Appendix: The Rate-of-Return Band with Uneven Depreciation
For the 60-month interest rates,
Let us assume that the home currency appreciates (depreciates) gradually at an even rate until the exchange rate reaches the edge in month t+60. That is, St+12n = St(S/St)n/5, n = 1, 2, .., 5 (
Suppose we define the lower (upper) bound on the rate of return under the alternative assumption that the exchange rate appreciates (depreciates) to the lower (upper) edges of the exchange rate band already in one year and then stays there. That is, St+12n = S, n = 1, 2, .., 5) (
These expressions can be rewritten according to
These equations are easy to solve numerically.
Bertola, Giuseppe, and Lars E.O. Svensson “Stochastic Devaluation Risk and the Empirical Fit of Target Zone Models,” Working Paper (Stockholm: Institute for International Economic Studies, 1990).
Grønvik, Gunnvald, “The Forward Foreign Exchange Market: Is the Growth of Bank Lending a Result of a Large Demand for Kronor on the Forward Market? or the Reverse?” (in Norwegian), Socialøkonomen: 6 (Oslo) (1986), pp. 19–29.
Hedman, Malin, “Foreign Exchange Management in Some Swedish Corporations? An Empirical Study January-October 1985” (in Swedish), Occasional Paper No. 3 (Stockholm: Sveriges Riksbank, 1986).
Krugman, Paul, “Target Zones and Exchange Rate Dynamics,” Quarterly Journal of Economics (Cambridge), Vol. 105 (1990, forthcoming).
Porter, Michael G., “A Theoretical and Empirical Framework of Analyzing the Term Structure of Exchange Rate Expectations,” Staff Papers, International Monetary Fund (Washington), Vol. 18 (1971), pp. 613–642.
Ringström, Olle, “The Exchange Rate Index: An Instrument for Monetary and Exchange Rate Policy,” Sveriges Riksbank Quarterly Review: 4 (Stockholm: Sveriges Riksbank, 1987), pp.16–26.
Svensson, Lars E. O. (1990a), “The Foreign Exchange Risk Premium in a Target Zone with Devaluation Risk,” Working Paper (Stockholm: Institute for International Economic Studies, 1990).
Svensson, Lars E. O., (1990b), “The Term Structure of Interest Rate Differentials in a Target Zone: Theory and Swedish Data,” Working Paper (Stockholm: Institute for International Economic Studies, 1990).
Part of the work for this paper was done while I was a Visiting Scholar at the Research Department at the International Monetary Fund. I thank the Research Department for its hospitality. This paper was initiated after a discussion with Gunnvald Grønvik, Norges Bank, who gave me the idea to compute rate-of-return bands. I am also grateful for comments from Michael Dooley, Bernard Dumas, Thomas Franzén, Morris Goldstein, John Hassler, Lars Hörngren, Peter Norman, Peter Sellin, Ingrid Werner, participants in a seminar at Sveriges Riksbank, and, in particular, Mats Persson who gave me the idea to also plot expected future exchange rates. I thank John Hassler for expert research assistance, Molly Akerlund and Rosalind Oliver for editorial and secretarial assistance, and Sveriges Riksbank and the Finance Department of the Stockholm School of Economics for providing data. Remaining errors and obscurities are my own.
Swedish capital controls the last few years have generally been considered very ineffective, with many ways of circumventing them. The capital controls have been gradually dismantled, in particular since January 1989, and since June 1989 they are for all practical purposes abolished.
Mats Persson has in lecture notes independently examined the credibility of the Swedish target zone by computing expected future exchange rates.
Porter (1971), in a remarkable paper pointed out to me by Michael Dooley and Morris Goldstein after the first version of this paper was written, shows how the implicit expected time path of exchange rates between two currencies can be inferred from the term structure of interest rate differentials, under the assumption of uncovered interest rate parity. Porter also presents an empirical analysis of the term structure of Canadian-U.S. interest rate differentials and the implied expected time path of future CAD-USD exchange rates.
Porter makes the assumption of no uncertainty and perfect foresight, and observes that with uncertainty a foreign exchange risk premium will enter and violate uncovered interest rate parity. In contrast, the present paper does not assume certainty and perfect foresight. Instead the paper relies on Svensson (1990a) which shows that the foreign exchange risk premium is likely to be small for relatively narrow target zones, also when there is devaluation risk. Therefore uncovered interest rate parity should indeed be a good approximation for narrow target zones.
Svensson (1990a) demonstrates that any foreign exchange risk premium in a relatively narrow target zone should be small, also when there is devaluation risk. Therefore uncovered interest rate parity should be a good approximation for narrow target zones.
Because of a labor market conflict in the banking industry Swedish interest rates are missing for January 1990. Therefore the January Swedish interest rates are set equal to the average of the December and February rates.
For October 1987-January 1988 it has been suggested that what we can call “institutional friction” (for instance a bias in corporations’ and banks’ accounting practices), rather than devaluation risks, may have reduced the amount of interest arbitrage. See Hedman (1986) for a report on some special aspects of foreign exchange management by Swedish corporations. For instance, corporations may prefer to clear their books before the end of the year in order to get a good solidity rating, since accounting rules do not allow the netting of foreign currency liabilities against domestic currency assets.
For longer term bonds with yearly coupons the rate-of-return band (4a) and (4b) still applies if the exchange rate depreciates/appreciates gradually so as to reach the edge if its band at maturity. Alternatively, we can assume that the exchange rate moves to the edge of the exchange rate band already in one year. That assumption causes a slightly wider rate-of-return band than the one given by (4a) and (4b), but the change is small. See Appendix for details.
See, however, footnote 1 on p. 7.
See Svensson (1990b) for a rigorous derivation of the expected depreciation within the band for arbitrary horizons.
For a Poisson process with a probability intensity λ per unit of time, the expected length of time to the next event is 1/λ units of time.
Bertola and Svensson (1990) formulate a model with explicit stochastic devaluation risk and indicate a more precise way of extracting the expected rate of devaluation from data on exchange rates and interest rate differentials.
The role of institutional friction should also be considered, cf. footnote 1 on p. 7.