The Forward Rate as an Unbiased Estimator of the Future Spot Rate - Example

International Finance Contents Introduction 3 Discussion 3 Forward Rate Determination 4 Spot rate Determination 5 The Forward Rate as an Unbiased Estimator of the Future Spot Rate 5 Conclusion 10 References 11 Introduction In today’s world any kind of international transaction has foreign exchange risk associated with it due to change in currency exchange rates. One technique to lowering the foreign exchange risk is entering into forward contract and thus the relation between forwards exchange rate and corresponding future spot rate is important for policy makers, portfolio managers and investors. It is often hypothesized that Forward rates are unbiased estimator of corresponding future spot rates. According to Chiang (1988), the forward rate is an unbiased estimator of the future spot rate because the forward rate fully reflects the available information on exchange rate expectations. But there has been other empirical research which shows that the forward rates are not good estimators of corresponding spot rates. This is known as forwards rate bias puzzle. This paper looks at the relation between the forwards exchange rate and corresponding future spot rate. This report uses many theories to solve this puzzle. Discussion International transactions involve goods or commodities and services and are moved from one country to other and their prices plays a major role in the demand, the entire balance of payments, trade balance and the exchange rate (the relative price of currencies) between the two countries that are comprised in trade. Also, there are financial transactions in assets, and capital inflows and outflows between nations in which nations in which their mobility depends on interest rates and the exchange rate between the two economies involved. All these transactions are the result of decisions by traders (i.e. importers and exporters) shipping goods across countries, by investors transferring funds among nations to find higher returns and lower risks, by speculators expecting some speculative profits, and by arbitrageurs earning arbitrage profits and manipulating the international funds market (Pilbeam, 2013, p. 121). From transactions that dictate international trade flows and capital movements, we can see some specific principles among interest rates, prices, and exchange rate that are part of the international parity conditions among these variables due to market efficiency, full information, and the ultimate objectives of participants, which is maximization of their objective functions: market value, profit, utility, wealth, welfare, power etc (Copeland, 2008, p. 162). Forward Rate Determination The forward rate is calculated by the difference of interest rate between the country where the foreign currency is used to eliminate arbitrage opportunity, referred to as foreign country, and the country where the base currency is used, referred to as base country. An investor has the choice of either holding a domestic currency at an annualized rate of interest or holding foreign currency at rate of interest. One option to the investor is to accumulate units of domestic currency. Other option is to convert the domestic currency into foreign currency at spot exchange rate into units of foreign currency. Then he can invest it into foreign assets for time t and accumulate units. If the investors has the opportunity to cover against any kind of exchange rate and covert the foreign currency to domestic currency at time t when the Forward exchange rate , then in equilibrium when no arbitrage opportunities exist, the following conditions must be met (Eiteman, Stomehill, Mofferr and Pandey, 2007, p. 191). Hence the forward rate is calculated at Spot rate Determination In 1944, during the time of Bretton Woods conference exchange rate was fixed. If is the exchange rate, behaves like a geometric Brownian motion and it follows a stochastic differential equation In the above equation is Wiener constant. After solving the above equation the spot rate comes out to be The Forward Rate as an Unbiased Estimator of the Future Spot Rate According to the unbiased forward rate expectation hypothesis the forward rate which is prevailing on day zero will fully reflects available information. This hypothesis assumes that forwards rates available on day zero are an unbiased estimate of the future spot rates. The word unbiased indicates that on average the difference between forward rate and actual spot rate is zero. The above hypothesis is based on market efficiency. Market efficiency implies that current prices reflect all available information. Thus if the market is pricing a forward contract to be a given value, this forward price reflects all expectation of economic agents regarding the spot rate to prevail in future. In the case that forward rate don’t predicts the future spot rate unbiasedly then speculators can make profit from the bias by taking one position in the spot market and the opposite position in the forward market (Moosa and Bhatti, 2009, p. 67). The unbiased forwards rate hypothesis states that given the conditions of rational expectations, efficiency, full information and risk neutrality, the forwards exchange rate is an unbiased estimator of the expected future spot exchange rate. The hypothesis claims that market expectations of the economic fundamentals that determine exchange rates are reflected in the forwards exchange rate. Without introducing a foreign exchange risk premium (due to the assumption of risk neutrality), the