Interest rates and risk premia in the stock market and in the foreign exchange market

Interest rates and risk premia in the stock market and in the foreign exchange market

Interest Rates and Risk Premia in the Stock Market and in the Foreign Exchange Market ALBERTO GIOVANNINI AND PHILIPPE Graduate School of Business, ...

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Interest Rates and Risk Premia in the Stock Market and in the Foreign Exchange Market ALBERTO GIOVANNINI

AND

PHILIPPE

Graduate School of Business, Columbia Universig,

Sew

JORION* York,

NY

10027,

USA

This paper documents common empirical regularities in the foreign exchange market and in the US stock market. We find that increases in interest rates are associated with predictable increases in the volatility of returns in both markets, and that expected returns both in the stock market and in the foreign exchange market are negatively correlated with nominal interest rates. We show that not taking into account the time variation of second moments may seriously affect tests of asset pricing models. Using a

numerical example based on the static capital asset pricing model, we are able to produce fluctuations in risk premia similar to those observed empirically. Finally we show that the overidentifying restrictions of the latent variable capital asset pricing model are not rejected when betas are assumed

to be correlated

with

nominal

interest

rates.

This paper discusses important empirical regularities which are common to both the US stock market and the foreign eschange market. It is motivated by two observations. First, the widespread evidence on the presence of risk premia in the foreign eschange market calls for empirical work on a wider spectrum of assets: this work is bound to offer insights on factors affecting equilibrium in the world capital market. Second, given significant correlations between US stock market returns and returns in the foreign exchange market, joint tests should be more powerful than the currently available work which looks at the two sets of assets separately.1 The presence of widespread regulations which preclude foreign investors from many foreign stock markets prevented us from including them in our sample.’ We show that excess returns on the US stock market and foreign currency deposits display very similar empirical regularities. Then we discuss the implications of our findings for tests of asset pricing models. This paper is organized as follows. Section I contains a set of empirical tests that characterize the stochastic properties of nominal dollar excess returns, which we * We thank participants in the Columbia Finance Seminar and the XBER Summer Institute for useful comments. Michael Adler and David Hsieh also provided helpful suggestions. .\I1 errors remain our responsibility. This research was supported by rhe Center for the Study of Futures XIarkers at Columbia University.

108

Interest Rates and Risk Premia

in the Stork iblarket and Foreign

Exchange

‘llarket

call risk premia. The section documents that risk premia in the foreign exchange market and the stock market have a number of common characteristics: (a) their conditional distributions are correlated with nominal interest rates, in such a way that an increase in interest rates is associated with a predictable increase in volatility of risk premia, for both foreign currency deposits and the stock market; (b) Fama and Schwert’s (1977) observation applies also to the foreign exchange market: exante dollar escess returns are negatively correlated with dollar interest rates both in the stock market and in the foreign exchange market. Finally, (c) dollar risk premia in the stock market and foreign currency deposits appear to move together over time. of the observed non-constancy of Section II explores the implications conditional second moments of risk premia. By performing illustrative numerical calculations, we show that the so-called ‘puzzle ofescess volatility’ of risk premia in the foreign exchange market, which Frankel (1986) claims cannot be esplained by the mean-variance international portfolio model, might in fact be not a puzzle when variation of conditional second moments is explicitly accounted for. In addition we show that the rejection of the Hansen and Hodrick (1983) tests of the intertemporal capital asset pricing model reported by Hodrick and Srivastava (1984) could also be due to their assumption of constant conditional second moments. Section III contains a few concluding remarks.

I. Stochastic

Properties

of Risk

Premia

This section offers evidence consistent with the presence of non-constant variances in the conditional distributions of rates of return in the stock market and the of returns in the foreign exchange foreign exchange market. Homoskedasticity market has been recently rejected by a number of authors, including Cumby and Obstfeld (1984), and Hodrick and Srivastava (1984). In the stock market, Christie (1982) documents a significant positive association between individual stocks variances and nominal interest rates. Poterba and Summers (1984), and French, ef a/. (1985) offer additional evidence on the time variation of stock returns’ volatility. Tests of time-varying conditional second moments focus on the conditional variance of rates of return:

(1)

var(E,+,II,)

E E[(E,+,

-E(~I~,I~,))21~,1

where E,+, is the dollar return on any asset from time t to time tf 1, and I, is information available at the end of period t. One type of test due to White (1980) would make use of consistent estimates of E(&,_,II,), and then project on variables included in I,. This test is difficult to imple(s ,+,-JYLIL))~ ment in our contest, because the presence of rime-varying conditional second moments would invalidate standard models of espected returns. For the assets under study, however, surprises account for most of the variance of realized rates of return. Typically, models of risk premia esplain at most 4 per cent of the variance of realized returns, as confirmed by the tables following in the paper. Therefore we different simply project of,, on variables in I,. a In addition, \ve find no appreciably results using the residuals from a model in the nest section.” ‘Along similar lines, Hodrick and Srivastava (1984) reject homoskedasticity using the residuals from an unconstrained model where escess returns are projected on a set of foruard premia.

