expected and unexpected changes in nominal and real variables

8/3/2019 Expected and Unexpected Changes in Nominal and Real Variables

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Journal 01Intrrnationul ,Clonr~ and Finance (1990). 9, 91-107

Expected and unexpected changes in nominaland real variables-evidence from the capitalmarkets

WALTER WASSERFALLEN*Studt. Center Gerzensee, Foundation of Swiss National Bmk. Gerzensee,

Swicerhnd

The sources of variation in asset returns and exchange rates are empiricallyexamined with the help of data from five industrialized countries. A fourfoldclassification in real vs. nominal and expected vs. unexpected changes is used.The empirical work is based on the observed time-series characteristics ofstock returns, interest rates,exchange rates and rates ofinflation. Unexpectedchanges in real stock returns and real exchange rates have by far the largertvariance. Movements in risk premia on shares and forward foreign exchangeare also substantial. Fisher and Mundell effects are, however, relatively small.

The main purpose of this paper is to document important empirical regularitiescharacterizing asset returns and exchange rates for a number of industrializedcountries. The relative importance of various sources of variability is emphasized.The observed time-series properties of interest rates, stock returns, exchange ratesand rates of inflation, as well as corresponding restrictions across linear stochasticprocesses, are exploited for that purpose. Contrary to most of the literature, theabove variables are analyzed simultaneously, allowing a more detailedinvestigation of their stochastic characteristics. No attempt, however, is made toexplain the findings in terms of a structural model.Monthly observations for the United States, West Germany, France, GreatBritain, and Switzerland over the years 1975 to 1987 are used in the empiricalwork. The main results, which are independent of the country or the period underinvestigation, can be summarized as follows: The variances of unexpected changesin real stock returns and real exchange rates are by far the largest components ofoverall variability. Movements of risk premia embedded in stock returns andforward exchange rates are also important. Several sources of variation figuringprominently in the literature, however, appear to be negligible on purelyquantitative grounds. Examples are the variance of real interest rates as well asFisher and Mundell effects.The remainder of the paper is organized as follows. The basic model is outlinedin the next section. Empirical results for asset returns and exchange rates arepresented in Sections II and III, respectively. Some conclusions are offered at theend of the paper.

*The comments by Alan Stockman and two anonymous referees are gratefully acknowledged0261-5606,90,01/0092-16$03.00 c 1990 Butterworth & Co (Publishers) Ltd


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WALTER WASSERFALLEN 93I. The model

The model for stock returns and interest rates is presented first. The framework isthen adapted to deal with exchange rates. Only the basic building blocks are givenin the text. The formulae for the variance decomposition are derived in theappendix.Adding the distinction between real and nominal returns to the model suggestedby Geske and Roll (1983), we can decompose stock returns, SR, as follows:(1) SR, = E(R,-P,)+E(P,)+E(RP,)

+C(SR,-P,)-E(SR,-P,)l+CP,-E(P,)l,where SR - P and R -P are real stock returns and real interest rates on nominallyriskless assets, respectively; P is the rate of inflation; RP denotes the aggregate riskpremium embedded in stock returns; and E symbolizes expectations formed at thebeginning of period t.The interest rate on nominally riskless assets, R, is set at the beginning of theperiod, implying that unexpected inflation and unanticipated changes in realinterest rates are of opposite sign but equal in magnitude. To avoid additionalidentification problems, the expected real interest rate is not decomposed into ariskless real rate and a premium accounting for inflation risk. The nominal interestrate can, therefore, be written as(2) R, = E(R,-P,)+E(P,) = (R,-P,)+P,.The third observable magnitude in equation (1) is the rate ofinflation. P, which isthe sum of an expected and an unexpected part.The model is completed by specifying stochastic processes for the variouscomponents of stock returns. For this purpose, the observed time-series propertiesof SR, R, and P are exploited. The evidence in this respect is presented anddiscussed in the next section. In addition, markets are assumed to be efficient.It is well known that interest rates on nominally riskless assets areapproximately generated by random walks. In this case, their components asdistinguished in equation (2) must also follow random walks.2 Consequently, theexpected nominal return on stocks, E(SR,) = R, + E(RP,), contains a unit root aswell. The expected risk premium is also assumed to follow a random walk, becausethe first difference of E(SR,) would be serially correlated, if E(RP,) were governedby a different stochastic process3 This is, however, inconsistent with the observedbehavior of stock returns. Market efficiency implies that unexpected componentsof real stock returns and inflation are purely random and independent ofexpectations. The following system of stochastic processes results:(3) E(R,-P,) = E(R,_, -P ,_,)+a l,_,,(4) E(P,) = E(P,_,)+d,_,,(5) E(RP,) = E(RP, _ 1) + ~73, 1,(6) [(SR,-P,)-E(SR,-P,)] = El , ,(7) CP,---E(P,)l ~21,where al, ~2, a3, ~1, and ~2 are random innovations which are serially uncorrelatedand only contemporaneously cross-correlated. Expectations depend only upon


