Relationship Between Discrete And Continuous Models of the Term Structure

N. J. Macleod

.tna Life and Casualty Company

February 9, 1989

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Abstract

This paper presents a unified approach to the development of continuous and discrete (binomial) equilibrium models of the term structure of interest rates. The principal benefit of such an approach is that, once a direct correspondence between continuous and discrete models has been established, it becomes a straightforward matter to adapt any continuous model to binomial form. This allows us to take advantage of the computational simplicity and flexibility associated with binomial models, while retaining the desirable features of the continuous models (such as the ability to generate realistic term structure behavior from simple spot rate dynamics). A specific model constructed along these lines is described in [2].

The development of the continuous model is standard. Following the argument given in [3] and [1], the term structure equation is derived from a Taylor series expansion (which relates the change in the price of a bond over a short interval to its yield, duration and convexity), by specifying the form of the stochastic process governing the behavior of the instantaneous spot rate, and by requiring the term structure at any time to be free of opportunities for riskless arbitrage.

The treatment of the binomial model, on the other hand, is novel in that it shows how the differential coefficients which appear in the term structure equation may be identified with the elements of a single binomial transition. The term structure equation in its binomial form emerges as a simple relation between the current price of a zero-coupon bond of arbitrary maturity, and the prices it may assume after a single transition. Once the stochastic process governing the behavior of the one-period spot rate has been specified, the relation can be used to generate a lattice of arbitrage-free term structures from a lattice of one-period spot rates.

Organization of the paper

The paper is organized into four sections. Section 1 begins with the mathematical theory underlying conventional immunization techniques. Section 2 goes on to show that by specifying the form of the stochastic behavior of the instantaneous spot rate and requiring the term structure at any time to be free of opportunities for riskless arbitrage, it is possible to extend that theory and to derive an equation for the term structure.

The term structure equation is a partial differential equation which relates the behavior of the term structure to the behavior of the instantaneous spot rate. It is possible to solve the equation provided that the stochastic process describing the behavior of the instantaneous spot rate has been specified in sufficient detail, although it is likely that it will be necessary to use numerical methods to achieve a solution.

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The third section shows how the term structure equation may be derived within a binomial framework. It goes on to show how, in its binomial form, the term structure equation may be used to generate arbitrage-free term structures from a lattice of one-period spot rates.

Section 4 shows how the properties of the model are related to conventional explanations of term structure behavior. For simplicity, the binomial model is used, but the same considerations apply to the continuous class of models. It is shown that in a predetermined interest rate environment the model reduces to the expectations hypothesis. This is natural because in a deterministic environment the expectations hypothesis follows directly by requiring the environment to be arbitrage-free. When interest rates are not predetermined, however, longer bonds entail some price risk, so we would expect yields on longer bonds to reflect some sort of risk premium, and this is shown to be incorporated into one of the model parameters.

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1. Conventional Immunization

Consider a portfolio which generates cash flow C(s) at time s, and suppose that the cash flows are fixed.

If the continuously compounded yield of the portfolio at time t is k, say, then the price of the portfolio at time t is given by

.0

P(t,k) ~ -k(a-t)

C(s)e a -

(1.1)

Suppose that over the tille interval [t, t + A tJ the portfolio yield ahifts to k + .A k.

Then the price of the portfolio at tille t + ~t is

P(t+A t,k+ Ak) ~ -(k+Ak)(s-t-At)

- Lc(s)e a - t

which can be written

ow

P(t+At,k+6k) ) -k(s-t) [k~t - Ak(s-t - bt) ]

LC(s)e e a - t

(1. 2)

The change in the price of the portfolio between t and t + 6. t is given by

AP - P(t +At, k +Ak) - P(t,k)

i.e. 00

L -k(s-t) [kll.t -6k(s-t -~t)] I:l P - C(s)e (e -1)

s - t

(1. 3)

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We can evaluate AP by expending elk .6t -~k(s-t -At») as a aeries. By chooain& e aufficiently ahort time interval ~t, we can limit the number of terma necesaary for a given degree of approximation. In fact bt ia usually chosen ao that terms s.aller than ~t can be ignored. With this convention,

e[k.t.t - Ak(s-t -At)] -1 - kAt -Ak(s-t -.6t) + llk/1t -Ak(s-t -At»)2 + .,. 2

leads to the approxi.ation

elkAt - Ak(s-t - At») -1 z kL\ t - (a-t)A k + 1 (a-t)2Ak2 2 (1.4)

The term in ~k2 has been retained, even though at first sight it appears to be a second order term and hence to be negligible. The reason that it is not negligible relative to terms of order At is that whereas the mean change in yields over an interval of length At is proportional to At, the atandard deviation of that change varies with the aquare root of 6t. If the mean and standard deviation of Ak are r At and fS'm, it follows that the mean value of Ak2 is liven by

ao that

(l.5)

neglectin& higher order terms.

6k2 1a therefore of the ._ order of _gnitude a. Il t.

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From (1.3) and (1.4),

of -k(s-t) L" -k(a-t) f- -k(s-t) ~P ~ kLlt Le(s)e . -Ak (a-t)C(a). + IL11t2 L (a-t)2C(s)e

a-t a-t 2 a-t

(1.6)

Dividing thro~gh by P (- P (t,k» and noting that

II -h

f- -It(a-t) kL C(a).

a - t

~ -k(s-t) - '-- (s-t)C(a). a - t

.:;;- -k(s-t) L.. (a-t)2C(a). • - t

III ~t P ~t

+

(1. 6) becomes

+ (1. 7)

(1.7) can be obtalned directly from the Taylor series expanslon of the f~nctlon P(t+ Ll t, k+ Ak). In the derlvatlon given above, the Taylor aerl •• expaneion of ek.o. t -Ak(s-t -Ak) has been a~bstit~ted for the Dore general expansion of P(t+.o.t, k+Ak) in order to make the proced~re clear. Although (1.6) holds only for fixed cash flows, (1.7) holds for cash flows which vary with k.

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(1.7) is frequently expressed as

M - kAt - D6 k + 1 c 6 k2 P 2

where D -.:l H p ak

and c-

(1.8)

are referred to respectively as the duration and convexity of the portfolio P

Application to Immunization

Suppose we have a portfolio of liabUity cash flows whose present value is PL(t,k), and that we must choose assets in that amount to support the liabilities.

Applying (1. 8) to the asset and liabUity portfolios in turn gives

(1. 9)

for the assets

(1.10)

for the liabilities

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If we are able to choose the assets such that

then

(1.11)

and since PA - PL, (1.11) implies that a small shift in yield ~k, whether positive or negative, will result in an increase

in the value of the asset portfolio relative to the liability portfolio. (Applying (1.7) to the net portfolio P - PA - PL shows that the assets are chosen such that the current value P(t,k) of the net portfolio, P, is a local minimum relative to k) .

Comments

The treatment given above relies on certain implicit assumptions. Specifically,

the same yield k is assumed to apply to all cash flows, regardless of when they occur

the same change in yield applies to all cash flows.

In other words, we have assumed parallel shifts in a flat yield curve. It should be noted, though, that no assumption has been made regarding the behavior of k, other than that it is subject to change.

The next section shows that making an explicit assumption about the stochastic behavior of interest rates, and requiring the term structure at any time to be free of opportunities for riskless arbitrage enables us to use (1.7) to derive an equation for the term structure.

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2. The term structure equation

Consider a default-free zero-coupon bond which matures for $1 at time s. Assume that the price of the bond, P(t,s,r>, at time t(t S s) depends on the assessment at time t of the behavior of the instantaneous spot rate reT) in the time interval t S T S s.

Assume that the behavior of the instantaneous spot rate can be described by • stochastic process of the form

dr - f(r,t)dt + g(r,t)dz (2.1)

where dro- ret + dt) - ret)

and dz is normally distributed with mean 0 and standard deviation ~

The change in the instantaneous spot rate over a sufficiently short time interval can be separated into a deterministic component f(r,t)dt, and a stochastic component g(r,t)dz. (It may be helpful to think of the move from r to r+dr as occurring in two steps: r ---. r + fdt --. r + fdt + gdz - r + dr).

From (2.1), dr is normally distributed with mean and variance given by

E(dr) f(r,t)dt (2.2)

and Var(dr) [g(r,t)J2 dt (2.3)

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From equation (1.7) the change in the price of the zero-coupon bond between t and t+dt can be written

dP - -il dt + -4l dr + 1 ~2p (dr]2 h h 2~

(2.4)

Note that in (2.4), the instantaneous spot rate r has replaced the yield k of equation (1.7).

From (2.1),

(dr]2 (fdt + gdz)2

f 2dt2 + 2fgdzdt + g2(dz)2 (2.5)

Since we are considering a very short time interval, we may neglect terms of order higher than dt. The first two terms on the right hand side of (2.5) are negligible, therefore. Since dz is normallL distributed with mean value 0 and standard deviation Vdt', the random variable (dz2) has a chi-square distribution with one degree of freedom. The mean and variance of (dz2) are given respectively by

dt (2.6)

and 2dt2 (2.7)

The variance of (dz2) is negligible and so we can treat (dz 2) as equal to its expected value, dt.

Then, froll (2.5)

(dr]2 (2.8)

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Substituting fro. (2.1) and (2.8) into (2.4) gives

dP - LU + f....il +.lg2 ~2p ]dt + at ~r 2 ~

(2.9)

which can be written as

dP - PflJdt + P '#'dz (2.10)

where

fIJ - lr...,U + f-1l + 192~ ] p ~t ar 2 ~r2 (2.11)

and

"f/ - 1Ul (2.12) P clr

(2.10) shows that the change in price of the zero-coupon bond between t and t+dt may also be separated into a deterministic component and a stochastic component and that the mean and variance of the fractional change in price are given by

and

E(!tf) - fldt P

Var(!tf) P

(2.13)

(2.14)

The development is completed by requiring the bond prices at any time to preclude opportunities for riskless arbitrage.

Consider an investor who at time t buys WI units of a default free zero-coupon bond which matures for $1 at time sl and simultaneously sells W2 units of a default free zero-coupon bond which matures for $1 at time s2.

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Denoting the prices of the bonds by PI and P2 respectively and applying (2.10) to each bond in turn,

(2.15)

(2.16)

The value of the investor's portfolio is W, where

and changes in value by an amount dW between t and t+dt, where

i.e.

from (2.15) and (2.16).

Suppose the investor chooses V1 and V2 such that

and

Substituting from (2.19) and (2.20) into (2.18),

dV - (V ~2'h 'f'2 • Y1.

• V 'h1l2 )dt + (O)dz 'f2 • If1

• 82 •

(2.17)

(2.18)

(2.19)

(2.20)

(2.21)

From (2.21) it is apparent that if the amounts WI and W2 are chosen according to (2.19) and (2.20), the stochastic component of dW (I.e., the coefficient of dz in (2.18) ) is zero. The investor is therefore guaranteed a return given by

(2.22)

over the interval [t, t+dtJ

If there are to be no opportunities for riskless arbitrage, this return must be equal to r(t)dt where ret) is the instantaneous spot rate at time t.

That is,

(2.23)

Rearranging (2.23) gives

(2.24)

Since the bonds which make up the investor's portfolio were chosen arbitrarily, the argument can be extended to all default free zero-coupon bonds. and it follows that at any particular time the ratio

GCt.s.r) - rCt) 'f(t,s,r)

must take the same value for all bonds. In other words, to exclude opportunities for riskless arbi­trage, the ratio may vary with time and with the

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value of the instantaneous spot rate, but may not vary according to a bond'. maturity date.

This condition may be expressed by requiring that at any time every default free zero-coupon bond satisfy the relation

get,s,r) - ret) - Q(r,t) 'I'(t,s,r)

(2.25)

where Q may vary with r and with t, but may not vary with s.

Term structures which satisfy (2.25) are said to be in equilib­rium.

substituting for p and'+' from (2.11) and (2.12) in (2.25) gives

(f-gQ)-Al + Ig2~ - rP _. 0 h 2 h 2

(2.26)

(2.26) is the term structure equation. If we specify the functions f(r,t) and g(r,t) it can be solved for P(t,s,r), subject to the boundary condition P(s,s,r) - 1. The term structure can then be obtained from the relation

R(t,s,r)

Comments

~ loge P(t,s,r) Sot

(2.27)

where R(t,s,r) is the yield to maturity at time t of a bond which matures for 1 at time s.

The development of the term structure equation may appear somewhat abstract. However, it can be summarized by noting that it is achieved by incorporating a description of the stochastic behavior of the short term rate into the basic expression for the change in value of a bond over a short time

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interval, subject to the requirement that the term structure remain in equilibrium at ,all times.

The next section derives the term structure equation within a binomial framework. The assumptions regarding the behavior of the short-term rate are perhaps less realistic than in the continuous case, but they are also simpler and allow a more elementary analysis. While there are various ways in which one-factor binomial models can be constructed, the methods are all ultimately equivalent. The development in the next section is presented in a form directly analogous to that of the contin­uous model, in the hope that the continuous development may be clarified by reference to the development of the binomial model.

3. Development of the binomial model

Divide the projection period into discrete time intervals, each of length Ilt.

Assume that over any single interval [t, t + ~tl the one-period spot rate may move from its current value ret) to one of two subsequent values, and that the stochastic process governing this behavior takes the form

Ar f(r,t) /:::.t + g(r,t) AI (3.1)

where tn ret + At) - ret)

and L:\I is a random variable whose distribution is given by

P[ 61 -wI 1/2 (3.2)

P[ AI ./"At I 1/2 (3.3)

The random variable ~I has a mean value of zero and a standard deviation of ~ ; it is the discrete analogue of the random variable dz of the continuous model, which is normally distri­but~ith a mean of zero and standard deviation

\}dt' .

The mean and variance of ~r are given by

E(l~r)

Var( A r)

f(r,t) 6t

[g(r,t)1 2 6,t

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(3.4)

(3.5)

Consider a default-free zero coupon bond which matures for $1 at time B. Let T - s-t be the time to maturity of the bond, and denote the current price of the bond by P(T). Assume that over the interval [t, t+~t}, the price of the bond moves to

Po (T- ~t) if ~ I - -JAt'

or PI (T- At) if 61 - .[At

That is, we assume that the change in the price of the bond over a single interval is determined by the value taken by ~I for that transition. The change in price is thus related directly to the change in r (since each is governed by ~I), and is given by

b,P PO(T-l!>t) P(T) if 61 -~ and by

b,P - Pl(T- bt) P(T) if bI -w (3.6) and (3.7) may be stated as a single equation

tH - 1/2[Pl(T- At) + PO(T- ~t) 1

+ 1/2[Pl(T- ~t) - PO(T- At) 1

In order to simplify the notation, let

P - P(T)

Po - PO(T - ~ t)

Pl - PI (T - At)

- P(T)

l!>I/~

(3.6)

(3.7)

(3.8)

There are two components of the change in the price of the bond between t and t + At.

1. The change in price attributable to the change in time. Between t and t + ~ t the time to maturity shortens from T to T - A t and in the absence of any change in r, we would expect the price of the bond to increase by an amount P x r6t, reflecting interest accrual over the interval.

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Using the notation generally reserved for continuous phe­nomena, applied to the interval [t, t + At],

U At - P x r At ~t

(3.9)

2. The change in price attributable to the change in r over the interval, which is simply the extent to which the price of the bond at time t + A t differs from P(1 + rAt)

Using the continuous notation,

H fa - Po - P(1 + rAt) h

ifAI--~l:lt' (3.10)

and

H llr - PI - P(1 + rAt) h

if LlI - Jl:lt' (3.11)

Since Ar - fAt + gAl, (3.10) and (3.11) imply that

and

H (f At - g {At) - Po - P(1+r Llt) ~r

U (f At + g W) - PI - P(l + r 6 t) ~r

From (3.12) and (3.13),

- -1- (l(Po + PI) - P(l + rAt)} At 2

-~ 1 (PI - PO) 2

Substituting from (3.9), (3.14) and (3.15) into (3.8),

- [Af + d.E) At + gll.1I dt ~r ~r

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(3.12)

(3.13)

(3.14)

(3.15)

(3.16)

The similarities between (3.16) and (2.9) are apparent; the term 1 g2 ~2p dt 1. absent from (3.16) because ~2p (or more

2 fi! ~

properly it. discrete analogue) can be taken to be zero over a single discrete transition.

(3.16) may be written in the form

where

and

l':lP - PjJ6t + P If' AI

fI - l(ll + dl) P ~t ~r

'f - Ig.ll p ~r

(3.17)

(3.18)

(3.19)

Proceeding from first principles, then, we have arrived at essentially the same description of the change in price over a single time increment of a default-free zero coupon bond of arbitrary maturity that we established in the continuous case. (3.11), (3.18) and (3.19) correspond directly to (2.10), (2.11) and (2.12).

In order to set up an equation for the term structure in such a way that riskless arbitrage is not permitted, we can apply precisely the same argument used to derive the term structure equation in the continuous case.

As for the continuous model, 'the necessary and sufficient condition for excluding riskless arbitrage is that at any time t, the ratio

Il(t.s.r) - ret) 'f'(t,s,r)

must take the same value for all default-free zero coupon bonds. That is, the prices of all such bonds must satisfy the relation

(3.20)

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where Q may vary with .r and t. but may not vary with T. the time to .aturity of the bond.

Substituting for' and 'fJ from (3.18) and (3.19) into (3.20) live.

U + (f - sQHZ - rP - 0 (3.21) ~t ~r

(3.21) i. the discrete analogue of (2.26). It relat.s the behavior of the price of a default-free zero coupon bond of arbitrary matur­ity to the behavior of the short-term rate when the term structure is constrained to remain in equilibrium.

In order to express (3.21) in terms of r. p. Po and Pl. substitute for lI. fRl. and gll from (3.9). (3.14) and (3.15) to obtain

h ~r ~r

P - 1 (Pl 1(1 - Q.[At) + Po 1(1 + Q.fAt») 1 + rLh 2 2

(3.22)

If we let

n - 1(1 - Q~). (3.22) can be written (in full notation) as 2

peT) 1 (IfPl(T -At) + (1 -TT) Po (T -f.t») (3.23) 1 + r(t)At

(3.23) expresses the price of an arbitrary default-free zero coupon bond at any time as a (discounted) weighted average of the possible values it may take at the end of the next time increment. The term structure will remain in equilibrium provided that the prices of bonds of all maturities are linked to their subsequent values by this relation at all times. The relation allows us to generate equilibrium term structures as follows.

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(i) Specify the functions f(r,t) and g(r,t). Apply (3.1) to the current short-term rate r(t) to obtain the two possible values of ret + At). Apply (3.1) to each of these values to obtain the four possible values of ret + 2At), and so on to create a lattice of short-term rates.

(ii) At each point on the lattice, P(O) - 1. That is, the price of a maturing bond is $1 at all times, irrespective of the value of the short-term rate. This condition corresponds to the boundary condition P(s,s,r) - 1 of the continuous model.

(Iii) At any point on the lattice, the price of a bond which matures at the end of the next time increment is obtained via (3.23) from the corresponding short-term rate and (trivi­ally) from the lattice of prices of maturing bonds.

P( At) 1 ( lIP1(O) + (1 - 'IT ) Po(O») 1 + rAt

i.e. ,

P( At) 1 (3.24) 1 + r6t

since Pl(O) - POcO) - 1

(iv) The price of a bond which matures at the end of two increments is obtained from the corresponding short-term rate, and the possible (one period) prices of the bond at the end of the next time increment.

P(2 At) 1 (nPl(6t) + (1 -If) POe At») (3.25) 1 + rAt

and so on.

(v) The yield to maturity on a zero coupon bond with time to maturity T is given by

R(T) [peT») -l/T _ 1 (3.26)

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Comments

(3.23) allows us to create. lattice of equilibrium term structures from a lattice of short-term rates by expressing the price of any default free zero coupon bond at any time as a linear combination of the value. it .ay take at the end of the next time increment. (Since we don't know which of the values the price will take at the end of the increment, it is natural that the current price at any time should reflect both of them).

The close parallels demonstrated between the continuous model and the binomial model should clarify certain aspects of the continuous development, and provide some insight into the role of the various components of the term structure equation (2.26). The relation (3.23) can be regarded roughly as an algorithm which generates solutions to that equation.

The assumption that the term structure will remain in equilibrium at all times 1s central to the development of the models. It may be argued that the assumption is not realistic in that opportunities for arbitrage do occur from time to time in the real market. There are two important reasons for incorporating equilibrium into any model of the term structure, however.

(i) When opportunities for risk-free gain occur in the market, arbitrageurs act to take advantage of them. Such actions tend to restore equilibrium. In physical terms, equilibrium is .IYhl&.

(ii) Disequilibriums in the market generally occur in a non­systematic fashion. They are noise in the system of market prices. Strategies developed using non-equilibrium models, on the other hand, rest on the implicit assumption that there exist systematic opportunities for risk-free gain.

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4. The model and conventional explanations of term structure behavior

This section discusses the relationship of the model to certain conven­tional theories which have been advanced to explain the observed behavior of the term structure.

Adherents of the expectations hypothesis essentially view the configura­tion of the term structure at any time as a reflection of investors' expectations of future interest rate behavior. The theory in its strict form (which will be defined later) is considered to be incomplete in that it fails to recognize that long maturity bonds are more sensitive than bonds of shorter maturity to changes in interest rates.

Proponents of the liquidity theory argue that because of the greater risk on longer bonds, yields on those bonds should generally be higher than yields on shorter bonds. The typical shape of the yield curve should therefore be upward-sloping, and other shapes can be regarded as temporary aberrations_

These views of term structure behavior are complementary rather than incompatible. We will show that the model Is consistent with the expec­tations hypothesis modified by the liquidity theory. Specifically, when interest rates are predetermined, (that is, when the stochastic component of interest rate behavior is removed), the model reduces to the strict form of the expectations hypothesis. This is reasonable; when interest rate behavior is not subject to uncertainty, longer bonds entail no greater risk than bonds of shorter maturity. When the stochastic element of interest rate behavior is reintroduced, however, the parameter comes into play and governs the degree to which yields on longer maturity bonds compensate the investor for the extra risk assumed.

The treatment of these ideas will emphasize intuition rather than mathe­matical rigour.

In its binomial form, the .odel il defined by

6r - f(r,t),1t + g(r,t)AI (4.1)

and

P(t,T) pet, At)( nPl(t+£:\t,T- At) + (l-n)PO(t+At,T-At)J (4.2)

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where

P(t,T)

PO(t+At,T- At)

and TT

is the price at time t of a default free zero coupon bond which matures for $1 at time t+T

is the price of that bond at time t+ A t if A r takes the value fAt - g~At'

is the price of the bond at time t+ A t if A r tak.. the value fAt + g J At'

is a parameter which in general may vary with r and t but which may not vary with T.

In what follows we will take l)t - 1 (that is, we will measure time in units of 6 t), to simplify the notation.

Suppose that g - 0 so that the short-term rate r(t) evolves in a deterministic sequence r(t), r(t+1), r(t+2), etc. Then

PO(t+1,T-1) - P1(t+1,T-1) - P(t+1,T-1), say, (4.3)

for all t, T,

and the price of the bond evolves over time according to the following deterministic sequence.

Time t t+l t+T-1 t+T

Price P(t,T) P(t+l,T-1) P(t+T-l,l) 1

From (4.2) and (4.3),

P(t,T) - P(t,l)[ np(t+l,T-l) + (1- n) P(t+l,T-l)]

- P(t,l) P(t+1,T-1)

- P(t,l) P(t+1,1) P(t+2,T-2)

i.e., P(t,T) - P(t,l) P(t+1,1) P(t+2,l) •.. P(t+T-1,l)

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(4.4)

Since P(t,l) - --1--­l+r(t)

P(t,T) -~ x l+r(t)

for all t, (4.4) may be written

1 X _ ..... llo-~ X • • • x _-..1"'--__ l+r(t+l) 1+r(t+2) l+r(t+T-l)

(4.5)

The effective yield on P(t,T) is R(t,T), where

(l+R(t,T»)T 80 that

(l+R(t,T»)T - (l+r(t») [l+r(t+l») . [l+r(t+T-l»)· (4.6)

The expectations hypothesis states that if the short-term rate at time t, r(t), is expected to follow the sequence r(t), r(t+l), .. r(t+T-l), then the term structure at time t will reflect those values according to

(l+R(t,T»)T (l+r(t»)(l+r(t+l») ... [l+r(t+T-l») (4.7)

When interest rates are predetermined, then, the model is equivalent to the expectations hypothesiS. It is apparent from (4.7) that according to the expectations hypothesis , investing in a T-period bond gives exactly the same result as investing sequentially in T one-period bonds. In a deterministic environment, this equivalence reflects the equilibrium condition; it eliminates the possibility of riskless arbitrage between bonds of differing maturities.

There is empirical evidence to support the contention that when interest rates are subj ect to uncertainty, investors demand a yield on longer bonds which is somewhat greater than would be predicted from the expectations hypothesis . While it is difficult to assign exact values to investors' expectations of future interest rates, we can see from (4 . 7) that at any time rates are expected to decline, the yield curve should be inverted. Observation, on the other hand suggests that there are frequently times when (as far as can be ascertained) rates are generally expected to fall, yet the yield curve remains upward-sloping.

Returning to (4.2), note that generally (taking g to be a positive function)

PI < Po because the value PI corresponds to the higher of the possible values of r(t+l) while Po corresponds to the lower value .

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Consider the limiting case where TT - 1 at all times. From (4.2),

P(t,T)

i.e., P(t,T)

Pl(t,l) Pl(t+l,T-l)

Pl(t,l) Pl(t+l,l) Pl(t+2,T-2)

Pl(t,l) Pl(t+l,l) Pl(t+2,1) .. Pl(t+T-l,l) (4.8)

From (4.8), when n - 1, the price P(t,T) of each zero-coupon bond is determined as though the short-term rate is certain to take the higher of the two possible values at each transition. In this extreme case, the term structure will be identical to that given by (4.6) when the sequence of short-term rates follows the uppermost path through the lattice, regardless of the fact that the actual probability of the short-term rate taking the higher value at any transition is 1/2, not 1. Similarly, values of n close to 1 will generate term structures which behave as though rates are expected to rise, and which will therefore tend to resist inversion.

Since in the general case IT may vary with rand t, the model can accommodate fairly subtle variation of the risk premium component of yields. The objective of the foregoing discussion is simply to establish that IT is the governing variable in this respect.

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References

1. P.P. Boyle. "Immunization Under Stochastic Models of the Term Structure" Journal of the Institute of Actuaries, Vol. lOS, pp. 177-187

2. N.J. Macleod and J.D. Thomison. "A Discrete Equilibrium Model of the Term Structure" ARCH1988,l, pp. 69-109

3. O. Vasicek. "An Equilibrium Characterization of the Term Structure" Journal of Financial Economics 5 (1977), pp. 177-188.

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