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Combining equations (9.22) and (9.23):

E , H ! ˜ \- r, = AE*COV(H˜:\, R;˜)

Hence the excess HPY depends upon the covariance between the holding perio

the bond and the return on the market portfolio. It is possible that this covaria

varying and that it may be serially correlated, implying serial correlation in

HPY. Notice that the CAPM version of the determination of term premia doe

that expected holding period returns on 10-year bonds should necessarily be

those on 5-year bonds. To take an extreme (non-general equilibrium) examp

10-year bonds have a negative covariance with the market portfolio, while 5-

have a zero or positive covariance, then the CAPM implies that expected ex

on 10-year bonds should be below those on 5-year bonds.

The standard one-period CAPM model assumes the existence of a risk-free

(1972) developed the zero-beta CAPM to cover the case where there is no ris

and this results in the following equation:

Here the expected HPY on the n period bond equals a weighted average o t f

return on the market portfolio and the so-called вЂ˜zero-beta portfolioвЂ™. A zero-be

is one which has zero covariance with the market portfolio. The next chapte

empirical results from the CAPM applied to bonds under the assumption of

beta, but empirical results from CAPM models which incorporate time vari

betas of bonds of different maturities are not discussed until Chapter 17.

The above taxonomy of models might at first sight seem rather bewilderin

main all we are doing is repeating the analysis of valuation models that we

stocks in Chapters 4 and 6. We have gradually relaxed the restrictive assump

return required by investors to hold bonds (i.e. k,) and naturally as we do so

become more general (see Table 9.1) but also more difficult to implement

Indeed, Table 9.1 contains two further hypotheses which will be dealt with b

Market Segmentation Hypothesis

The market segmentation hypothesis may be viewed as a reduced form or mar

rium solution of a set of standard asset demand equations. To simplify, suppo

only two risky assets (i.e. bonds B1 and B2) and the proportion of wealth hel

B2 is given by their respective demand functions

- r, H; - r )

( B ˜ / w =˜ I ( H ?

)˜

+ B2)/W

(TBIW) = 1 - (Bi

and need not concern us. If we now assume that the supply of B1 and B2 is

then market equilibrium rates of return, given by solving (9.26), result in e

the form

- r = G i [ B I / W ,B 2 / W ]

H;

He2 - r = G 2 [ B l / W ,B 2 / W ]

Hence the expected excess HPYs on the two bonds of different maturities

the proportion of wealth held in each of these assets. This is the basis of

segmentation hypothesis of the determination of excess HPYs.

Tests of the market segmentation hypothesis based on (9.28) are often a gro

plification of the rather complex asset demand functions usually found in th

literature in this area. In general, the demand functions (9.26) contain many

pendent variables than holding period yields; for example, the variance of r

wealth, price inflation and their lagged values and lags of the dependent va

appear as independent variables (see Cuthbertson (1991)). Hence the reduced

librium equations (9.28) should also include these variables. However, tests

segmentation hypothesis usually only include the proportion of debt held i

maturity вЂ˜nвЂ™ in the equation to explain excess holding period yields.

Preferred Habitat Hypothesis

The preferred habitat theory is, in effect, agnostic about the determinants o

premium. It suggests that we should only compare вЂ˜returnsвЂ™ on governmen

similar maturities and one might then expect excess holding period yields to m

together.

9.2.1 Theories Using Spot Yields

The term structure of interest rates deals with the relationship between the

bonds of different maturities. The yields in question are spot yields and therefo

tually, the analysis applies to pure discount bonds or zero coupon bonds. Pu

government bonds for long maturities do not exist, nevertheless as we noted e

yields can be derived from a set of coupon paying bonds (McCulloch, 199

measure of the return on a bond is the yield to maturity and our analysis o

structure in terms of spot yields can be applied with minor modification (and

of approximation) to yields to maturity.

For the moment we proceed as follows. First, we examine how the term

relationship can be derived from a model of expected one-period HPYs and th

our derivation of the RVF for stock prices in Chapter 4, since it involves

difference equation in one-period rates. This derivation requires the use of co

compounded rates. We then examine the economic behaviour behind the term

before demonstrating how it may be rewritten in a number of equivalent way

+ T!"'

