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at

If a = 0 then this reduces to a model with a constant value for 'U. Using wee
:
data on US stocks, Chou et a1 (1989) first estimate equation (18.33) with a con
of a and a GARCH(1,l) model for a :



The regression over subperiods reveals that a does vary over time. Using
walk model for a, the time variation in a is found to be substantial (whe
monthly returns data). There is also persistence in a since a1 is about 0.84 a
:
0.13, and both coefficient estimates are very statistically significant. Havin
their time series for at, from (18.34), Chou et a1 investigate whether at de
number of macroeconomic variables that might be thought to influence x,, /3,U
equation (18.31)).
From (18.31) one can see that if W is constant the unobservable variables x
as cross-products of the form (zro:)where zr = x, or /: Hence we can appr
I.
general relationship (18.32) as



When (18.36) is estimated together with the GARCH(1,l) model for the time v
element a, Chou et a1 find that for zr = rate of inflation, the coefficient q 2 is
:
significant and fairly stable. Unfortunately for the CAPM, however, the coeff
a becomes negative and statistically insignificant. (Other variables such as t
:
of interest and the ratio of the value of the NYSE stock price index to consu
tried as measures of XI, but are not usually statistically significant or stable.)
The above study while ingenious in its statistical approach does not yield
tional insight into the behaviour of asset returns other than to show that the b
the (own) variance and inflation.
0;



18.3 SUMMARY
The main conclusions from the empirical work that uses a version of the CAP
of asset shares are:
Expected excess returns are only weakly related to changes in asset s
when time varying covariances are introduced into the CAPM. In a
CAPM restrictions l = C appear to fail even when Il and C are ass
l
time varying.
Estimates of h (or p) tend to be very imprecise and statistically we can
h = 0. The CAPM therefore fails and risk neutrality applies.
The results differ somewhat depending on what is chosen to constitute
portfolio. For example, some researchers use only domestic assets or
bond holdings, rather than a wider set of assets.
Equity returns may well depend on time varying covariances but not in
tive form given in the standard CAPM. Also, there is some evidence (a
shaky) that the statistical significance of any time varying risk premia e
be due to a few dramaticperiods of turbulence.
Additional assumptions such as differential transactions costs, heterogene
tations formation and differential taxation across investors are required
CAPM, in the face of the evidence presented above.
Tests of the CAPM which use the excess market return ER:, - r, as the
able may be subject to less measurement error than using the xit as RHS
Hence tests on the former which we discussed in the previous chapter m
informative about the validity of the CAPM.
L Risk Premia and the Bond Mark
In Chapter 10 when examining empirical evidence on the term structure the
premium on bonds was allowed to vary depending only on the term to ma
gives rise to various hypotheses applied to holding period yields (HPYs), spo
yields to maturity. This chapter deals exclusively with the determination of
HPYs on bonds of various maturities, where the term premium is allowed t
time. The procedure is as follows.
We begin with studies which deal with the short end of the term structure,
choice between six-month and three-month bills. We look first at studies
rather ad-hoc measures of a time varying risk premium, and then move on
fairly simple ARCH models of conditional variances.
Next we turn to the modelling of time varying term premia on long-term
begin with the study by Mankiw (1986) who attempts to correlate variou
of risk with the yield spread. We then return to the CAPM. As a benchm
how far (variants of) the CAPM with time invariant betas can explain ex
on long-term bonds. Here the risk premium may vary with the excess
benchmark portfolio such as the market portfolio and the zero beta portfo
The CAPM does not rule out the possibility that conditional variances and
in asset returns are time varying (i.e. the betas are time varying). We theref
some empirical work that utilises a set of ARCH and GARCH equation
these time varying risk premia on long-term bonds.
Finally we examine the possible interdependence between the risk premi
and that on stocks.


19.1 TIME VARYING RISK: PURE DISCOUNT BON
In Chapters 10 and 14 we saw that the expectations hypothesis applied to z
(or pure discount) bonds at the short end of the maturity spectrum received s
ical support both from single equation studies and studies using the VAR m
However, empirical anomalies still remain (e.g. violation of the VAR param
+
tions under the null of the EH RE). For example, Simon (1989) using wee
three-month (13 weeks) and six-month bills ran the following regression:
+
hence he rejects the EH RE (see Section 10.2). In this section we examin
Simon (1989), Jones and Roley (1983) and a pioneering study by Engle et a
of which attempt to apply models involving time varying risk premia to the b
In the article referred to above, Simon (1989) tries to improve the model by
a time varying risk premium. The expected excess one-period holding period y
when investing in six-month bills rather than three-month bills is defined as“