unbiasedness can be seen from the following hypothesis Here = the forward exchange rate at time n (n = 1, 3, 6, 9, 12,), = the expected spot rate exchange at time n (i.e. the number of months from now), the ln of the forwards exchange rate, = the ln of the spot rate, and the forward discount or premium of the two currencies. The assumptions are that all relevant information is quickly reflected in both the forward exchange markets and the spot market and the forward discount or premium of the two currencies (Baillie and McMahon, 1989, p. 83). The assumptions for the above equations are that all relevant information is quickly reflected in both the forwards exchange markets, transaction costs are low and instruments denominated in different currencies are perfect substitutes for one another. But, it has been found that the forward exchange rate sometimes under predicts or over predicts the future spot rate. But on an average it is safe to assume that it is approximately equal to the future spot rate. Hence forward rate is often seen as an unbiased estimator of the corresponding future spot rate. There is a test through which we can test the forward rate hypothesis through regression analysis. If then, (the forward rate last period is an unbiased estimator of the current spot rate) and further the current forward rate is an unbiased estimator of the future spot rate. Figure 1: The Forward Rate as an Unbiased Estimator of the Future Spot Rate (FRUPFSR) The above graph shows the unbiased forward rate where the one year forward discount of the US dollar is an unbiased estimator that the dollar will depreciate by the same amount ( ) during the next year with respect to euro (Pilbeam, 2010, p. 251). The unbiased estimator is often seen as a puzzle by finance researchers. Empirical evidence for co integration between the future spot rate and the forward spot rate is mixed. But many researchers have proven the above hypothesis wrong by conducting regression analysis of the realized changes in spot exchange rates on forward premiums. Researchers have often found negative slope coefficients. They have tried to offer many numerous rationales for such failures. One such rationale centres around the relaxation of risk neutrality though it is still assuming rational expectations like a there might exists foreign exchange risks premium which can account for differences between the expected future spot rate and the forward rate. The following equation indicates the forward rate as being equal to an expected future spot rate and a risk premium : In 1984, Fama concluded that large positive correlations of the difference between the current spot exchange rate and the forward exchange rate signals variations over time in the premium component of the forward-spot differential or in the forecast of the expected change in the spot exchange rate. Fama recommended that in the regressions coefficient of the slopes of the difference between forward rate and future spot rate and the estimated gap in spot rate on the forward-spot differential are different from zero. This implies that there are variations over time in both components of the forward-spot differential which is the difference in usual change in premium and spot rate (Sarno and Taylor, 2002, p. 94). Finally it was found that presence of large variations in estimated gaps in spot rate can be evaluated by aversion coefficients which were inadequately high. Other researchers have found that the unbiased hypothesis has been rejected in both the cases in case there is evidence of risk premia which varies over time and case. Where risk premia are constant. There are other rationales for failure of the forward rate unbiased hypothesis. These include considering the conditional bias which is taken as an exogenous variable explained by a policy aimed at smoothing interest rates and stabilizing exchange rates, or considering that an economy allowing for discreet changes could facilitate excess returns in the forward market (Kallianiotis, 2013, p. 67). Conclusion This report takes a look at the forward rate bias puzzle. This hypothesis indicates that the Forward Rate is an Unbiased Estimator of the Future Spot Rate. It has been shown that forward exchange rate unbiasedness hypothesis holds true in many. But other researchers have shown that the following hypothesis has been rejected in many cases. Many of the forward exchange rates are biased from both foreign currency investors’ and dollar investors which supports their theoretical argument. But economist has demonstrated that forward rate serves as a useful proxy for expected future spot exchange rates. References Baillie, R. T. and McMahon, P. C. 1989. The Foreign Exchange Market: Theory and Econometric Evidence, Cambridge: Cambridge University Press. Copeland, L. S. 2008. Exchange Rates and International finance, 5th edition, Harlow: Prentice Hall / Financial Times. Eiteman, D.K., Stomehill, A.I., Mofferr, M.H. and Pandey, A. 2007. Multinational Business Finance, 10/E. New Delhi: Pearson Education India Kallianiotis, J.N. 2013. International Financial Transactions and Exchange Rates: Trade, Investment, and Parities Moosa, I. A. and Bhatti, R. H. 2009. The Theory and Empirics of Exchange Rates, Hackensack, NJ: World Scientific. Pilbeam, K. 2013. International Finance, 4th edition, Basingstoke: Palgrave Macmillan Pilbeam, K. 2010. Finance & Financial Markets, 3rd edition, Basingstoke: Palgrave Macmillan. Sarno, L. and Taylor M. P. 2002. The Economics of Exchange Rates. Cambridge: Cambridge University Press. Read More