Stochastic prices and tests of efficiency of foreign exchange markets

Stochastic prices and tests of efficiency of foreign exchange markets

165 Economics Letters 6 (1980) 165- 170 North-Holland Publishing Company STOCHASTIC PRICES AND TESTS IGN EXCHANGE MARKETS * OF EFFICIENCY OF FORE-...

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165

Economics Letters 6 (1980) 165- 170 North-Holland Publishing Company

STOCHASTIC PRICES AND TESTS IGN EXCHANGE MARKETS *

OF EFFICIENCY

OF FORE-

Jacob A. FRENKEL University of Chicago, Chicago, IL 60637, USA National Bureau of Economic Research, New York, NY 10012, USA

Assaf RAZIN Tel Aoio Unioersity, Tel-Aviv, Israel Received

2 February

1981

This paper shows that typical specifications of efficiency tests of foreign exchange markets are strictly valid only when individuals are risk neutral and prices are non-stochastic. Empirically, however, the neglect of these factors is shown to be of little significance.

Recent studies of efficiency of foreigh exchange markets examined the relationship between forward exchange rates and subsequent realizations of the spot rates. Typical regressions in such studies were of the form S, =a+bF,_,

+t,,

(1)

where S, denotes the spot exchange rate at period t (the price of foreign exchange in terms of domestic currency), F,_, denotes the forward exchange rate at period t - 1 for a contract to be delivered at period t, and where E, denotes a serially uncorrelated error term with a zero mean [see, e.g., Frenkel(1977,1981), Levich (1978,1979), Hakkio (1979)]. In this context the requirement that F,_, is an unbiased forecast of S, implies that in eq. (1) the constant term is zero and the slope coefficient is unity.

* This research is part of the NBER’s program in International Studies. The views expressed are those of the authors and not necessarily those of the NBER. J.A. Frenkel acknowledges an NSF grant SOC78-14480 for financial support.

016%1765/81/0000-0000/$02.50

0 North-Holland

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Stochastic prices and tests of efficiency

The formulation in eq. (1) includes only variables which pertain to the foreign exchange market; specifically, this equation does not contain variables which pertain to the commodity markets such as price levels. In this note we examine the circumstances under which such a formulation is valid, and we show that the omission of prices from the exchange rate equation is strictly valid only when individuals are risk neutral and when the covariance between prices and exchange rates is zero. Using a very simple model we derive parity conditions relating prices to spot and forward exchange rates and we then examine the empirical relevance of the role of prices. It should be emphasized that the importance of the attitude towards risk has been recognized in the literature [e.g., Stockman (1978) Frankel (1979), Kouri (1977) Fama and Farber (1979)] but the emphasis on the relationship between exchange rates and prices in this context may not have received sufficient attention. Consider the following two-period model of an individual who maximizes expected utility. Assume that the individual holds domestic and foreign short-term nominal bonds (denoted by B and B* respectively), as well as forward contracts of foreign exchange (denoted by A). Denote the first period consumption by C, and the second period consumption by C,(a) where LXdenotes a state of nature in the second period. C,(cu) can be written as

Cd4 =

RB+S2(a)R*B*+[SZ(a)-F,]A P2(4

9

(2)

where R denotes the return on domestic bonds in terms of domestic currency (i.e., one plus the domestic rate of interest), R* denotes the return on foreign bonds in terms of foreign currency, S,(a) denotes the spot exchange rate in period two in state a and P2( a) denotes the second period price level in state (Y.This formulation assumes that in the second period income consists only of the return on the portfolio. Formally, the individual’s problem can be written as RB+S2(a)R*B*+[S2(a)-F,]A max

tY(C,) +PE P,(ff)

C,,B,B*,A

subject

to

P,C, + B + S, B* = W,,

(3)