The holding period in this case is three months (13 weeks). Given our definit
the RE forecast error qr+13 may be written:

= yr+13 - Etyr+13
qr+13

Over a three-month holding period, the six-month bill constitutes the risky
its selling price in three months™ time is unknown (and is, of course, directly
Etrr+13 in (19.2) since after three months has elapsed the six-month bill ˜beco
with only three months to maturity). If we denote the risk premium as 0, then
yield on the risky six-month bill (held for three months) is determined by:

+ 0,
= 4)
ErYr+l3

Simon assumes that the risk premium is proportional to the square of the exc


Substituting (19.5) in (19.4) using (19.2) and the RE condition (19.3) and
we have:
+
A13rr+13 = a -k bl(Rr - rr) -k b2Er(2Rr - r - rr+13)2 &+13
r
where we expect 61 = 2, b2 < 0. (Strictly bl = 2 should be imposed so th
variable is a holding period yield.) Equation (19.6) has to be estimated by IV
the errors in variables problem introduced when we replace the expected valu
by its ex-post value. Because of the use of overlapping data the error term
MA(12) and may also be heteroscedastic. Simon (1989) ˜corrects™ for these p
using an estimation technique known as two-step two-stage least squares (see C
Simon™s most favourable result is:
+ 1.6 ( R - r), - 0.47 (2R, - r, - rf+l)2
= 0.06
(0.06)
(0.18) (0.34)
1972-1979 (weekly data), R2 = 0.48, ( - ) = standard error
The last term indicates the presence of a time varying risk premium and the co
the yield spread (R, - r , ) is not statistically different from 2. A similar result to
is found for the 1961-1971 period but for the post-1982 period (ending in 19
premium term is statistically insignificant and there is some variability in the
six-month over three-month US Treasury bills depends on a time varying ris
6,. Here 6, is measured by a weighted average of the change in the absolute v
short rate which again is a rather ad-hoc measure of risk(*)




Using weekly data for Fridays over the period January 1970-September 1979 a
that the three-month return exactly matches the holding period of the six-month
they find
+ 0.97r, + 0.756,
2Rr - rt+13 = -1.0
(0.86) (0.08) (0.31)
R2 = 0.79, SEE = 0.90
1970-1979,
The coefficient on r, is not statistically different from unity so that th
equation (19.9) can be rewritten in terms of the excess yield yt+13 (see (19
and Roley also test to see if any additional variables at time t influence
yield. They find that an unemployment variable and the stock of domest
three- or six-month bills are statistically insignificant. However, they find th
foreign holdings of US Treasury bills are just statistically significant at c
significance levels (suggesting some market segmentation). Hence, 0, does
an ˜exhaustive explanation™ of excess yields and strictly speaking the RE
efficiency assumption is violated. However, too much weight should not be
the result that elements of the complete information set 52, are statistically
because if one undertakes enough permutations of additional variables, some
to be found to be significant at conventional significance levels (i.e. Type I er
Shiller et a1 (1983, page 199) in commenting on the Jones-Roley measure
premium 6, in (19.8) note that their proxy is not well grounded in any econo
Shiller et a1 also find that when the Jones-Roley sample period is extended to
period 1979- 1982, when the Federal Reserve targeted the monetary base and th
of short rates increased sharply, then 6, is much less significant. Also et is f
statistically insignificant if the ratio of the flow of short debt to long debt
equation (19.9) (see Shiller et a1 (1983), Table 4) which suggests some form
segmentation rather than ˜risk™. Hence, the excess HPY on six-month over t
bills does not appear to be a stable function of the ad-hoc risk premium 0, use
and Roley.

ARCH Model
The problem with the above studies is that the risk premium is rather ad hoc
given by the conditional variance of forecast errors. In a pioneering study
(1987) utilise the ARCH approach to model time varying risk in the bill m
expected excess yield E,y,+l of long bills over short bills is assumed to be
var(y,+llQ,) in (19.10) may be replaced by:



Hence not surprisingly the conditional variance of the excess yield is the s
conditional variance of the rational expectations forecast error. Thus (19.1
rewritten:
+ +
Yf+l = B Wfa,2,,) E f + l
Equation (19.12) has the intuitive interpretation that the larger the variance of
errors, the larger the ˜reward™ in terms of the excess yield that agents requi
willingly to hold long bills rather than short bills. Thus in periods of turbulenc
is high) investors require a higher expected excess yield and vice versa.
Equation (19.12) may be viewed as being derived from an ˜inverted™ me
model of asset demands, where there is only one risky asset. The demand f
the risky ˜six-month™ asset is:



where c is a coefficient of risk aversion. If asset supplies ˜A™ are fixed and
constant (or slowly varying) then equilibrium gives
2
= AcErq+,
&Yf+l

which is similar to (19.12) with /?= 0 and 6 = Ac. In this model there i
risky asset, namely the return over three months on the six-month bill.
equation (19.12) may also be viewed as a very simple form of the CAPM w
is measured by the conditional variance of the single risky asset. The corre
between the CAPM and the mean-variance model of asset demands has bee
previous chapters.
To make (19.12) operational we require a model of the time varying varia
et a1 assume a simple ARCH model in which of+l depends on past (square
errors:
r4 1

