Article

A Theoretical and Empirical Framework for Analyzing the Term Structure of Exchange Rate Expectations1

Author(s):
International Monetary Fund. Research Dept.
Published Date:
January 1971
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THIS PAPER develops a simple framework for analyzing the expected future time path of the exchange rate between two countries. It is clearly a matter of considerable importance to be able to obtain clues as to expectations regarding not merely the short-run expected value of the exchange rate but also the expected time path of the exchange rate over a substantial future period of time. By using the information embodied in the term structures of interest rates in two countries that have integrated capital markets, and by making the assumption that exchange rate expectations are uniform and held with certainty, it is possible to infer the implicit value of the expected exchange rate for varying periods into the future. In other words, given that the required assumptions hold, one can map the term structures of international interest rate ratios into the implicit expected time path of the exchange rate between two countries. Relaxation of the assumption of certainty of exchange rate expectations requires significant modification of the theory, with foreign exchange “risk” now explaining a portion of the international yield ratios.

One interesting aspect of the analysis is that it suggests a way of analyzing the relative stability of exchange rate expectations under alternative exchange rate systems. By comparing the variation in the term structure of international interest rate ratios over time, it is possible to obtain quantitative estimates of the variation in the combined value of the implicit exchange rate expectations and foreign exchange risk premiums, by time horizon, as these estimates are obtainable regardless of the exchange rate system in operation. However, for the approach to be meaningful, the assumption of integrated capital markets, and of market-determined interest rates, must be satisfied—assumptions that may hold for very few countries. In particular, it is required that there be no factors such as divergent tax rates that might lead to interest rate differentials between countries. Furthermore, interest rates must respond to exchange rate expectations, and not vice versa. This latter condition requires, among other things, that yields not be controlled by the authorities—a situation that will be particularly unlikely at the short end of the market.

The second part of this paper is an empirical analysis of the term structure of Canadian-U. S. interest rate ratios. The approach developed is essentially exploratory, and consequently the empirical results should be treated with great caution. Bond yield ratios, by maturity, are examined and discussed in relationship to the theory on exchange rate expectations. Several, rather primitive, models of the formation of these exchange rate expectations are tested and found to be moderately satisfactory and interesting for periods of both flexible and fixed exchange rates. The predictive power of the exchange rate expectations implicit in the term structure of yield ratios is also tested, the conclusion being that for Canada, over the period of flexible exchange rates, the market had some limited ability to predict exchange rate changes over two-year periods but not for other time horizons.

Part I

The model

The basic idea is simple and amounts to a restatement of the interest rate parity theory,2 except that we shall not be dealing with the explicit forward exchange market but with the implicit forward market as revealed in interest rate differentials between the two countries. The essential point is that we assume, as a condition of equilibrium, that there is equality in the expected “total” yields from investing in n-period bonds in two countries having integrated capital markets but different currencies. If nominal bond yields of maturity n differ between countries, then the assumption of equal expected total yields immediately gives the implicit value of the expected exchange rate. By varying the maturity we obtain values for the implicit expected exchange rate for a range of time horizons.

The algebraic statement is as follows:

where RntW,RntX are the yields on n-period bonds denominated in the foreign and domestic currencies, respectively, at time period t.

Kt+nE is the expected price at the end of n periods (i.e., expected exchange rate) of one unit of W’s currency in terms of X’s currency.

It is possible to express the expected change in the exchange rate over a time horizon of n periods as a function of the current market rates for one-period forward exchange contracts over the next n periods. To demonstrate this, it will be useful to apply the framework developed for the one-country analysis of the term structure of interest rates.3

Suppose that we define the current market rate in country X, for one-period loans for any future period, say, period n, as follows:

This definition provides us with a measure of the implicit one-period forward interest rate for period n, as rnx is the rate necessary to equate the total expected yield from both n - 1 and n-period bonds over an n-period time horizon. By deriving a similar relationship for country W, and dividing, we obtain the following:

By substituting (1) into (3)—for both periods n and n – 1—we obtain

where KnE is the implicit expected change in the exchange rate in period n. Suppose that we also write (2) in the following form: 4

We may now obtain the following relationship:

This last result demonstrates that the international yield ratio on bonds of maturity n can, under the ideal conditions specified above, be interpreted as the geometric mean of the compounded expected changes in the exchange rate over the next n periods. In equilibrium these expected exchange rates would correspond to the forward exchange rates, given that the forward market was developed for such a wide range of maturities.

To summarize, if expectations are uniform and held with certainty, and if market equilibrium prevails, we can infer the expected exchange rate for varying periods into the future from the ratio of international bond yields of various maturities. The assumption is that the equilibrium yield ratios that are observed at the end of a particular day can be equilibrium yields only if the differentials are consistent with exchange rate expectations. The best way to illustrate this is through some examples of particular states of expectations, with the term structure of yield ratios and the implicit expected exchange rate path being shown side by side. (See Table 1.) These examples are useful illustrations of the sort of information that could, ideally, be extracted from international term structures of interest rates.

Table 1.Four Cases Showing Graphic Relationship Between the Term Structure of International Interest Rate Ratios and the Expected Time Path of the Exchange Rate1
Observed International Bond Yield Ratios, by Maturity, at time tImplicit Exchange Rate Expectations for Varying Time Horizons
Case 1: Stationary exchange rate expectations
RnX=RnW (for all n) implies Kt+nE=Kt
Case 2: Expected change of the exchange rate to a new level
1+RnX1+RnW=ean implies Kt+nE=Ktea
Case 3: Constant expected rate of change of exchange rate (i.e., steady depreciation or appreciation)
1+RnX1+RnW=a implies Kt+nE=Ktan
Case 4: Expected asymptotic convergence to a new value of the exchange rate
1+RnX1+RnW=(e)an[11n] implies Kt+nE=Ktea(11n)

It is assumed that n > 1, where the unit is the shortest quoted maturity. The letter e represents the base of natural logarithms.

It is assumed that n > 1, where the unit is the shortest quoted maturity. The letter e represents the base of natural logarithms.

In Case 1 interest rates are equal in the two countries, for all maturities up to n, the implication—in terms of our model of certain exchange rate expectations—being that the exchange rate is expected to remain stationary at its current “spot” value over the next n years.

In Case 2 it is expected that in the next period, t + 1, the exchange rate will move to a new value, Ktea, at which value it will remain for all n periods into the future.5 The interest rate differentials between the two countries reflect this expectation, since they diminish with maturity. This diminution with maturity, n, is accounted for by the fact that the yield differential is compounded for n periods, and hence the expectation of a once-and-for-all shift to a new, stable exchange rate value reflects itself in diminishing yield differentials between the countries as the maturity of the bond increases.

In Case 3 the exchange rate is expected to depreciate or appreciate by a steady percentage each year. Thus, a constant yield differential between countries of, say, 1 per cent on all maturities implies an expected steady rate of change in the exchange rate of 1 per cent per annum.

Case 4 is particularly interesting, since it illustrates an expectation of a gradual convergence to a new, stationary value of the exchange rate. We see the expected exchange rate moving smoothly from its spot value and converging, asymptotically, to its new equilibrium value. The consistent term structure of yield ratios is the humped pattern indicated in the graph in Table 1.