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91 Changes in nmninul and real rariahlesinformation available at the beginning of period t, whereas unexpected termscapture the content of news arriving during period t. The above framework impliesthat all observable magnitudes are generated by random walks with added noise.As shown in the appendix, rather imprecise empirical estimates are the unpleasantbut unavoidable consequence.The variances and covariances of the various random terms in equations (3) to(7) are the basic magnitudes of interest in this paper. Given the random walkproperty of expected magnitudes, the proposed variance decomposition must becarried out in the stationary first differences of the variables. The formulae relatingthe elements in the basic covariance matrix to observable magnitudes are derivedin the appendix.It turns out that the covariance matrix of the innovations in the anticipatedmagnitudes, al, ~2, and a3, can be determined without further assumptions. Thenine expressions involving ~1 and ~2, however, can only be identified with the helpof additional restrictions. Covariances already known from above are assumedto be equal to economically similar cross-effects. Specifically, it is hypothesizedthat Cov(nl,~2)=Cov(n2,~l)=Cov(al, a2), Cov(a3,sl)=Cov(al, n3), andCov(a3, ~2) = Cov(n2, a3).J The first two equalities involve dependencies betweeninnovations in real returns and inflation. The third and fourth deal with therelationships of the risk premium with these two magnitudes.The framework developed above can easily be adapted to decompose thevariance of exchange rates, in which case the spot rate, S, plays the role of stockreturns. The interest rate on the nominally riskless asset and the rate of inflation arereplaced by the respective differences between two countries, DR and DP. Theresulting structure is the same as proposed by Fama (1984) with the distinctionbetween nominal and real magnitudes added. The analogues to equations (1) and(2) become(8) S, = E(S,-DP,)+E(DP,)+[(S,-DDP,)-E(S,-DP,)]+[DP,-E(DP,)],(9) DR, = E(DR,-DP,)+E(DP,) = (DR,-DP,)+DP,

= FP, = E(S, - DP,) + E(DP,) + E(RPF,).S - DP denotes the rate of change in the real exchange rate. Contrary to stockreturns, no separate risk premium must be accounted for with respect to relativechanges in exchange rates in equation (8). The interest differential, DR, inequation (9) is not only equal to the sum of the expected real interest differential,E(DR-DP), and the expected inflation differential, E(DP), but also to the forwardpremium, FP, through covered interest parity. its elements are seen to be theexpected growth in the real exchange rate, E(S- DP), the expected inflationdifferential, E(DP), and the anticipated risk premium contained in the forwardexchange rate, E(RPfJ5The stochastic structure is formulated in analogy to stock returns, taking thetime-series properties of the observed variables S, DR, and DP into account. Therespective estimates presented in Section III indicate that S is approximately whitenoise whereas DR is heavily autocorrelated. Consequently, expected magnitudesare again assumed to follow random walks and unanticipated variations are whitenoise as implied by market efficiency. 6 The resulting structure becomes(10) E(S,-DP,) = E(S,_l-DP,_l)+bl,_l,


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WALTER WASSERF ALLEN 95

(11)(12)(13)

E(DP,) = E(DP,_,)fb,_,,E(RPF,) = E(RPF,_,)+b3,_,,C(S,-DP,)-E(S,-DP,)] = pl,,

(14) [DP, - E(DP,)] = 112,.As above, the basic stochastic terms bl, b2, b3, /11, and ~2 are taken to be seriallyuncorrelated and only contemporaneously cross-correlated. The variance decom-position proceeds in the same way as for asset returns. The covariance matrix ofbl,b2, and b3 can be determined without further restrictions. Equivalent assumptionsconcerning the equality of covariances are made, namely Cov(b1, j(2)=

Cov(b2, Ltl) = Cov(bl, b2), Cov(b3, pl) = Cov(b1, b3), and Cov(b3, p2) = Cov(b2, b3).Further details are given in the appendix.Some further magnitudes of interest derived from the basic terms are alsoinvestigated. Essentially, the actual change of the considered variables, X,--X,_ r,the change in expectations, E(.u,) - E(x, _ 1 , the expected change, E(x,) - x, _ I, andthe unexpected change, X,-E&), are distinguished. Furthermore, estimates ofFisher and Mundell effects are obtained. Fisher effects are defined as thecovariances of changes in R, E(SR), DR, and E(S) with expected changes in P orDP, respectively. Mundell effects are defined as the covariances of expectedchanges in real interest rates, real stock returns, real interest rate differentials, andgrowth rates of real exchange rates with expected changes in inflation or inflationdifferentials, respectively. The formulae are also given in the appendix. Notefurther that the Fisher effect, as generally defined, is the sum of one, the pureFisher effect, and the Mundell effect.

