If there were to be no expected changes in exchange rates then the interest rates in different countries would be equivalent to one another. If this were not the case then there would be arbitrage opportunities. Where we allow for changes in the exchange rate, covered interest rate parity condition describes a no-arbitrage relationship between spot and forward exchange rates and the two nominal interest rates associated with these currencies. This implies that forward premiums and discounts in the foreign exchange market offset interest differentials to eliminate possible arbitrage that would arise from borrowing at the low-interest rate currency and lending at the high-interest-rate currency, while covering for the foreign exchange risk.

Interest rate parity is a critical equilibrium relationship in international finance. However, it does not always hold perfectly, as we will see. The availability of borrowing and lending opportunities in different currencies allows firms to hedge transaction foreign exchange risk with money market hedges. We demonstrate that when interest rate parity is satisfied, money market hedges are equivalent to the forward market hedges of transaction exchange risk. Moreover, we can use interest rate parity to derive long-term forward exchange rates. Knowledge of long-term forward rates is useful in developing multiyear forecasts of future exchange rates, which are an important tool in the valuation of foreign projects.1

If an investor chooses not to cover (or hedge) the exchange risk on a foreign money market investment, the return is uncertain and will be high if the foreign currency appreciates or low if the foreign currency depreciates. Our discussion of uncovered interest rate parity condition in the foreign money market uses some basic statistical methods that are commonly used to explain empirical evidence about investment returns in all asset markets. This allows for us to test what is termed the unbiasedness hypothesis. We conclude this chapter by considering the empirical evidence that relates to this hypothesis.

# 1 Covered Interest Parity

The covered interest rate parity (CIP) condition suggests that the interest rate differential between two currencies approximately equals the percentage spread between the currencies’ forward and spot rates on international money markets. If this is not the case, traders have an opportunity to earn arbitrage profits.

To appreciate the intuition behind this, recall that forward exchange rates allow investors to contract to buy and sell currencies in the future. Or stated more formally, a forward contract is an agreement between two parties to buy or sell a security (or currency) at a pre-specified price on a pre-specified date in the future. The party agreeing to buy (sell) the underlying in the underlying asset is said to “buy (sell) a forward”, and will be taking a long (short) position in the asset.

Hence, interest rate parity conditions refer to the equality of the returns on comparable money market assets when the forward foreign exchange market is used to eliminate foreign exchange risk. Or alternatively, it relates the spot and forward exchange rates and the nominal interest rates denominated in the two currencies.

Interest rate parity holds if markets are efficient and there are no government controls to prevent arbitrage. In the absence of these conditions, traders could make an extraordinary profit via covered interest rate arbitrage (where the term covered implies that the use of a forward ensures that the investment is not exposed to transaction foreign exchange risk).

To consider a hypothetical example of where an arbitrage opportunity would exist, suppose the following conditions are given:

• We can borrow R 100,000 from the bank
• USD interest rate is 4% p.a. and ZAR interest rate is 12% p.a.
• Spot rate is USDZAR 14.00 and 1-year forward is USDZAR 18.00

The arbitrage opportunity that would exist in this case would be realised by performing the following transactions:

• Convert from rands to dollars (R 100,000 $$/$$ 14 = $7,142.86) • Earn dollar interest on the principal ($7,142.86 $$\times$$ 1.04 = $7,428.57) • Cover the foreign exchange risk by engaging in a forward contract to sell the dollar interest plus principal ($7,428.57 $$\times$$ 18 = R 133,714.29)
• After a year we will owe the bank (R 100,000 $$\times$$ 1.12 = R 112,000)
So our profit will be R 133,714.29 - R 112,000 = R 21,714.29 for every R 100,000 that is borrowed.

If interest and exchange rates were as presented in the above example, then many banks and investors would borrow rands, convert to dollars, invest the dollars, and sell the dollar interest plus principal in the forward market for rands. This arbitrage activity would quickly eliminate the profit opportunity, since the additional demand to borrow rands would drive up the rand interest rate. In addition, the sale of rands for dollars would lower the spot exchange rate (closer to the R18.00 level). The lending of dollars would also lower the dollar interest rate, and the forward purchase of rands with dollars would raise the forward exchange rate (closer to the R14.00 level). Each of these movements would reduce the arbitrage profits that are present at the current prices.