L i=O J
where the W i are declining (arithmetic) weights set at wi = (4/10,3/10,. . .,
Cwj = 1, and a and a1 are to be estimated. We can ˜get a handle™ on the sign
0
on a by assuming past forecast errors E;-; have been constant (= o2 say
1
circumstances we would envisage agents expecting next period™s forecast erro
be equal to this constant value 0 2 .For the latter to hold in (19.15) we requi
a = 1. Equation (19.12) given (19.15) is often referred to as an ˜ARCH in me
1
A representative result from Engle et a1 (1987) using quarterly data, 1959-19
+ 1.64
g? , = 0.0023
,+
(1.0) (6.3)
The excess yield responds positively to the expected variance of forecast
0.687) and the variance of forecast errors depends on past forecast errors a
coefficient a is somewhat greater than unity. Engle et a1 then include the y
1
(R, - r,) in equation (19.16a) since in earlier studies this had been found to
+
the excess yield at t 1. The idea here is that the yield spread might no
yt+l once allowance is made for time varying risk and hence the RE assump
then not be violated. Unfortunately, both o+ and (R - r), are statistically si
:l
explaining yf+l and therefore the RE information efficiency assumption is s
in this model.
The results of Engle et a1 appear to demonstrate strong effects of the
variance on equilibrium returns. Tzavalis and Wickens (1993) demonstrate tha
is sensitive to the data period chosen and in particular whether the period
volatility in interest rates in 1979-1982, when monetary base targeting was in o
included. Broadly speaking Tzavalis and Wickens (using monthly data) reprodu
al™s results using a GARCH(1,l) model but then include a dummy variable DV
value unity over the months 1979(10)-1982(9) and zero elsewhere. They fin
+
DV, is included, the degree of persistence in volatility falls, that is a1 a2 in t
process (17.13) is of the order of 0.3 rather than 0.9 and the dummy variab
significant and positive. In addition, the expected HPY is no longer influen
conditional variance, that is we do not reject the null that 6 = 0 in equation (
dummy variable merely increases the average level of volatility in the 1979-1
and the reasons for such an exogenous shift has no basis in economic theory
based on intuitive economic arguments one might still favour the model
dummy variable and take the view that persistence is high when volatility is h
there may be a threshold effect. In periods of high volatility, volatility is highl
and influences expected equilibrium returns. In contrast, in periods of low
persistence is much lower and the relatively low value of the conditional va
not have a perceptible impact on equilibrium returns. Intuitively the above seem
(cf. the effect of inflation on money demand or consumption in periods of hi
inflation) but clearly this non-linear effect requires further investigation.


1 . TIME VARYING RISK: LONG-TERM BOND
92
This section begins with the study by Mankiw (1986) who tries to explain the
of the excess HPY on long bonds in terms of a time varying term premium
examines the behaviour of the CAPM with constant betas as a benchmark for
which allow betas to be time varying.
Under the EH with constant term premium the excess one-period holding p
(H, - r,) on a bond of any maturity should be independent of information
A number of researchers have found that this hypothesis is resoundingly re
(3.59)
(2.27) (3.04)
1.51
Yield Spread 4.99 3.40
(1.40)
(1.58) (1.62)
Summary Statistics
R2 0.002
0.086 0.034
2.23
Durbin Watson 2.17 1.96
23.9 28.1
Standard Error of Estimate 20.4
˜ ˜ ˜ ˜˜ ˜˜




Data from OECD, Main Economic Indicators, various issues.
(a) The dependent variable is the excess holding period return between long and short bonds, H
yield spread is defined as R, - rf, where RI is the long rate and rf is the short rate. Standar
parentheses.
Source: Mankiw (1986).