ALBERTO

GIOV.INNINI

109

PHILIPPE JORION

AND

Thus, given

the importance of the innovation component, the results seem robust to adjustments in expected values. The data are described in the Appendis. We collected neekly spot rates, one-week Eurocurrency rates and stock market returns on a value-weighted index. Using weekly data rather than monthly data increases the power of our tests: our longest sample period, August 1973 to December 1984, includes 596 observations. All tests were repeated (but are not reported) for monthly observations, with similar results, except of course for larger standard errors. As we found that correlations between the stock market and eschange rates change significantly after October 1979, the sample was broken in two subperiods: August 1973 to October 1979 and October 1979 to December 1984.s We test homoskedasticity for the following variables: (a) The realized rate of return on a foreign currencv Eurodeposit relative to a Eurodollar deposit (of maturity one week). Because of interest rate parity, this \-ariable is sometimes referred to as the rate of return on an open position in foreign currency, i.e., the rate of return on a for\vard contract: E:- , =s,+,--,+iF-i,

=s,+,

-1.

where sr is the (log of the) exchange rate, expressed as the dollar price of one unit of interest rate; i, foreign currency; i? is the (log of one plus) the foreign Eurodeposit the (log of one plus) the Eurodollar rate; f, the (log of the) forward rate on oneperiod contracts. (b) The realized rate of return on the US stock market in excess of the Eurodollar rate over the same period: E ,-I

=

T/+1

--I,

The squared returns are regressed on a constant, the nominal rate, and the foreign currency Eurodeposit interest rate: (2)

E:,, = a,, +a,i,

Eurodollar

interest

+a&+ +rl,,,

This specification has been chosen to verify whether a positive relationship between nominal rates and volatility of returns similar to the one reported by Christie (1982) can be observed with aggregate data. Alerton (1973) also postulated nominal interest rates as state variables in his capital asset pricing model. It is likely, however, that nominal interest rates may not-be the only variables which agents use to forecast volatilities of rates of return. The presence of omitted variables might bias test statistics in our regressions. We thus allow for a more general process for the residuals in the regressions, and compute the \-ariancecovariance matris of estimates as suggested by White (1980). A consistent estimate of the variance-covariance matrix of the projection coefficients is: (3)

V(B) = I-(X)X)-QX’X)-’

where fi is the estimated vector of projection coefficients, and X_ is the matrix of included right-hand-side variables in the projection equation. fl is a consistent with e representing the projection residuals. estimate of lim,-_, [(l/T)E(X’ee’X)], We follow Hansen (1982) and Cumby, eta/. (10_3), and estimate 6 in the following fashion. First, we estimate e^from OLS, then fl is taken from the spectral density matrix of the vector stochastic process [X,‘;,] evaluated at frequency zero.6

110

Interest Rates and Risk Premia in the Stock ~Llarket and Foreign

Exchange

.Ilarkrt

The results are presented in Table 1, which reveals a number ofinteresting facts. The hypothesis of constant conditional variances of rates of return on forward contracts is rejected for all currencies, although for each currency coefficients var) quite substantially in the subsamples. The null hypothesis is also rejected for the US stock market risk premium, both in the whole sample, and in the first subsample. Table 1 also reports marginal significance levels for the null hypothesis, computed using Hansen’s (1982) generalized method of moments (GMM) outlined above. The rejection of the null appears about as often as before. The magnitudes of interest rates coefficients are comparable across assets. Finally, the table shows that all significant coefficients are positive. This remarkably uniform result suggests that higher conditional variances are associated with higher nominal interest rates. The evidence of course needs to be interpreted carefully, however, as we lack an explicit model for the time-variation of conditional distributions, and therefore cannot spell out the conditions for. consistency of the estimated parameters in Table 1. We are now equipped to further characterize the stochastic properties of risk premia by performing a series of consistent tests which esploit the rational expectations assumption, and explicitly account for heteroskedasticity in the data. Under rational expectations the realized rate of return in excess of the risk-free rate on any asset can be decomposed into a ‘risk premium’ and a ‘surprise.’ The risk premium is the expected rate of return from holding an asset, and as such can be forecast using information available at the beginning of the holding period. The surprise in the rate of return arises from neiv information Lvhich by definition cannot be forecast, and is thus orthogonal to the expected rate of return. Given our notation, the risk premium of a foreign currency deposit over the US stock market is defined as: (4) where As, =s,,, -s,. Similarly, the risk premium of a foreign currency a Eurodollar deposit, i.e., the rate of return on a forward contract, E[~,+,l1,]

(5) Finally,

(6)

the risk premium

deposit is:

over

= E[As:+i,?-i,lI,]

of the stock

market

over a dollar

deposit

is:

E[&,+,l~,l= E[r:_, -i,lI,]

Expressions (4) to (6) are not observable. The assumption of rational expectations allows us to project ex-post rates of return differentials on information available at time t in order to uncover some of the stochastic properties of these risk premia.’ With OLS estimation, point estimates are consistent, to the extent that the rational expectations assumption holds true, and that variables included in the projection equations are a satisfactory approsimation of the information set.s OLS standard errors, however, are biased because of heteroskedasticity. Therefore, the tests in the following tables are computed using the Generahzed Slethod of Moments. The first natural candidate for capturing variation in risk premia is lagged values of the realized differences in returns <7>

Eti, = b,, +b,E! +Q,_,

+q,.

GIOVANNINI AND

ALBERTO

T.*BLE 1. Tests

of time-varying

PHILIPPE

volatility

EL, = a,, +u,it+aX 2,

I a,

73-84 73-79 79-84

-

-

14.26**

2.E-6**

mark

J3-84 73-19 79-84

15.19** 11.75** 19.09**

-3.29 -1.78 - 6.49

7.E-9** 0.051 l.E-4**

guilder

73-84 73-79 79-84

10.97** 6.36 17.19**

3.19 5.99* -4.99

Swiss franc

73-84 73-79 79-84

16.26** 16.23* 22.68**

UK pound

74-84 74-79 79-84

9.3g** 1.26 11.63*

Italian

J3-84 73-79 79-84

10.68**

French

franc

German

Dutch

lira

Japanese

us stock market

yen

of excess

returns.

+rl,_, P-ml for test a, = aZ =0 GXIbI OLS

Period

Asset

111

JORION

R’

I ‘(E?)

4.E-3**

0.09’

27.06

9.E-5** 3.E -3** 7.E -3**

0.061 0.01s 0.063

15.51 13.42 17.25

9.E-lO** 0.002** l.E-4**

2.E-6** 2.E-3** 4.E -3**

0.06s 0.030.063

12.39 9.78 14.82

-3.18 -4.61 -1.07

2.E-6** 0.040* 4.E-5**

l.E-4** 0.015* 4.E-3**

0.0-H 0.030 0.0-3

28.44 27.95 28.62

-0.29 3.56 - 5.35

4.E--3** 0.275 0.046*

3.E-3** 0.208 0.203

0.030.010 0.026

13.21 5.38 20.35

0.011*

0.048

9.70

J.16

2.s3**

- 1.98 -

0.001+*

73-84 73-19 79-84

4.55

5.81

0.049*

0.031*

0.022

73-84 73-79 79-84

15.05* 63.16** 10.16

-

O.OlG* 5.E-5** 0.198

3.E-4** 3.E-3** 0.119

0.01~:~ 120.29 0.03 162.35 JO.81 0.006

-

-

?LTo&s: Significance at the 5 per cent and 1 per cent levels denoted by * and **, respectively. GAISI method of estimation allows for heteroskedastic error terms. Tests of significance of slope coefficients obtained by OLS. Periods are: Aug. 3, t973-Dec. 28, 1984 (596 observations), except for the VK, where data stait on 12 July 19’4 (546 observations). First subperiod ends on Oct. 5, 19-O (323 observations). Second subperiod begins on Oct. 12, 1979 (273 observations). All variables measured in percent per week. I’(&‘) refers to the variance of the dependent variable.

Here, 6, and b, are nor-zero only if risk premia are autocorrelated.g Table 2 reports the results of such tests. We find significant coefficients for the guilder and the yen. Cumby and Obstfeld (1981) also find a strong rejection of the null for the guilder for the period 1974-80. They reject for other currencies as well, but their BoxPierce statistics may be biased because of heteroskedasticity. When adjusting for this effect, we find that the null was rejected less often than before.rO For all other currencies and the stock market, autocorrelation of expected returns is not detectable. Notice that in the regressions where we include the return differential between foreign currency deposits and the stock market, all of the autoregressive

112

Interest Rater and Risk Premia in the Stock ,\iarket and Foreign

Exchange

Xlarket

ALBERTO

GIOVANNIN~.IND

PHILIPPE

JORION

113

coefficients are insignificant. Constant terms, escept in one case, are also insignificant. Given the evidence in the previous section, the most likely source of variation in risk premia should be the level of interest rates. Interest rates, or more precisely differences of interest rates (forward premia), have been used estensively as predictor variables in tests of risk premia in foreign exchange markets. The typical regression equation in these studies is:”

(9 Because

<8b>

Ar, +i,: --i, f E,+I = c,, +c,(i. of interest

rate parity,

(8)

is usually

-i?)

written

+r?,_,. as

=c”+c,(f,--.J+rlf+l.

(JlCl-fJ

the hypothesis of absence of risk premia between Based on this equation, Eurodollars and Eurocurrency deposits implies c,, =c, =O; this hypothesis is rate differentials, reveal usually rejected. Thus forward premia, i.e., interest substantial information in the movement of risk premia in the foreign eschange market.‘* In our tests, we allow for different coefficients on the domestic and foreign interest rates. Me estimate

0)

E ,_I

-

c

0

+c,i. +c$.* $_‘/._I

with E,.+, defined in (4) to (G), and report the results in Table 3. The section on the left contains projections of realized returns in the foreign exchange market on dollar and foreign interest rates. Remarkably we find that, in nearly all coefficients on the US spite of potential problems of multicolinearity, interest rate are negative whereas all the coefficients on the foreign interest rate are positive, across subperiods and currencies. The joint test of zero slope coefficients is rejected in many cases. Thus increases in foreign eschange risk premia are reliably associated with decreases in US interest rates and increases in foreign interest rates. The bottom section of the table contains the projection of risk premia in the stock market on the Eurodollar rate. The results confirm Fama and Schwert’s (1977) finding that nominal stock market returns and, afortiorz’, returns in excess of the risk-free rate are negatively correlated with beginning of period interest ratesI The similarity of these results with those of the Eurocurrency markets is quite striking: both in the stock market and in the foreign exchange market, expected dollar returns in escess of dollar interest rates are negatively correlated with dollar interest rates. The pervasiveness of this empirical regularity seems to suggest that its theoretical explanations should derive from general models, rather than specific features of the individual markets.14 The right-hand-side section of Table 3 shows the results obtained projecting the differential between stock market and foreign eschange market returns on nominal interest rates. In no case is the slope coefficient significant, whereas a number of significant constant terms are observed over the period 1979-84, suggesting a constant positive risk premium in the stock market. Thus, the available evidence (both from Table 2 and Table 3) suggest that KS-CINDY’ returns in the foreign eschange market and the stock market move together over time.

stock market

73384 73 -70 79 84

-0.50

us

-0.27

73-84 73-79 7% 84

Japanese

yen

73-84 73-79 79-84

Italian lira

- 0.43 -0.56* -0.83

73-84 73-79 79-84

0.34 0.26 --0.18

0.07 0.01 - 0.24

0.25 0.20 -0.19

0.08

UK pound

73-84 73-79 79-84

73-84 73-79 79-84

73-84 73-79 79-84

C;,

73-79 73-79 79-84

guilder

mark

franc

Period

Swiss franc

Dutch

German

French

Asset

0.98 0.32 1.86 3.32** 1.83 5.17**

- 2.74* -1.32 - 1.30

- 2.20, 1.55 - 2.69

- 2.07 (,.oo* +

1.89**

4.35** 3.61** 5.71**

-4.19** - 2.80* -4.06*

- 2.48

0.56 2.40 1.53 5.23

-3.49** -2.14 -3.74

A c2

- 2.08

* Cl

Eurodollars

1:, +,

TABLE 3. Dependence =

0.004**

._

0.025*

0.01)1** 0.089 0.004**

0.039* 0.708 0.267

2.E-7** 2.E-5** 0.021*

0.019* 0.361 0.178

0.067

I’-Vd

0.0489

0.0287

0.0238 0.0176 0.0318

0.0143 0.0016 0.0083

0.0450 0.0415 0.0236

0.0189 0.0067 0.0135

0.0366

I<’

0.55* 0.81* 1.24*

-1.18

-1.31*

- 0.60 -0.83 -1.44*

- 0.21 -0.53 -1.41*

- 0.47 -0.79 -1.36*

-0.26 -0.61 -1.21*

-1.17*

6,

minus

interest

deposits

+‘I, +,

on nominal

Eurocurrency

+c,i, +I.$:

Test c, =cz =o

(‘,I

of risk premia

0.88 3.63 5.1 I

3.12

~

minus

2.03

1.16

0.80 0.34 0.94

0.26 1.22 -1.52

4.16* 3.45 2.64

-0.12 1.54 -2.09

0.59

Stock market

market - 3.00* - 5.78 -5.18*

Stock

3.10

2.93

1.47 5.19 4.11

0.54 3.96 5.07*

-1.08 3.09 2.68

rates.