II. Empirical results for asset returnsThe data used in the empirical work include interest rates for one-monthEuromarket deposits, stock market indices, and seasonally unadjusted consumerprice indices for the United States, West Germany, France, Great Britain, andSwitzerland. End-of-month observations over the period 1975 to 1987 areselected, yielding a sample size of 156 observations. Returns and rates of inflationare continuously compounded and expressed as per cent per month. Variances andcovariances are estimated from the stationary first differences of the originalobservations. All calculations are also performed for several subperiods. Theresults are robust over time and are, therefore, not included below.Relevant stochastic characteristics for all variables are presented in Table 1. Themeans and standard deviations of the levels of the indicated series are shown as wellas first-order autocorrelation coefficients for the levels and the first differences.Higher-order serial correlation coefficients for the differenced series (not shown)are generally not significantly different from zero, a feature implied by the modeloutlined in the previous section.Interest rates exhibit high serial correlation that falls only slowly with increasinglags. The first differences generally exhibit some weak negative dependence,indicating slight overdifferencing. More formal tests for unit roots confirm thisfinding. The autocorrelations in the levels, however, decay more slowly than wouldbe implied by an autoregressive process of order one. The chosen random walkapproximation therefore appears to be justified. Moreover, small deviations from


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96 Changes in nominal and real rariablesTABLE 1. Stochastic characteristics of returns.

United WestStates Germany France

GreatBritain Switzerland

Interest ratesMeanStand. dev.r, (level)r, (diff.)

0.14 0.41 0.92 0.92 0.300.21 0.20 0.33 0.21 0.1802 0s 0.77 02 0a02 _o.zo :E -0.29 -0.29

Stock returnsMeanStand. dev.rl (level)r, (diff.)Inflation ratesMeanStand. dev.r, (level)r, (diff.)

0.63 0.48 0.92 0.98 0.404.60 5.36 5.99 6.54 4.290.12 -0.01 0.17 0.01 0.14

___0.41 2 -0.0. -0.42 33

0.51 0.27 0.69 0.78 0.240.36 0.30 0.38 0.77 0.360 .73 0.54 0.52 0.48 0.18

+J.lJ -0.28 3.0 -0.3 +I5JStock returns minus interest ratesMean -0.12 -0.02Stand. dev. 4.63 5.33r, (level) 0.13 -0.01r, (diff.) -0.4! IO.49

0.00 0.07 0.105.98 6.53 4.300.17 0.01 0.15

=0__36 T.P.42 -0.42Real interest ratesMean 0.23Stand. dev. 0.36r, (level) !Er, (diff.) -=0.2J

0.20 0.23 0.13 0.060.30 0.42 0.74 0.37U? 02.5 0.43 0.20-_To.27 ~~ 90 -0.47

Real stock returnsMean 0.11Stand. dev. 4.61rl (level) 0.14r, (diff.) zO.4J

0.18 0.23 0.20 0.165.37 6.00 6.52 4.320.00 0.17 0.03 0.14

:z -0.2 To.41 I?L43IVotes:Variables are measured monthly and expressed as per cent per month. The estimation periodextends from 1975 to 1987 (156 observations).Nominal interest rates: Euromarket rates for l-month deposits (end of month).Nominal stock returns: Relative changes in stock market indices (end of month).Inflation rates: Relative changes in consumer price indices.Real interest rates: Nominal interest rates minus inflation rates.Real stock returns: Stock returns minus inflation rates.rl (level): First-order autocorrelation coeflicient of the levels of the series.r, (diff.): First-order autocorrelation coelficient of the first-differenced series.Underlined autocorrelation coeflicients are signilicantly ditTerent from zero on the 5 per cent level(standard error of r, =0.08).


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WALTER WASSERFALLEN 97

the random walk would only have a weak influence on the empirical resultsbecause short-term variations are the focus of the analysis.Stock returns behave very differently. The volatility is much higher andautocorrelation is almost absent. Recently, some researchers have argued thatstock prices exhibit slight mean-reverting behavior and may therefore be describedby a stationary stochastic process. * Here it is maintained that even stock returnsmust contain a unit root because they implicitly include the interest rate on thenominally riskless asset. Therefore, second differencing of stock prices or firstdifferencing of returns is necessary to obtain a stationary time series. Takentogether, the observed behavior of interest rates and stock returns indicates thatunexpected movements probably dominate the return generating process.Inflation rates are quite heavily serially correlated but have a comparatively lowstandard deviation. Due to the high variability ofstock returns, the characteristicsof relative returns, SR - R, and real stock returns, SR- P, are almostindistinguishable from the properties of stock returns.The results of the variance decomposition are given in Table 2. The entries arepercentages ofunexpected variations in real stock returns, ~1. This way, the relativeimportance of various sources of movements in returns can be easily evaluated. Forall countries, the variance of unexpected real stock returns clearly dominates. Noother value in the covariance matrix of the basic terms distinguished in the model isnearly as high. Variations in the expected risk premium on shares are also quiteimportant for most countries. The high variances ofactual and expected changes innominal and real stock returns are no surprise because the respective expressionscontain ~1.

