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9.3 SUMMARY

Often, for the novice, the analysis of the bond market results in a rather bewild

of terminology and test procedures. An attempt has been made to present the

a clear and concise fashion and the conclusions are as follows:

The one-period HPY on coupon paying bonds consists of a capital gain plu

0

payment. This is very similar to stocks. Hence the rational valuation fo

may be viewed as consisting of a risk-free rate plus a term premium.

The вЂ˜returnвЂ™ on zero coupon bonds of different maturities is measured by the

0

Theories of the term structure imply that continuously compounded long

specific linear weighted average of (continuously compounded) expected f

rates. This linear relationship applies as an approximation if one uses (co

spot yields and a similar approximate relationship applies if one uses yields

The yield to maturity, although widely quoted in the financial press, is a

0

misleading measure of the вЂ˜returnвЂ™ on a bond particularly when one wishes

the shape and movements in the yield curve.

Alternative theories of the term structure are, in the main, concerned wi

0

term premia are (i) zero, (ii) constant over time and for all maturities, (i

over time but differ for different maturities, (iv) depend on the proportion

held in вЂ˜long debtвЂ™. These assumptions give rise to the pure expectatio

esis, the expectations hypothesis, the liquidity preference hypothesis and

segmentation hypothesis, respectively. In addition, term premia may be ti

(this is discussed in Chapter 14).

The key element in the expectations hypothesis is that the long rate is

0

(geometric) weighted average of expected future short rates which can b

mated by a linear relationship. Agents are risk neutral and equalise expec

over all investment horizons, using all available relevant information to pre

short rates. Hence, no abnormal profits can be made by switching between

shorts and the EMH under risk neutrality holds.

The expectations hypothesis (plus RE) applied to HPYs implies that the ex-

0

holding period yield (Hj;), - rt) is independent of information at time t, Q

The expectations hypothesis applied to (continuously compounded) spot yie

0

that the variance of the perfect foresight long rate

n

i

should exceed the variance of the actual long rate R,.

The expectations hypothesis implies that the spread S!n9m) between the n-pe

0

yield R,вЂ˜вЂњвЂ™and the m-period spot yield R:вЂќ) is an optimal predictor of both

change in the long rate and future changes in short rates and gives rise to the

regression tests:

where the perfect foresight spread

s- 1

i=O

are now in a position to examine illustrative empirical results, in the next cha

ENDNOTES

1. Strictly the sequence of future one-period required returns should be deno

+

k(n - 1, t l), etc. but for ease of exposition k,, k , + l , etc. are used.

2. Any complications that arise from Jensen's inequality are ignored.

3. When market determined spot rates exist then riskless arbitrage ensures t

the bond is given by (9.3). Note that rs!') is the spot rate applicable betw

+

+

t and t i , whereas kf+; is the one-period rate of return between t i an

The two formulae give the same price for the bond when, for example, f

+ r s ( l ) )= (1 + ko)

(1

+

(1 + rs(2))2= (1 + ko) E,(1 kl)

*

which implies

where kl is the one-period rate applicable between periods 1 and 2. So

will recognise (iii) as the pure expectations hypothesis. So-called general

models of interest rate determination also deal with these issues but su

are beyond the scope of this book (e.g. Cox et a1 (1981) and Brennan an

(1982)).

4. In discussing the various theories of the term structure, problems that

Jensen's inequality are ignored. In fact, the EH expressed in terms of the

HPY and in terms of the long rate as a weighted average of expected sho

not generally equivalent. However, after linearisation the equivalence hold

et a1 (1981), McCulloch (1990) and Shiller et a1 (1983)).

5 . For notational simplicity the subscript 'c' to denote continuously compo

is not used. Later it is noted that, subject to an element of approximatio

formulae hold for both continuously compounded rates and for discrete c

rates. This analysis also goes through for HPYs for rn > 1 periods, tha

n , m (with n / m an integer). See Shiller et a1 (1983).

6. In terms of continuously compounded rates the same analysis yields

where Rt(n),r,+; are continuously compounded rates. Taking logarithms

relationships (ignoring Jensen's inequality) holds exactly for these c

compounded (spot) rates:

Moving (1) two periods forward, the six-period bond becomes a four-per

Applying the law of iterated expectations to (2), that is Ef(E,+2R,+,)=

have:

+

E,Rj$ = (1/2)(E,RI?2 ErR::):)

Substituting for the RHS of (3) in (1):

8. Use will be made of continuously compounded rates with a maturity v

since this makes the algebra more transparent. We have:

where we have used ln(1) = 0. The (logarithmic) HPY on holding the

+

bond from t to t 2 is

The safe return on the two-period bond is

Equating (4) and (5) and rearranging we obtain equation (9.44) in the tex

+ E,RI:)\ + EfRi:\]

Rj6) = (1/3)[Ri2)

9.

Substracting Rj2вЂ™ from both sides and rearranging:

The RHS of (3) is equal to that is equation (9.47b) in the text.

Empirical Evidence on the

Term Structure

The previous chapter dealt with the wide variety of possible tests of the EMH

market which can be broadly classified as regression-based tests and variance bo

These two types of test procedure can be applied to spot yields, yields to mat

prices and holding period yields (HPY). This chapter provides illustrative (

exhaustive) examples of these tests. At the end of this chapter some broad c

are reached on whether such tests support the EMH but the reader should

these conclusions as definitive, given the wide variety of tests not reported.

the EMH requires an economic model of expected (excess) returns on bon

failure of the EMH may be due to having the wrong economic model of (e

returns. In general the theories outlined in the previous chapter and the tests d

this chapter assume a time invariant term premium - models which relax this

are presented in Chapter 17. The stationarity of the variables used in particu

of importance, as noted when applying variance bounds and regression test

prices. The issue of stationarity is noted in this chapter but is discussed in mo

Chapter 14 on the VAR methodology. After presenting a brief account of som

facts for bond returns and the quality of data used, the following will be disc

We investigate the term structure at the short end of the market, name

three-month and six-month bills. These are вЂ˜pure discountвЂ™ or вЂ˜zero coup

and this enables us to use quoted spot rates of interest.

We examine bonds at the long end of the maturity spectrum. In particular

the relationship between actual long rates and the perfect foresight lon

undertake the appropriate variance bounds and regression based tests. We

these tests using bond prices.

Again for long maturity bonds, we examine the variance bounds inequaliti

period HPYs and examine whether the zero-beta CAPM can explain the

of HPYs.

A word about notation in this chapter. When using the symbol R, it is no

explicitly to distinguish between spot rates and yields to maturity as this

should be clear from the context/data being discussed. Where no ambiguity wi

superscript вЂ˜nвЂ™ on R, and H, will be dropped in order to simplify the notation

reverse is the case.

Table 10.1 presents results for Germany across the whole term structure

to maturity rises monotonically as the term to maturity increases (column 1)

yield spread (R!"' - r f ) (column 7). The one-month holding period return H: I:

excess holding period return (H!:), - r f )both increase with maturity (colum

and the volatility of the price of 'long-maturity bonds' is greatly in excess

short-maturity bonds (column 4). Thus for Germany the results are quite strai

It appears that in order to willingly hold long-term bonds both a higher y

(Rj"' - r l ) or a higher excess holding period yield (Hf:\ - r f ) is required

over the 1967-1986 period and this is broadly consistent with the liquidity

hypothesis. However, agents who hold long bonds in order to obtain a high

expected excess HPY also experience increased 'risk', as measured in terms of

standard deviation (of bond prices). For example, when holding a 10-year b

than a 5-year bond the investor receives an average additional HPY of about

(= 1.62/1.34) but the standard deviation increases by about 43 percent (= 24

(Table 10.2, columns 5 and 6). As we shall see in Chapter 17, the above evid

15

14

13

12

11

h9

E

88

b

e 7

6

5

4

3

69112 74412 79/12 8

Figure 10.1 Government Securities Yields: Germany (Secondary Market - Percent p

Reproduced by permission of Joseph Bisignano

236

Table 10.1 Summary Market Yield and One-Period Holdin

Holding Period

Market Yield Return

Maturity Mean Variance Mean Varia

˜˜

Three-mont h

- -

6.78 8.06

Interbank

26.4

6.90

6.82 3.97

1 Year

62.1

7.27 2.92 7.44

2 Year

108.

7.76

7.5 1 2.44

3 Year

162.

7.66 7.97

2.17

4 Year

218.5

7.76 8.12

1.97

5 Year

293.0

8.24

1.83

7.84

6 Year

326.

1.71

7.89 8.31

7 Year

7.93 373.8

1.61 8.36

8 Year

423.0

8.39

1.53

7.95

9 Year

7.96 569.