Table 2 gives a numerical example of Case 4 with the interest rate differentials being 0.5 per cent, 1 per cent, and 0.7 per cent for one-year, two-year, and three-year bonds, respectively. The implied total expected changes in the exchange rate for one-year, two-year, and three-year time horizons are 0.5 per cent, 1.9 per cent, and 2.0 per cent, respectively—implying that the exchange rate is expected to change by 0.5 per cent in the first year, 1.4 per cent during the second year, and 0.1 per cent in the third year. Thus, there is expectation of a gradual, if uneven in this instance, adjustment of the exchange rate to a level that is 2 per cent different from its current spot value.

Table 2.Case 4, Showing Expected Asymptotic Convergence of the Exchange Rate to a New Value Over Three Years
Compounded

Ratios
Expected Change

in Exchange Rate

Over n Periods

(per cent)
Maturity of

Bond in

Years, n
RntXRntw(1+RntX1+Rntw)n(Kt+nEKt)
(per cent)
14.54.01.0050.5
26.05.01.0191.9
36.76.01.0202.0

Application to fixed and flexible exchange rate systems

The foregoing analysis is, at first sight, more applicable to systems of flexible exchange rates than it is to fixed exchange rates. However, even though an exchange rate is “fixed,” the consensus of expectations in the market may be for a move in the par value of the exchange rate. In particular, it may be interesting to see to what extent, and how far in advance, particular devaluations that occurred had been implicit in the corresponding international term structure of interest rates. Additionally, it may be interesting to see the extent to which the exchange rate is expected to move within the “band” around the par value. The wider the band, the greater the scope for the application of the term-structures technique to fixed exchange rate systems. Whereas the explicit forward exchange market may indicate the expected exchange rate in the very short run, the term-structures approach can be used for more distant time horizons and for determining the path of adjustment that may be expected.

Canadian examples of the four cases

The degree of integration of Canadian and U. S. capital markets makes the study of Canadian-U.S. Government bond yield ratios, by maturity, of interest in this context. Instances of all the four classes of yield ratios cited above have been observed for Canadian-U. S. yield ratios.6 Case 1 (stationary expectations) occurred frequently in 1955 and 1956, although, interestingly enough, not in the period of fixed rates, 1962–70; Case 2, with a > 0 (expected once-and-for-all depreciation of the exchange rate to a new level), was observed in the second and third quarters of 1959; Case 3 (constant expected rate of change in the exchange rate) has occurred with great frequency throughout the last two decades. However, many factors may contribute to the yield ratio curve being flat, as depicted in Case 3. For example, taxation and foreign exchange risk may cause a pattern of constant differentials.7 Case 4, with a > 0 (expectations of asymptotic depreciation), was observed early in 1954 and in 1957. More recently, in 1970, examples of Case 4, with a < 0 (asymptotic appreciation), have been observed in the months prior to and after the flotation of the Canadian exchange rate in 1970. Thus, there is at least superficial evidence that the term-structures approach is of interest. However, as previously mentioned, foreign exchange risk and taxation differences, for example, may make direct use of such curves questionable.

Relevance to the crawling peg

In principle, the term-structures approach should indicate the expected speed and the extent of movements in the exchange rate under the crawling peg system. If there were complete certainty that the maximum allowable rate of adjustment of the exchange rate would not be changed, this would set bounds on the term structure of the relevant international interest rate ratios. If there were a firm expectation of the exchange rate moving a total of 6 per cent over a three-year period, this would show up, approximately, as a yield differential of 2 per cent between the three-year bond yields in the two countries. Consequently, the fact that the exchange rate is not allowed to move by more than, say, 2 per cent a year does not prevent expected changes of greater magnitude revealing themselves in the market at a point in time.

The effect of exchange rate uncertainty on the term structure of yield differentials

By its very nature, uncertainty as to the expected time path of the exchange rate is a difficult element to incorporate into the analysis. Nevertheless, it is possible, using an approach similar to that adopted by Markowitz8 and Tobin,9 to gain some feeling for the way in which uncertainty, or foreign exchange risk, will manifest itself in the term structure of international yield differentials. The approach summarized here, and developed more fully in the Appendix, is given increased plausibility by the fact that statistical estimates of the impact of foreign exchange risk show that a proxy for foreign exchange risk, as defined below, does help to explain yield ratios, the degree of significance of the risk coefficient increasing with the maturity of the bonds in question.

The method adopted is to assume that there is a debtor country, X, whose capital market is small in relation to the world capital market, W, and that the equilibrium bond yield in X is dictated by the market conditions in W, but that yields in X and W are not necessarily equal, because investors in W adjust for three main factors: (1) exchange rate expectations; (2) foreign exchange risk; (3) the correlation (or interdependence) between yields in the two countries.10

Factor (1) has been discussed in an earlier part of the paper. Factor (2) says that, quite apart from expectations of a particular movement in the exchange rate, there is a risk of fluctuations in the exchange rate. If we assume that lenders in W are averse to risk, this will cause a yield differential between securities in X and W to compensate for the risk. The size of the risk premium will depend on the amount of risk that is perceived by the lenders in W, as well as on the extent of their aversion to risk.

Factor (3) adjusts yield differentials for the extent to which yields in X and W have been correlated over time. If the yields have been only weakly correlated, then, given the “own” variances of X and W yields, the mixed portfolio of X and W offers lower risk, since the fluctuations in the yields of X and W will tend to offset each other. On the other hand, if the yields in X and W have been highly correlated, the mixed portfolio is correspondingly less attractive, and this will tend to cause larger differentials than would appear otherwise.

Because of the difficulties involved in obtaining a time series of correlation coefficients, factor (3) has not been introduced into this paper. However, statistical tests incorporating a (admittedly arbitrary) measure of foreign exchange risk are included in the empirical tests of the formation of interest rate ratios under flexible exchange rates.

Correlation of Canadian and U.S. yields

Before discussing the formation of exchange rate expectations, it will be useful to examine briefly the connections between U. S. and Canadian yields over periods of differing exchange rate regimes. To obtain some idea of the extent to which the United States and Canada have interdependent capital markets, correlation matrices of yields are given in Table 3, for both flexible and fixed exchange rates.11 The diagonal elements indicate the correlation of the yields on matching maturities in the United States and Canada; the off-diagonal elements refer to correlation between yields of differing maturities. Looking down a column enables us to determine the Canadian maturity whose yield is most highly correlated with the yield of a particular U.S. maturity. The notable feature is not so much the ranking of magnitude but the fact that (a) within each period the coefficients are high and remarkably similar across maturities, and (b) there is little difference in the correlation coefficients between the periods of differing exchange rate regimes.

Table 3.Correlation Coefficients of Canadian Yields (1 + Ric) with U. S. Yields (1 + Rjus), Wherei, j = 3 Months and 1, 2, 3, and 10 Years
United States3 Months1 Year2 Years3 Years10 Years
Canada
A. 1953–60, Flexible Exchange Rates
3 months0.8770.8780.9010.8900.856
1 year0.8750.8770.8920.8850.848
2 years0.8700.8760.8980.8970.871
3 years0.8620.8640.8840.8880.878
10 years0.8200.8420.8680.8830.936
(Average, 0.877)
B. 1962–68, Fixed Exchange Rates
3 months0.7540.7740.7890.7750.776
1 year0.7790.8120.8270.8200.818
2 years0.7730.8190.8380.8360.844
3 years0.7890.8350.8550.8520.865
10 years0.8080.8610.8850.8860.930
(Average, 0.824)

The matrices in Table 4 show the correlation coefficients of differing maturities of Canadian-U. S. yield ratios. Our hypothesis regarding the yield ratios for particular maturities is that they embody exchange rate expectations and foreign exchange risk premiums for particular time horizons. The matrices in Table 4 reveal that, on average, there was slightly higher intertemporal correlation under fixed rates than under flexible rates, the exceptions being at the long end of the maturity scale. It is clear that more variability in the implicit expected exchange rate changes was evident under flexible rates than under fixed exchange rates, but quantitatively the differences are not very great.