J.A. Frenkel, A. Ruin

161

/ Stochastic prices and tests of efficiency

where P,, C,, S, and W, denote respectively prices, consumption, spot exchange rate, and the individual’s resources in the first period. The utility function U( .) is assumed to be concave thus reflecting aversion to risk. In the limiting case we will assume that individuals are risk neutral so that U’( .), the marginal utility of second period consumption, is constant. In eq. (3) p denotes the subjective discount factor and as is evident, the holdings of A, B and B* need not all be positive. For example, if the individual is a net seller of forward contracts A would be negative. It is also relevant to note that the formulation in (2) assumes that forward contracts entail no margin requirement so that the interest cost is zero. The first-order conditions imply that the ratio of the forward to the spot exchange rate must equal the ratio R/R*: F,/S,

which, known

= R/R*.

by subtracting interest parity

(4)

one from both condition

sides of eq. (4), yields

the well-

F, - S, _ i, - i: s,

l+i:’

where i and i* denote respectively the domestic and the foreign rates of interest. The first-order conditions for the maximization of (2) also imply that

where U’(a) denotes the marginal utility of second period consumption in state CL Condition (5) relates the current forward exchange rate to the future spot rate. As is evident this relationship involves the marginal utility of second period consumption US well US the second period price level. It is noteworthy however that in the special case of full certainty eq. (5) reduces to F, = S,, which states that the forward

(5a) rate in period

1 equals the realized

second

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J.A. Frenkel, A. Razin / Stochasric prices and tests of efficiency

period’s spot rate. In the general case, however, this equality need not hold. In order to examine the relationship between the current forward rate, F,, and the expected future spot rate, E{S,( (II)}, it is convenient to use the definition of a covariance [according to which cov(x, y) = Exy ExEy] and to rewrite eq. (5) as

Eq. (6) provides the base for the analysis of the role of prices. Consider first the case in which prices follow a fully deterministic path, so that P2(~) = P2. Under these circumstances the difference between F, and E{S,( LX)}depends on the attitude towards risk [i.e., on the sign of U”(e)] as well as on the initial asset position (i.e., on the signs and magnitudes of B* and A). For example, if the individual is a net holder of foreign bonds and of forward contracts of foreign exchange [i.e., if B* and A are positive so that a rise in &((Y) raises second period consumption] and if he is risk averse [i.e., U”( .) < 01, then the covariance term in (6) would be negative and the forward rate would be smaller than the expected future spot rate. This example illustrates the relevance of the attitude towards risk and the net asset position which were emphasized in the literature. In order to highlight and isolate the role of prices, consider next the special case for which the individual is risk neutral so that V’( .) = 0. Under these circumstances the second term on the right-hand side of (6) becomes cov[ S,( (u), l/P,( a)]/E{ l/P,(a)}, and thus

Eq. (7) reveals that even when individuals are risk neutral, the forward exchange rate need not equal the expected future spot rate when the price level is stochastic (unless the covariance between it and the exchange rate is zero). Consequently, estimates of the constant term in eq. (1) need not be zero even under conditions of risk neutrality and, therefore, the deviations between the estimates of the constant term and zero may not be interpreted as measures of the risk premium. The foregoing discussion emphasized that as an analytical matter the attitude towards risk, the initial asset positions and the stochastic pattern

J.A. Frenkel, A. Razin / Stochustlc

prices and tests of efficiency

169

of prices may induce a discrepancy between the forward exchange rate and the expected value of the future spot rate. As an empirical matter the relevant question is whether these considerations are of a sufficient quantitative significance so as to make them important for the interpretation of estimates of the constant term in eq. (1). To deal with the empirical question we have estimated eq. (1) for various exchange rates using monthly data from the 1920s and the 1970s. Data sources for the 1920s and the 1970s are the same as in Frenkel (1977 and 1981), respectively. The estimates of the constant terms are denoted by ci and are reported in table 1. In order to conform with the formulation of eq. (7) we also report in table 1 estimates of the constant terms of a restricted form of eq. (1) in which the coefficient on the lagged forward rate is contrained to be unity. These estimates are denoted by 6’. As is evident in all cases the constant terms do not differ significantly from zero. The discussion of eq. (7) demonstrated however that when prices are stochastic the constant term need be zero even when individuals are risk neutral. In order to examine the relative importance of this

