ñòð. 12 
var(v:+,) = elâ€™gel
=A â€™ W
cov(rl:+, t)f+l)
7
var(u,+l) = elâ€˜AWAâ€™el
P h = (Aâ€™WA)/(elâ€™AWAel)
Once the â€˜Aâ€™ parameters of the VAR and the covariance matrix W have been estimated
variances and covariances can easily be calculated. One can use OLS to estimate e
in the VAR individually, but Campbell suggests the use of the Generalised Method
(GMM) estimator due to Hansen (1982)to correct for any heteroscedasticity that ma
in the error terms. The GMM point estimates of parameters are identical to the ones
OLS, although the GMM variancecovariance matrix of all the parameters in the m
â€˜correctedâ€™ for the presence of heteroscedasticity (White, 1984).
The standard errors of the variance statistics in (43)(48) can be calculated as foll
the vector of all parameters in the model by 8 (comprising the nonredundant elements
and the heteroscedasticity adjusted variancecovariance matrix of the estimate of thes
by v. Suppose, for example, we are interested in calculating the standard error of Ph
a nonlinear function of 8 its variance can be calculated as:
The derivatives of P h with respect to the parameters 8 can be calculated numerically. T
error of P h is then the square root of var(Ph).
ENDNOTES
1. Note the change in notation in this chapter: 6, is not the discount fac
used in earlier chapters.
2. The usual convention of dating the price variables P , as the price at the
period is followed. In Campbell and Shiller (1989) and Shiller and Belt
price variables are dated at the beginning of period, hence equation (16
6, = d,l  p r but Campbell (1991) for example, uses â€˜end of periodâ€™ v
3. Here r, represents any economic variables that are thought to influence th
oneperiod return. In some models r, is the nominal riskfree rate wh
CAPM, for example, r, would represent a conditional variance. Note tha
chapters k, was used in place of r,.
4. As noted in Chapter 15 the Wald statistic is not invariant to the form o
linear) restriction even though they may be algebraically equivalent.
5. In matrix form the restriction may be expressed as follows usin
and (16.17b):
6, = elâ€˜z,
E,(rd,+l)= e2â€˜Az,
E,(k+l  G + l ) = E,[&  P&+1  4 +ll
which is independent of zf only if the term in square brackets is zero.
easily seen to be given by equation (16.20).
6. Clearly (16.26) can also be obtained from (16.8) and then using p: = d
7. Since a VAR of any order can be written as a firstorder system (the
form) the analysis of the 2 x 2 case is not unduly restrictive.
8. Equation (16.45) arises by successive substitution. For example
+
h2t = hlf &+l
which using (16.4) gives
One can see that the intermediate values of 6, in this case &,+I, do not ap
only 6, and &+i appear in the expression for hi,.
9. The algebra goes through for any model of expected returns (e.g. when
are constant or depend on consumption growth).
10. Equation (16.48) collapses to the infinite horizon RVF (16.12) as i goe
FURTHER READING
The VAR methodology is relatively recent and hence the only major source
material is to be found in Shiller (1989) in Sections I1 and I11 on the stoc
markets. Mills (1993) also provides some examples from the finance literat
articles employing this methodology are numerous and include Cuthbertson (1
bertson et a1 (1996) on UK and German shortterm rates, and Engsted (1993
short rates. Recent examples of the cointegration approach for billsbonds are
et a1 (1996) and Engsted and Tanggaard (1994a,b).
Campbell and Mei (1993), Campbell and Ammer (1993) and Cuthbertson
extend the Campbell (1991) variance decomposition approach to disaggre
returns and macroeconomic factors.
PART 6
I
I
I 2
Time Varying Risk Premia
One of the recent growth areas in empirical research on asset prices has
modelling of time varying risk premia. To an outside observer it may seem
financial economists have only recently focused on the most obvious attribute
stocks and longterm bonds, namely that they are risky and that perceived
likely to vary substantially over different historical periods. As we have see
chapters the consumption CAPM provides a model with a time varying ris
but unfortunately this model does not appear adequately to characterise the d
returns and asset prices. In part, the reason for the delay in economic models â€˜c
with the perfectly acceptable intuitive idea of a time varying risk premium w
of appropriate statistical tools. The recent arrival of socalled ARCH models
models in which the risk premium depends on time varying variances and co
be explored more fully. As we saw in Chapter 3, the basic CAPM plus an
that agentsâ€™ perceptions of future riskiness is persistent results in equilibr
being variable and in part predictable. With the aid of ARCH models, the va
CAPM can be examined under the assumption that equilibrium returns for
stocks depend on a time varying risk premium determined by conditional va
covariances.
Chapter 17 is concerned primarily with testing the oneperiod CAPM mod
returns, and will look at how persistence in the risk premium can, in princi
the large swings in stock prices which are observed in the data. However, e
the degree of persistence in the risk premium may be sensitive to the inclusi
economic variables in the equation for stock returns, such as the dividend
the riskfree interest rate and the volume of trading in the market. In earl
we also noted that the presence of noise traders may also influence stock retu
Chapter 17 examines how robust is the relationship between expected return
varying variances, when additional variables are included in the returns equa
Chapter 18 begins by noting the rather close similarities between the me
model of asset demands encountered in Chapter 3 and the oneperiod CAPM.
together a strand in the monetary economics literature, namely the meanvari
with the CAPM model which is usually found in the finance literature. W
explore how the basic CAPM can be reinterpreted to yield the result that
returns depend on a (weighted) function of variances, covariances and asset sh
to that in Chapter 17 which uses the standard form of the CAPM.
Chapter 19 examines the validity of the basic CAPM applied to the bond m
particular the determination of the oneperiod holding period yield on bills (z
bonds) and longterm bonds using ARCH models to examine the role of ti
risk premia.
The reader will have noted that we do not proposed to analyse explicitly t
impact of time varying risk premia in the FOREX market, in particular on th
This is because in this strand of the literature foreign assets are treated as
general portfolio choice problem. The return to holding foreign assets equals t
the local currency plus the expected change in the exchange rate. The change
rate is therefore subsumed in the â€˜returnsâ€™ variables. Similarly, the (conditiona
and covariance of the exchange rate are subsumed in those for the returns. In
international CAPM implicitly models the expected change in the exchange r
(time varying) covariances associated with it. Of course data availability on
to various types of foreign asset may limit the scope of the analysis.
L Risk Premia: The Stock Marke
This chapter begins with a summary of the empirical analysis undertaken
(1989) who looked at possible sources for the time varying volatility found
stock returns. He examined how far the conditional volatility in stock returns
its own past volatility and also on the volatility in other economic variables (fun
such as bond volatility and the volatility in real output. The remainder of thi
concerned with the measurement and influence of risk premia on stock return
prices. If perceptions of risk are persistent then an increase in risk today w
perceptions of risk in many future periods. The discount factors in the rationa
formula (RVF) for stock prices depend on the risk premium. Hence if risk is
a small increase in perceived risk might cause a large fall in stock prices. T
the basic intuition behind the PoterbaSummers (1988) model to explain the
stock prices. The Poterba Summers model is discussed under various assump
the precise form one might assume for the time varying risk premium.
For the market portfolio, the CAPM indicates that risk is proportional to the
variance of forecast errors, but the model gives no indication of how â€˜riskâ€™ m
over time. ARCH and GARCH models assume that a good statistical repres
movements in risk is that â€˜risk tomorrowâ€™ is some weighted average of â€˜risk
periods. The CAPM plus any ARCH models provide an explicit model for th
return on stocks which depends on a time varying risk premium. It seems re
ask whether this â€˜joint modelâ€™ is sufficient to explain stock returns or whethe
variables (e.g. dividend price ratio) remain a statistically important determinan
The model of Attanasio and Wadhwani is discussed, which addresses this mo
test of the EMH, together with how the â€˜smart money plus noisetraderâ€™ theore
of De Long et a1 (1990) may be implemented and this also provides a han
behaviour of the serial correlation in returns found by Poterba and Summers
others in the earlier work described in Chapter 6. Analysis of the above mo
us to present in a fairly intuitive way a variety of ARCH models of conditiona
To summarise, the key aims in this chapter are:
to examine the economic variables that might influence changes in st
0
volatility over time
to measure the degree of persistence in the risk premium on stock retu
0
impact on changes in stock prices
to ascertain the importance of time varying risk premia in determining sto
0
I
[g
P =Er
r Yr+jDt+j
j= 1
where rr = riskfree rate, r p , = risk premium. Stock price volatility therefore
the volatility in future dividends and discount rate (and any covariance betw
The return R,+l to holding stocks depends on future price changes and hence th
of returns depends on the same factors as for stock prices. Expected future div
in principle depend on many economic variables, indeed on any variables tha
the future profitability of companies (e.g. inflation, output growth). The di
depends on the riskfree rate of return r, and on changing perceptions of the
stocks, rp,. Schwert (1989) does not ask what causes volatility in stock return
to establish on a purely empirical basis what economic variables are correlate
volatility in returns. He is also interested in whether volatility in stock return
other economic variables. It may be the case for instance that changes in s
volatility lead to changes in the volatility of fixed investment and output. If
deemed to be undesirable, one might then wish to seek ways to curb stock pric
Schwert examines conditional volatilities, that is the volatility in stock retu
tional on having obtained the best forecast possible for stock returns. If the b
=  E l R r + l ) is the
for stock returns is denoted E(R,+1IQr) then
forecast error. If ErE,+l = 0 then the conditional variance of the forecast erro
is var(e,+llQ2,)= E,(R,+l  E,R,+1)2. obtain a measure of var(E,+I) or
To
error we need to model the â€˜bestâ€™ forecasting scheme for R , + l .
Schwert uses a fairly conventional approach to measuring conditional vo
assumes that the best forecast of monthly stock returns R,+l is provided by an
(we exclude monthly dummies):
j=O
Schwert finds that the (absolute values) of the residuals &+I from (17.2) ex
correlation. Hence there is some predictability in itself, which he mod
further autoregression:
j= 1
As we shall see in Section (17.2,2), equation (17.3) is a form of autoregressive
heteroscedasticity (ARCH) in the forecast errors. From the ARCH regression
estimates of the pj. The predictions from (17.3), that is
S
j=l
rates, the growth in industrial production (output) and monetary growth.
The sum of the pj in (17.3) is a measure of persistence in volatility. It
of (the sum of the) pj that is important and not the number of lagged value
determining the degree of persistence. To see this let s = 1 and let p1 = p. B
substitution in (17.3) we have (ignore the fact that E is an absolute value):
Thus, given a starting point at time t (i.e. e t ) then Et+n is determined by
(moving) average of the white noise error terms ut+j. If p is close to unity
then a positive â€˜shockâ€™ at time t (i.e. ut > 0) will still have a strong effect o
after n periods, since pnâ€™ is still fairly large. For example, for p = 0.95 a u
time t (i.e. ut = 1) has an impact on ˜ ˜ + 1 2 0.54 after 12 months and an imp
of
of 0.3 after 24 months. However, if p = 0.5 the impact of ut on ˜ ˜ + 1 2very is
A high value of p therefore implies a â€˜long memoryâ€™. For p close to unity, sho
t continue to influence Et+n in many future periods and even though these
+ +
outâ€™ (i.e. the effect on E at t j is greater than at t j  l),nevertheless they
many periods and â€˜die outâ€™ extremely slowly. Since Et+n is a long moving av
errors U t  j (if p is close to 1) then even though each ut+j is white noise, never
will exhibit long swings. Once E˜ becomes â€˜largeâ€™ (small) it tends to stay la
for some considerable time.
All the series examined by Schwert for the US are found to exhibit pe
volatility over the period 18591987 with each Zpj in the region of 0.80.
in principle, the persistence in stock return volatility is mirrored by persist
volatility of the fundamentals.
We now turn to the possible relationship between the conditional volatili
prices Et+l and the conditional volatility of the economic fundamentals &it
conditional volatility of output, bond rates, inflation, etc. Schwert runs a re
stock return volatility li?tl on its own lagged values and also on lagged values
for the fundamental variables. Any â€˜reverse influenceâ€™, that is from stock v
volatility in fundamentals such as output, can be obtained from an equation
volatility as the dependent variable. In fact Schwert generally estimates the s
volatility equation together with the â€˜reverse regressionsâ€™ in a vector autoregres
system.
Schwertâ€™s results are mixed. He finds little evidence that volatility in econo
mentals (e.g. output, inflation) has a discernible influence on stock return vo
the impact is not stable over time). However, there is a statistically significant
interest rate and corporate bond rate volatility on stock volatility. Also some
volatilityâ€™ economic variables do influence the monthly conditional stock retur
These include the debt equity ratio (leverage) which has a positive impact, as d
ARCH model for volatility.) Stock volatility is also shown to be higher du
sions than in economic booms. Examination of the results from the â€˜reverse
reveals that there is some weak evidence that volatility in stock returns has
explanatory power for the volatility in output.
It must be said that much of the movement in stock return volatility in
study is not explained by the fundamental economic variables examined. Th
values in Schwertâ€™s report regressions are usually in the region of 0.00.3. H
of the monthly conditional volatility in the forecast errors of stock return
â€˜unobservablesâ€™. It is possible that the presence of â€˜fadsâ€™ due to the actio
traders in the market may be associated with these unmeasurable elements of
volatility. Although stock return volatility cannot be adequately explained by
the economic fundamentals considered by Schwert, this in itself does not throw
on the relative importance of the smart money versus noisetrader view of t
of the stock market.
17.2 THE IMPACT OF RISK ON STOCK RETURN
We have seen in the previous section that the variance of the forecast erro
is highly persistent and hence predictable. Future values of the variance
errors depend on the known current variance. If the perceived risk premium
is adequately measured by the conditional variance, then it follows that the
premium is, in part, predictable. An increase in variance will increase the per
iness of stocks in all future periods. Since the risk premium is an element of t
factor, which determines the stock price (i.e. DPV of future dividends), then
in the risk premium could lead to a change in the discount factors for all fu
and hence a large change in the level of stock prices.
17.2.1 The PoterbaSummers Model
Poterba and Summers (1988) investigate whether changes in investorsâ€™ per
risk are large enough to account for the very sharp movements in stock pric
actually observed. First, a somewhat stylised account is presented of the centr
of Poterba and Summers. The market portfolio of stocks is taken to be the
market index. Stock prices may vary if forecasts of expected dividends are r
the discount factor changes: Poterba and Summers concentrate on the behav
latter. The discount factor 8, can be considered to be made up of a riskfree
++
risk premium, that is & = 1/(1 rr r p r )and the stock price is determined b
equation (17.1):
Poterba and Summers argue that the growth in future dividends is fairly pred
they concentrate instead on the volatility and persistence of the risk prem
investors thought that the risk premium would increase tomorrow, then this w
all the discount factors for all future periods. If only the next periods risk pre
increases, this effects only the Sr+1 terms on the right hand side of equation
not the other discount factors 61+2, St+3, etc. However, now consider the case
is persistence in the risk premium. This means that should investors believe
premium will increase tomorrow, then they would also expect it to increase in
periods. Hence a change in risk premium today has a large effect on stock pr
rpr increases then 6t+l, 6r+2, 6r+3, etc. are all expected to increase.
The key element in Poterba and Summersâ€™ view of why stock prices are hig
is that there can be â€˜shocksâ€™ to the economy in the current period whic
perceived risk premium for many future periods. Thus the rational valuati
for stock prices with a time varying and persistent risk premium may expla
observed movements in stock prices. If this model can explain the actual mo
stock prices, then as it is based on rational behaviour, stock prices cannot be
volatile.
Poterba and Summers use a linearised approximation to the RVF and th
to show exactly how the stock price responds to a surprise increase in th
riskiness of stocks. The response depends crucially on the degree of persistenc
by agentsâ€™ perceptions of changes in the conditional volatility of the forec
stock returns. If, on average, the degree of persistence in volatility is 0.5, then
in volatility of 10 percent is followed (on average) in subsequent periods b
increases of 5, 2.5, 1.25, 0.06, 0.03, etc. percent. Under the latter circumstan
and Summers calculate that a 1 percent increase in volatility leads to only a
fall in stock prices. However, if the degree of persistence is 0.99 then the
would fall by over 38 percent for every 1 percent increase in volatility: a size
Given that stock prices often do undergo sharp changes over a very short peri
then for the PoterbaSummers model to â€˜fit the factsâ€™ we need to find a hig
persistence in agentsâ€™ perceptions of volatility.
Poterba Summers: Empirical Results
Poterba and Summers treat all variables in the RVF in real terms and con
the real stock price of the market portfolio which uses the S&P index over
19281984. As Poterba and Summers wish to focus on the impact of the ris
on P, they assume
(i) real dividends grow at a constant rate g, so that the â€˜unobservableâ€™ Et(D
+
to (1 g)â€™Dt and
(ii) the real riskfree rate is constant (rt = r).
The CAPM (Merton, 1973 and 1980) suggests that the risk premium on
portfolio is proportional to the conditional variance of forecast errors on eq
Eta:+l
consumersâ€™ relative risk aversion parameters which are assumed to be consta
and Summers assume (and later verify empirically) that volatility can be rep
ARMA model(s) and here we assume an AR(1) process:
+ slat2 + V t + l
2
O>ai>l
a,+, =
where v,+l is a white noise error. The latter provides the mechanism by whi
(randomly) switch from a period of high volatility to one of low volatility a
from positive to negative. If a1 is small, the degree of persistence in volatili
For example, if a = 0, 0: is a constant (= ao) plus a zero mean random e
1
hence exhibits no persistence. Conversely if a1 is close to 1, for example a1
if vt increases by 10, this results in a sequence of future values of a2 of 9, 8
in future periods and hence a is highly persistent. If a follows an AR(1) p
: :
so will rp,:
+
rpt = Aao Aa1rpfi
To make the problem tractable Poterba and Summers linearise (17.1) aroun
value of the risk premium p . This linearisation allows one to calculate the
response of Pt to the percentage change in a:
;
Thus the response of Pt to a change in volatility a increases with the degree of
:
al. Poterba and Summers compute an unconditional volatility measure for t
of monthly stock returns based on the average daily change in the S&P comp
over a particular month. Hence for month t:
m
1sE./m
i=
:
i= 1
where Sri = daily change in the stock index in month t and m = number of t
in the month. They then investigate the persistence in ; using a number of
?
t
ARMA models (under the assumption that 5; may be either stationary or non
For example, for the AR(1) model they obtain a value of a1 in the range 0.6
also use estimates of a2 implied by option prices which give estimates of
forwardlooking volatilities. Here they also find that there is little persistence in
The values of the remaining variables are
(i) 7 = average real return on Treasury bills = 0.4 percent per annum = 0.
per month,
(ii) g = average growth rate of real dividends = 0.01 percent per annum
percent per month,
(iii) â€˜p = mean risk premium = average value of (ERm  r ) = 0.006 per mo
implies a value of A = rp/a2 of 3 5 .
.)
movements in stock prices.
17.2.2 Volatility and ARCH Models
Chou (1988) notes that there are some problems with the PoterbaSumm
ical estimation of the time varying variance 0 He notes that the Poterb
.
:
measure of the variance remains constant within a given horizon (i.e. a mo
then assumed to vary over longer horizons (i.e. the AR model of equation
It
02). is therefore not a correct measure of the conditional variance. Chou
PoterbaSummers analysis using an explicit model of conditional variances
GARCH(1,l) model which is explained briefly below and more fully in Chap
also that the Poterba Summers estimation technique being a twostep proce
inconsistent estimates of the parameters (see Pagan and Ullah (1988)). Chou, a
and Summers, assumes expected returns on the market portfolio are given by
(plus RE):
(&+1  rr) = h[E,o;+J+ E f + l
where R,+l = return on the market portfolio, r, = riskfree rate. Taking exp
(17.9) it is easy to see that the best forecast of the excess return depends
forecast of the conditional variance
The conditional forecast error is therefore
 ErRr+l = &f+l
&+l
and the conditional variance of the forecast error is

We assume Er+1 N ( 0 , U:+,) and hence has a time varying variance. Accor
CAPM the expected excess return varies directly with the time varying variance
errors: large forecast errors (i.e. more risk) require compensation in the form
expected returns. It remains to describe the time path of the conditional vari
+
assumes a GARCH(1,l) model in which the conditional variance at t 1 is
average of last periods conditional variance o and the forecast error (square
:
The GARCH(1,l) model is a form of adaptive expectations in the second mo
distribution. The best guess of o + at time t (i.e. Erof+l)is given by the RHS
:,
+
The expected value of the variance for time t 1 is:
The ai are constrained to be nonnegative so that the conditional variance is a
+
negative. If (a1 a2) = 1 then a change in the current variance of has a
+
effect on all future expectations. If a1 a2 < 1 then the influence of o on :
+
away exponentially. Thus (a1 a2) measures the degree of persistence in the
+
+
1 012 2 1 then the unconditional variance ao/[l  (a1 ay^)] is
variance. If a
and we have a nonstationary (explosive) series in the conditional variance.
(17.9) + (17.13) taken together are often referred to as a 'GARCH in mean' or
model. Chou (1988) estimates these two equations simultaneously, using the
likelihood method. His data is for weekly returns (TuesdayTuesday closing pr
NYSE value weighted stock price index (with dividends assumed to be reinv
the period 1962 1985 (1225 observations). The price index and weekly returns
in Figures 17.1 and 17.2. The crashes of 1974 and 1982 are clearly visible in
and periods of tranquillity and turbulence in returns are noticeable in Figure
Chou finds that the estimate of the market price of risk (or index of r
aversion) A over various subperiods is not well determined statistically and
being statistically insignificant. However, it has plausible point estimates in the
(Poterba and Summers obtain a value of 3.5 and Merton (1973) finds a value o
+
value of a 1 a is very stable over subperiods and is around 0.98 indicating
2
1000
l2O01
J
800
400
200
of I I
I I I
1 I
1
1962 1965 1968 1971 1974 1977 1980 1983 1985
Figure 17.1 NYSE: Stock Index. Source: Chou (1988). Reproduced by permission o
and Sons Ltd
Figure 17.2 NYSE: Weekly Stock Returns. Source: Chou (1988). Reproduced by p
John Wiley and Sons Ltd
Figure 17.3 NYSE: Variance of Stock Returns. Source: Chou (1988). Reproduced by
of John Wiley and Sons Ltd
persistence in the conditional variance (Figure 17.3). It follows from our previ
sion that observed sharp falls in stock prices can now be explained using the R
+
PoterbaSummers framework. Indeed, when a1 a2 = 1 (which is found to
acceptable on statistical grounds by Chou) then stock prices move tremendou
elasticity d(ln P,)/d(ln a;) can be as high as (minus) 60.
+ +
values of N . He then estimates o&+l= a a1q& Vr+l for these various v
0
He finds that a1,the degree of persistence, varies tremendously increasing from
for N = 5 (working) days to a = 0.6 for N = 20 days (i.e. one month) and t
1
for N = 250 days (i.e. one year). This suggests that the PoterbaSummers m
not have correctly captured the true degree of persistence. Note, however, t
experiment should really have used a closer measure of o2 to that used by P
Summers, namely
U , = I=(P N ) 2 / N
R
;
For daily data over one month as used by Poterba and Summers, E will be cl
but over longer horizons (i.e. N increases), is likely to be nonzero because
bear markets. Hence the above measure is more representative of the Poterba
approach and may not yield such sensitivity in the estimate of a1, as found
experiment. However, Chouâ€™s use of a GARCH model is preferable on U prio
as it correctly estimates a measure of conditional variance.
As a counterweight to the above result by Chou consider a slight mod
Chouâ€™s model as used by Lamoureux and Lastrapes (1990). They assume that
volatility is influenced both by past forecast errors (GARCH) and by the volum
(VOL) (i.e. number of buyhell orders undertaken during the day):
They model dairy returns (price changes) and hence feel it is realistic to assum
expected returns p :
+
They estimate (17.17) (17.18) for 20 actively traded companies using abo
of daily data (for 1981 or 1982). When y is set to zero they generally find a si
+
to Chou, namely strong, persistent GARCH effects (i.e. a 0.80.95).
1a 2
the residuals Et+l are nonnormal and hence strictly these results are statistic
given the assumption (17.19). When VOL, is added they find that a = a2 = 0
1
and the residuals are now normally distributed. Hence conditional volatility
determined by past forecast errors but by the volume of trading (i.e. the pe
VOL accounts for the persistence in okl). They interpret VOL as measuring
of new information and therefore conjecture that in general GARCH effec
studies are really measuring the persistence in the arrival of new informatio
this data set, the Chou model is shown to be very sensitive to the specificat
of introducing VOLr into the GARCH process. Note that volatility (risk) i
varying but its degree of persistence is determined by the persistence in tradi
There are some caveats to add to the results of Lamoureux and Lastrapes. In
as their model uses data on individual firms a correct formulation of the CA
include the covariance between asset i and the market return. This makes th
17.2.3 CAPM, Noise lkaders and Volatility
The empirical analysis above has highlighted the potential sensitivity of
equations to assumptions about the equilibrium model determining returns
precise parameterisation of any time varying conditional variances. Below we
explore the ability of the CAPM to explain equilibrium asset returns but use
specifications of the GARCH process. We then examine a model in which n
have an influence on equilibrium returns.
The CAPM and Dividends
The study by Attanasio and Wadhwani (1990) starts with the empirical obser
from previous work, it is known that the expected excess return on an aggr
market index (which is assumed to proxy the market portfolio) depends on t
periodâ€™s dividend price ratio. The latter violates the EMH under constant expe
returns. Previous work in this area often assumed a constant risk premiu
sometimes interpreted the presence of the dividend yield as indicative of a ti
risk premium. Attanasio and Wadhwani suggest that if we explicitly mod
varying risk premium then we may find that the dividend yield (= Z,) does n
expected returns. If so, this would support the CAPM version of the EMH. If
the CAPM is the correct equilibrium pricing model:
then we expect h > 0 and 6 = 0. The time varying conditional variance of
return is given by the variance of E r + l which is assumed to be determined by th
GARCH( 1,2) model (with the dividend yield added)
+ W O ; + a24 + + nz,
2
a, = a0
,,
where ai and TC are constrained to be nonnegative. The GARCH model imp
+
expected variance for period t 1, that is E,o:+l, depends on a weighted ave
periods of the variance 0 and the two most recent forecast errors squared (
:
addition, equation (17.21) explicitly allows the dividend yield to affect the
variance  this is not a violation of the CAPM and the EMH. Using monthly d
1953November 1988 on an aggregate US stock price series, a representati
(Attanasio and Wahwani 1990, Table 2, page 10):
+ 0.552,  4.05rf + 22.30:+,
[ R c l  Y , ] =  0.035
(0.025) (0.39) (1.05) (11.3)
53(1)88(11),R2 = 0.059, (.) = standard error
+ 1.5(102)˜f+ 2.2(102)˜;1 + 0.870; + 5.31022
= ao
2
Or+,
(2.9. 102) (4.0.102) (0.06) (2.4. 1Ov2)
equilibrium returns (under the assumption of RE).
In the GARCH equation (17.23), the dividend yield Z , has a statistically
effect on the conditional variance and this would explain why previous resea
assumed a constant risk premium found Z, significant in the CAPM retur
(17.22). Note that there is also considerable persistence in the conditional var
++ a = 0.91.
3
a1
y
Noise Traders, Risk and Serial Correlation in Stock Returns
Noise traders are now introduced into the market who (by definition) do no
asset decisions on fundamental value. Positive feedback traders buy after a pr
sell after a price fall (e.g. use of â€˜stoplossâ€™ orders or â€˜portfolio insurersâ€™). Th
to positive serial correlation in returns, since price rises are followed by an
demand and further price rises next period. Negative feedback traders pursue t
strategy, they â€˜buy lowâ€™ and â€˜sell highâ€™. Hence a price fall would be followed
rise if these traders dominated the market. (The latter would also be true fo
who assign a constant share of market value wealth to each asset, since a p
asset i will lead to a fall in its â€˜value shareâ€™ in the portfolio and hence lead to
purchases and a subsequent price rise.) The demand for stocks by noise trade
proportion of the total market value of stocks) may be represented
N, = YR,1
with y > 0, indicating positive feedback traders and y < 0,indicating negativ
traders and R,1 is the holding period return in the previous period. Let us
the demand for shares by the smart money is determined by a (simple) me
model (Section 3.2).
s, = [Et& a]/@,
where S , = proportion of stock held by smart money, a = expected rate o
which demand by the smart money is zero, p, = measure of the perceived
shares. We assume p f is a positive function of the conditional variance 0; of
(i.e. p = ˜ ( 0 )Thus the smart money holds more stock the higher the expe
˜).
and the smaller the riskiness of stocks. If the smart money holds all the s
S, = 1 and rearranging (17.25) we have the CAPM for the market portfolio:
return (E,R,  a ) and depends on a risk premium, which is proportional to
tional variance of stock prices, p r = m2.Equilibrium in the market require
to be held:
S,+N,=l
Substituting (17.24) and (17.25) in (17.26) rearranging and using the RE
+
R, = E,R, E, we have:
+ 802 + ( ˜ + yi02)Rti +
o
R, = a Et
The direct impact of feedback traders at a constant level of risk is given by the
However, suppose yo is positive (i.e. positive serial correlation in R,) but y1
+
Then as risk 0; increases the coefficient on R,1, namely yo no:, could
and the serial correlation in stock returns would move from positive to n
risk increases. This would suggest that as volatility increases the market bec
dominated by positive feedback traders, who interact in the market with the sm
resulting in overall negative serial correlation in returns.
Sentana and Wadhwani (1992) estimate the above model using US
18551988, together with a complex GARCH model of the time varying
variance. Their GARCH model allows the number of nontrading days t
conditional variance (French and Roll, 1986) although in practice this is not
statistically significant. The conditional variance is found to be influenced d
by positive and negative forecast errors. Ceteris paribus, a unit negative fo
leads to a larger change in conditional variance than does a positive forecast
The switch point for the change from positive serial correlation in returns
serial correlation is q? > (yo/y1) and they find yo = 0.09, y1 = 0.01 and
point is a > 5.8. Hence when volatility is low stock returns at very sho
:
(i.e. daily) exhibit positive serial correlation but when volatility is high retu
negative autocorrelation. This model therefore provides some statistical supp
view that the relative influence of positive and negative feedback traders ma
the degree of risk but it doesnâ€™t explain why this might happen. As is becom
in such studies of aggregate stock price returns, Sentana and Wadhwani also f
conditional variance exhibits substantial persistence (with the sum of coeffici
GARCH parameters being close to unity). In the empirical results 8 is not
different from zero, so that the influence of volatility on the mean return on
works through the nonlinear interaction variable yla;Rtl. Thus the empirica
not in complete conformity with the theoretical model.
17.3 SUMMARY
In the past 10 years there has been substantial growth in the number of empir
examining the volatility of stock returns, particularly those which use ARCH an
processes to model conditional variances and covariances. Illustrative exam
work have been provided and in general terms the main conclusions are:
Only a small part of the conditional volatility in stock prices is explai
0
volatility in economic fundamentals or by other economic variables (such
of gearing).
There is considerable support for the view that the conditional variance of t
0
errors of stock returns are persistent although ARCH models being purely
not provide theoretical reasons why this is so.
market index are influenced by agentsâ€™ changing perceptions of risk. Hence
with time varying conditional variances would appear to be an improvem
assumption that equilibrium returns are constant. Nevertheless the CAPM
varying returns does not provide a complete explanation of equilibrium r
the volume of trading may influence expected returns).
It is possible that noise traders as well as smart money influence the expe
on an aggregate stock market index, even after making allowance for ti
risk premia.
7the CAPM 1
This chapter is concerned with two main topics: first, the relationship between
variance model of asset demands and the CAPM, and second, tests of the CAPM
a formulation based on asset shares. The strengths and weaknesses of the latt
are assessed in relation to the tests of the CAPM outlined in the previous c
focus of the empirical work is on testing a set of implied restrictions on the
of the CAPM and on the importance or otherwise of time varying risk premi
18.1 THE MEANVARIANCE MODEL
There is a strong tradition in monetary economics of optimising models wh
determine an investor's desired demand for individual assets in a portfolio. T
generalises on the results from the meanvariance model of asset demands
discussed in Section 3.2. The MVMAD (or one version of it) predicts tha
equilibrium demand for a single risky asset (as a share of total wealth) depen
of expected returns and the variances and covariances of the forecast errors
which represent the riskiness of these assets. Hence we have
= pl E (E,&+'  rr)
;
'
X:
where x; = (n x 1) vector of n risky asset shares, C, = (n x n ) matrix of for
of returns comprising the variancecovariance terms (aij), E,Rr+1  rr is
vector of expected excess returns, rr = riskfree rate and p = coefficient of r
+
aversion. The demand (share) of the (n l)th riskfree asset is derived from
constraint
n
i=l
The MVMAD (or rather one version of it) assumes that investors choose t
values of asset shares XT in order to maximise a function that depends on expe
risk and a measure of the individual's aversion to risk. The latter sounds very
the objective function in the CAPM. The obvious question is whether these
in the literature are interrelated. The answer is that they are but the corresp
not oneforone. The MVMAD is based on expected utility theory. One can a
The budget constraint for endofperiod wealth is:
The first term on the RHS of (18.4) is initial wealth, the second term is the retu
the portfolio multiplied by initial wealth and the final term is the receipts from
riskfree asset. If asset returns are normally distributed and we assume a consta
risk aversion utility function then maximising (18.3) subject to (18.4) reduces
where E&'+, = Cxj(ErRj,+l) = expected return on the portfolio, 0;= vari
portfolio and p = risk aversion parameter. There is much debate in the as
meanvariance literature (see, for example, Cuthbertson (1991b) and Courakis
an overview) about whether specific utility functions can give rise to asset d
desired asset shares xi*, which are independent of initial wealth. Another co
is whether the model expressed in terms of maximising endofperiod utility
wealth (or real returns) yields a similar functional form for asset demand func
the problem is conducted in terms of endofperiod nominal wealth (or nomin
The reader will have noted from the CAPM that it is crucial that asset de
equilibrium asset returns are independent of the level of initial wealth. If
returns are not independent of initial wealth then the theory is somewhat ci
equilibrium asset returns then depend on initial wealth which itself depends
asset returns. Hence if the meanvariance model of asset demands is to y
similar to the CAPM it must (at a minimum) yield desired asset demands in d
Aj that are proportional to initial wealth and hence asset shares that are i
of wealth. To choose a particular functional form for the utility function w
such asset demand functions is not straightforward and the current state of th
suggests that this can only be achieved as an approximation and results are on
small changes around the initial point.
We will sidestep these theoretical issues somewhat and assume that, to a
approximation, a version of the mean variance model of asset demands base
and (18.4) does yield asset demands in terms of equilibrium asset shares of the f
Frequently this type of model assumes that the 'constant' p in (18.1) is the co
relative risk aversion. In general p does depend on the level of wealth but the
made is that for small changes in the variables p may be considered to b
constant. Having obtained the meanvariance asset demand functions it is t
assumed that desired asset supplies xs (e.g. for government bonds, equity
bonds and foreign assets) are exugenous. Market equilibrium is given by xs =
xr denote the market equilibrium (which also equals the actual stock of assets o
and inverting (18.1) we have
over all individuals, which will in general depend on the distribution of weal
It is worth noting that the MVMAD only deals with the demand side of
If supply side decisions are governed by a set of economic variables zf (e.g
government bonds is determined by the budget deficit and corporate bonds b
gearing position of firms) then market equilibrium becomes:
Hence in general
(ERt+1  rt) = P ˜ X " Z t )
and equilibrium expected returns depend not only on {Oi,} but also on a set o
Equation (18.7) is the basis of the socalled structural or portfolio appro
zf.
determination of asset returns which occurs frequently in the monetary econo
ture. In this approach a set of asset demand and asset supply equations are es
then these resulting equations are solved algebraically to yield the reduced form
for expected and actual returns given in (18.7) (Friedman, 1979). Alternatively
economists estimate the reduced form equation (18.7) directly: that is they reg
or relative returns on a whole host of potential economic variables represent
(18.7) (e.g. budget deficit, changes in wealth, inflation).
The CAPM Revisited
At present the reader must be somewhat bewildered since the RHS of (18.6)
be nothing like the CAPM equation which for asset i is:
where E&"+, = expected return on the market portfolio and
h = (EfRzl  rf)/cri = market price of risk
However, (18.6) does reduce to something close to (18.8) as can be seen in th
illustrative case of three risky assets. Writing (18.6) in full:
where o,?, 2 = 021. It follows that
01
0ji
Thus the standard C M M (18.8) and the meanvariance model of asset de
exogenous supply (18.10) are equivalent providing p = A. Now unfortunately
not quite the same thing. However, both p and A are similar in that they are bo
of risk. In broad terms p measures the curvature of the individualâ€™s utility fun
l / A measures the additional return on the market portfolio per unit of marke
is about as far as we can go in terms of drawing on the common elements
approaches.
To summarise, the MVMAD with an exogenous supply of assets yields
for equilibrium asset returns that is very similar to that given by the CAPM
differ in their measure of overall risk. The MVMAD uses a measure of risk
depends on the curvature of the utility function while the CAPM uses a m
which depends on the additional return on the market portfolio to compensate
risk (0;).
Both the meanvariance asset demand approach and the standard CAPM are
problems. The investor is only concerned with (wealth) at the end of the first
all future periods are ignored. However, Lintner (1971) has shown that in a
minimisation problem of the form (18.5) that h is equal to the harmonic m
agentâ€™s coefficient of relative risk aversion.
Summary: MVMAD and the CAPM
We have concentrated on the relationship between the MVMAD which app
monetary economics literature and the CAPM of the finance literature. The m
sions are:
(i) The MVMAD is usually couched in terms of maximising the expect
utility from endofperiod wealth. Under certain restrictive assumption
form of the utility function this may be reduced to a maximand (or m
terms of portfolio variance and portfolio expected return as in the CAP
(ii) Unlike the CAPM, the MVMAD has asset demands that depend on th
erences of individuals and are in general not independent of initial wea
(iii) The MVMAD is usually inverted to give an equation for asset return
CAPM) on the assumption that asset supplies are exogenous. The latter
assumption unlikely to hold in practice.
(iv) Both the CAPM and the MVMAD (with exogenous asset supplies) i
asset returns depend on (a) cov(Ri, Rm)or equivalently (b) Ex,, where
(v) The weakness of the MVMAD is that it requires the researcher to assume
utility function and equilibrium prices depend on individualsâ€™ preferen
distribution of wealth, represented by the average coefficient of risk
across agents in the market.
18.2 TESTS OF THE CAPM USING ASSET SHAR
We now want to explore an alternative method of presenting and testing the CA
is often discussed in the literature. For each asset i we have:
using the definition of cov(R;, Rm)we have:
For illustrative purposes consider again the case of n = 3 assets then (18.15) i
Rlf+l  r
r
or in matrix form
E,R,+l  rr = hEx,
First assume the 0 i j are constant (and note that a ,= 0 j i ) . The CAPM then p
;
the expected excess return on asset i depends only on a weighted average
asset shares Xi, (which are equilibrium/desired asset shares in the CAPM). We
how economists often like to test the implications of a theory by testing res
parameters. Are there any restrictions in the system (18.16)? Writing out (18.
Assume for the moment that h is known so that we can use (Rj,+l  r r ) / h as
variables in (18.18). Suppose we run the unrestricted regressions in (18.18)
six â€˜unrestricted coefficientsâ€™
In matrix notation
E,R,+1  rr = h ( n â€™ x , )
If the CAPM with constant is true then from (18.18) we expect
aij
is symmetric.
Restrictions on the VarianceCovariance Matrix
Tests of the above restrictions are rarely conducted since there is another mo
set of restrictions for the CAPM implicit in (18.18) which logically subsume
restrictions. If we add the assumption of rational expectations:
where Qr is the information set available at time t. Then from (
variance covariance matrix of forecast errors is:
Substituting (18.23) in (18.16) we obtain a regression equation in terms of ac
excess returns
+
(R,+1  r,) = AXx, Er
where 6; = (Elr, c z r , â‚¬2,). However, from the derivation of the theoretical
know that
I = ( O i j } where aij = E[(Ri  ERi)(Rj  E R j ) ]
:
But under RE comparing (18.25) and (18.23) we see that
I: = E(66â€™)
+
Thus the CAPM RE imposes the restriction that the estimated parameters
regression of the excess returns on the X i [ (i.e. equation (18.19)) should equal
sion estimates of the variancecovariance matrix of regression residuals E
unrestricted set of equations (18.19) can be estimated using SURE or maxim
hood and yield a value for Il which does not equal E ( â‚¬ â‚¬ â€™ ) .The regression c
recomputed imposing the restrictions that the nij equal their appropriate E (
will worsen the â€˜fitâ€™ of the equations but if the restrictions are statistically acc
â€˜fitâ€™ shouldnâ€™t worsen appreciably (or the estimates of the l l i j alter in the
regressions). This is the basis of the likelihood ratio test of these restrictions.
The elements nij are determined exclusively by the estimates of the E
therefore the regression also provides (â€˜identifiesâ€™) an estimate of A, the m
of risk. The estimate for A (or p) can be compared with estimates obtained
studies. Also note that because the regression package automatically imposes E
symmetric, then, under the restriction that Il = E(&), Il is also automatically
Difficulties in Estimating the â€˜Asset Sharesâ€™ Form of the CAPM
The above restriction ll = E(&&â€™) undoubtedly provides a â€˜strong testâ€™ of the
in practice we face the usual difficulties with data, estimation and interpretatio
these are as follows:
The returns Rif+l in the CAPM are holding period returns (e.g. over
and these are not always available, particularly for foreign assets (e.g. s
use monthly Eurocurrency rates to approximate the monthly holding p
on government bonds).
The shares Xit according to the CAPM are equilibrium holdings at m
Often data on Xir for marketable assets (e.g. for government bonds) are on
at the issue price and not at market value.
There is the question of how many assets one should include in the emp
Under the CAPM investors who hold the market portfolio hold all asset
real estate, land, gold, etc.). Clearly, given a finite data set, to include a la
of assets would involve a loss of degrees of freedom and multicollinearit
would probably arise.
If either any important asset shares are excluded or if the asset shar
are measured with error then this may result in biased parameter es
principle the measurement error problem can be mitigated by applying
instrumental variables. A measure of omitted variables bias can be asc
trying different sets of Xit and comparing the sensitivity of results obt
similar vein, omitted variables bias might show up as temporal instab
parameter estimates (Hendry, 1988). Often in empirical studies these â€˜r
to the estimation procedure are not undertaken.
Under the assumption of constant aij (the socalled static CAPM) it is invar
that the restrictions ll = E(&&â€™) rejected. Also as the variability in the Xi
are
rather low relative to the variability in (Rit+l  r f ) ,the fit of these CAPM r
is very poor (Frankel, 1982, Engel and Rodrigues, 1989, Giovannini and Jo
and Thomas and Wickens, 1993). Finally these studies find that h (or p) is oft
(rather than positive) and is extremely poorly determined statistically. Indeed
does not reject h(or p ) = 0 statistically. The CAPM then reduces to the hyp
expected excess returns are zero (or equal a constant)
In the context of international assets with a known own currency nominal
Eurocurrency rates) this amounts to uncovered interest parity (for a = 0). Th
i
expected excess returns are found to depend on asset shares Xir the restricti
parameters llij do not conform to the static CAPM (i.e. constant aij) and th
equations is rather poor.
ARCH model
+
A2
(for all i and j )
o ˜ ˜ ,PO ˜P l & $ ,
=+
Thomas and Wickens (1993) find little evidence of time varying variances and
for a diverse set of assets which include foreign as well as domestic assets. H
a subset of assets, for example only â€˜nationalâ€™ assets or only equities, are incl
CAPM model then ARCH effects are found to be present. Other empirical s
do not test for ARCH effects but assume they may be present and then rec
above model and test the time varying covariance restrictions Il, = E ( E i E j ) f
As far as estimation is concerned the introduction of ARCH effects can
horrendous computational requirements as the number of parameters can be
For example, with seven risky assets each single O i j t + l can in principle de
the other seven lagged values of O i j r for the other assets (and a constant).
firstorder ARCH model for asset 1 only we have:
Equation (18.28) for asset 1 includes individual terms for 0 1 1 , 0 1 2 , 0 1 3 , . . ., 01
similar equations are needed for 0 2 j , 03jâ€™ etc. (of course remembering that O i j
higherorder ARCH processes additional lagged terms also need to be estim
is the computational task (and loss of degrees of freedom) that usually only
ARCH processes are used and sometimes only own lagged covariances are e
the ARCH process:
+
2
( O i j ) t + 1 = ˜1 eij<a$)t
Since we are now allowing the C matrix in (18.24) to vary over time (as we
naturally the time varying covariance CAPM â€˜fitsâ€™ better but again the CAPM
that (schematically) Il, = C, is invariably rejected (e.g. Giovannini and Jo
and Thomas and Wickens (1993). Also it is often the case that the point es
(or p ) is negative and again the null hypothesis that is zero is easily accepted
is a rejection of the CAPM.
An alternative method of â€˜modellingâ€™ the time varying variances in th
to assume that the variances O i j are (linearly) related to a small set of mac
variables Zir. Engel and Rodrigues (1989)â€™ who use the government debt of s
as the â€˜complete setâ€™ of assets in the portfolio, assume Zjr is either â€˜surp
prices or surprises in the US money supply (the data series for these â€˜su
residuals from ARIMA models). They find that both surprises in oil prices an
money supply help to explain the relative rates of return on this international
government bonds. However, the CAPM restriction that C, = Il, is still reje
or p is statistically zero).
There is an additional potential weakness of the above reported tests of
This is that they only consider returns of over one month. Now while in p
horizon returns, for example, for 3, 6 months or even 12 years. In this case
ARCH process might also be a better representation than for onemonth retur
All the above studies only consider the oneperiod CAPM model. The c
CAPM and the MVMAD yield expressions for equilibrium returns under the
that agents are concerned only with oneperiod returns (or next periods wealth
of the MVMAD). Merton (1973) in his intertemporal CAPM has demonstrate
constant preferences (e.g. constant relative risk aversion parameter p) and som
restrictions, the CAPM yields a constant price of risk (A):
for the entire market portfolio. However, in Merton's model the price of
constant for components of the market portfolio (Chou et al, 1989, page 4)
therefore that in an intertemporal CAPM one would not expect the 'price
be constant for returns on subsets of the portfolio. However, the latter is p
assumption made in much empirical work. Put another way, a weakness of
empirical studies is that they assume that the portfolios chosen are good repr
of the market portfolio (i.e. the 'errors in variables' problem is not acute)
section discusses a model which relaxes the assumption that the price of risk
and hence attempts to tackle this potential omitted variables problem.
Time Varying Price of Volatility
The CAPM for asset i may be written
where we shall refer to Qr as the price of volatility and allow it to be time var
term is chosen because the price of risk h is usually assumed to be a constant.)
Chou et a1 (1989) suppose we approach the problem of the potential mism
of the market portfolio by assuming it consists of an observable stock return
unobservable component with return RU.The variances are 0 and o, and the
:
:
between the stock return and the unobservable return is usu. The CAPM fo
portfolio (we omit t subscripts on the oij terms):
+ (1  xr)0sul
E&+,  rf = Qt[xra,"
w r + (1  ˜ r ) # I a , 2
=
= utera,"
+
where #' = (osu/o:)r, = xr (1  xr)B,", and xr is share of equities in the
Or
portfolio. Hence, 3 is the unobserved but possibly time varying CAPM be
/:
the stock portfolio and the unobserved part of the true market portfolio.
equation (18.31) becomes:
+
R;+,  rt = q r 8 p ; Er+l
+
R:+,  r, = aaS Et+,
where a is constant only if
(i) x, = 1, that is the true market portfolio consists only of stocks, or
=
(ii) that is the (conditional) covariances between the 'omitted
(asu), (a:),,
the included assets (stocks) are equal for all time periods, and
(iii) ' U r the price of volatility is constant.
Any time variation found in a, in an equation of the form (18.33) will be
breakdown of one or all of the above assumptions. A number of studies find
varying own variance U: can help to explain movements in own expected ex
but 'the linkage' is unstable (i.e. a is time varying). This is consistent with
variables interpretation of equation (18.33). A purely statistical method of mod
variation in a which is reasonably general is to assume a, follows a random w
+ U,
= at1
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 crossproducts 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 sixmonth and threemonth bills. We look first at studies
rather adhoc 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 longterm
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 longterm 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 longterm 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
threemonth (13 weeks) and sixmonth 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 oneperiod holding period y
when investing in sixmonth bills rather than threemonth bills is defined asâ€œ
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 threemonth holding period, the sixmonth bill constitutes the risky
its selling price in three monthsâ€™ time is unknown (and is, of course, directly
Etrr+13 in (19.2) since after three months has elapsed the sixmonth bill â€˜beco
with only three months to maturity). If we denote the risk premium as 0, then
yield on the risky sixmonth 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 expost value. Because of the use of overlapping data the error term
MA(12) and may also be heteroscedastic. Simon (1989) â€˜correctsâ€™ for these p
using an estimation technique known as twostep twostage least squares (see C
Simonâ€™s most favourable result is:
+ 1.6 ( R  r),  0.47 (2R,  r,  rf+l)2
= 0.06
(0.06)
(0.18) (0.34)
19721979 (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 19611971 period but for the post1982 period (ending in 19
premium term is statistically insignificant and there is some variability in the
sixmonth over threemonth 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 adhoc measure of risk(*)
Using weekly data for Fridays over the period January 1970September 1979 a
that the threemonth return exactly matches the holding period of the sixmonth
they find
+ 0.97r, + 0.756,
2Rr  rt+13 = 1.0
(0.86) (0.08) (0.31)
R2 = 0.79, SEE = 0.90
19701979,
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 sixmonth 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 â€˜exhaustive explanationâ€™ 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 JonesRoley 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 JonesRoley 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 â€˜riskâ€™. Hence, the excess HPY on sixmonth over t
bills does not appear to be a stable function of the adhoc 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 â€˜rewardâ€™ 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 â€˜invertedâ€™ me
model of asset demands, where there is only one risky asset. The demand f
the risky â€˜sixmonthâ€™ asset is:
where c is a coefficient of risk aversion. If asset supplies â€˜Aâ€™ 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 sixmonth 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 meanvariance 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 â€˜get a handleâ€™ 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 periodâ€™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 â€˜ARCH in me
1
A representative result from Engle et a1 (1987) using quarterly data, 195919
+ 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 19791982, when monetary base targeting was in o
included. Broadly speaking Tzavalis and Wickens (using monthly data) reprodu
alâ€™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 19791
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 nonlinear effect requires further investigation.
1 . TIME VARYING RISK: LONGTERM 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 oneperiod 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 oneperi
period yield on the longterm 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 zerobeta 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 zerobeta portfolio of bonds and Rm
period return on a market portfolio of bonds. The two hypotheses to be teste
HA : p(")# 0, = 0 (zerobeta CAPM)
U(")
H i : p(") = 0 # 0 (consumption CAPM)
U(")
ñòð. 12 