number of different countries. For example, Mankiw (1986) finds that for th
UK, Canada and Germany (Table 19.1), the excess HPY on long bonds dep
yield spread. When the data for all the countries are pooled he obtains:
- rjt) = -3.28 + 2.04 (Ri, - rjt)
(Hjr+l
(2.01) (0.66)
+
-
where ( . ) = standard error and H , + l = (P,+1 P, C ) / P , is the one-peri
period yield on the long-term bond. He conjectures, as many have done, that
( R, - r,) could be a proxy for a time varying (linear additive) term premium



and 8, depends on (R - T ) ˜ .If one had a direct measure of the risk premi
one could include this in (19.17) along with the spread and if 8, is correctly
one would expect the coefficient on (R - r), to be zero. However, Mankiw
if 8r is subject to measurement error this would bias the coefficient estimate
regression. He decides instead to investigate the regression

+ s l (- r), + v,
˜
8, = so

using alternative variables to measure risk. Since 8, appears on the LHS of
parameters 6i are not subject to measurement error. If 61 # 0, the argument wo
(R - r), influences (H- r), in (19.17) because (R - r), is correlated with th
dently measured risk premium 8,. As alternative measures of 8, Mankiw (1986)
absolute value of the percentage change in the price of the long bond, (ii) the
between the HPY and the growth of consumption as suggested by the consumpt
(see Chapter 4) and (iii) the covariance between the HPY and the excess re
market portfolio of stocks. In (iii) the tentative view is that the excess return o
market might be a good proxy for consumption growth for investors who a
hold bonds (i.e. predominantly the wealthy). Hence (iii) can also be loosely
in terms of the consumption CAPM.
where g;+l = C,+l/C, and C , = real consumption. Mankiw therefore assum
(conditional) expected values for consumption growth and holding period
constant and equal to their sample values.
Unfortunately for all four countries studied, Mankiw finds that for risk m
and (iii) the coefficient 61 in (19.19) is statistically insignificantly different fro
risk measure '(ii)' the null that 61 = 0 is not rejected for three out of four co
in the one case where 61 is significant it has the wrong sign (i.e. it is negat
Mankiw finds that these proxy variables for a time varying term premium do
the expectations hypothesis.
Bisignano (1987) examines whether the zero-beta CAPM with the addition
term in consumption (i.e. consumption CAPM) can provide an acceptable statis
nation of holding period yields. Here, excess HPYs are time varying bec
variation in 'market returns' and consumption covariability: however, the beta
with term to maturity. The equation to be estimated is:



where RZ = holding period return on a zero-beta portfolio of bonds and Rm
period return on a market portfolio of bonds. The two hypotheses to be teste
HA : p(")# 0, = 0 (zero-beta CAPM)
U(")

H i : p(") = 0 # 0 (consumption CAPM)
U(")


The market return R:+, is measured by the Saloman Brothers world bond por
and the zero-beta portfolio comprises a portfolio of short-term assets (such tha
lation between Rm and RZ is zero). From equation (19.21), we see that if we
risk premium 6, as E , ( H ( " )- Rz)r+l then


For Germany Bisignano (1987, Table 27) reports the following results for fi
10-year bonds:
+
= 0.74R;+, 0.26Rz1 - 125.6ccI:)1
(10.7) (3.8) (2.3)
78(2)-85(11), R2 = 0.182, DW = 1.78, ( . ) = t statistic
+
= 0.64R:+, 0.367Rz1 - 246.5CCI:q'
H/
I:
(5.75) (3.2) (2.1 )
78(2)-85(11), R2 = 0.133, DW = 2.2, ( . ) = t statistic
There is therefore some support for this 'two-factor model', namely that both
return R?+, and the consumption covariability term CC,+1 help in explain
holding period yields. However, the consumption term is relatively less well
is statistically significant and rises with term to maturity n. For example, fo
p(') = 0.21 (t = 4.2) and p(*O) = 1.28 ( t = 5.4) (Table 23) indicating greater
risk for long bonds than for short bonds. Thus there is some support here for th
CAPM as an explanation of time varying excess HPYs on bonds even whe
are assumed to be time invariant. Below Bisignano's model is extended by a
conditional covariance between Hi") and RY, that is p("), to vary not only w
maturity but also over time.

ARCH and GARCH Models
In the CAPM risk is measured either by the asset's beta or by its covarianc
market portfolio. ARCH or GARCH processes can be used to model time v
premia. After setting out the model we look at some recent illustrative empir
using this approach. We shall see that implementation of ARCH models ofte
estimating a large number of parameters and in fact the estimation procedure is
non-linear which can create additional (convergence) problems. Because of the
difficulties particularly when working with a finite data set, we shall see that
have simplified the estimation problem in various ways. First, there is us
limitation on the number of asset returns considered (e.g. just returns on dom
rather than domestic and foreign bonds) and second, the parameters of the GAR

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