0.012* 0.069 0.010*

Eurodollar

0.379

0.285

0.406 0.446 0.146

0.904 0.396 0.124

0.090 0.084 0.190

0.846 0.393 0.236

0.186

Test c, = c2 = 0 I’-val

0.0091 &)I 27 0.0230

0.0090

0.0108

0.0031 0.0083 0.0149

0.0004 0.0072 0.0169

0.0084 0.0165 0.0134

0.0005 0.0068 0.0123

0.0187

ALBERTO

GIOVANNINI

II. Implications

ASD

for Asset

PHILIPPE

JORION

Pricing

Models

113

Among the empirical regularities documented in the previous section, the nonconstancy of conditional second moments has potentially the most far-reaching implications for empirical tests of asset pricing models. This section discusses such implications. Asset pricing models have generally fared rather poorly in recent empirical tests. The dynamic capital asset pricing model of Lucas (1978) and Breeden (1979) extended to an international setting by Stulz (1981) has been applied to the data by various authors. Hansen and Singleton (1982) test and reject the model using data on the US stock market. In the foreign exchange market, the model is rejected bl Hodrick and Srivastava (1984). The international version of the Sharpe-LintnerlZIossin static asset pricing model illustrated in Dornbusch (1983) is rejected b! Frankel and Engel (1984). All these tests assume constant conditional second moments. Among tests of capital asset pricing models which admit time variations of second moments, Campbell (1985) assumes that rates of return on assets are perfectly correlated with the rate of return on a benchmark asset, thus implying that conditional expected returns are proportional to conditional variances of returns. Cumby (1985) applies the Gibbons-Ferson (1983) test to the foreign exchange market, and assumes that conditional covariances between rates of return on foreign assets and the rate of change of real consumption move together over time for all currencies. These specific constraints have been rejected. Further empirical evidence on international asset pricing models has been provided by Fama (1984) and Hodrick and Srivastava (1986), who demonstrate that variations of risk premia account for the largest fraction of the total variation of foreign exchange forward premia. This empirical fact is regarded as a puzzle by Fama (1984) and Frankel(1986), but has been reconciled, at least theoreticall!-, with the dynamic capital asset pricing model by Hodrick and Srivastava (1986). In this section we explore the potential of time-varying conditional second moments to explain the observed variation of risk premia. X’e first illustrate the puzzle of rhe volatility of risk premia and show that, by allowing for the observed time-variation of conditional variance, the predictions of the static ChPhI model are of the same order of magnitude as the empirical data. Then we run tests similar to those of Hansen and Hodrick (1983) and Hodrick and Srivastava (198-L), and show that by releasing the constraint that conditional second moments are constant we are unable to reject the null hypothesis, which is otherwise rejected when second moments are taken to be constant.

II.A.

The ‘Puzzle’

Since the forward premium (S-s) forecast E[A.s] and a risk premium,

V(f--I)

of Risk Prentih I 7nlatiiig

can be decomposed p, we have

= V(E[As])

+2 COV(E[AJ],P)

Further, because of rational expectations, the innovation orthogonal to the forward premium, and therefore,

<11>

Cov(f---s,

As) = Cov(E[As] = l’(E[As])

+p,

into

E[As]).

rate

+ I,‘(p). in the eschange

E[As] +(AJ-E[AJ]))

+Cov(p,

an exchange

rate is

116

Eliminating (12)

Interest Rates and Risk Premia

Cov(p,

E[As])

in the

from (10)

I,‘(p) = I ‘(f-s)

-2

Stock Market and Foreign Exchange Market

and (1 l> yields Cov(f-s,

As) -I- Y(E[As]).

Since the covariance term is empirically reliably less than zero, Fama (1981) points out that the variance of the risk premium I,‘(p) must be greater than the variance of the expected movement in the exchange rate Il(E[As]). This observation has become known as the ‘puzzle’ of risk premia volatility. The variance of the risk premium is unknown, but relationship (12) can be used to derive a lower bound on L/(p) (13)

V(p)

>/ V(f-J)(l--2P),

where ; is the regression coefficient of As on (j-s). Let us now illustrate the orders of magnitude involved with a simple numerical esample. For comparative the computations will be performed using the German mark, and purposes, monthly data over the period August 1973-December 1984. Similar results hold for weekly data, but are not reported here. With returns expressed inprr centpcr month, the standard deviation of the forward premium is 0.206, and the regression coefficient 7 is -2.0. Therefore, the lower bound on the variance is I’(p)2 (0.206)‘(1 -2( -2.0)) =0.21. l5 Using the static mean-variance model, Frankel (1986) argues that variations in asset supplies cannot explain time-varying risk premia. To see this point, consider the simplest expression for a risk premium, derived from a mean-variance model with one risky asset only. We can urite, as in Dornbusch (1983): (14)

p = c+pxa”,

where c is a constant, p the coefficient of relative risk aversion, h: the net supply of the asset under consideration, and 0” the conditional variance of the asset escess return. Assuming that x is the sole source of variation in p, we have (15)

V(P)

= (PO’)‘I++

As in Frankel(l986), take p =2, & =(3.28)‘, and a large standard deviation of asset supply, say 0.01 .16With these numbers, the variance of the premium is L’(p) =0.05, which is much lower than the previously obtained lower bound 0.21. Asset pricing theory however does not require stationarity in espected returns and covariances.*T Here, time variation in \-ariances implies: (16)

I’(p)

= (px)? V(0)

With monthly data espressed in per cent, the standard deviation of the squared escess return is 20.84 over the same period as above. This yields a variance of 434.26. The predictable component of this variance can be obtained from regression (2). Kith an R-square of 0.063, the variation in the second moment and s=O.l, say, (16) yields attributable to interest rates is 27.6. With p=2 V(p) = 1.10. This is well above our lower bound of 0.21 implied by the data. Thus this simple numerical illustration shows that esplicitly accounting for the time variation of conditional second moments can overturn what appeared to be a gross empirical failure of the mean variance model.