A formal expression that summarizes the relationship between the interest rates denominated in two different currencies and the spot and forward exchange rates between those currencies may be derived as follows. If we assume that there are no arbitrage opportunities or barriers to trade in the respective international financial markets, and after ignoring the effects of transaction costs, we may make use of the following expression:

$\begin{eqnarray} \nonumber \frac{F_t (1 + i^{\star}_t )}{S_t} = 1 + i_t \tag{1.1} \end{eqnarray}$

where $$F_t$$ refers to the one-period forward exchange rate (domestic currency per foreign currency), $$S_t$$ refers to the spot exchange rate (domestic currency per foreign currency), $$i_t$$ refers the domestic currency interest rate for one period, and $$i^{\star}_t$$ refers to the foreign currency interest rate for one period.

Note that the representation in equation (1.1) would suggest that the amount that we need to repay the domestic bank, $$[1 + i_t]$$, will equal the amount of domestic currency that is converted to foreign currency to earn foreign interest and converted back to domestic currency at some point in the future, $$[1/S_t] \times [1 + i^{\star}_t ] \times F_t$$. Hence, a domestic money market investment and a foreign money market investment have the same domestic currency returns as long as the foreign exchange risk in the foreign money market investment is covered with a forward contract.

With the aid of a little algebra, the forward exchange rate would then be given by:

$\begin{eqnarray} F_t = S_t \frac{1 + i_t}{1 + i^{\star}_t} \tag{1.2} \end{eqnarray}$

Or alternatively, when we derive

$\begin{eqnarray*} \frac{F_t}{S_t} = \frac{1 + i_t}{1 + i^{\star}_t} \end{eqnarray*}$

and after subtracting one from both sides

$\begin{eqnarray} \nonumber \frac{F_t}{S_t} - \frac{S_t}{S_t} &=& \frac{1 + i_t}{1 + i^{\star}_t} - \frac{1 + i^{\star}_t}{1 + i^{\star}_t} \\ \therefore \frac{F_t - S_t}{S_t} &=& \frac{i_t - i^{\star}_t}{1 + i^{\star}_t} \tag{1.3} \end{eqnarray}$

Hence, the left-hand side of (1.3) is the forward premium or discount rate on the foreign currency and that the numerator of the right-hand side is the interest rate differential between the domestic and foreign currencies. It is often said casually that interest rate parity requires equality between the interest rate differential and the forward premium or discount in the foreign exchange market. For simple interest rates, the expression of interest rate parity in (1.3) demonstrates that this statement is an approximation because it ignores the term $$[1 + i^{\star}_t]$$ in the denominator on the left-hand side. However, as this term is usually close to $$1$$ (particularly in developed world economies) the approximation is reasonably plausible, especially when the maturity is short.

## 1.1 Deviations from Covered Interest Rate Parity

As the settlement procedures in the external currency markets are identical to the settlement procedures in the forward markets, and because transaction costs are small, banks operating in this market should arbitrage away all deviations from covered interest rate parity. In fact, it is often the case that banks use interest rate parity to quote forward rates in outright forward transactions.

Prior to the financial crisis that began in 2007, documented violations of interest rate parity were exceedingly rare. Because prices move quickly within the day, careful analysis of the issue requires time-stamped data. Akram, Rime, and Sarno (2008) assembled such data from Reuters for the pound, euro, and yen, all versus the dollar, for a short period from February 13 to September 30 of 2004. They detected multiple short-lived deviations from covered interest rate parity that provided possible arbitrage profits. Nevertheless, the deviations tended to persist only for a few minutes and represented a tiny fraction of all possible transactions. Hence, unless you are a trader in a bank, it is safe to assume that covered interest rate parity holds, at least in normal times.

Note also that when dealing with emerging market currencies you should account for the additional risk factors before seeking to derive any arbitrage profits from potential deviations from the CIP condition. These risks may include degree of default (or credit) risk, exchange controls, and political risk. For example, those institutions with a low credit rating would need to pay higher rates of interest. In the case that we considered, if the 4% rate of interest in U.S. was provided by an institution that had a lower rating than the S.A. institution then the profit that we derived may be seen as compensation for taking on additional credit risk.

Another problem with assessing the validity of interest rate parity is caused by exchange controls. Governments of countries periodically interfere with the buying and selling of foreign exchange. They may tax, limit, or prohibit the buying of foreign currency by their residents. In addition, they may also tax, limit, or prohibit the inflow of foreign investment into their country. Such exchange controls have been applied in several low income and emerging market economies at various points in time. Note also that even if no exchange controls are currently present, foreign investors may rationally believe that a government will impose some form of exchange controls or taxes on foreign investments in the future. The possibility of any of these events occurring is called political risk.

# 2 Uncovered Foreign Money Market Investments

If an investor chooses not to hedge (or “cover”) the exchange risk on a foreign money market investment, the return is uncertain and will be high if the foreign currency appreciates or low (and potentially negative) if the foreign currency depreciates. Returning to our previous example, we can consider the following payoff from an uncovered trade that is speculative in nature.