Table 4.Correlation Coefficients of(1+Ric1+Rius)with(1+Rjc1+Rjus), Wherei, j = 3 Months and 1, 2, 3, and 10 Years
i3 Months1 Year2 Years3 Years
j
A. 1953–60, Flexible Exchange Rates
3 months1.000
1 year0.7591.000
2 years0.6250.9251.000
3 years0.5060.8610.9451.000
10 years0.2610.5750.7510.847
(Average, 0.705)
B. 1963–68, Fixed Exchange Rates
3 months1.000
1 year0.9131.000
2 years0.8230.9591.000
3 years0.8750.9100.9071.000
10 years0.5990.7300.6970.675
(Average, 0.809)

An aspect of the correlation matrices in Table 4 that is of particular interest is the uniformly high correlation between the one-year, two-year, and three-year ratios. Thus, in terms of our hypothesis, this implies that the one-year, two-year, and three-year implicit forward exchange rates are closely related, in contrast to the relationships between the three-month and ten-year implicit expectations. One could infer from this that there is a fairly firm notion of a “medium-term” exchange rate expectation, and that this expectation is rather distinct from the very short-term, i.e., three-month, and very long-term, i.e., ten-year, expectations. We will return to this matter when we discuss the regressions of the actual exchange rates on the implicit expected exchange rates for the various time horizons.

Part II

The formation of exchange rate expectations

So far, the term structure of international bond yield ratios, at a point in time, has been interpreted as embodying exchange rate expectations for varying future time horizons. The question to be discussed now is whether these yield ratios can be explained in terms of simple models of the formation of exchange rate expectations. The approach taken is to set up quite general models of the formation of exchange rate expectations and to test them for the periods of the alternative exchange rate systems.

For the period of flexible exchange rates we assume that the yield ratio of a particular maturity, n, reflects expectations as to the exchange rate in n periods of time and an allowance for the degree of foreign exchange risk that is perceived. We make three alternative hypotheses about the actual form of the expectations function under flexible exchange rates.

For the period of fixed exchange rates we assume that the risk premium the market attaches to Canadian dollar yields, as opposed to U. S. yields, is a function of the difference between the expected level of Canadian foreign exchange reserves and the “desired” level (the “desired” level being defined as the level at which the risk premium is zero). We will now consider the two periods and exchange rate systems separately.

Flexible exchange rates

Our basic hypothesis is that the Canadian-U. S. yield ratio is explained by the expected change in the exchange rate and the level of foreign exchange risk, i.e.,

where Kt+nE is the exchange rate, in Canadian cents per U. S. dollar, expected in period t + n.

Kt is the spot exchange rate.12

St is the measure of foreign exchange risk.

The expected signs of b and c are positive, since a rise in either Kt+nE or St should cause a rise in the Canadian-U. S. yield ratio.

Kt+nE will be represented by three alternative hypotheses, namely,

All these hypotheses are somewhat naïve, in that they assume that past behavior of the exchange rate is the raw data from which expectations are formed. It is clearly desirable to have more complete hypotheses; however, the objective of this study is to show that the hypothesis that Canadian-U. S. interest rate ratios reflect exchange rate expectations cannot be completely rejected, and for this purpose the simple models are adequate.

(A) The first expectations function assigns geometrically declining weights to past values of the exchange rate, with the value of λ being determined by the regression procedure, the constraint being the specific pattern assigned to the weights.13 For tests incorporating (A) and (B) we have chosen the period i to be three months, since we are using quarterly data.

(B) This hypothesis allows a negative coefficient of expectations as a possibility, since Ψ may take negative values.14

(C) This function says that people have an expectation that the rate will return to a “normal” value, and that this value is K*. Thus, for Canada, if there were a belief that the exchange rate were to return to parity with the United States, then K* would be equal to 1.0.15

The measure we use for St is the standard deviation of the previous 12 monthly average values of the Canadian-U. S. exchange rate around their mean, i.e.,

On the basis of the regression results we reject the model incorporating hypotheses (A) and (B) but we do not completely reject (C). The estimates for the equations incorporating hypothesis (C), “persistent” exchange rate expectations, are given in Table 5.

Table 5.Canadian-U. S. Interest Rate Ratios as Explained by Exchange Rate Expectations and Foreign Exchange Risk, 1953-60 1(Hypothesis C)
(1+Rnc1+Rnus)t=(a+bK*)bKt+cSt+(μt+bμct)
Maturitya + bK*bcR¯2DW
3 months1.15570.1568-0.0016515.1%
(2.12)(2.07)(0.42)1.22
1 year1.09390.09300.003197.5%
(1.37)(1.31)(0.86)1.91
2 years1.12050.12250.0061720.6%
(2.06)(2.03)(1.97)0.87
3 years1.10630.10800.0068622.7%
(2.01)(1.90)(2.33)0.91
10 years1.05510.05390.0039821.3%
(1.75)(1.66)(2.35)0.97

The figures in parentheses are t-coefficients. For the constant term, a + bK*, it is the number of standard errors from 1.0. St is measured in units of percentage points. For example, if St = 1, the standard deviation of the exchange rate over the past 12 months was 1 per cent.

The figures in parentheses are t-coefficients. For the constant term, a + bK*, it is the number of standard errors from 1.0. St is measured in units of percentage points. For example, if St = 1, the standard deviation of the exchange rate over the past 12 months was 1 per cent.

The “persistence” hypothesis, (C), cannot be entirely rejected for maturities other than one year. The coefficients of the foreign exchange risk variable are positive and significant for maturities of longer than one year.16 This would appear to indicate that the moderate amount of fluctuation that did occur in the Canadian exchange rate was not sufficient to affect short-term yield ratios but was sufficient to cause adjustment of long-term yields.

If we make the assumption that there were no constant factors explaining yield differentials, i.e., if a = 1, then we can solve for the implicit values of K*.17 The results are also given in Table 5, and the values of K* are seen to approximate 1. This suggests that the market did reflect expectations of eventual parity with the U. S, dollar.

Implicit Values for K*,1 by Maturity (the “persistently” expected value of K on the assumption that a = 1.0) 2
MaturityK*
3 months0.9928
1 year1.0090
2 years0.9835
3 years0.9842
10 years1.0020

Kt and K* are measured in Canadian cents per U. S. dollar.

If we hypothesize that a = 1, then from our estimates of (a + bK*) and b we can solve for K*. These values of K* were computed before rounding the estimates of (a + bK*) and b.

Kt and K* are measured in Canadian cents per U. S. dollar.

If we hypothesize that a = 1, then from our estimates of (a + bK*) and b we can solve for K*. These values of K* were computed before rounding the estimates of (a + bK*) and b.