Rather surprisingly, covariances in general amount to less than 1 per cent of thevariance in ~1. Even the Fisher and Mundell effects, figuring prominently in theliterature, contribute very little to the explanation of varying returns. Some general

TABLE 2. Variance decomposition of returns.United WestStates Germany France GreatBritain Switzerland

VariancesNominal interest ratesActual changeNominal stock returnsActual changeChange in exp.Expected changeUnexpected changeChange in expected riskpremium (a3)Inflation ratesActual changeChange in exp. (a2)Expected changeUnexp. change (~2)

0.07 0.04 0.23 0.03 0.07240.87 202.84 271.90 252.72 233.4446.04 5.99 68.58 51.29 35.12142.18 103.43 172.26 149.83 135.0998.70 99.41 99.64 102.89 98.35

42.57 4.30 69.57 46.86 36.670.46 0.28 0.64 1.71 1.560.29 0.13 0.04 0.50 0.010.10 0.07 0.35 0.62 0.770.36 0.21 0.29 1.09 0.79


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98 Chanyrs in nominal anti red cariablesTABLE 2. continwf

VariancesReal interest ratesActual changeChange in exp. (al)Expected changeReal stock returnsActual changeChange in exp.Expected changeUnexp. change (~1)

CovariancesBasic terms(al,a2)=(al,s2)=(a2,sl)(al,a3)=(a3,&1)(al,cl)(02, (13) = ((13, E2)(a2, ~2)(cl, E2)Mttnrfell ejJectInterest fatesStock returnsFisher ejfctInterest ratesStock returnsRelatice real returnsExpectedUnexpected

UnitedStates

WestGermany France

GreatBritain Switzerland

0.46 0.32 0.77 1.71 1.640.21 0.14 0.08 0.50 0.080.10 0.11 0.48 0.62 0.85

243.16 203.92 212.67 246.38 236.8646.09 5.98 68.60 51.26 35.12

143.16 103.92 172.67 146.38 136.86100.00 100.00 100.00 100.00 100.00(15.34) (28.56) (21.99) (35.68) (13.43)

-0.26 -0.13 0.05 -0.49 - 0.021.60 0.78 -0.53 1.96 -0.82

-0.19 -0.25 - 1.52 0.49 -0.03-2.01 -0.59 -0.38 - 1.56 0.96

0.28 0.13 0.00 0.49 0.00-0.82 - 0.40 -0.33 0.91 - 1.25

-0.08 -0.08 -0.30 -0.61 0.19- I .08 -0.53 -0.28 - I .39 - I.28

0.01-0.47

0.00- 0.20

0.350.40

-0.04-0.04

I .070.33

0.022.01

-0.97-0.91

0.01-0.44

0.600.82

1.491.25

Notes:Variables are measured monthly and expressed as per cent per month. The estimation period extends from1975 to 1987 (156 observations). Entries are percentages of the variance of unexpected changes in real stockreturns, a, (absolute values given in parentheses).Nominal interest rates: Euromarket rates for l-month deposits (end of month).Nominal stock returns: Relative changes in stock market indices (end of month).Intlation rates: Relative changes in consumer price indices.Real interest rates: Nominal interest rates minusinflation rates.Real stock returns: Stock returns minus inflation rates.Mundell effect: Covariance ofexpected changes in real interest rates and real stock returns, respectively, withexpected changes in inflation rates.Fisher effect: Covariance of changes in nominal interest rates and expected nominal stock returns.respectively, with expected changes in inflation rates.Relative real returns: Covariance ofexpected and unexpected changes in real interest rates. respectively, withcorresponding changes in real stock returns.Exp.: Expected or expectations.


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WALTER WASSERF ALLEN 99statements can nevertheless be made. Expected real interest rates and stock returnsare negatively related to expected inflation, implying that the sign of the Mundelleffect is theoretically correct. The Fisher effect, that is the relationship betweenexpected nominal returns and anticipated inflation, should however be positivewhich is the case for interest rates but not for stock returns. With the exception ofGreat Britain, expected and unexpected variations in real stock returns and realinterest rates are furthermore positively related. These results are in generalagreement with the literature.It is quite often maintained that the Fisher effect is closely fulfilled for nominallyriskless assets and that interest rates can therefore be taken as readily observableproxies for expected inflation. This procedure would be valid if the variance ofchanges in expected real interest rates, a 1, is much smaller than the variance of changesin expected inflation, a 2. The results in Table 2 show that this condition is generallynot fulfilled.

