стр. 24 |

If we have an n-period coupon paying bond and market determined spot

for all maturities, then the market price of the bond is determined as:

+

where M = maturity (redemption) value and Vi = Cf+i/(l for i = 1,

+ +

and Vn = (Cr+n M n ) / ( l r ˜ ; ! $ ) ˜ . market price is the DPV of future co

The

maturity value) where the discount rates are spot yields. If the above formula do

then riskless arbitrage profits can be made by вЂ˜coupon strippingвЂ™. To illustrate

consider a two-period bond and assume its market price Pj2вЂ™ is less than V1

current market price of two, zero coupon bonds with payouts of Ct+l and (C1

will be V1 and V2, respectively. The coupon paying bond can be viewed as a

+

coupon bonds. If Pj2вЂ™ < V1 V2 then one could purchase the two-year coupo

sell a claim on the вЂ˜coupon paymentsвЂ™ in years 1 and 2, that is Ct+l and (Ct

to other market participants. If zero coupon bonds are correctly priced then th

could be sold today for V1 and V2, respectively. Hence an instantaneous ris

+

of ( V l V2 - PjвЂќ) can be made. In an efficient market, the increased dema

two-year coupon paying bond would raise Pj2),while sales of the coupons wou

prices of one and two-year zero coupon bonds. Hence this riskless arbitrage

to the restoration of the equality in (9.3).

Holding Period Yield (HPY) and the Rational Valuation Formula (RVF)

As with a stock, if one holds a coupon paying bond between t and t + 1 th

made up of the capital gain plus any coupon payment. For bonds this measure

is known as the (one-period) holding period yield (HPY).

Note that in the above formula the n-period bond becomes an (n - 1) period

one period. (In empirical work on long-term bonds (e.g. with n > 10 years) r

often use P: in place of PI:;вЂ™вЂ™ since for data collected weekly, monthly o

!

:

they are approximately the same.)

At any point in time there are bonds being traded which have different time

maturity or вЂ˜term to maturityвЂ™. For example, a bond which when issued had

to form expectations of P,+l and hence of the expected HPY. (Note that fut

payments are known and fixed at time t for all future time periods and are usua

so that C , = C).

Rational Valuation Formula (RVF) for Bonds

In our most general model for determining stock prices in Chapter 4 it was a

expected one-period return required by investors to willingly hold the stock

over time. The required one-period return (or HPY) on an n-period bond w

and hence for bonds we have:

This equation can be solved forward (as demonstrated in Chapter 4) to give

bond price as the DPV of future known coupon payments discounted at th

one-period spot returns k,+i. For an n-period bond:

where A = the redemption price and it is usually assumed that C,+j = C (a c

4

Note that the only variable the investor has to form expectations about is h

one-period return k,+i in (all) future periods. Also no transversality conditio

be imposed since after iterating forward for n periods the expected price is e

known redemption value, M .

As with our analysis of stocks we can вЂ˜splitвЂ™ the per period required retu

one-period risk-free (interest) rate r, and a risk premium. However, when ta

the government bond market the risk premium on an n-period bond is frequen

to as the termpremium, 7вЂ˜;вЂќ) Hence(2):

+ ++

where r,+i is the one-period rate applicable between t i and t i 1. Becau

ment bonds carry little or no risk of default, the only вЂ˜riskвЂ™ attached to such b

eyes of the investor arises because they have different terms to maturity; henc

вЂ˜term premiumвЂ™. One aspect of risk arises if the investor wishes to liquidate h

before the maturity date of the bond; the investor faces вЂ˜price uncertaintyвЂ™. Al

if the investor holds the bond to maturity there is no (nominal) price uncertai

rate at which future coupon payments can be reinvested is uncertain; this is

as вЂ˜reinvestment riskвЂ™. Note that (9.7) is an identity which is used to defin

premium and there is as yet no behavioural content in (9.7).

J

L i=O i=O

j=1

The variance bounds test is then var(P,) < var(PT). We can also test the E

this explicit assumption for k,+i) by running the regression

where Qr is any information available at time t . Under the null of the EM

a = b = c = 0.

Strictly speaking one cannot have both (9.3) and (9.6) determining the price

since there is only one quoted market price. Rather, equation (9.3) determine

of the bond via riskless arbitrage. Equation (9.6) may then be viewed as det

set of expected one-period returns that yield the same bond price given by (9.

preferences working via the supply and demand for bonds of different mat

establish a set of equilibrium one-period rates k,+i. Of course, when there doe

a set of market determined spot rates for all maturities n then equation (9.6) m

tually be viewed as determining bond prices. Since for n > 2 years, zero cou

are usually not available, equation (9.6) can be used legitimately to examine

of long-term bonds. The reason equations (9.3) and (9.6) give different expr

the price of the bond is that they are based on different behavioural assumpti

derivation of (9.6) agents are only concerned with the sequence of one-perio

returns k,+i whereas the bond pricing equation (9.3) involving spot rates rsf

under the assumption of (instantaneous) riskless arbitrage(вЂќ1.

Yield to MaturitylRedemption Yield

For coupon paying bonds the rate which is quoted in the market is the yield

(YTM). Investors know the current market price of the bond P,, the stream

coupon payments C, the redemption value of the bond (= M )and its maturity d

ussume that the coupon payments at different horizons are discounted using

discount rate 1/(1 + R : ) . Note that RT has a subscript t because it may vary

(but it does not vary in each period in the DPV formula). If we now equate t

the coupon payments with the current market price we have

The bond may be viewed as an investment for which a capital sum P, is paid

and the investment pays the known stream of dollar receipts (C and M )in the

constant value of R: which equates the LHS and RHS of (9.10) is the вЂ˜inter

returnвЂ™ on this investment and when the investment is a bond Rr it is referred to

to maturity or redemption yield on the (n-period) bond. Clearly one has to ca

each time the market price changes and this is done in the financial press which

report bond prices, coupons and yields to maturity. It is worth noting that th

for each bond (whereas (9.3) involves a sequence of spot rates). The yield to m

an n-period bond and another bond with q periods to maturity will generally b

at any point in time, since each bond may have different coupon payments

course the latter will be discounted over different time periods (i.e. n and q). I

see from (9.10) that bond prices and redemption yields move in opposite dir

that for any given change in the redemption yield R; the percentage change in

long bond is greater than that for a short bond. Also, the yield to maturity form

reduces to P, = C/R;"for aperpetuity (i.e. as n -+ 09).

Although redemption yields are widely quoted in the financial press they a

what ambiguous measure of the 'return' on a bond. For example, two bon

identical except for their maturity dates will generally have different yields t

Next, note that in the calculation of the yield to maturity, it is implicitly as

agents are able to reinvest the coupon payments at the constant rate R: in all fut

over the life of the bond. To see this consider the yield to maturity for a two-p

given by (9.10) rearranged to give:

The LHS is the terminal value (in two years' time) of $P,invested at the con

+ +

alised rate R;".The RHS consists of the amount (C M ) paid at t 2 and

+ +

C( l R r ) which accrues at t 2 after the first year's coupon payments have

vested at the rate R;. Since (9.10) and (9.11) are equivalent, the DPV formu

that the first coupon payment is reinvested in year 2 at a rate R:. However, th

reason to argue that investors always believe that they will be able to reinves

coupon payments at the constant rate Ry. Note that the issue here is not tha

have to form a view of future reinvestment rates for their coupon paymen

they choose to assume, for example, that the reinvestment rate applicable on

bond, between years 9 and 10 say, will equal the current yield to maturity

20-year bond.

There is another inconsistency in using the yield to maturity as a measure o

on a coupon paying bond. Consider two bonds with different coupon payme

CiL:, C::: but the same price, maturity date and maturity value. Using (9.1

imply two different yields to maturity R:, and R;,. If an investor holds both of t

in his portfolio and he believes equation (9.10) then he must be implicitly ass

+ ++

he can reinvest coupon payments for bond 1 between time t j and t j 1

R:, and at the different rate Ri, for bond 2. But in reality the reinvestment ra

++

t + j and t j 1 will be the same for both bonds and will equal the one-

rate applicable between these years.

In general, because of the above defects in the concept of the yield to mat

curves based on this measure are usually difficult to interpret in an unambigu

(see Schaefer (1977)). However, later in this chapter we see how the yield

may be legitimately used in tests of the term structure relationship.

in value of its liabilities. It is easiest to examine the concept of duration using

uously compounded YTM, which we denote as y. Duration is a measure of t

time one has to wait to receive coupon payments. For a zero coupon bond th

in n years the duration is also n years. However, a coupon paying bond ma

years has a duration less than n since some of the cash payments are receiv

n. Let us determine the price response of a coupon paying bond to a small ch

YTM. The price of the bond with coupons Ci (where Cn also includes the

price) is:

n n

i= 1 i= 1

where PV; = C;exp(-yt;) is the present value of cash flows Cj. Differenti

respect to y and dividing both sides by P:

If we now define duration D as:

n

D= ˜˜[Pv˜/PI

i= 1

then d PIP = -Ddy. From the definition of D we see that it is a time weight

of the present value of payments (as a proportion of the price). If we know th

of a bond then we can calculate the capital gain or loss consequent on a small

the yield. The simplest case of immunisation is when one has a single liabili˜y

DL. Here immunisation is most easily achieved by purchasing a single bond

has a duration equal to DL. However, one can also immunise against a sing

with duration DL by purchasing two bonds with maturity D1 and 0 2 such that

2

i= 1

and wi are the proportions of total assets held in the two bonds

Clearly this principle may be generalised to bond portfolios of any size. Ther

tations when using duration to immunise a portfolio of liabilities, the key o

that the calculations only hold for small changes in yields and for parallel sh

(spot) yield curve. Also since D alters as the remaining вЂ˜lifeвЂ™ of the bond(s)

immunised portfolio must be continually rebalanced, so that the duration of

liabilities remains equal (for further details see Fabozzi (1993) and Schaefer (

The RVF can also be written in terms of real variables. In this case, the no

price P, and nominal coupon payments are deflated by a (nominal) goods pric

they are then measured in real terms. The required real rate of return which w

is determined by the real rate of interest (i.e. the nominal one-period rate less th

one-period rate of inflation Efnr+l) the term premium:

and

+ +

k: = (rt - Efnr+1) T , = Ef(rrf) Tt

where rr, = real rate of interest (and is assumed to be independent of the rate o

It should be fairly obvious that the nominal price of the bond is influenced

degree by forecasts of future inflation. Coupon payments are (usually) fixed

terms, so if higher expected inflation is reflected in higher nominal discount

the nominal price will fall (see equation (9.6)). More formally, this can be

+

easily by taking a typical term in (9.6), for example the term at t 2 where w

+

+ +

(1 h+i)= (1 k;+j)(l rTi

T t)+

where k,? is the real rate of return (discount factor). Equation (9.13) can be re

Hence for constant real discount factors k;+i the nominal bond price depends

on the expected future one-period rates of inflation over the remaining life of

Let us now demonstrate that the real bond price is independent of the rate o

+

as one might expect. If we define the real coupon payment at t 2 as CT+2 =

+ +

where Ir+2 is the goods price index at t 2, and note that Ir+2 = (1 nr+l)(

then from (9.14) the real bond price has a typical term:

Hence as one would expect, the real bond price depends only on real variables

if the variables are all measured in real terms or all in nominal terms, the term

is invariant to this transformation of the data.

9.2 THEORIES OF THE TERM STRUCTURE

We now examine various theories of the term structure based on different a

made about the required rate of return k,. These are summarised in Table 9.1

with theories of the term structure based on the one-period HPY as a mea

return on the bond. In subsequent sections we then consider how these same t

be implemented using spot yields and the yield to maturity(4). Our aim is to

RI"' - E,(r,+,(s)= 0

E,H!:\ - r, = 0

2. Expectations Hypothesis or Constant Term Premium

(i) Expected excess return equals a constant which is the same for all maturities

or (ii) The term premium 'T' is a constant and the same for all maturities

RI"' - E,(r,+,rs)= T

E,Hi:), - r, = T

3. Liquidity Preference Hypothesis

(i) Expected excess return on a bond of maturity n is a constant but the value o

constant is larger the longer the period to maturity

or (ii) The term premium increases with n, the time period to maturity

ErHj:), - rr = T'"' RI"' - Er(r,+,rS)= T'")

where T(")> T("-'). . ., etc.

4. Time Varying Risk

(i) Expected excess return on a bond of maturity n varies both with n and over

(ii) The term premium depends on the maturity n and varies over time

RI")

- E,(r,+,/s) ˜ ( nz,)

E,Hj:), - r, = ˜ ( nz,) = ,

,

where T ( ) is some function of n and a set of variables z,.

5. Market Segmentation Hypothesis

(i) Excess returns are influenced at least in part by the outstanding stock of asse

different maturities

(ii) The term premium depends in part on the outstanding stock of assets of diffe

maturities

RI"' - E,(r,+,!s) T(zj"')

E,H!:{ - r, = T(zj"') =

where z:") is some measure of the relative holdings of assets of maturity 'n'

proportion of total assets held.

6. Preferred Habitat Theory

(i) Bonds which mature at dates which are close together should be reasonably c

substitutes and hence have similar term premia

clearly the tests implied by the various hypotheses: actual empirical results are

in the next chapter and more advanced tests in Chapter 14.

