizon t + Īø (that is, if the actuarial

rate of return r(t + Īø ) is equal to the rate r at issue), t + Īø is equal to the duration D of

the bond.

The reinvestment and sale values at the moment t + Īø are given respectively by:

t

r)t+Īøā’k

At+Īø = Ck (1 + r +

k=1

T

r)ā’k+t+Īø + R(1 + r + r)ā’T +t+Īø

Bt+Īø = Ck (1 + r +

k=t+1

11

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

Bonds 125

Their sum is therefore equal to:

T

t+Īø

r)ā’k + R(1 + r + r)ā’T

At+Īø + Bt+Īø = (1 + r + r) Ck (1 + r +

k=1

Let us express the powers shown in this expression using a Taylor development to the

ļ¬rst degree:

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

(1 + r +

This allows the following to be written:

At+Īø + Bt+Īø

= [(1 + r)t+Īø + (t + Īø )(1 + r)t+Īøā’1 r + O(( r)2 )]

T

Ck [(1 + r)ā’k ā’ k(1 + r)ā’kā’1 r + O(( r)2 )]

Ā·

k=1

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

t +Īø

= (1 + r)t+Īø 1 + r + O(( r)2 )

1+r

T

Ck (1 + r)ā’k + R(1 + r)ā’T

Ā·

k=1

T

k Ā· Ck (1 + r)ā’kā’1 + T Ā· R(1 + r)ā’T ā’1

ā’ r + O(( r)2 )

k=1

t +Īø DĀ·P

= (1 + r)t+Īø 1 + r + O(( r)2 ) Ā· P ā’ r + O(( r)2 )

1+r 1+r

t +Īø ā’D

= (1 + r)t+Īø P Ā· 1 + r + O(( r)2 )

1+r

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

At+Īø + Bt+Īø = P Ā· (1 + r(t + Īø ))t+Īø

t +Īø ā’D

is zero, that is, if t + Īø = D, which we

This is only possible if the coefļ¬cient

1+r

wished to prove.

Example

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 Ā· (1.09Dā’1 + 1.09Dā’2 + 1.09Dā’3 + 1.09Dā’4 + 1.09Dā’5 ) = 644.98

100 100 1100

BD = + + = 1104.99

1.096ā’D 1.097ā’D 1.098ā’D

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

1000 Ā· (l + r(D))D = 1104.99 + 644.98

We therefore ļ¬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

4.2.2.6 Approximation of a bond price

Applying the Taylor formula to the expression P (r + r) gives

P (r + r) = P (r) + P (r) Ā· r + O(( r)2 )

D Ā· P (r)

= P (r) ā’ r + O(( r)2 )

1+r

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

D

P (r + r) ā P (r) 1 ā’ r

1+r

ā P (r)[1 ā’ Dm Ā· r]

Example

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

modiļ¬ed duration:

5.868

Dm = = 5.335

1.1

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

8

1.108ā’t + 1000 Ā· 1.108ā’8 = 958.536

P (0.108) = 100 Ā·

t=1

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

12

that we have given (demonstrated using the Taylor formula) is only valid for an inļ¬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) ā 1000 Ā· (1 ā’ 5.335 Ā· 0.008) = 957.321

The relation P (r + r) ā’ P (r) ā ā’Dm P (r) Ā· 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

4.2.3.1 Deļ¬nition

The duration (or modiļ¬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 ļ¬rst-degree relation:

r

P (r + r) ā P (r) Ā· 1 ā’ D

1+r

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 deļ¬ned by the relation:

T

t (t + 1)Ct (1 + r)ā’t + T (T + 1)R(1 + r)ā’T

t=1

C= T

Ct (1 + r)ā’t + R(1 + r)ā’T

(1 + r)2 Ā·

t=1

Example

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 Ā· [1 Ā· 2 Ā· 1.1ā’1 + 2 Ā· 3 Ā· 1.1ā’2 + Ā· Ā· Ā· + 8 Ā· 9 Ā· 1.1ā’8 ] + 1000 Ā· 8 Ā· 9 Ā· 1.1ā’8

C= = 38.843

1.12 Ā· {100 Ā· [1.1ā’1 + 1.1ā’2 + Ā· Ā· Ā· + 1.1ā’8 ] + 1000 Ā· 1.1ā’8 }

P(r)

r

Figure 4.3 Price of a bond

128 Asset and Risk Management

4.2.3.2 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 (t + 1) Ā· Ct (1 + r)ā’tā’2 + T (T + 1) Ā· R(1 + r)ā’T ā’2

P (r) =

t=1

T

ā’2

t (t + 1) Ā· Ct (1 + r)ā’t + T (T + 1) Ā· R(1 + r)ā’T

= (1 + r)

t=1

= C Ā· P (r)

P (r)

This allows us to write: C = .

P (r)

4.2.3.3 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) Ā· r+ ( r)2 + O ( r)3

2

D Ā· P (r) C Ā· P (r)

= P (r) ā’ r+ ( r)2 + O ( r)3

1+r 2

This gives us the approximation formula

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

2

Example

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

valuation:

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

2

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 DETERMINISTIC STRUCTURE OF INTEREST RATES13

4.3.1 Yield curves

The actuarial rate at the issue of a bond, as deļ¬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 ļ¬rst factor is of course very difļ¬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 simpliļ¬cation

is justiļ¬ed by the fact that a classic bond is a simple ā˜superimpositionā™ of zero-coupon

securities, which will be valuated by discounting of the various ļ¬nancial ļ¬‚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))ā’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 ļ¬‚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

13

A detailed presentation of these concepts can be found in Bisi` re C., La Structure par Terme des Taux dā™intĀ“ rĖ t, Presses

e ee

Universitaires de France, 1997.

14

This justiļ¬es the title of this present section, which mentions ā˜interest ratesā™ 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 ļ¬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.

4.3.2.1 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))ā’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 ā’ 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 ā’ 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:

ļ£±

ļ£² r(1) = R0 (1)

(1 + R0 (s))s

ļ£³ 1 + r(s) = s = 2, 3, . . .

(1 + R0 (s ā’ 1))sā’1

In the same way, we have:

P0 (s ā’ 1)

r(s) = ā’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, . . .}.

Example

Let us consider a spot rate structure deļ¬ned for maturity dates 1ā“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) = 1.075ā’5 = 0.6966

r(5) = 1.0755 /1.0734 ā’ 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.

15

Bonds 131

0.08

0.07

0.06

0.05

0.04

0.03

0.02

0.01

0

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

4.3.2.2 Continuous model

If the time set is [0; +ā], we retain the same deļ¬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 : esĀ·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 deļ¬ne the instant term

interest rate by:

s

1

r(t) = lim r(t, u) du

sā’t+ s ā’ t t

We can readily obtain, as above:

s+ s

d

1 + R0 (s + s)

s

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

d

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) ā’ 1 + R0 (s)

s

d

r(s, s + s) Ā· 1 + R0 (s) + O( s) =

s

After taking the limit, this becomes:

d

(1 + R0 (s))s s

d d

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

s

d

1+ R0 (s)

This relation, which expresses the spot rate according to the instant term rate, can easily

be inverted by integrating:

1s

R0 (s) = r(u) du

s0

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:

s

ā’ r(u) du

ā’sĀ·R0 (s)

P0 (s) = e =e 0

Note

For a ļ¬‚at rate structure (that is, R0 (s) independent of s), it is easy to see, by developing

the relation [s Ā· R0 (s)] = r0 (s), that

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

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

4.3.3.1 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 ā’ 1; t] is termed r(t).

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

16

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))ā’(sā’t)

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

This will invert readily to

ļ£±

ļ£² r(t + 1) = Rt (t + 1)

(1 + Rt (s))s

ļ£³ 1 + r(s) = s = t + 2, t + 3, . . .

(1 + Rt (s ā’ 1))sā’1

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

Pt (s ā’ 1)

r(s) = ā’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)) Ā· (1 + r(t + 2)) Ā· . . . Ā· (1 + r(s))]ā’1

ā’1

(1 + r(t)) Ā· (1 + r(t + 1)) Ā· (1 + r(t + 2)) Ā· . . . Ā· (1 + r(s))

=

(1 + r(t))

(1 + Rtā’1 (s))ā’(sā’t+1)

=

(1 + Rtā’1 (t))ā’1

Ptā’1 (s)

=

Ptā’1 (t)

This result can easily be generalised, whatever u may be, placed between t and s (t ā¤

u ā¤ 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.

Example

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 ļ¬nd easily:

P0 (5) 0.69656

P2 (5) = = = 0.7915

P0 (2) 0.88001

ā’1

R2 (5) = P2 (5) 5ā’2 ā’ 1 = 0.7915ā’1/3 ā’ 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) = ā’1= ā’ 1 = 0.0830

P0 (5) P2 (5)

4.3.3.2 Continuous model

The prices Pt (s) and the spot rates Rt (s) are deļ¬ned as for the static models, but with

an observation at moment t instead of 0. The instant term rates r(t) are deļ¬ned in the

same way.

It can easily be seen that the relations that link the two are:

r(s) = [(s ā’ t) Ā· Rt (s)] s āt

s

1

Rt (s) = r(u) du

s ā’t t

Meanwhile, the relations that link rates to prices are given by:

s

ā’ r(u) du

ā’(sā’t)Ā·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, deļ¬ned by: r(t) = lim Rt (s).

sā’t+

2. The instant term or forward rate, deļ¬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).

uā’s+

In a general model, this forward rate must be used to ļ¬nd the price and spot-rate

structures:

s

ā’ ft (u) du

Pt (s) = e t

s

1

Rt (s) = ft (u) du

s ā’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 identiļ¬ed: ft (s) = r(s). It is therefore only in this context that

we have:

s

ā’ r(u) du

Pt (s) = e t

s

1

Rt (s) = r(u) du

sā’t t

Pu (s)

Pt (s) =

Pu (t)

4.4 BOND PORTFOLIO MANAGEMENT STRATEGIES

4.4.1 Passive strategy: immunisation

The aim of passive management is to neutralise the portfolio risk caused by ļ¬‚uctuations

in interest rates.

4.4.1.1 Duration and convexity of portfolio

Let us consider a bond portfolio consisting at moment 0 of N securities (j = 1, . . . , N ),

each characterised by:

ā¢ a maturity (residual life) Tj ;

ā¢ coupons yet to come Cj , t (t = 1, . . . , Tj );

ā¢ a repayment value Rj ;

ā¢ an actuarial rate on issue rj ;

ā¢ a price Pj .

The highest of the maturity values Tj will be termed T , and Fj,t the ļ¬nancial ļ¬‚ow

generated by the security j at the moment t:

ļ£±

ļ£² Cj if t < Tj

= CTj + Rj if t = Tj

Fj,t

ļ£³

if t > Tj

0

The duration of the j th security is given by

Tj T

ā’t ā’Tj

t Ā· Fj,t (1 + rj )ā’t

t Ā· Cj,t (1 + rj ) + Tj Ā· Rj (1 + rj )

t=1 t=1

Dj = =

Pj

Tj

ā’t ā’Tj

Cj,t (1 + rj ) + Rj (1 + rj )

t=1

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 ļ¬nancial ļ¬‚ow generated by the portfolio at moment t totals:

N

nj Fj,t (1 + rj )ā’t

j =1

Its price totals: N=1 nj Pj .

j

The duration of the portfolio can therefore be written as:

T N

nj Fj,t (1 + rj )ā’t

tĀ·

t=1 j =1

DP = N

nk Pk

k=1

Tj

N

nj

t Ā· Fj,t (1 + rj )ā’t

= N

j =1 t=1

nk Pk

k=1

Tj

t Ā· Fj,t (1 + rj )ā’t

N

nj Pj t=1

= Ā·

Pj

N

j =1

nk Pk

k=1

N

= Xj Dj

j =1

nj Pj

Where: Xj = represents the proportion of the j th security within the portfolio,

N

k=1 nk Pk

expressed in terms of capitalisation.

The same reasoning will reveal the convexity of the portfolio:

N

CP = Xj Cj

j =1

4.4.1.2 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 ļ¬xed must therefore:

ā¢ Put together a portfolio with duration H .

ā¢ After each (signiļ¬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 ā˜pursuedā™.

Of course these alterations to the portfolio composition will incur transaction charges,

which should be taken into consideration and balanced against the beneļ¬ts supplied by

the immunising strategy.

Note

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

Example

Let us take once again the rate structure shown in the previous section (Table 4.4).

Let us suppose that the investor ļ¬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

ā

1 ā’ 0.8800

= 0.1364 and 1.1364 = 1.066

0.8800

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

17

If the rate curve is ļ¬‚at and remains ļ¬‚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 ļ¬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 ā’ 0.6966

= 1.1719

0.6966

ā

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)) Ā· (1 + r(5)) = 1.0821 Ā· 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 speciļ¬cally the spot rate used

to calculate the resale price, ļ¬‚uctuates, the investor will be exposed to the interest-rate

ļ¬‚uctuation risk. This ļ¬‚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.

Example

With the same information, if after the purchase of a security with maturity in ļ¬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 ā’ 0.6966

= 1.1396

0.6966

ā

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

We therefore have

4.5 STOCHASTIC BOND DYNAMIC MODELS

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 ļ¬xed), using:

18

Ho T. and Lee S., Term structure movement and pricing interest rate contingent claims, Journal of Finance, Vol. 41,

No. 5., 1986, pp. 1011ā“29.

19

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. 419ā“40.

Bonds 139

ā¢ the absence of arbitrage opportunity;

ā¢ 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:

ā¢ The Brennan and Schwarz model,23 which uses the instant term rate r and the long

rate l as variables.

ā¢ 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 ā’ r.

ā¢ The Richard model,26 which uses the instant term rate and the rate of inļ¬‚ation.

ā¢ 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 ā˜stochas-

tic durationā™.

4.5.1 Arbitrage models with one state variable

4.5.1.1 General principle

It is once again stated (see Section 4.3) that the stochastic processes of interest to us

here are:

ā¢ 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) = eā’(sā’t)Rt (s)

20

Merton R., Theory of rational option pricing, Bell Journal of Economics and Management Science, Vol. 4, No. 1, 1973,

pp. 141ā“83.

21

Vasicek O., An equilibrium characterisation of the term structure, Journal of Financial Economics, Vol. 5, No. 2, 1977,

pp. 177ā“88.

22

Cox K., Ingersoll J. and Ross J., A theory of the term structure of interest rates, Econometrica, Vol. 53, No. 2, 1985,

pp. 385ā“406.

23

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. 133ā“55.

24

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.

25

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. 413ā“24.

26

Richard S., An arbitrage model of the term structure of interest rates, Journal of Financial Economics, Vol. 6, No. 1,

1978, pp. 33ā“57.

27

Ramaswamy K. and Sundaresan M., The valuation of ļ¬‚oating-rate instruments: theory and evidence, Journal of Financial

Economics, Vol. 17, No. 2, 1986, pp. 251ā“72.

28

The attached CD-ROM contains a series of Excel ļ¬les that show simulations of these stochastic processes and its rate

curves for the various models, combined together in the ā˜Ch4ā™ ļ¬le.

140 Asset and Risk Management

ā¢ 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

s

1

rt = lim Rt (s) = lim ft (u) du

sā’t+ s ā’ t

sā’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 ļ¬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 coefļ¬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 ItĖ formula to the function Pt (s, rt ) leads to the following, with simpli-

o

ļ¬ed notations:

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

2

= Pt (s, rt ) Ā· Āµt (s, rt ) Ā· dt ā’ Pt (s, rt ) Ā· Ļt (s, rt ) Ā· dwt

Here, we have:

ļ£±

Pt + Pr a + 1 Prr b2

ļ£“

ļ£² Āµt = 2

P

Pr b

ļ£“

ļ£³Ļ = ā’

t

P

(Note that Ļt > 0 as Pr < 0). The expression Āµt (s, rt ) is generally termed the average

instant return of the bond.