III. Empirical results for exchange ratesThe presentation of the empirical estimates follows the same pattern as inSection II. Exchange rates, observed at the end of the month, are measured asSwiss Francs per unit of foreign currency, including the US dollar, the Germanmark, the French franc, and the British pound. The data for interest rates andinflation are the same as above. Exchange rates have been flexible over theobservation period that covers the years 1975 to 1987, as in Section II. Rates ofchange are again continuously compounded and expressed as per cent per month.The results for subperiods are the same and are therefore not discussed.The stochastic characteristics of the different variables are shown in Table 3. Inalmost all respects, the results are strikingly similar to the ones found for assetreturns. In accordance with the theoretical framework presented in Section I, serialcorrelation at lags beyond one (not shown) is generally absent.Interest rate differentials, defined as the Swiss minus the corresponding foreigninterest rate, are highly serially correlated but have a comparatively low variance.

TABLE3. Stochastic characteristics of relative changes in exchangerates.


GreatBritain

Nominal interest rate differentials (forward premia)Mean -0.44 -0.17 -0.62Stand. dev. 0.22 0.13 0.29rl (level) 0.90 0.83 0.68r, (diff.) 0.14 -0.22 -0.29Relativ e changes in nomina l exchange ratesMean -0.41 -0.17 -0.57Stand. dev. 3.89 1.74 2.10rI (level) -0.01 @ 0.19rl (diff.) -0.58 -0.34 -0.41

-0.620.22m

-0.28

-0.583.270.13

-0.41


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100 Changes in nominal ant i real rarinhlesTABLE 3. continued


GreatBritain

fnj7ation diffrrentialsMean -0.21 - 0.03 -0.45 -0.54Stand. dev. 0.43 0.35 0.50 0.80r, (level) 0.31 0.15 0.17 0.39rt (diff.) ZL?L% -0.43 -0.52 yo.37Relat ke changes in exchange rates minus int erest rat e ti ijerentialsMean 0.04 0.00 0.05 0.04Stand. dev. 3.94 1.77 2.09 3.34r, (level) 0.01 0.22 0.19 0.16r, (diff.) -o:!Ec -0133 +?.40 ro.40Red int erest rat e diflerentia lsMean -0.17 -0.14 -0.17 -0.07Stand. dev. 0.45 0.37 0.53 0.11r, (level) 0.35 0.17_ 0.27 0.33rl (diff.) g.g -XL!!.42 Y-o.48 +L36Relat ice changes in real exchange ratesMean -0.13 -0.14Stand. dev. 3.96 1.77rl (level) -0.03 0.17r, (diff.) -0.59 z-

-0.12 -0.032.14 3.410.12 0.14

-M ~$IJ

Notes:Variables are measured monthly and expressed as per cent per month. Theestimation period extends from 1975 to 1987 (156 observations).Nominal interest rate differentials: DitTerence between Euromarket rates onl-month deposits for Swiss francs and corresponding foreign currency (endof month).Nominal exchange rates: Spot rates, measured as Swiss francs per unit offoreign currency (end of month).Inflation differentials: Diflerence between Swiss and corresponding foreigninflation rates, measured by consumer price indices.Real interest rate differentials.: Nominal interest rate differentials minusinflation differentials.Relative changes in real exchange rates: Relative changes in nominalexchange rates minus inflation differentials.rl (level): First-order autocorrelation coetlicient of the levels of the series.rl (diff.): First-order autocorrelation coefticient of the iirst-differencedseries.Underlined autocorrelation coeflicients are significantly different from zeroon the 5 per cent level (standard error of rl =0.08).

Autocorrelation in the differenced variables again suggests slight overdifferencing,which is confirmed by unit root tests. For the same reasons as in Section II, therandom walk assumption is nevertheless maintained.Relative changes in spot exchange rates behave very much like stock returns.The standard deviation is however considerably smaller. First differencing appears


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WALTER WASSERFALLEN 101unnecessary to achieve stationarity, indicating that unexpected movementsdominate. Note further that the Swiss franc on average appreciated relative to theother currencies as indicated by the mean.Inflation differentials, measured as the Swiss minus the respective foreigninflation rate, exhibit comparatively low volatility, implying that thecharacteristics of real and nominal exchange rates are almost identical.The outcome of the variance decomposition is contained in Table 4. Theestimation is carried out in the stationary first differences of the variables. Theprocedures are essentially the same as in Fama (1984), who however works withpartly nonstationary observations and concentrates exclusively on nominalmagnitudes. The entries in Table 4 are percentages of the variance of ~1, theunanticipated component of relative changes in real exchange rates.The results are comparable to Fama (1984), but the distinction between nominaland real magnitudes provides interesting additional insights. Unexpectedvariations in the growth of real exchange rates clearly dominate as a source ofvariability. Changes in the anticipated growth of real exchange rates, 61, and in theexpected risk premium embedded in the forward exchange rate, b3. are alsosubstantial.