Table 1 Efficiency

Period

Feb. 1921May 1925

June 1973July I979

tests and stochastic

Exchange

rate

prices:

monthly

data. a

Price index

Constant

Cov(S,l/P)

ci

ir’

Dollar/Pound

0.1589 (0.1626)

0.0158 (0.0116)

Wholesale Material

Dollar/Franc

0.0038 (0.0035)

-0.0004 (O.OOQ7)

Wholesale Material

0.0002 0.0005

Dollar/Pound

0.09 15 (0.0489)

0.0024 (0.0064)

Wholesale Cost of living

0.0234 0.0197

Dollar/Franc

0.035 1 (0.0116)

0.0005 (0.0008)

Wholesale Cost of living

-0.0000 0.0002

Dollar/DM

0.0083 (0.0142)

0.0011 (0.00 17)

Wholesale Cost of living

-0.0056 - 0.005 1

E(l/P) -0.0051 -0.0179

a The constants ci and d’ are respectively the OLS estimates of n and n’, from the exchange rate equations S, =a+bF,_, +eI and S, -I$, =a’+“,; standard errors of the estimates are in parentheses.

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J.A. Frenkel, A. Razm / Stochastrc prices and tests of efficiency

factor we have also computed the constant term that would have been implied under these circumstances. These estimates, which are reported in table 1 under the heading cov( S, l/P)/E( l/P), were obtained by using alternative series of the U.S. price indices. From eq. (7) it follows that when individuals are risk neutral these estimates should be equal to - ci’. The results in table 1 show that for the 1920s this hypothesis is in accord with the data (particularly for the material price index) and, that the estimates of the constant terms of eq. (7) are relatively small. For the 1970s the relationship between the two parameter estimates is somewhat less pronounced (particularly for the dollar/pound exchange rate) but in general the magnitudes of the various estimates seem to be very small. The principal conclusion that can be drawn from this note is that as an analytical matter the typical specification of efficiency tests of foreign exchange markets is strictly valid only when individuals are risk neutral and prices are completely independent of exchange rates; as an empirical matter, however, the neglect of these factors does not seem to be of great quantitative importance.

References Fama, E.F. and A. Farber, 1979, Money, bonds, and foreign exchange, American Economic Review 69,639-649. Frankel, J.A., 1979, Diversifiability of exchange risk, Journal of International Economics 9, 379-396. Frenkel, J.A., 1977, The forward exchange rate, expectations and the demand for money: The German hyperinflation, American Economic Review 67, 653-670. Frenkel, J.A., 198 1, Flexible exchange rates, prices and the role of ‘news’: Lessons from the 1970’s, Journal of Political Economy 89, forthcoming. Hakkio, C.S., 1979, Expectations and the foreign exchange market, Unpublished Ph.D dissertation (University of Chicago, Chicago, IL). Kouri, P.J., 1977, International investment and interest rate linkages under flexible exchange rates, in: Ahber, ed., The political economy of monetary reform (Allanheld, New York). Levich, R.M., 1978, Tests of forecasting models and market efficiency in the international money market, in: Frenkel and Johnson, eds., The economics of exchange rates: Selected studies (Addison-Wesley, Reading, MA). Levich, R.M., 1979, The efficiency of markets for foreign exchange, in: Frenkel and Dombusch, eds., International economic policy: Theory and evidence (Hopkins University Press, Baltimore, MD). Stockman, A., 1978, Risk, information and forward exchange rates, in: Frenkel and Johnson, eds., The economics of exchange rates: Selected studies (Addison-Wesley, Reading, MA).