To consider a hypothetical example of a speculative trade, suppose that we are given:

• Assume that we can borrow R100,000 from the bank
• USD interest rate is 4% and ZAR interest rate is 12%
• Spot exchange rate is USDZAR 14.00

The investment opportunity that would exist in this case would be realised by performing the following transactions:

• Convert from rands to dollars (R100,000/14= $7,142.86) • Earn dollar interest on the principal ($7,142.86 1.04 = $7,428.57) • Sell the dollar principal plus interest at the spot exchange rate in 1 year: ($7,428.57 $$S_{t+1}$$ = Rand proceeds after 1 year)
The return on investment is going to be determined by the future spot rate.

If we denote the USDZAR current spot rate as $$S_t$$ and the future spot rate as $$S_{t+1}$$, then the above calculations can be summarised as follows to provide the rand return, $$r_{t+1}$$, from investing R$$1.00$$ in the dollar money market

$\begin{eqnarray} r_{t+1} = \frac{1}{S_t} \times \big[1 + i^{\star}_t \big] \times S_{t+1} \tag{2.1} \end{eqnarray}$

To measure the excess return on this investment (i.e. the return over and above what he could earn risk free domestically) we could calculate $$exr_{t+1}$$ as

$\begin{eqnarray} exr_{t+1} = \frac{S_{t+1}}{S_t} \times \big[1 + i^{\star}_t \big] - \big[1 + i_t \big] \tag{2.2} \end{eqnarray}$

We are then able to calculate the future exchange rate for which we would break even between the dollar and the domestic money market investments, which we denote $$S^{BE}$$, after setting (2.2) to zero,

$\begin{eqnarray} S^{BE} = {S_t} \times \frac{1 + i_t}{1 + i^{\star}_t} \tag{2.3} \end{eqnarray}$

which is essentially equal to (1.2).

## 2.1 Profits and Losses from Forward Market Speculation

It is worth noting that as forward contracts represent contractual obligations between parties, a positive payoff to one party would represent a loss to counterparty. Hence, speculators on the foreign exchange markets are involved in what is termed a zero-sum-game. For example, consider the case where Mr. Buy buys dollars forward with rands from Ms. Sell. The payoff to Mr. Buy would be:

$\begin{eqnarray*} S_{t+1} - F_t \end{eqnarray*}$

Similarly, as Ms. Sell sells dollars forward for rands from Mr. Buy, her payoff would be:

$\begin{eqnarray*} F_t - S_{t+1} \end{eqnarray*}$

These profits and losses are displayed in Figure 1 as a function of $$S_{t+1}$$.2 Note that the rand profit of the person buying foreign currency forward is the rand loss of the person selling foreign currency forward, and vice versa.

Figure 1: Payoffs from FX Forwards

# 3 Currency Speculation and Profits and Losses

The uncertainty about future exchange rates makes currency speculation risky. To quantify our uncertainty about future returns, we use conditional probability distributions of the underlying variables. In terms of notation, we will index today as being time $$t$$, and remember that the conditional probability distribution of the spot exchange rate for some time in the future describes the conditional probabilities associated with all the possible exchange rates that may occur at that time conditioned on all the information that is available today.

The collection of all information that is used to predict the future value of an economic variable is typically called an information set. Also, recall that we refer to the expected mean of this probability distribution as the conditional expectation of the future exchange rate. We denote the conditional expectation at time $$t$$ of the future spot exchange rate at time $$t+1$$, as $$\mathbb{E}_t \big[S_{t+1}\big]$$.

If we assume that this exchange rate may be represented by a normal distribution then we could quantify the expected losses and profits in our previous example.3

The probability of incurring a trading loss

Suppose we expect the rand to depreciate relative to the dollar by 5% over the next year. Then, the conditional expectation of our future spot rate in 1 year is

(USDZAR 14.00) $$\times$$ (1 - 0.05) = (USDZAR 14.70)

After we convert our rands into dollars at the current spot and earn foreign interest, our return in dollars would be,

R100,000 / (USDZAR 14.00) $$\times$$ 1.04 = $7,428.57 which makes the conditional expectation of our uncertain rand return equal to$7,428.57 $$\times$$ (USDZAR 14.70) = R109,200

If we think that the rate of appreciation of the dollar relative to the rand is normally distributed. From the symmetry of the normal distribution, we know that there is a 50% probability that we will do better than the rand investment and there is a 50% probability that we will do worse.