One might expect the size of the b coefficients to become smaller as maturity increases, since a given expected change in the exchange rate will, because of the compounding factor, cause the annual interest rate to adjust by less, the longer the maturity. Seen in this light the ten-year coefficient is too large (0.0539), compared with the two-year coefficient (0.1225), for example. If the expected adjustment is to take place immediately, then the ten-year coefficient should be slightly less than one fifth of the two-year coefficient. However, if the market anticipates a less than immediate adjustment to the “normal” value, one would expect this to make the short-term coefficients smaller, since the adjustment will not be complete by the time the security matures. Thus, our results are consistent with expectations of a gradual path of adjustment to the “normal” value of the exchange rates.18

Fixed exchange rates

The basic hypothesis advanced is that the yield ratios will not equal 1 on account of expectations of (a) movements in the exchange rates, (b) controls on capital flows,19 and (c) general uncertainty as to the future value of the exchange rate. These expectations are assumed to manifest themselves in the yield ratios between the two countries. The historical paths of the exchange rate clearly will not be the basis for forming such expectations, since the variability of the exchange rate is highly constrained and subject to official interference. Our source of expectations data is to be foreign exchange reserves. Trade figures are not used, on the grounds that exchange rate stability is not dependent on trade balance in each period but rather on the ability of a country to meet the total claims denominated in foreign currency.

The basic hypothesis is

where Rnc,Rnus are the Canadian and U.S. yields on n-period bonds; Ft+nE, is the expected level of reserves in period t + n; F* is the level of reserves at which U. S. and Canadian yields would be equalized but for the existence of constant factors that might cause a to be unequal to 1. The expected sign of b is negative, since the hypothesis is that a smaller risk premium will be charged by U. S. lenders, the higher the value of Ft+nE,.

We assume that for a particular state of expectations as to reserves, characterized by a value for Ft+nE, there is a unique interest rate ratio

that is viewed as consistent with the state of expectations. Capital flows into, or out of, Canada until the Canadian bond yields adjust in such a way that the desired and actual ratios are the same. Expectations are assumed to be based on published foreign reserves data (i.e., Ft is the level of reserves at the end of the month prior to the month from which yield data were taken), and the capital flows that are caused by those expectations do not affect published data—a fact that enables us to avoid the problem of simultaneity.

The alternative hypotheses we try for Ft+nE, are

and

These models are similar to those used for the flexible exchange rate period, the difference being that we do not have a model allowing expectations of “normal” values for Ft+nE.

The hypothesis incorporating (B) could not be rejected, whereas hypothesis (A) was rejected. The results for (B) are given in Table 6. The sign of b was negative, as was hypothesized, in all cases, although for the three-month maturity the coefficient was not significant for any reasonable confidence level. Since bΨ is positive, our finding is that the value of Ψ, the coefficient of expectations, ranges from -0.32 to -0.89, the most significant values being in the range -0.63 to -0.89. Consequently, a rise in actual reserves is seen to cause “expected” future reserves to fall, although less than proportionately.

Table 6.Canadian-U.S. Interest Rate Ratios as Explained by Expectations of Future Foreign Reserves, 1962–68 1(Hypothesis B)
Equation: (1+Rnc1+Rnus)t=(abF*)+bFt+bΨ(FtFt1)+μt
Maturitya - bF*b(×104)bΨ(×104)R¯2(percent)D-W
3 months1.0434-0.14650.047830.40.84
(3.02)(3.01)(1.17)
1 year1.0365-0.11910.075545.21.56
(4.13)(3.63)(2.73)
2 years1.0279-0.08440.075541.31.80
(3.54)(2.89)(3.06)
3 years1.0265-0.07400.049724.91.06
(3.01)(2.26)(1.80)
10 years1.0255-0.05940.019227.51.97
(4.16)(2.60)(1.62)

The figures in parentheses are t-coefficients. For the constant term, a - bF*, it is the number of standard deviations from 1.0.

The figures in parentheses are t-coefficients. For the constant term, a - bF*, it is the number of standard deviations from 1.0.

We find that F* increeases steadily with the maturity of the bonds. Thus, Ft+nE must equal $4.3 billion for yields to be equal for a ten-year bond, whereas Ft+nE must equal only about $3.0 billion for yields to be equal for three-month treasury bills. The suggestion is that it takes a larger value of expected reserves to bring Canadian long-term yields into equality with U. S. long-term yields than it does for short-term yields.

Implicit Values of Desired Reserves and Coefficient of Expectations(Reserves are measured in millions of U.S. dollars)
MaturityΨF*(a = 1)
3 months-0.332,962
1 year-0.633,065
2 years-0.893,306
3 years-0.673,581
10 years-0.324,301

Over all, the hypothesis does well, in that 25 per cent to 45 per cent of variation in the yield ratios is explained by the model. Clearly, there are other exogenous forces affecting both reserves and yields. However, the force of the foregoing results is that despite these other forces, or “noise,” we still find a significant connection between yield ratios and our proxy for exchange rate expectations and risk.

Yields as a monetary instrument?

An obvious criticism of these results is that they may indicate that Canadian bond yields are used as instruments and are assigned to external balance, with Canadian yields being increased as the level of reserves falls. However, it would not appear reasonable, in the Canadian case, to view bond yields as an instrument; rather, the amount of open market sales of bonds and other monetary policies should be viewed as instruments. Accordingly, the main channel of the authorities for varying the yield differential with the United States is, according to our hypothesis, through policies that affect the degree of confidence in the Canadian dollar vis-à-vis the U. S. dollar—our proxy for this degree of confidence being expected reserves. It is entirely possible that a “tight” money policy (such as was pursued in 1969) may be largely offset by capital inflows and trade surpluses, the end result being higher reserves and lower interest rates relative to the United States—the lower yields being the result of greater confidence in the Canadian dollar. Thus, the suggestion is that under conditions of extremely high mobility of capital interest rates can be varied, indirectly, by affecting exchange rate expectations. In this instance, the traditional relationship between the eventual changes in interest rates and the degree of tightness of monetary policy may be reversed, with interest rates rising after acts of monetary ease and falling after acts of monetary tightness.20

Interest rate ratios as predictors of exchange rate movements, Canada, 1953–60

The previous section attempted to explain yield ratios in terms of simple proxies for exchange rate expectations and foreign exchange risk. Another obvious exercise is to see if yield ratios had any predictive power as to actual future changes in exchange rates over the period of flexible exchange rates. For estimation purposes, the following equation has the compounded interest rate ratio as the independent variable. This predictive test is quite separate from the test of the hypothesis explaining the yield ratios.

where Kt is the current exchange rate;

  • Kt+n is the actual exchange rate in n periods’ time;

  • Rnc is the current Canadian yield for government securities of n periods to maturity;

  • Rnus is the corresponding U. S. yield;

  • n is both the maturity of the security and the time horizon of the exchange rate expectation.

We will test the hypotheses that b = 1, a = 0. The regression is run for various values of n, thus testing the model for various time horizons and maturities.21

Before discussing the results, it should be recalled that the yield ratios for the various short-term and medium-term maturities were highly correlated (see Table 4); this may imply that the predicted exchange rate changes over the different time horizons are also highly correlated. However, we know, before estimating the equations, that actual exchange rate changes over the various time horizons were not highly correlated, and hence we know that it is unlikely that more than one of the maturities would have any significant predictive power. Consequently, it is not surprising to find in what follows that only one maturity has significant predictive power.

The results of the estimates for the four maturities are given in Table 7. The result that conforms to theory is the two-year equation, with the interest rate term being significant at the 1 per cent level, the R¯2 being 32 per cent.22 Thus, if we were to take this result on its own, we could justifiably maintain that the market is capable of foreseeing future exchange rate developments two years in advance. The results for one-year and three-year time horizons are satisfactory insofar as the sign of b is concerned, but the coefficients are statistically insignificant.

Table 7.Canadian-U. S. Interest Rate Ratios as Predictors of Future Exchange Rates, 1953–60 1
3 monthsKt+1/4Kt=3.75(3.37)2.749(2.47)[1+R1/4c1+R1/4us]t1/4R2¯=0.169DW=1.56d.f=30
1 yearKt+1Kt=0.57(0.82)+0.425(0.61)[1+R1c1+R1us]tR2¯=0.014DW=0.58d.f=26
2 yearsKt+2Kt=0.026(0.09)+0.957(3.23)[1+R2c1+R2us]t2R2¯=0.322DW=1.07d.f=22
3 yearsKt+3Kt=0.765(2.65)+0.22(0.78)[1+R3c1+R3us]t3R2¯=0.033DW=0.75d.f=18

D-W is the Durbin-Watson statistic; d.f. = degree of freedom; Kt is the number of Canadian cents per U. S. dollar, at time t; the figures in parentheses are the t-coefficients.

D-W is the Durbin-Watson statistic; d.f. = degree of freedom; Kt is the number of Canadian cents per U. S. dollar, at time t; the figures in parentheses are the t-coefficients.

The most disquieting result from the point of view of the hypothesis advanced in this paper is the three months’ regression. The regression indicates a significant negative relationship between the Canadian-U. S. interest rate ratio and the change in the exchange rate over the next three months. The implication of this result is that if the Canadian Treasury bill rate exceeds the U. S. rate, this will be followed, three months hence, by an appreciation of the value of Canada’s currency, and hence by a fall in Kt+1/4Kt.23

The R¯2 of 0.32 for the two-year predictions is high, since we are predicting changes in the exchange rate. Additionally, the period 1953–60 was not free of significant disturbances; in particular, there were major open market operations in 1954, 1955, 1957, and 1959. Also, there were bank rate adjustments as well as other monetary actions over this period. Open market operations that are pursued over a substantial period of time should, according to the theory of the working of monetary policy under flexible exchange rates, affect exchange rates and cause (positively correlated) errors in prediction. Thus, it is despite these disturbances that some significant predictive power is found in the two-year yield ratios.24

The explicit forward exchange market

It is interesting that the explicit three-month forward exchange rate is also a bad predictor of exchange rate movements, as is indicated in the following regression for 1953–60.

where K90F is the 90-day forward exchange rate ruling in period t.25

The relative variance of implicit exchange rate expectations under periods of flexible and fixed exchange rates, Canada, 1953–60 and 1962–68

Table 8 shows the average values of the Canadian-U. S. yield ratios and the relative variance of these ratios over two distinct periods of time. The first period, 1953–60, was one in which both interest rates and exchange rates were market determined, whereas in the period 1962–68 the exchange rate was fixed. In terms of our theory, the relative variance statistics reveal the extent to which our implicit exchange rate expectations and risk premiums varied under the two exchange rate regimes.

Table 8.Means and Relative Variances of Canadian-U.S. Yield Ratios, 1+Rc1+Rus, 1953–60 and 1962–681
MeanRelative Variance (× 10-4)
Maturity1953–601962–681953–601962–68
3 months1.003851.004910.37110.3712
1 year1.006081.005410.30190.2593
2 years1.006311.006030.25340.1955
3 years1.006751.007240.23080.1433
10 years1.005911.009820.07460.0493

From end-of-quarter data. The relative variance is the variance divided by the mean.

From end-of-quarter data. The relative variance is the variance divided by the mean.

The interesting point is that although the variances (both absolute and relative) are small for both periods they are not significantly different, even for large confidence intervals, such as 10 per cent.26 The inference is that there was no significant difference in the variability of exchange rate expectations under the alternative exchange rate regimes. It is true that for all but the three-month implicit time horizons the relative variances were higher under flexible exchange rates; however, the point is that the variances were not significantly higher from a statistical point of view.

Summary

It has been argued that under conditions approximating perfect capital mobility the term structure, and level, of interest rates in a particular country will differ from those in the external capital market because of a lack of certainty that the exchange rate will remain stationary.

To illuminate the connections between international interest rate structures and expected time paths of exchange rates, some interesting relationships between hypothetical anticipated time paths of the exchange rate and the resultant term structures of international yield ratios were graphed and discussed briefly. It was noted that empirical counterparts to the theoretical examples had been observed in Canada during the past two decades.

Various hypotheses regarding the formation of exchange rate expectations were tested against Canadian-U. S. data to see whether international interest rate ratios, by maturity, could reasonably be interpreted as reflecting anticipations regarding the time path of the exchange rate. For the period of flexible exchange rates that was studied, 1953–60, it was found that the Canadian-U. S. yield ratios could reasonably be interpreted as reflecting expectations of the exchange rate returning to a “normal” value. Additionally, there was some evidence that aversion to foreign exchange risk explained a portion of the yield ratios. For the period of fixed exchange rates that was studied, 1962–68, it was found that Canadian-U. S. yield ratios could be meaningfully interpreted as reflecting expectations as to the future level of foreign exchange reserves. The expected level of Canadian reserves was interpreted as the proxy for the risk premium attached to claims denominated in Canadian currency.

It was noted that for quarterly observations over the period 1953–60 the Canadian-U. S. yield ratios for two-year maturities were correlated, to a significant degree, with exchange rate changes over the subsequent two-year period. However, this was not true for other maturities and time horizons, and hence the term-structures approach yielded reasonable predictions only for two-year time horizons.

An interesting policy implication of the analysis is that it suggests that, to the extent that interest rates are affected by expectations as to exchange rates, acts of monetary ease (tightness) that have substantial effects on reserves will be followed eventually by relatively higher (lower) domestic interest rates. Consequently, there is a likelihood that excessive monetary expansion in an open economy will eventually cause interest rates to rise. Whereas in a closed economy interest rate increases will be observed because of anticipations of inflation, in an open economy the upward pressure on interest rates stems from anticipations of depreciation of the exchange rate.

To conclude, it should be recalled that the empirical analysis in this paper is essentially exploratory and should, therefore, be interpreted cautiously. The term-structures approach, developed in an earlier section of the paper, demonstrates connections between the term structures of interest rates in various countries and the implicit expectations as to the time path of the exchange rate. Given the importance of dynamic considerations pertaining to exchange rate systems, this is clearly a matter that deserves much more complete analysis than has been possible in this paper.

APPENDIX

International Interest Rate Ratios as Behavior Toward Foreign Exchange Risk and Exchange Rate Expectations

This paper has examined the term structure of international interest rate ratios in terms of a model that assumed certainty of expectations as to the future path of both interest rates and exchange rates. We now relax these assumptions. Although we simplify matters by dealing with only one maturity, the results are easily generalized to any number of maturities. The analysis applies the conventional Tobin-Markowitz27 approach to portfolio selection under conditions of risk to international portfolio selection. However, rather than solve for the optimal proportion of foreign bonds, the approach taken is to assume that the capital market of one of the two countries is tiny in relation to the other.

Thus, only an infinitesimal proportion of the large country’s portfolio will be held in terms of the small country’s bonds. For equilibrium to exist in the small capital market, there must be no gain, at the margin, to investors in the large country switching into the small country’s bonds.28 Determining the conditions necessary for equilibrium in the small country gives us a relation between yields in the two countries, with the actual differentials being a function of the expected exchange rate, the yields’ variances, the variance of the exchange rate, and the associated covariances.

The model

Let the large country be W, the small country X. Both countries have their own currencies. The expected value, at the end of the period, of X’s currency to people in W is (1 + ΔK), where ΔK is the expected appreciation in X’s currency. Thus, the expected value to people in W of holding X’s bonds is equal to (1 + Rx) (1 + ΔK), where Rx is the expected “own” yield, in X’s currency, of X’s bonds. People in W are faced with the choice of three securities: (1) a domestic bond offering an expected yield Rw, with variance σW2; (2) a riskless bank deposit offering a yield RD with a zero variance; and (3) a foreign bond offering an expected yield (1 + Rx) (1 + ΔK) and a variance σX2. W’s currency is used to measure σX2.

The expected, end-period, value of the portfolio of the representative investor in W is

where u is the portfolio’s expected value

  • xW is the proportion of bond W in W’s portfolio

  • xX is the proportion of bond X in W’s portfolio

  • 1 - xW - xX is the proportion of W’s portfolio in bank deposits in W

  • RD is the deposit rate in W.

The portfolio variance, σ2, may be written

where σW2 is the variance of the yield on W’s bonds,

  • σX2 is the variance, to W, of the yield on X’s bonds,

  • ρWX is the correlation coefficient of the two yields.

We assume that there is one utility function in the dominant economy W.

where Uu > 0, Uσ < 0.

The assumed negativity of Uσ implies risk aversion.

The conventional maximization problem is to choose xW, xX in such a way that U is at a maximum. However, our small country assumption tells us that xX → 0. Our method is to set UxW=UxX=0 and to solve for the values of xW and Rx that are consistent with the first-order conditions. In other words, what value of Rx must there be in order that there can be no gain to W through acquiring X’s bonds? Our first-order conditions are as follows:

It can be seen by inspection that the second-order conditions for U to be a maximum, namely,

and

will hold, provided that Uσ < 0 and ρwx < 1. In other words, the assumptions of risk aversion and less-than-perfect correlation are required.

Our small country assumption implies that xX → 0. If we set xX = 0 we may solve for the value of xw and RX that solve (10) and (11), these being the values that will maximize utility. The solutions are

and

It can be seen from (12) that the proportion of W’s domestic bonds in the optimum portfolio is independent of ΔK, RX, σX2, and ρWX. This follows directly from the small country assumption that xX → 0. Thus, economy X has no impact on economy W. However, the equilibrium yield in X depends on the yields in W and the expected change in the exchange rate (ΔK), together with the variance-covariance terms.

It will simplify matters to create a new variable s

where

Substituting (14) in (13) we obtain29

It can be seen that the yield RX, measured in W’s currency, is simply a weighted average of RW and RD. We assume that the correlation coefficient, ρWX, is positive, and hence s > 0. It is possible that s > 1, since σW may be small in relation to σX because σX incorporates an additional source of variance—the variance in the value of X’s currency. We will now consider various special cases of equation (15).

Perfect capital market integration (s = 1)

If σW = σX, i.e., if the yields have the same variance, and if ρWX = 1, i.e., if the yields are perfectly correlated, then it follows that

This is the familiar relationship from the previous section concerning expectations under conditions of certainty. In this instance any difference in yields RW - RX implies certain expectations of movements in the exchange rate. The deposit rate, RD, has absolutely no effect on Rx.

The deposit rate in W is zero

If the return to the riskless asset is zero, i.e., RD = O, then

If we regard s as predominantly a measure of the foreign exchange risk involved in acquiring X’s bonds, with s = 1 being the value for which the bonds in W and X offer equivalent “risk,” then it is clear that an increase in foreign exchange risk will increase s, and so cause Rx to increase. Thus, it is clear that part of the fluctuation in the relative magnitudes of Rw and Rx will be due to changes in the value of foreign exchange risk. If s were a measurable variable over time, and if we had data on Rw and Rx, then we could infer the value of the expected change in the exchange rate, ΔK. It is clear that, in general, we cannot infer from Rx > Rw that ΔK is negative, since Rx > Rw is compatible with ΔK = 0, providing that s > 1. Hence, we must be careful to avoid making inferences from changes in interest rate differentials direct to changes in exchange rate expectations. Only if s is constant will changes in the differential imply expectations of changes in the exchange rate, and it should be recalled that we are assuming RD = 0 in order to proceed to even this conclusion.

The effect of changes in the yield ratios in W

In this case we have RD = aRw, and we will assume that a < 1. We may substitute in equation (15) to obtain

The point of this case is to show that, for a given risk factor s and a given exchange rate expectation ΔK, it is possible for Rx to change merely because of changes in either Rw, RD, or a.

A rise in Rw will always induce a rise in Rx, although the size of the effect will depend on the sizes of a and s. Suppose, for example, that ΔK = 0, then

Now if s = 1, then dRXdRW=1, i.e., the yields of the two countries will move together. However, if s > 1, owing to the foreign exchange risk attached to X’s currency, then fluctuations in yields in W will be magnified in X. Thus, for the United States and Canada, if we know that s > 1, we would expect that even though the exchange rate, foreign exchange risk, and a are all constant, the yield differentials will fluctuate, since if Rw shifts then Rx will shift by a greater amount. This result demonstrates once more how careful one must be in making firm deductions about changes in expectations and risk, based on evidence of changes in yield differentials between countries.

It is also clear that if s < 1, then (given s, ΔK, and a) dRXdRW<1.

A change in the degree of interdependence between the markets in W and X

We know that s=σXρWXσW and that 1+RX=11+ΔK[sRW+(1s)aRW+1].

Suppose that for some reason the correlation coefficient, ρwx, falls, while the values of σx, σw are unchanged. This will cause a fall in s, and, for given values of K, Rw, and a, this will cause a fall in Rx. What this means is that the variance of a given portfolio of bonds from W and X is now reduced, and hence X’s bonds are more attractive; the price is bid up, and Rx falls. We see again how the yield differential Rw - Rx may vary without the expected value and variance of X’s currency altering.

Conclusion

The analysis given above relaxes the condition of exchange rate certainty set out in the first portion of the text. International interest rate ratios under conditions of foreign exchange risk are shown to bear quite complicated relationships to the various expectational and risk variables. The unwieldy nature of the microrelationship has forced us to adopt in the text a highly simplified linear version of the “true” relationship.

Cadre théorique et empirique pour une analyse de l’évolution probable de la structure des taux de change

Résumé

La présente étude élabore un cadre simple en vue d’analyser à un certain moment l’évolution probable du taux de change des monnaies de deux pays. Dans l’hypothèse d’une situation de certitude absolue et d’intégration parfaite du marché des capitaux et en étendant la théorie de la parité des taux d’intérêt à un nombre n de périodes, il est possible d’établir un rapport entre la structure des rapports de taux d’intérêt dans les deux pays et l’évolution probable des taux de change. Autrement dit, si dans deux pays les rendements d’obligations comparables à échéance de 1 à 10 ans sont déterminés librement, il est alors possible, en utilisant la relation de la parité des taux d’intérêt, de déterminer l’évolution espérée des taux de change au cours des douze mois suivants et jusqu’à dix ans. Des graphiques reproduisent quatre séries de prévisions de taux de change et les structures connexes des rapports entre les taux d’intérêt internationaux. Ces graphiques représentent : 1) des prévisions de taux de change stationnaires; 2) la prévision que le taux de change sera modifié une fois pour toutes; 3) la prévision d’une modification continue du taux de change et 4) la prévision d’une convergence asymptotique vers une nouvelle valeur du taux de change.