8.40

1.49

10 Year

(a) Holding period yield less three-month interbank rate.

Reproduced by permission of Joseph Bisignano.

(column 7). For the USA, results are even more non-uniform (Table 10.3).

HPY (column 5) for the USA shows no clear pattern and although the yiel

higher at the long end than the short end of the market, it does not rise mon

However, for very short maturities between two months and 12 months McCul

has shown that for US Treasury bills the excess holding period yield HI;:

monotonically from 0.032 percent per month (0.38 percent per annum) fo

0.074 percent per month (0.89 percent per annum) for n = 12 months. (Alth

is a вЂ˜blipвЂ™ in this monotonic relationship for the 9-/10-month horizon, see

(1987)). Thus, for the UK there is no additional reward, in terms of the excess

the one-month return) to holding long bonds rather than short bonds althou

a greater yield spread on long bonds. The latter conclusions also broadly a

USA for long horizons but for short horizons there is monotonicity in the ter

(McCulloch, 1987) which is consistent with the LPH. There are therefore

differences between the behaviour of вЂ˜returnsвЂ™ in these countries.

A much longer series for yields to maturity on long bonds for the USA

corporate bonds which carry some default risk) and on a perpetuity (i.e. the Co

for the UK are given in Figures 10.2(a) and 10.2(b), together with a represen

rate. The data for yields on the two long bonds appear stationary up until about

yields rise steeply. Currentperiod short rates (for both the USA and the UK

be more volatile than long rates so that changes in the spread Sr = (RI - r t ) a

dominated by changes in rr rather than in Rr.

According to the expectations hypothesis the perfect foresight spread R T

weighted (moving) average of future short rates r, and hence should be

series than r,. It appears from Figures 10.3(a) and 10.3(b) for the USA an

respectively, that R вЂ˜tracksвЂ™ R, fairly well, as we would expect if the expe

T

liquidity preference hypotheses (with a time invariant term premium) hold. H

require more formal test procedures than the quick вЂ˜data analysisвЂ™ given abov

Quality of Data

The вЂ˜qualityвЂ™ of the data used in empirical studies varies considerably. This

comparisons of similar tests on data from a particular country done by different

or cross-country comparisons somewhat hazardous.

For example, consider data on the yield to maturity on вЂ˜five-year bondsвЂ™.

be an average of yields on a number of bonds with four to six yearsвЂ™ matu

maturities between five years and five years plus 11 months. The data used by

might represent either opening or closing rates (on a particular day of the m

may be вЂ˜bidвЂ™ or вЂ˜offerвЂ™ rates, or an average of the two rates. The next issu

вЂ˜timingвЂ™. If we are trying to compare the return on a three-year bond with the

investment on three one-year bonds, then the yield data on the long bond for

be measured at exactly the same time as that for the short rate and the investme

should coincide exactly. In other words the rates should represent actual deali

which one could undertake each investment strategy.

238

Table 10.2 Summary Market Yield and One-period Holding

Holding Period

Market Yield Return

Maturity Mean Variance Mean Varia

- -

8.45 10.44

91 Day

9.09

9.45 906.

5 Year 8.60

9.89 1579.

8.80

9.99

10 Year

2608.

8.62

10.14 9.96

20 Year

˜˜ ˜ ˜

Reproduced by permission of Joseph Bisignano.

Empirical Evidence on the Term Structure

Table 10.3 Summary Market Yield and One-p

H

Market Yield

Sample

Period Maturity Mean Variance Mea

Jan. 1960-Aug. 1986 9.16

6.32

3 Month

Jan. 1960-Aug. 1986 6.50

6 Month 6.49

8.72

Jan. 1960-Aug. 1986 1 Year 6.97

6.99 10.14

Jan. 1978-Aug. 1986 10.79

2 Year(a) 11.01

5.79

Jan. 1960-Aug. 1986 7.28

3 Year 7.12

9.35

Jan. 1960-Aug. 1986 9.14

7.40 7.09

5 Year

July 1969-Aug. 1986 9.08

7 Year(a) 8.94

6.38

7.48

Jan. 1960-Aug. 1986 8.94

10 Year 6.77

Jan. 1960-Aug. 1986 9.04

7.50 6.28

20 Year

Mar. 1977-Aug. 1986 4.16

10.67

30 Year 10.84

Reproduced by permission of Joseph Bisignano.

11

10

9

8

7

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