Let us now consider two ļ¬xed maturity dates s1 and s2 (> t) and apply an arbitrage

reasoning by putting together, at the moment t, a portfolio consisting of:

ā¢ The issue of a bond with maturity date s1 .

ā¢ 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 = ā’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 :

dVt

= rt Ā· dt + 0 Ā· dwt

Vt

Instead of r(t) as in Section 4.3, for ease of notation.

29

30

See Appendix 2.

Bonds 141

By differentiating the value of the portfolio, we have:

dVt = ā’Pt (s1 )(Āµt (s1 ) dt ā’ Ļt (s1 ) dwt ) + X Ā· Pt (s2 )(Āµt (s2 ) dt ā’ Ļt (s2 ) dwt )

= [ā’Pt (s1 )Āµt (s1 ) + XPt (s2 )Āµt (s2 )] Ā· dt + [Pt (s1 )Ļt (s1 ) ā’ XPt (s2 )Ļt (s2 )] Ā· dwt

The arbitrage logic will therefore lead us to

ļ£±

ļ£“ ā’Pt (s1 )Āµt (s1 ) + XPt (s2 )Āµt (s2 ) = r

ļ£“

ļ£² t

ā’P (s ) + XP (s )

t t

1 2

ļ£“ Pt (s1 )Ļt (s1 ) ā’ XPt (s2 )Ļt (s2 )

ļ£“

ļ£³ =0

ā’Pt (s1 ) + XPt (s2 )

In other words:

XPt (s2 ) Ā· (Āµt (s2 ) ā’ rt ) = Pt (s1 ) Ā· (Āµt (s1 ) ā’ rt )

XPt (s2 ) Ā· Ļt (s2 ) = Pt (s1 ) Ā· Ļt (s1 )

We can eliminate X, for example by dividing the two equations member by member,

which gives:

Āµt (s1 ) ā’ rt Āµt (s2 ) ā’ rt

=

Ļt (s1 ) Ļt (s2 )

Āµt (s) ā’ rt

This shows that the expression Ī»t (rt ) = is independent of s; this expression

Ļt (s)

is known as the market price of the risk.

By replacing Āµt and Ļt with their value in the preceding relation, we arrive at

b2

Pt + (a + Ī»b)Pr + P ā’ 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, deļ¬nes the price process. This equation

must be resolved for each speciļ¬cation of a(t, rt ), b(t, rt ) and Ī»t (rt ).

4.5.1.2 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 =

Ī± Ā· dt + Ļ Ā· dwt with Ī± and Ļ being constant and the market price of risk being zero

(Ī» = 0).

The partial derivatives equation for the prices takes the form:

Ļ2

Pt + Ī±Pr + P ā’ rP = 0.

2 rr

31

Merton R., Theory of rational option pricing, Bell Journal of Economics and Management Science, Vol. 4, No. 1, 1973,

pp. 141ā“83.

32

This is in fact the ļ¬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

Ī± Ļ2

Pt (s, rt ) = exp ā’(s ā’ t)rt ā’ (s ā’ t) + (s ā’ t)3

2

2 6

The average instant return rate is given by

Ļ2

Pt + Ī±Pr + P

2 rr = rt Ā· P = r

Āµt (s, rt ) = t

P P

which shows that in this case, the average return is independent of the maturity date.

The spot rate totals:

1

Rt (s, rt ) = ā’ ln Pt (s, rt )

sā’t

Ī± Ļ2

= rt + (s ā’ t) ā’ (s ā’ 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 ļ¬nish as a

negative for distant maturity dates; this is much less logical.

Note

If one generalises the Merton model where the market price of risk Ī» is a strictly positive

constant, we arrive at an average return Āµt 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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