L’analyse tient compte de l’incidence de l’incertitude présentée par la future valeur du taux de change et l’appendice présente une analyse plus rigoureuse du modèle dans un petit pays où règne un climat d’incertitude.

La partie de cette étude consacrée à la méthode empirique analyse les rapports des taux d’intérêt entre le Canada et les Etats-Unis sous forme de modèles simples de formation des prévisions des taux de change à la fois en périodes de taux de change fixes et de taux de change flexibles. On a constaté qu’au cours de la période 1953–60 (années où le taux de change était flexible) le comportement des rapports des taux de change entre le Canada et les Etats-Unis pouvait fort bien être interprété comme traduisant la prévision d’un retour à une valeur «normale» du taux de change. Certains indices, peu concluants, sembleraient indiquer que le risque de change (représenté par l’écart type entre les cotations du taux de change au cours des précédents douze mois) a exercé une influence sur les rapports de rendement. Au cours de la période 1963–68, années où le taux de change était fixe, on s’est aperçu que les rapports de rendement pouvaient être interprétés comme représentant une prime de risque qui est fonction du niveau escompté des réserves internationales. On a noté que les données sur les rapports de rendement entre le Canada et les Etats-Unis n’indiquent pas que les prévisions de taux de change aient en quoi que ce soit varié davantage en période de taux de change flexibles.

Un marco teórico y empírico para analizar la estructura cronológica de las expectativas de tipos de cambio

Resumen

En este trabajo se elabora un marco sencillo para analizar, en un momento dado la línea cronológica que se espera que siga el tipo de cambio entre las monedas de dos países. En condiciones de perfecta certidumbre y perfecta integración de mercados de capitales, una extensión de la teoría de la paridad de tipos de interés desde el período uno al n, permite obtener una relación entre la estructura cronológica de las razones entre los tipos de interés de los dos países y la línea cronológica prevista para el tipo de cambio. Es decir, si ambos países tienen bonos comparables con vencimiento de uno a diez años, cuyos rendimientos son libremente determinados, entonces, utilizando la relación de paridad de los tipos de interés, se puede determinar la línea cronológica consistente del tipo de cambio esperada a lo largo de un período de uno a 10 años. Se presentan ejemplos gráficos de cuatro grupos de expectativas del tipo de cambio con las correspondientes estructuras cronológicas de las razones internacionales de tipos de interés. Los ejemplos son: 1) expectativas de tipo de cambio estacionario; 2) previsión de que el tipo de cambio pasará a un nuevo nivel, pero una sola vez; 3) previsión de un ritmo constante de variación del tipo de cambio; y 4) previsión de convergencia asintótica hacia un nuevo valor del tipo de cambio.

Se modifica el análisis para dar cabida al efecto de la incertidumbre sobre el valor futuro del tipo de cambio, y el apéndice contiene un análisis más riguroso del modelo de un país pequeño en condiciones de incertidumbre.

En una sección empírica se analizan las razones de tipos de interés entre Canadá y Estados Unidos, en términos de modelos sencillos de la formación de las expectativas de tipo de cambio para períodos que comprendan tanto tipos de cambio flexibles como fijos. Se demuestra que durante el período 1953–60 (a lo largo del cual los tipos de cambio fueron flexibles) podría interpretarse acertadamente que el comportamiento de las razones de tipos de interés entre Canadá y Estados Unidos reflejaba expectativas de una vuelta a un valor “normal” constante del tipo de cambio. Hay una ligera evidencia de que el riesgo del cambio extranjero (medido mediante la desviación típica de las cotizaciones del tipo de cambio correspondientes a los doce meses anteriores) afectó a las razones de rendimiento. En un período de tipos de cambio fijos, 1963–68, se vio que podría interpretarse que las razones de rendimiento incorporaban una prima de riesgo en función del nivel previsto de las reservas en divisas. Se hizo notar que los datos sobre las razones de rendimiento Canadá-Estados Unidos no indican que las expectativas de tipo de interés sean más variables, en grado significativo, con tipos de cambio flexibles.

Mr. Porter, economist in the Financial Studies Division of the Research Department, is a graduate of the University of Adelaide (Australia) and Stanford University (California). He formerly taught at Simon Fraser University (British Columbia, Canada) and visited the University of Essex (England) on a Canada Council Research Grant in 1969–70.

The author is deeply grateful for the comments and suggestions of staff colleagues and R. I. McKinnon and E. S. Shaw of Stanford University, and to Douglas Fisher and J. Michael Parkin for their remarks and criticisms at a seminar at the University of Essex in 1970. He wishes to emphasize that responsibility for the present text is his alone.

See, for example, John Maynard Keynes, A Tract on Monetary Reform (London, 1923), pp. 115–39. A review of the recent literature may be found in Lawrence H. Officer and Thomas D. Willett, “The Covered-Arbitrage Schedule: A Critical Survey of Recent Developments,” Journal of Money, Credit and Banking, Vol. II (1970), pp. 247–57.

See J. R. Hicks, Value and Capital (Oxford University Press, Second Edition, 1946), pp. 141–52; David Meiselman, The Term Structure of Interest Rates (Englewood Cliffs, N. J., 1962).

We may write (2) as follows:

Now R0x=0, since zero-period loans do not involve interest, and R1x=r1x, since the current forward rate for a one-period loan will equal the actual market one-period rate. By substituting (2), the above expression may be written

which is equivalent to (5) above.

If a = 0, then we obtain Case 1 as a special example of Case 2.

The data referred to were obtained from the sources listed below. These same data were used in the econometric work discussed below.

a. Interest rates. We examined the yields on Canadian and U.S. Treasury bills and bonds. (The U. S. Treasury Bulletin monthly yield curves and treasury bill quotations provided the basic U. S. data. Canadian data are from the Bank of Canada, Statistical Summary.) Quarterly, end-of-month, quotations were obtained for a range of maturities in both countries, and these yields were then plotted, one graph being drawn for each time period. A continuous curve was then drawn and used to obtain theoretical yields of securities of identical maturity for both countries. In some instances the theoretical yields obtained were sensitive to the manner in which the curve was drawn; however, the errors are likely to have been small in relation to the interest rate differentials. The three-month data are not completely satisfactory, since the Canadian Treasury bill market is less highly developed than the bond market.

b. Exchange rates. The absence of published daily exchange rate quotations led us to use the average monthly quotations from the Bank of Canada, Supplement to the Statistical Summary. However, the daily quotations from the internal records of private banks were also used. (Herbert G. Grubel, Forward Exchange, Speculation, and the International Flow of Capital, Appendix A, pp. 167–82, Stanford University Press, 1966.) Grubel’s data commence in 1955; in the results using exchange rate data over the period 1953–60, the monthly average data were used for 1953–54 and the Grubel data for 1955–60. Estimating the same regressions using only the monthly average data does not significantly change any of the results. This is so because changes in the monthly average of daily quotations are close approximations to the changes in the end-of-month quotations over the same periods.