ALBERTO

GIOVANNINI

ASD

PHILIPPE

11;

JORION

Variable Capital Asset Prick,, Xlodel

ZZ.B. The Latent

This section evaluates the implications of time-varying second moments for empirical tests of the international CAP&I. In the discrete time, intertemporal asset pricing model equilibrium returns are determined as follows. Assume a single representative agent with time-separable utility over (infinite) lifetime consumption. The first-order conditions for espected utility maximization imply: <17>

E[@o,,;+, RI_,)lZ;l

= 1

where R:_, is one plus the nominal dollar return on assetj, on a holding period is the marginal rate of substitution of money between t from t to t+l, andQ,,,+, and t+l.ts Define the return on a benchmark portfolio, R:_,, as: <18>

RF+, =Q,,,:+,

and the risk-free (19)

R:,,

The first-order conditions following two relations:

E[R:+,-R:_,IZ:]

<21> (22)

(17),

wla/,,+,I41

= with

the definitions



and (19>,

yield

the

E[R:‘+,R:+,I~,l = F~b.,+,~~,1

(20)

Nest,

.!Z%xJZ/l

rate of return:

<20)

and <21) E[R;+,

_

= -cov[R:+,,

R:+,IZ,]/E[R:‘+,IZ,]

can be used to obtain: R,v_:(r,,

=

co~r[R:+IY Rj’,!ILl

ELK,, -R:+,lLl

var[R:_,lZI]

Equation (22) is used in empirical tests of the intertemporal CAPhI with two additional assumptions. The first is that the ratio cov[R:,,, R:‘+,IZ,]/var[R._!IZr] is a constant equal to fi/ for each asset j. The second assumption is that is an unobservable variable for which the following projection E]R:+, -R/+,11,] equation holds: (23)

E]Rf+,

- R:_,IZ,]

where x, is a set of K observable variables into <22), and replacing the left-hand-side we obtain: (24)

= xzj +H,

and t(, is i.i.d. normal. Substituting variable in (22) with its realized

(23) value,

R:,, - R:+, = ,@rp,+i;+,

The first term is orthogonal to 2: by where c,+, =/?/tl, +[R{?, -E(R:+,IZ,)]. construction, whereas the second term is orthogonal to all variables in I;, including x,, under rational espectations. Let <24) hold for K assets _j= 1, . . . , N. Then there are NK orthogonality conditions (K orthogonalit)conditions for each asset, N assets) but only I\ + S1 parameters to estimate.lg As suggested bv Hansen (1982), the SK -(_\‘+K1) orthogonality conditions not used in the estimation of the parameters are overidentifying restrictions that can be tested. Hansen’s J-statistic measures the extent to which the LYK - (S + K - 1) inner products of right-hand-side variables and residuals are close to zero, as predicted bv the theorv.“O The top portion

of Table 4 reports

the results’of

such a test. We limit oursell-es

to

118

Interest Rates and Risk Premia

in the Stock Market

and Foreign

Exchange

Market

the four currencies for which the longesr sample period is available. Escess rates of return are the differences betueen the dollar returns on one-ueek Eurodeposits in UK pounds, German marks, Dutch guilders and Swiss francs, and the one-week Eurodollar deposit rate; the excess rate of return on the US stock market is defined similarlv. The system is estimated with the generalized method of moments of Hansen (1982) and Cumby, rt al. (1983), assuming, consistently with the findings of Section II, that the variance-covarinnce matris of residuals is conditionall! l The variables included in the vector q, are one-week nominal heteroskedastic.’ Eurodeposit interest rates on all five currencies and a constant term. This test is similar to that carried out by Hansen and Hodrick (1983) and Hodrick and Srivastara (1984), although these authors use different data sets, which do not include the stock market, and different regressors. The former projecting on lagged forecast errors are unable to reject the latent variable model while the latter, projecting on forward premia, strongly reject the model estimated o\-er a longer time period. In line with the latter results, the table shows that the overidentifying restrictions of the model are strongly rejected. These tests are internally consistent only if the heteroskedasticity can be traced to the error term from the projection equation (23) of the benchmark portfolio. As Hodrick and Srivastava (1984) suggest, rejection may be due to non-constancy of betas. In order to allow for time-varying second moments we posit the following model :

<22b) (23b) (25)

E[R:+, _R:+,/1,]

=

cov[R’-”RL,II.] W:‘,, - R,!‘+ //1,1 var[R:_!jl:] - R;_,]I,]

