ñòð. 41 |

calculation is then repeated using h,+l in the VAR representation. The AR(1) model

returns is:

E,+lhf+2= BE,hf+l + U f + l

(Ef+1 - Er)hr+z = Uf+1

Leading (21) one period and taking expectations at time t we have:

EA+3 = BE,hf+2 = B2E1hf+l

and similarly:

+

Et+lhr+3 = BEr+lhr+Z = B(BErhr+l ur+l)

Subtracting (23) from (24) we obtain:

(E,+1 - Et)hr+3 = B&+1

In general, therefore, we can write:

(Et+l - ˜ ˜ h t + j = lPi-'ur+1

+

Using the definition of news about future returns, in (15) and (16) and using (25), we

j=l

Hence the variance of discounted unexpected returns is an exact function of the vari

period unexpected returns:

var(q;+l) = [P/U - PS,l2 var(u1+1)

Using (27), the measure of persistence Ph in (19) is seen to be

Ph = P/(l - p g ) % 1/(1 -

Hence if B is close to unity which we can interpret as a high degree of persistence i

model, then Ph will also be large. Since pr+l- Efpt+l = -Q!+˜ (when qd = 0) and q

pB)]u,+l % [ l / ( l - /l)]u,+l then for the AR(1) case pl+l - E,P,+˜= Ph . u,+1. Hence

increase in uf+l leads to a Ph percent increase in qh and hence a P h percent unexpected

For the AR(1) case it can now be shown that even if we can only explain a sma

of the variability in one-period returns h,+l (i.e. returns are difficult to forecast), yet i

persistent, then news about returns can be very important in explaining stock price

short, the more persistent are expected returns, the more important is the variance of

future returns var(qf+l) in explaining unexpected returns v:+˜ - E,h,+l (or unexpected

or losses, P f + l - EfPf+l).

R2 can be defined as the fraction of the variance of stock returns which is predictable

R2 = var(E,ht+l)/ var(h+1)

1 - R2 = var(vF+,)/ var(h,+l)

R2/(1 - R2) = var(E,hf+1)/var(vF+,)

Also from (20) the variance of E,h,+l is:

var(Erhr+1) = var(u,+1)/(I - B2)

Substituting (31) in (30) and solving for var(u,+l), we obtain:

var(u,+l) = (1 - ˜ 2 ) v a r ( v ˜ + l ) R 2-(R2)

/1

The left-hand side of (33) is one of the components of the variance decomposition

interested in and represents the importance of variance of discounted expected future re

to variance of unexpected returns (see equation (17)).

For monthly returns, a forecasting equation with R2 RZ 0.025 is reasonably represe

variance ratio (VR) in (33b) for B = 0.5 or 0.75 or 0.9 is VR = 0.08 or 0.18 or 0.49,

Hence for a high degree of persistence but a low degree of predictability, news about f

can still have a large (proportionate) effect on unexpected returns var(v;+,).

3. The VAR Model, Variance Decomposition and Persistence

The above univariate case neglects any interaction between news about expected retur

about dividends, that is the covariance term in (17). At a minimum an equation is requir

dividend growth. The covariance between the forecast errors (i.e. news) for dividend

those for returns can then be examined, and other variables in the VAR that it is th

help in forecasting these two fundamental variables can be included. It is possible th

the expected return along with some other forecasting variables in the context of a VA

carry out the variance decomposition for this multivariate case.

This section assumes the (m x 1) vector zr+l contains hr+l as its first element. The ot

+

in z1+1 are known at the end of period t 1 and are used to set up the following VAR

*

zr+1 = A Z r + Wf+l E(wf+lw;+l)=

where A is the companion matrix. The first element in wr+l is v:+˜.First note that:

+

= AJ+â€˜Zt AJw,+l

â‚¬r+lZf+j+l

E,z,+,+l = AJ+â€™z,

Subtracting (37) from (36) we get:

(Er+1 - Er)zr+j+l = ˜ â€™ w t + l

Since the first element of zr is hr, if we premultiply both sides of (38) by elâ€™ (where el

row vector containing 1 as its first element with all other elements equal to zero) we

(Er+1 - Er)hr+j+l = elâ€™Aâ€™wr+l

and hence:

j=l j=l

= elâ€™pA(1- pA)-lwr+l = Aâ€™wr+l

where Aâ€™ = elâ€™pA(1- pA)-â€™ is a non-linear function of the parameters of the VAR. S

element of wr+l is v:+˜, using (16) and (40) we can write:

+ pAU - pA)-â€™lwr+l

d

= ef[I =h

vr+1 +l

It can be seen from (40) and (41) that both unexpected future returns and unexp

dividends can be written as linear combinations of the VAR error terms where each

multiplied by a non-linear function of the VAR parameters. Setting j = 1 in (39) an

we obtain:

W r + 1 - Er)hr+z = 4 + l = elâ€™Awr+l

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 variance-covariance 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 non-redundant elements

and the heteroscedasticity adjusted variance-covariance matrix of the estimate of thes

by v. Suppose, for example, we are interested in calculating the standard error of Ph

a non-linear 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

one-period return. In some models r, is the nominal risk-free 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 first-order 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 short-term 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 long-term 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 so-called 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 one-period 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 risk-free 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 one-period CAPM.

together a strand in the monetary economics literature, namely the mean-vari

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 one-period holding period yield on bills (z

bonds) and long-term 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 Poterba-Summers (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 noise-traderâ€™ 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 = risk-free 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 risk-free 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 pn-â€™ 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 1859-1987 with each Zpj in the region of 0.8-0.

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 0.0-0.3. 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 noise-trader 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 Poterba-Summers 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 risk-free

++

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

ñòð. 41 |