Actual, or anticipated, taxation of foreign assets and liabilities can cause yields to diverge between countries for reasons that may have little to do with exchange rate expectations. This particular complication is unlikely to have been important for Canada in the period of flexible rates (see Gerald K. Helleiner, “Connections Between United States’ and Canadian Capital Markets, 1952–1960,” Yale Economic Essays, Vol. 2, Fall 1962, pp. 351–400), although it may have been important since December 1960, the date on which the Canadian Government introduced a 15 per cent withholding tax on interest paid on foreign holdings of provincial and government bonds.

Harry Markowitz, Techniques of Portfolio Selection (New York, 1959).

J. Tobin, “Liquidity Preference as Behavior Towards Risk,” The Review of Economic Studies, Vol. XXV (February 1958), pp. 65–86.

The analysis requires that country X borrow from country W, and the results would have to be modified if it is assumed that X is a creditor country.

The Canadian exchange rate was floated in October 1950. We have chosen to focus on interest rate differentials from 1953, on the grounds that until 1953 the Canadian short-term security market was extremely thin (R. M. Macintosh, “Broadening the Money Market,” Canadian Banker, Autumn 1954) and that interest rates were pegged in the United States up until the 1951 accord. Over the period 1953–60, the effect of taxation policy in the United States and Canada on the yield differences on government bonds is regarded to have been negligible (Helleiner, op. cit.). In December 1960 the Canadian Government introduced a 15 per cent withholding tax on interest paid on foreign holdings of provincial and government bonds, and in 1961 the Government began to intervene actively in the foreign exchange market, in contrast to its passive role in the previous decade. For these reasons, our end point for the series of interest rate differentials was chosen as December 1960, thus providing us with the period 1953–60, a period that was relatively free from large-scale intervention in either the securities or foreign exchange market. (For an up-to-date survey of Canadian monetary policy over the whole period, see Gordon Boreham, Eli Shapiro, Ezra Solomon, and William L. White, Money and Banking: Analysis and Policy in a Canadian Context (Canada, 1969), pp. 748–73, and Bank of Canada, Annual Reports, 1950–69. Paul Wonnacott, The Canadian Dollar, 1948–1962 (University of Toronto Press, 1965), is also a useful reference. An excellent and comprehensive econometric study of the Canadian economy may be found in Rudolf R. Rhomberg, “A Model of the Canadian Economy under Fixed and Fluctuating Exchange Rates,” The Journal of Political Economy, Vol. LXXII (February 1964), pp. 1–31. A study of speculation and exchange rate stability may be found in William Poole, “The Stability of the Canadian Flexible Exchange Rate, 1950–1962,” The Canadian Journal of Economics and Political Science, Vol. XXXIII (1967), pp. 205–17.)

The exchange rate was pegged again in May 1962. In order to avoid the particular effects of the transition to the fixed exchange rate and the uncertainty attached to the June 1962 election, we have commenced our second period on July 31, 1962 and continued it to October 1968.

The exchange rate data refer to average monthly figures. The interest rate ratios were computed using yield curves based on end-of-month data.

The following example illustrates hypothesis (A):

If λ = 0.4, the weight given to the current observed value of the exchange rate (i = 0) is 0.6; the weight for the previous period (i = 1) is 0.4 × 0.6 = 0.24 and the weights for the preceding periods are (0.4)2 × 0.6 = 0.096, when i = 2; (0.4)3 × 0.6 = 0.0384, when i = 3; and (0.6)k × 0.4 when i = k. The sum of these weights approaches 1.0 as i increases.

This expression may also be written Kt+nEKt=ψ(KtKti)+μ2; i.e., the expected change in the exchange rate over the next n periods is equal to ψ times the change over the last i periods.

There have been various references to this sort of behavior. For example, Rhomberg, in a discussion of the stability of the Canadian flexible exchange rate (op. cit., p. 5), makes reference to expectations of the exchange rate moving away from “unusually high or unusually low” values.

We cannot reject the hypothesis of positive serial correlation in the error terms, and hence our t-ratios are biased upward. The presence of positive serial correlation is not surprising, since acts of policy, such as open market operations, are likely to account for part of the error terms, and these acts will inevitably be positively correlated over time.

This assumption would seem consistent with evidence contained in the study by Helleiner, op. cit.

However, it is clear that a much more complete study would be required before anything definite could be said about the adjustment process.

The recent economic history of Canada makes item (b) unlikely to be of any great importance.

This situation is analogous to recent arguments and evidence pertaining to closed economies. The argument, in the closed economy case (Milton Friedman, “The Role of Monetary Policy,” The American Economic Review, Vol. LVIII, March 1968, pp. 1–17), is that sustained monetary expansion leads, eventually, to anticipations of inflation, and these anticipations cause nominal interest rates to rise—possibly above the level ruling prior to the acts of sustained monetary expansion.

We have omitted to adjust the yield ratios for the foreign exchange risk factor. By assuming that (Kt+nEKt) is orthogonal to the risk factor, St, it is possible to use the residuals from the regression of the yield ratios on St as unbiased estimates of Kt+nEKt, and then to test the power of these residuals to predict subsequent exchange rate changes. It turns out that the results from proceeding this way are only slight improvements on the results given in Table 7.

Although the standard errors are biased downward owing to significant positive serial correlation.

It is clear that we could interpret this result as indicating disequilibrium in the Canadian Treasury bill market in the short run; i.e., that the high value of the Canadian Treasury bill rate causes capital inflow, which causes Kt+3 to fall. This was found to be so in the study by Rhomberg, op. cit. The presumption is, therefore, that in terms of our analysis the Canadian Treasury bill rate, R1/4c,t, is not an equilibrium market rate. It will be recalled that the Canadian Treasury bill market is less developed than is the bond market, and that the treasury bill market is more subject to the influence of monetary policy than are the longer-term yields. If we regard long-term interest rate differentials as being based on longer-run factors that are more stable than those determining the short-run interest rates, then these difficulties would not apply to the longer maturities.

However, the significance attached to the estimates of any particular equation decreases with the number of regressions run (i.e., the wider the range of maturities and time horizons), since the probability of finding, by chance, a maturity with a yield ratio that correlates with subsequent exchange rate changes increases with the range of maturities tried.

If there were perfect covered interest arbitrage, K90FKt would be approximately equal to 1+R1/4c1+R1/4us, in which case these results, with appropriate adjustment, would correspond to the three-month result, using yield ratios as the independent variable. In fact, arbitrage is not perfect, and this explains the divergence between the two sets of results.

The test of significance is an F-test on the ratios of the two variances.

See Tobin, op. cit., and Markowitz, op. cit.

With respect to the mounting literature on portfolio models of international capital flows, interest rates, in the countries between which capital flows, are assumed to be exogenous and independent of the capital flows. This is a serious weakness that we avoid by invoking the small country assumption and solving for the interest rates consistent with stock equilibrium.

The results shown above can easily be generalized to any number of assets in the dependent country, X. If RFi, RFj are the yields on any two assets in X, then we can write down the following results from our generalized first-order conditions:

Thus, the difference between yields in X depends on the difference in the variances and correlation coefficients with yields in W. The constant multiplicative factor is a function of the spread between the risky and the riskless assets (RWRD), the variance in W, and the expected value of X’s currency.

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