E[R,,

cov[R:+,, R:+#,l

= “;/ frr,

= p.’ = 8,; +p;il

+&!

var[ R:_, 1I,]

Thus, <25) is a deterministic model for the Bs which includes constant bs as a special case. Our model implies the following system of regression equations: (26)

R:,, -R:+,

= (PA + B:i: + lj;+q,

+ 5:_,

to 2, given the In (26) the composite disturbances t!,, are still orthogonal normality assumption for ~,.a? With our data, the model (26) contains 11 overidentifying restrictions (30 orthogonality conditionsa less 19 parameters to be estimated).2J The l-statistic reported in the middle section of Table 4 indicates that the overidentifying conditions imposed by (26) are not rejected. as Furthermore, we carried out a Wald test of the hypothesis that pi =p =O for all is: the null hypothesis is strongly rejected for all assets. The non-rejection of the model with time-varying betas might be due to nonlinearities in the relationship between the conditional espectation of the benchmark portfolio risk premium and interest rates, rather than to time variation of the proportionality coefficients. hIode 3 in Table 4 is carried out to verify whether the rejection of (24) is due to misspecification in the projection equation <23>. We assume in this case that p! includes a constant, the levels of all five currencies’ interest rates, their squares, and all their cross products.‘6 &-hen betas are taken to be constant, the model is not rejected bp the data. Then, we espand the

t .k 4 ‘, P ‘,‘,

Constant,

.k .k .I 1, ) 1, f,

Constant,

.I f,

Constant,

.k ‘,

Constant,

2,

-

34

25

19

10

Parameters

The latent

variable

CAPM.

105

105

30

30

Number of Orthopnalit! conditions

71

80

11

20

Degrees of freedom

66.77 (0.620)

73.74 (0.675)

12.34 (0.339)

(0.0019)

43.14**

j-statistic (p-value)

3487** (0.000)

261.93** (0.000)

STK UK GE NE SW

STK UK GE NE SW

2342** 47** 32** 38** 22**

149.g** 7.1 * 10.3** 10.1 ** 10.2**

Wald test /I; = p; = 0 (p-value) Individual Joint test tests

Weekly d:~ta, from 12 July 1974 to 28 Dcccmbcr 1984 (547 observations). Uncler the null, thej-statistic is distributed chi-square with the indicated of degrees of frecclom. Rejection nt the 5 per cent and II) per cent level denoted by * and **, respectively. Marginal level of significance prenthrsrs. All test statistics computed using Generalized Method of Moments.

B/I +P:;, +P:;:

4.

No/es:

Pi

+/li;:

PA +P:;,

P:,

.3.

2.

1.

B:

Model

4.

R:,, --R/+,= /%W+t:+,

TABLE

between

number

(0.000) (0.000) (0.000) (0.000) (0.000)

(0.000) (0.029) (0.006) (0.006) (0.006)

120

Interest Rates

and Risk Premia in the Stock Market

and Foreign Exchange Market

model to allow for time-varying betas in addition to squared interest rates in the benchmark portfolio escess return (model 4 in TabIe 4). TheJ-statistic, as expected, is well within the acceptance range. Most importantly, Wald tests for the hypothesis that /?i =pi =0 for all assets do reject the null hypothesis convincingly. We thus conclude that the data are not inconsistent with a model uhere betas (i.e., the ratios between the conditional covariance of the benchmark return and the returns on foreign deposits and the stock market, and the conditional variance of the benchmark return) depend on nominal interest rates as in equation <25).

III.

Concluding

Remarks

This paper offers evidence which is relevant for the empirical literature on asset pricing models applied to both the foreign exchange market and the stock market. Fama and Schwert (1977) and Christie (1982) observed two interesting regularities in stock returns: risk premia are negatively correlated with nominal interest rates, and the volatility of stock returns is positivelv correlated u-ith nominal interest rates. We show that dollar risk premia on foreign currency deposits display the very same empirical features. In addition we are unable to contradict the hypothesis that espected dollar returns on the US stock market and foreign currency deposits move together over time. Our evidence suggests that tests of asset pricing models which assume constant conditional second moments of returns are likely to be seriously misspecified. Illustrative calculations which use our estimates imply that the link between nominal interest rates and conditional second moments could actually esplain the documented large variation in foreign exchange market risk premia. In addition, by postulating a general relationship between betas and interest rates we find that the specification of the dynamic asset pricing model of Lucas (1978) used in empirical tests by Hansen and Hodrick (1983) and Campbell (1985) is not rejected. These tests, however, should still be considered descriptions of the data, rather than formal tests of asset pricing models. In particular, we do not impose the parameter restrictions arising from the distribution of the estimated residuals. Domowitz and Hakkio (1985) and Bollerslev, et al. (1985) do impose such restrictions, using extension of the Autoregressive Conditional Heteroskedastic (ARCH) model due to Engle (1982). Unfortunately, we have found applications of these procedures to our multi-asset model still intractable computationallp.27 Our work indicates what we regard as two important research areas. On one side, theoretical work is needed on the link between time varying distributions, nominal interest rates and expected returns. On the other side, empirical tests of asset pricing models should esplicitly and consistently incorporate the reported time variation in conditional second moments. This is the subject of our current research. Notes These correlations are not reported in the paper to save space. Obstfeld (1985) presents a set of correlations that is very similar to ours. The use of foreign stock markets also rakes a few data problems. The US stock market, on the other hand, accounts for a significant fraction of aggregateUS wealth, and represents one the major alternatives for diversification of international portfolios. Cumby and Obstfeld (198-I) ha\-e also rejected homoskedasticiry under a null hyporhesis zero risk premia.