TABLE 4. Variance decomposition of relative changes in exchange rates.

UnitedStates

WestGermany France

GreatBritain

VariancesNom inal int erest rate differentials (forw ard premia)Actual change 0.06 0.31Change in expected risk

premium (b3) 1.09Relat ive changes in nom inal exchange rates

45.96

Actual change 187.65 153.11 159.29 161.17Change in exp. 1.03 41.93 30.09 25.37Expected change 93.00 55.28 65.93 67.81Unexpected change 94.65 97.83 93.36 93.36Inflation difjerentialsActual changeChange in exp. (b2)Expected changeUnexpected change (4)

1.58 6.52 9.07 6.730.18 0.93 0.22 1.810.73 2.79 5.09 2.500.85 3.73 3.98 -1.23

Real interest rate differentialsActual changeChange in exp.Expected change

1.58 6.52 9.07 6.730.28 0.62 0.44 1.950.73 2.80 5.09 2.50

Relat ice changes in real exchange ratesActual change 198.24Change in exp. (bl) 1.16Expected change 98.24Unexpected change (~1) 100.00

(16.44)

163.35 178.10 172.1335.40 23.67 26.8363.35 78.10 72.13

100.00 100.00 100.00(3.22) (4.52) (11.59)

1.3331.86

0.1727.52


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102 Changes in nom inal rmd red carirrblrsTABLE 4. continued

UnitedStates

WestGermany France

GreatBritain

CovariancesBasic terms(bl,b2)=(bl,~2)=(b2,~~1)(bl,b3)=(b3,pl)(bl,/d)(b2, b3)= (b3,$)(b2, $1(PI. jaMundrll efSeccInterest ratesExchange ratesFisher effectInterest ratesExchange rates

-0.12 2 . 8 00.97 - 40.06I .46 36.02

-0.06 - 3.730.18 0.93

-3.10 -2.80

-0.73 - 2.80-2.98 -3.11

0.00 0.00 0.66 0.002.55 2 . 8 0 2.88 -4.49

3.10 - 1.64- 27.65 - 26.23

22.79 27.35-2.88 -0.17-0.44 1.81-5.53 -5.26

- 4.42 - 2.42-8.63 -3.62

Notes :Variables are measured monthly and expressed as per cent per month. The estimation periodextends from 1975 to 1987 (156 observations). Entries are percentages of the variance ofunexpected variations in growth rates of real exchange rates. /it (absolute values given inparentheses).Nominal interest ratedifferentials: Difference between Euromarket rates on l-month depositsfor Swiss francs and corresponding foreign currency (end of month).Nominal exchange rates: Swiss francs per unit of foreign currency (end of month).Inflation differentials: Difference between Swiss and corresponding foreign inflation rates,measured by consumer price indices.Real interest rate differentials: Nominal interest rate differentials minus inflation differentials.Relative changes in real exchange rates: Relative changes in nominal exchange rates minusinflation differentials.Mundell etTect: Covariance of expected changes in real interest rate differentials and expectedchanges in growth rates of real exchange rates, respectively, with expected changes in inflationdifferentials.Fisher effect: Covariance of changes in nominal interest rate differentials and expected changesin growth rates of nominal exchange rates, respectively, with expected changes in inflationdifferentials.Exp.: Expectations.

In contrast to asset returns, some covariances are quite large. Interestingly, onlyreal magnitudes are involved in these cases. The negative covariance between bland b3 is the consequence of the rather variable risk premium that is part of therelatively stable interest rate differential. The covariances of bl and b3 with /rl areabout the same in magnitude but of opposite sign. It, therefore, appears that newinformation becoming available in a given period induces unexpected changes inreal exchange rates and also leads to substantial but opposite revisions in expectedreal exchange rates and risk premia. Terms involving cross-effects betweennominal and real variables, including Fisher and Mundell effects, are howeverrelatively unimportant. *Given these results, it is not surprising that the evidence on international parity


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WALTER WASSERFALLEN 103conditions, such as the equality of expected real interest rates across countries ore-u-ant e relative purchasing power parity, depends heavily on the variables used inthe respective tests. The results obtained by Cumby and Obstfeld (1984) andMishkin (1984) are good examples in this respect. Whenever the highly noisyexchange rate enters the empirical procedure, the underlying hypotheses areaccepted, whereas estimates based on interest and inflation differentials alonereject the same theoretical formulation. In any case, the deviations from the parityconditions mentioned above are small compared to unexpected movements in realexchange rates.

IV. ConclusionsThe sources of variations in asset returns and exchange rates are empiricallyexamined over the period 1975 to 1987 for the United States, West Germany,France, Great Britain, and Switzerland. A fourfold classification in real vs.nominal and expected vs. unexpected changes is used. Contrary to most of theexisting literature, the various magnitudes are analyzed simultaneously and moreattention is paid to the time-series properties of the different variables, especially toapparent nonstationarities. No attempt is made to explain the documentedempirical regularities with the help of a structural model.On the whole, the findings for asset returns and exchange rates are strikinglysimilar as well as consistent across countries and time. The main results can besummarized as follows:1. Unexpected changes in real stock returns and real exchange rates are by far thelargest components.2. Variations in risk premia contained in stock returns and forward exchangerates are also substantial. The often used assumption of time-invariant riskpremia is, therefore, inconsistent with the evidence.3. Relative changes in exchange rates, but not asset returns, exhibit quite largevariability in the expected real component. The covariance between theinnovations in the stochastic processes ofthe anticipated real exchange rate andthe risk premium on the forward market is negative and of comparable size.4. The variability of inflation rates, interest rates on nominally riskless assets, andall cross-effects between nominal and real variables are of relatively trivialmagnitude. Fisher and Mundell effects, often examined in the literature, fall inthis category.

The nature of the information inducing changes in asset prices and exchangerates is currently not well understood. In principle, both real factors, such aschanges in technology, tastes, risk aversion and regulatory practices, as well asnominal forces-for example, changes in the money supply process-can beresponsible for the observed variations. However, given current knowledge, mostof the observed volatility in asset markets is probably best treated as risk forpractical purposes.AppendixThe appendix contains three parts. The variance decompositions of asset returns andexchange rates are explained tirst. It is seen that all variables are generated by a randomwalk with added noise. Some general characteristics of this process are derivedsubsequently.


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WALTER WASERF ALLEN 10 5Variance decomposit ion of exchange rates

The same steps as above are taken to decompose the variance ofexchange rates. In this case,the model is given by equations (8) to (14) in the text. The observable stationary variablesare the first differences of S, DR, DP, S- DR, S- DP, and DR- DP, again denoted bycorresponding small letters. The elements of the covariance matrix involving bl, 62, and b3can be determined without further restrictions as(Al8) Var(b1) = g(s-dp),(A19) Var(b2) = Mp),(A2O) Var(b3) = y(s - dr),(A21) Cov (bl,b2) = OS[g(s)-g(s-&)-g(dp)],(422) Cov(bl,b3) = OS[g(dr-dp)-g(s-dp)-g(s-dr)],(A23) Cov(bZ,b3) = OS[g(dr)+g(.s-@)-g(s)-g(dr-dp)].Using the four restrictions in the text, the remaining live terms are sequentially given as(A24)(A25)(A26)

(A27)


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106 Cllclnyes in nom inal and real caridlesThe relative variance of expected and unexpected variations in I is implied by the

autocovariance function of .Y and therefore by the parameter 5. From Box and Jenkins(1976, pp. 123-124), the following relationships can be derived(A33) Corr(s) = Cov(.y)/Var(.u) = -CT/( I+ T')] ,(A34) [Var(6)/(Var(P)- Cov(ii, /I)] = [( 1 - T)~/T].The dependencies between the parameter r, the lirst-order autocorrelation in .Y.Corr(.u),and the covariance matrix of the basic stochastic terms 6 and /3 are seen to be highlynon-linear. The consequences are imprecise estimates of the various terms in the basiccovariance matrix. In the following illustrations of this important problem the covariancebetween 6 and B is assumed to be zero. Suppose 7 is 1.0 and consequently Corr(r) is - 0.50,a value commonly found for stock returns and exchange rates. In this case, X consists onlyof the unexpected component [I and is therefore overdifferenced. In the empirical uork, thesample size is 1.56 observations, implying a standard error for Corr(.u) of 0.08. A value forCorr(.u) of -0.34 is therefore still consistent with overdifferencing of X if tests are carriedout on the 5 per cent level. But under these conditions, T is about 0.4 and Var(ci) is aboutequal to Var(fl). Relatively small variations in the autocorrelation properties of the sampleobservations therefore imply substantial differences in the relative importance of varioussources of volatility.