Using the HPY

The theories based on the HPY include the expectations hypothesis, the liquid

ence hypothesis, market segmentation and the preferred habitat hypothesis. T

only in their treatment of the term premium. Each will be dealt with in turn.

The Expectations Hypothesis

If all agents are risk neutral and concerned only with expected return then the

one-period HPY (over say one month, or one quarter) on all bonds, no matter

This is the pure expectations hypothesis (PEH). The term premium T is z

maturities and the discount factor in the RVF (9.6) is simply kt+j = rf+,,the s

one-period risk-free rates. All agents at the margin are 'plungers'. For example

bond with three years to maturity has a HPY in excess of that on a bond with

maturity. Agents would sell the two-year bond and purchase the three-year

pushing up the current price of the three-year bond and reducing its one-pe

The opposite would occur for the two-period bond and hence all holding per

would be equalised, To (9.16) we now add the assumption of rational expectatio

+

E&::: qf;;, where E f ( q ˜ ˜ ) l l = ,0 and : is the (one-period) rational e

Q) q

forecast error:

(n ) (n )

H I + , - r, = v,+˜ (for all n)

+

Hence a test of the PEH RE is that the ex-post excess holding period y

have a zero mean, be independent of all information at time t(C2,) and should

uncorrelated.

It seems reasonable to assert that because the return on holding a long bo

period) is uncertain (because its price at the end of the period is uncertai

excess holding period yield ought to depend on some form of 'reward for ri

premium T!") .

+

E,H;I_; = r, ˜ j " )

Without a model of the term premium, equation (9.18) is a tautology. The sim

trivial) assumption to make about the term premium is that it is (i) constan

and (ii) that it is also independent of the term to maturity of the bond (i.e. Ty

constitutes the expectations hypothesis (EH) (Table 9.1). Obviously this yie

predictions as the PEH, namely no serial correlation in excess yields and th

should be independent of 52,. Note that the excess yield is now equal to the co

+

premium T and the discount factor in the RVF is k, = rf T .

Under RE and a time invariant term premium we obtain (see (9.18) with T

the following variance inequality.

2 var(rt)

var[H,(:;l

Thus the variance of the HPY on an n period bond should be greater than or

variance of the one-period safe rate such as the interest rate on Treasury bills

Liquidity Preference Hypothesis (LPH)

Here, the assumption is that the term premium does not vary over time but it d

on the term to maturity of the bond (i.e. TI"' = T'")). For example, bonds

periods to maturity may be viewed as being more 'risky' than those with a s

to maturity, even though we are considering a fixed holding period for both

might arise because the price change is larger for any given change in the yield

with longer maturities. Consider the case where the one-month HPY on 20-

The liquidity preference hypothesis asserts that the excess yield is a const

given maturity but for those bonds which have a longer period to maturity

premium will increase. That is to say the expected excess HPY on a 10-year b

exceed the expected excess HPY on a 5-year bond but this gap would rema

over time. Thus, for example, 10-year bonds might have expected excess return

above those on 5-year bonds, for all time periods. Of course, in the data, actu

excess returns will vary randomly around their expected HPY because of (

forecast errors in each time period. Under the liquidity preference hypothes

+

k, = r, T ( вЂќ )as the discount factor in the RVF and expected excess HPYs ar

Under rational expectations, the liquidity preference hypothesis predicts that ex

are serially uncorrelated and independent of information at time t . Thus, apart f

innocuous constant term, the main testable implications using regressions ana

PEH, EH and the LPH are identical. For the PEH we have a zero constant t

EH, we have T = constant and for the LPH we have a different constant for

of maturity n , as in (9.20).

Time Varying Risk

If the risk or term premium varies over time and varies differently for bonds

maturities then

+

k, = r, T ( n , z,)

where z, is a set of variables that influences investorsвЂ™ perceptions of risk.

most general model so far, but unless one specifies an explicit form of the func

model of expected excess HPYs is non-operational. Below is an illustration

CAPM provides a model of the term premium, and Chapter 19 examines the be

HPYs on bonds when the term premium is assumed to depend on time varying

and covariances.

CAPM and HPY

An obvious theoretical approach to explain the excess HPY on bonds and to

explicit form for the risk premium are the CAPM-type models which we ha

discussed when examining stock returns. Thus, for those who are familiar wit

theory, it should not come as a surprise that the term premium may vary both f

maturities and over time. The CAPM predicts that the expected holding period

any asset, which of course includes an n period bond, is given by

constant over time and in general one might therefore expect time variation

premium.

According to MertonвЂ™s (1973) model, the excess return on the market

proportional to the conditional variance of the forecast errors on the market p

стр. 24 |