of

of

ALBERTO

GIOVANNINI

AND

PHILIPPE JORION

121

4. These results are not reported here but are available on request. 5. Engel and Frankel (1984) and Huizinga and !vIishkin (1986) a Iso report that real inrerest rates behavior changes markedly after October 1979. 6. This adjustment was also performed by Hodrick and Srivastava (1984) and Cumby and Obstfeld (1984) in their tests of substitutability between Eurocurrency and Eurodollar deposits. 7. See Abel and hlishkin (1983). 8. This issue is discussed in Xlishkin (1984). 9. However, as some algebra can readily show, b, and bz underestimate autocorrelation of risk premia to the extent that the variance of unpredictable returns exceeds the variance of predictable returns. 10. Hsieh (1984) also points out that adjusting for heteroskedasticity could have a substantial effect on tests of risk premia. 11. See for instance Bilson (1981), Hansen and Hodrick (1980), Hansen and Hodrick (1983). Hodrick and Srivastava (1984), and Fama (1984). 12. Note that the dependent variable itself contains the interest rate differential, so that a l-alue of c, equal to minus one would indicate that exchange rate movements are completely unrelated to interest rate differentials. 13. This result was generalized by Solnik (1983). who found that returns on foreign equity markets are also negatively related to their own risk-free rates. 14. Like, for example, tax rates effects on the US stock market. See Feldstein (1980). 15. From Tables 1 and 2 in Fama (1984), the equivalent number would be 0.21 for -3-82. 16. Fama (1984) reports a value of & =9.7 per month for 1973-82. Since all returns are expressed in percent, the variance terms are multiplied by 10000. 17. E.g., see Constantinides (1980). 18. This is obtained by including money balances in the utility function or by imposing a tr.msactions constraint as in Lucas (1982). 19. Identification requires normalization of the betas. The stock market beta was taken ro be one. 20. This statistic is asymptotically distributed chi-square with NK -(S + K - 1) degrees U freedom. 21. Computations were based on a frequency domain procedure. 22. This implies that iii is stochastically independent of s,. 23. We use here six instruments by equation (the level of five interest rates plus a constant for a total of 30 orthogonality conditions. 24. The parameters add up as follows: six common parameters for the vector?,, three betas times four currencies, and for the stock market, one US interest rate beta coefficient only. As before. the constant beta is normalized to one for the stock market. 25. We also estimated the system with the levels and cross-products of interest rates as instruments. This yields 105 orthogonality conditions and thus 86 overidentifying restrictions. The resulting j-statistic was 71.46 with a p-value of 0.87. As before, model (26) is not rejected b:: the data. 26. In this case, there are 105 orthogonnlity conditions (21 right-hand-side vnriab!es times 5 equations) and 25 parameters to estimate. 27. Domowitz and Hakkio (1985) impose zero correlation of returns on different currencies’ deposits. Given that most of the variation of foreign exchange market return is accounted for by exchange rate movements, this assumption effectively implies that different eschange rites move independently uis-&ris the dollar. We find this hypothesis too restrictilse to impose on the data.

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Appendix Data Sources Daily observations on spot and one-month forward rates were obtained from DRI, for the period 27 July 1973-31 December 1984. Daily stock market returns are taken from the CRSP database, for the Value-Weighted market as constructed by CRSP. Weekly one-seek Eurocurrency rates hare been collected by hand from the Fitzam.zizL Tims. Data on the French franc, Italian lira and Japanese ven start onlv in July 1978; UK one-ueek rates start in July, 1974. hfterwards the few missing one-week rates have been replaced by one-month rates. -From these daily files, we constructed monthly and weekly files where prices are measured simultaneously in the foreign exchange and stock markets. That is, when prices were missing either in the foreign exchange or in the stock market, we went back one day until we found simultaneous quotes. Times data Exchange rates are recorded at 11:30am (EST), while the Financial correspond to the close of the London market, or 12:OO noon (EST). CRSP stock market returns are based on closing trade prices of all securities on the NYSE and on the AMEX, which close at 4:00 pm (EST).