We might also be interested in knowing the probability that we will lose some of our rand principal. At what future value of the spot exchange rate $$S_{t+1, \text{USDZAR}}$$ will we just get our R100,000 principal back? Let’s call this value $$\hat{S}$$, which satisfies,

$7,428.57 $$\times$$ $$\hat{S}$$ = R100,000 from which we find $$\hat{S}$$ = R100,000 /$7,428.57 = (USDZAR 13.46)

We can then calculate the probability that the future exchange rate will be lower than USDZAR 13.46. To perform such a calculation, we need to determine the standard deviation of the payoff on our investment. Suppose we think that the standard deviation of the rate of appreciation of the dollar relative to the rand over the next year is 10%.

Since 10% of USDZAR 14.00 is USDZAR 1.40, the standard deviation of the conditional distribution of the future spot exchange rate is R1.40. We can calculate the probability of losing money by creating a standard normal random variable. A standard normal random variable has a mean of $$0$$ and a standard deviation of $$1$$, which we denote with $$\mathcal{N}(0,1)$$. We can then calculate the distribution of the exchange rate with,

$\begin{eqnarray*} \frac{S_{t+1, \mathsf{USDZAR}} - (\mathsf{USDZAR \; 14.70})}{\mathsf{USDZAR \;1.40}} \end{eqnarray*}$

which has a mean of 0 and a standard deviation of 1. The value of the standard normal variable associated with a zero rate of return is then calculated as

$\begin{eqnarray*} \frac{(\mathsf{USDZAR \; 13.46})- (\mathsf{USDZAR \; 14.70})}{\mathsf{USDZAR \; 1.40}} = \mathsf{-0.8846} \end{eqnarray*}$ From the probability distribution of a standard normal variable, we find that there is a 18.82% probability that a $$\mathcal{N}(0, 1)$$ variable will be less than R13.46, or equivalently that $$S_{t+1,\mathsf{USDZAR}}$$ will be less than R13.46.

Note that one could think of the conditional probability distribution as reflecting the subjective beliefs of an individual investor, an importer or an exporter, would have about the uncertain future exchange rate. If we had made use of a larger standard deviation for the distribution the future exchange rate, it would be more disperse and the probability that we would lose some of our principal would be much greater.4

In the following section we discuss the theories surrounding the value for the conditional mean of the distribution.

# 4 Uncovered interest rate parity

Covered interest rate parity maintains that a domestic money market investment and a foreign money market investment have the same domestic currency return as long as the foreign exchange risk in the foreign money market investment is covered using a forward contract. Two related theories predict what may happen when exchange rate risk is, by contrast, not hedged. Uncovered interest rate parity maintains that the uncovered foreign money market investment, which has an uncertain return because of the uncertainty about the future value of the exchange rate, has the same expected return as the domestic money market investment. The unbiasedness hypothesis states that there is no systematic difference between the forward rate and the expected future spot rate and that, consequently, the expected forward market return is zero.

## 4.1 Uncovered Interest Rate Parity

If we take the expected value of the return from investing R$$1$$ in the dollar money market, we could use (2.1) to derive

$\begin{eqnarray*} \mathbb{E}_t \big[ r_{t+1} \big] = \frac{1}{S_t} \times \big[ 1 + i^{\star} ] \times \mathbb{E}_t \big[ S_{t+1} \big] \end{eqnarray*}$

Note that since the current spot rate, $$S_t$$, and the interest rate, $$i^\star$$, are in the time $$t$$ information set, the expectation applies only to the future exchange rate.

Uncovered interest rate parity is the hypothesis that the expected return on the uncovered foreign investment equals the known return from investing R1 in the domestic money market $$[1 + i]$$. If uncovered interest rate parity is true, there is no compensation to the uncovered investor for the uncertainty associated with the future spot rate, and expected returns on investments in different money markets are equalized. Equivalently, the speculative return on borrowing R1 and investing it in the dollar money market, $$exr_{t+1}$$ is expected to be zero, given current information.

For example, if the interest rate on the dollar investment is 4%, and the interest rate on the rand is 12%. Uncovered interest rate parity suggests that it would be naive to think that dollars therefore constitute a great investment. In fact, the higher yield on the rand investment implies that the market expects that the rand will depreciate by just enough that the expected dollar return from the currency speculation in the foreign market would make the rates of return equivalent.

This leads to the intuition that interest rate interest rate differentials across countries on bonds of similar credit risk should reveal an expected change in the value of one of the currencies. Hence an initially attractive interest rate differential should be met by an offsetting event so that you can make no more money on average from investing in one currency relative to another. