<< . .

. 26
( : 51)



. . >>

where c = 1/R.


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 ˜return™ 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 ˜return™ 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 ˜long debt™. 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,˜“™and the m-period spot yield R:”) 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 Rj2™ 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 ˜pure discount™ or ˜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 ˜n™ 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 ˜blip™ 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 ˜returns™ 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 ˜tracks™ 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 ˜data analysis™ given abov

Quality of Data
The ˜quality™ 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 ˜five-year bonds™.
be an average of yields on a number of bonds with four to six years™ 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 ˜bid™ or ˜offer™ rates, or an average of the two rates. The next issu
˜timing™. 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

<< . .

. 26
( : 51)



. . >>