Er[ln P:?;') - In P:"'] = r

r

Erh,'?:

Equation (9.29), although based on the HPY, leads to a term structure rela

terms of spot yields. For continuously compounded rates we have In PI"' = In

and substituting in (9.29) this gives the forward difference equation:

+ rr + ˜j"'

n ˜ ! " ) (n - I)E˜R:;;')

=

Leading (9.30) one period:

+ +T,+˜

( n - I)R!:;') = (n - 2)˜,+1˜j:;˜) r,+l ("-1)

Taking expectations of (9.31) using E,E,+1 = Et and substituting in (9.30)

+ E,(rr+l + rt) + E,("q;" + 7';"')

nRI") = (n - 2)E,R,'q;2'

Continually substituting for the first term on the RHS of (9.32) and noting

j)EtRf;;'' = 0 for j = n we obtain:

where

n-1

i=O

n-1

i=l

Hence the n-period long rate equals a weighted average of expected future sho

plus the average risk premium on the n-period bond until it matures, @In'. T

R:'"' is referred to as the perfect foresight rate since it is a weighted aver

outturn values for the one-period short rates, rr+i. Subtracting r from both side

r

we obtain an equivalent expression:

where

Equation (9.35) states that the actual spread SjnV1)

between the n-period and

rate, equals a weighted average of expected changes in short rates plus a term

Equations (9.33) and (9.35) are general expressions for the term structure relat

they are non-operational unless we assume a specific form for the term prem

The PEH applied to spot yields assumes investors are risk neutral, that is, the

ferent to risk and base their investment decision only on expected returns. T

variability or uncertainty concerning returns is of no consequence to their

decisions. In terms of equations (9.33) and (9.35) the PEH implies @(") =

We can impart some economic intuition into the derivation of (9.33) when w

zero term premium. To demonstrate this point we revert to using per-period

than continuously compounded rates, but we retain the same notation so that

are now per-period rates.

Consider investing $A in a (zero coupon) bond with n years to maturity. T

value (TV) of the investment is:

+ R,'"')"

TV, = $A(1

where R,'"' is the (compound) rate on the n-period long bond (expressed a

rate). Next consider the alternative strategy of reinvesting $A and any interes

a series of 'rolled-over' one-period investments, for n years. Ignoring transac

the expected terminal value E,(TV) of this series of one-period investments i

++

+

where rr+j is the rate applicable between periods t i and t i 1. The invest

long bond gives a known terminal value since this bond is held to maturity. In

series of one-year investments gives a terminal value which is subject to uncert

the investor must guess the future values of the one-period spot yields, rr+j

under the PEH risk is ignored and hence the terminal values of the above two

investment strategies will be equalised:

+ rr)(l + Errr+1)(1+ Etrt+2) - - + Errr+n-i)

(1 +I$"))" = (1 (1

The equality holds because if the terminal value corresponding to investment

bond exceeds the expected terminal value of that on the sequence of one-year in

then investors would at time t buy long bonds and sell the short bond. This w

in a rise in the current market price of the long bond and given a fixed mat

a fall in the long (spot) yield R,. Simultaneously, sales of the short bond wo

fall in its current price and a rise in r,. Hence the equality in (9.39) would

(instantaneously) restored.

We could define the expected 'excess' or 'abnormal' profit on a $1 investm

long bond over the sequence of rolled-over short investments as:

where Er(r;+j) represents the RHS of (9.39). The PEH applied to spotyields

implies that the expected excess or abnormal profit is zero. We can go through

+ Etrt+i + Errr+2+ -

RI"' = (l/n)[rt Errr+2]

*

In general when testing the PEH one should use continuously compounded

since then there is no linearisation approximation involved in (9.41)(@.

The PEH forms the basis for an analysis of the (spot) yield curve. For exam

from time t, if short rates are expected to rise (i.e. Errr+, > Errr+j-1) for all j

(9.41) the long rate R!")will be above the current short rate rr. The yield curve

of RI"' against time to maturity - will be upward sloping since R!"' > R!"

Since expected future short rates are influenced by expectations of inflation (Fi

the yield curve is likely to be upward sloping when inflation is expected to

future years. If there is a liquidity premium that depends only on the term to

and T(")> T("-l) > . . ., then the basic qualitative shape of the yield curve wil

described above. However, if the term premium varies other than with 'n', f

varies over time, then the direct link between R(") and the sequence of future

in broken.

ff

PEH: Tests Using Different Maturities

Early tests were based on equation (9.41) and include variance bounds and

based tests. So far, we have analysed the expectations hypothesis only in te

n-period spot rate RI"' and a sequence of one-period short rates. However, the

can also be expressed in terms of the relationship between R!"' and any rn-

R!*' (for which s = n/m is an integer):

s- 1

R!") = (l/s) E,Rj$m

i=O

Two rearrangements of (9.42)can be shown to imply that the spread Sr(n*m) (I?

=

is an optimal predictor of future changes in long rates and the spread is

predictor of (a weighted average of) future changes in short rates. We consi

these in turn using convenient values of (n, rn) for expositional purposes.

Spread Predicts Changes in Long Rates

Consider equation (9.42) for n = 6, m = 2:

this may be shown to be equivalent to(')

ErRj:\ - RI6' = (1/2)S!612'

Hence if Rj6' > Ri2' then the PEH predicts that, on average, this should be fol

+

rise in the long rate between t and t 2. The intuition behind this result can

+ /!3[S!6t2)/2]+ q1+2

- =

RI:\

where we expect = 1. Under RE, S,'6'2' independent of qt+l and therefo

is

(9.45) yields consistent estimates (although a GMM correction to the standa

required if there is heteroscedasticity in the error term, or if qt+2 is serially cor

to the use of overlapping observations). For any (n,m), equation (9.44) can

to be:

E t R("-m) - RI"' = [ m / ( n -

r+m

The Spread Predicts Future Changes in Short Rates

Equation (9.43) may be rearranged to give(')

E ˜ ( 6 * 2 )= ˜ 1 6 ˜ 2 )

*

tt

and AmZr= 2, - 2 r - m . The term Sj6'2'* the perfect foresight spread. Equa

is

implies that the actual spread S16*2' an optimal (best) predictor of a weight

is

of future two-period changes in the short rate Hence if Si6'2' > 0 then ag

on average that future (two-period) short rates should rise. Note that althou

an optimal predictor this does not necessarily imply that it forecasts the RHS

+

accurately. If there is substantial 'news', between periods t and t 4, then Si

a rather poor predictor. However, it is optimal in the sense that, under the ex

hypothesis, no variable other than S, can improve one's forecast of future

short rates. The fact that S , is an optimal predictor will be used in Chapter 1

examine the VAR methodology. The generalisation of (9.47) is

=

E,Sjn*m)* S p m )

where

s- 1

= (-

Sj"*")* c 1 i/s)A( m )R,+im

(m)

i= 1

and equation (9.48) suggests a straightforward regression-based test of the ex

hypothesis (under RE) based on:

S p ) *= a + /!3Sl"*m) + wt

Under the EH we expect /?= 1 and if the term premium is zero a = 0. Sinc

(by RE) independent of information at time t , then OLS yields unbiased estim

parameters (but G M M estimates of the covariance matrix may be required - s

Note, however, that if the term premium is time varying and correlated with

then the parameter estimates in (9.49) will be biased.

are equivalent. Also if (9.46) holds for all ( n , m) then so will (9.48). Howev

is rejected for some subset of values of (n,m) then equation (9.48) doesn't

hold and hence it provides independent information on the validity of the PE

In early empirical work the above two formulations were mainly undertak

2m, in fact usually for three- and six-month pure discount bonds (e.g. Treasu

which data are readily available. Hence the two regressions are statistically

Results from equations (9.46) and (9.48) are, however, available for other ma

n # 2m) and some of these tests are reported in the next chapter.

Variance Bounds Tests on Spot Yields

Shiller (1988) uses an equation similar to (9.33) to perform a very simple vo

for the expectations hypothesis. In the extreme case where agents have perfe

then future expected short rates would equal their ex-post outturn values. If w

perfect foresight long rate as:

+ q,+i) we hav

then under the PEH equation (9.33) and RE (i.e. rf+i = E,r,+i

+

Rf = R,

where

n-1

i= 1

and E(q,+l IQ,) = 0. Using past data we can construct the perfect foresight lo

each year of any sample of data. Under RE, agents' forecasts are unbiased an

sum of the forecast errors 0, should be close to zero, for large n . Hence fo

bonds we expect the perfect foresight rate R to track the broad swings in the

:

rate R, and as we shall see in the next chapter Shiller's evidence on US data i

with this hypothesis.

Clearly, as we found for stock prices, a more sophisticated method of testing

volatility is available. Under the RE the forecast error w, is independent of 52

it is independent of R,. Hence from (9.51) we have:

var[R:] 3 var[Rr]

since by RE, cov(R,, w , ) = 0 and var(w,) 2 0. These inequalities also apply if

a constant term premium T,'"' = T ( " ) .Thus, if (i) the expectations or liquidity

hypothesis (as applied to spot yields) is correct, (ii) agents have RE, (iii) the ter

depends only on n (i.e. is time invariant), then the variance of the actual long

be less than the variability in the perfect foresight long rate.

It is also the case that the expected (abnormal) profit should be independent

tion at time t . In this linear framework the expected abnormal profit is given by

market.) If interest rates are non-stationary I(1) variables then the unconditio

tion variances in (9.52) are undefined but the variance bound inequality can b

in terms of the variables in (9.48) which are likely to be stationary. Hence w

var(S:) b var(St)

and in a regression context we have

where we expect b = 1, c = 0 if the expectations hypothesis is true.

9.2.2 Using the Yield to Maturity

The tests described above are based on spot yields for pure discount (zero cou

either expressed as continuously compounded rates or as per-period rates. A

already noted the calculation of (approximate) spot yields from published data

maturity is often not undertaken. There is generally much more published da

to maturity so it would be useful if our tests of the term structure could be rec

of the yield to maturity. In a pioneering article this has been done by Shiller

the results are summarised below.

The basis of Shiller's approximations can be presented heuristically as fo

price of an n-period coupon paying bond is a non-linear function of the yield

(equation (9.10)) and therefore so is the holding period yield Hi:\.

where Rj") is the yield to maturity (i.e. the superscript 'y' is dropped) and

non-linear function. Shiller linearises (9.56) around the point

where E is the mean value of the yield to maturity. For bonds selling at o

(redemption) value this gives an approximate expression for the holding p

denoted fi!") where

where

and

yn = y(1 - y"-')/(l - y")

+R)

y = 1/(1

Equation (9.58) is an (approximate) identity which defines the HPY in terms o

to maturity. We now add the economic hypothesis that the excess holding p

From (9.58) and (9.59) we obtain a forward recursive equation for RI"' w

= r f + n - l , where r, is the one-period

solved with terminal condition

gives Shiller's approximation formula for the term structure in terms of th

maturity:

+ @"

= E,(R:)

where @ n = f ( y , y", 4") and Rf is the perfect foresight long rate. Hence t

maturity on a coupon bond is equal to a weighted average of expected f

rates rf+k where the weights decline geometrically (i.e. fl < yk-' < . . .). Th

is constant for any bond of maturity n and may be interpreted as the aver

one-period term premia, over the life of the long bond. For the expectations

@ n = @ for all n and for the pure expectations hypothesis = 0. However

alternative theories of the term structure the precise form for any constant term

is of minor significance and the term premium only plays a key role when we

is time varying (see Chapter 17).

The RHS of (9.60) with Errt+k replaced by actual ex-post values rt+k is

foresight long rate (yield to maturity) RI")* and the usual variance bound

regression tests of R!"' on R!")*may be examined. Shiller (1979) also shows

HI"),

using the approximation for the holding period yield the variance bound

var H!") 6 a2var(r,)

where a = (1, - y2)-1/2. Note that for E = 0.064 then y = 0.94 and a2 = 8.6,

variance of H:, can exceed that of the short rate. Equation (9.62a) puts an up

i)

(whereas (9.19) imposes a lower bound). If r, is non-stationary

on varfi!:;

var(r,) is undefined but in this case Shiller (1989) provides an equivalent ex

terms of the stationary series At-,, namely:

- r,) 6 c2var(Ar,)

var<&!!$

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