Notes1. The literature on these issues is huge. Space constraints prevent even short surveys,References are therefore made only to selected papers.2. This is an application of the constraints derived by Zellner and Palm (1974). Direct evidenceon this point is provided by Rose (1988), who investigates the time-series properties ofnominal interest rates and price indices for the United States.3. Poterba and Summers (1986) and French et nl. (1987) report some direct evidence on thispoint for the United States. It appears that the variance ofdaily returns on the Standard andPoors Composite Index. which should be closely related to the risk premium. is heavilyserially correlated. Based on an empirical analysis of stock returns alone, Conrad and Kaul(1988) argue that expected nominal stock returns are generated by a first-orderautoregressive process.4. Factors of proportionality other than one in these restrictions would not have a large effecton the empirical results because the quantitative importance of the covariance terms isrelatively small.5. Remember that a risk premium in the expected real interest rate, which is the equivalent toRPF, is neglected in equation (2).6. Wolff (1987) identifies models of the risk premium by focusing exclusively on the forecast

error resulting from the forward rate as a predictor of the future spot rate.7. The following market indices for stock prices are used: Dow Jones Industrials for the UnitedStates, FAZ for West Germany, CAC for France, FT ordinary for Great Britain. and SBV forSwitzerland.8. Examples are Fama and French (1988) and Poterba and Summers (1988). who use onlyinformation on stock prices. Note however that mean reversion can also result ifstock pricesare determined by a mixture of stationary and nonstationary components.9. Inflation rates exhibit slight seasonality which is probably the result of infrequent samplingand fixed index weights. The return generating process correctly ignores this feature.10. Geske and Roll (1983) provide a summary of the evidence and possible explanations.11. Huizinga (1987) observes weak mean reversion in real exchange rates. The comments madein Section II with respect to stock returns are again relevant.

12. The Fisher effect for exchange rates is better known as purchasing power parity.References

Box, GEORGE E.P., AND GWILYM M. JENKINS, Time Series Analysis, Forecasting and Control,San Francisco: Holden-Day Inc, 1976.


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WALTER WASSERFALLEN 107CONRAD, JENNIFER. AN D GAL;TA.M KAUL, Time-Var iat ion in Expected Returns, Jolcrnnl ofBusiness, December 1988. 61: 409-425.CUMBY. ROBERT E., AND MAURICE OBSTFELD, Intern ational Interest Rate a nd Price Level

Linkages u nder Flexible Exchan ge Rat es, A Review of Recent Evidence, in John F.O.Bilson and Richard C. Marston, eds, Exchange Rnte Theory nd Pructice, Chicago:University of Chicago Press. 1984, 121-151.FAMA, EUGE NE F., Forwa rd an d Spot E xchan ge Rates, Journnl of Monetary Economics,November 1984, 14: 319-338.

FAMA, E UGE NE F., AND KENNETH R. FRENCH, Perm anent and Temporary Components ofStock Prices, Journal of Political Economy, April 1988, 96: 246273.

FREN CH, KENNE TH R., G. WILLIAM SCHWERT, AN D ROBE RT STAMBAUGH, Exp ected St ockReturns and Volatility, Journal of Financial Economics, September 1987, 19: 3-30.

GESKE , ROBERT,AN D RICHARD ROLL, The Fiscal an d Moneta ry Linka ge between Stock Retu rn sand Inflation, Journnf of Finance, March 1983, 38: l-33.

HUIZINGA, JOHN, An Empirical Investigation of the Long-Run Behavior of Real ExchangeRates, Carnegie-Rochester Conference Series on Public Policy, Autumn 1987,27: 149-214.MISHKIN, FREDERIC S., Are Real Int erest Rates Equal Across Countries?, An Empirical

Investigation of International Parity Conditions, Journal of Finance, December 1984,39:1345-1357.POTERBA, J AMES M., AN D LAWRENCE H. SUMMERS, The Per sistence of Volatility an d St ock

Market Fluctuations, American Economic Review, December 1986, 76: 1142-l 151.POTERBA, J AMES, M., AN D LAWRENCE H. SUMMERS, Mean Reversion in Stock Prices, Evidenceand Implications, Journnl of Financial Economics, March 1988, 22: 27-59.ROSE, ANDREW K., Is the Real Interest Rate Stable ?, Journal of Finance, December 1988, 43:1095-l 112.WOLFF, CHRISTIAN C.P., Forwa rd Foreign Exchan ge Raies, Expected Spot Rates, an d Pr emia:

A Signal-Extra ction Approach, Journal of Finance. June 1987, 42: 395-406.ZELLNER, ARNOLD, AND FR AXZ PALM, Time Series Analysis and Simultaneous EquationEconometric Models, Journd of Econometrics, March 1974, 2: 17-54.

expected and unexpected changes in nominal and real variables

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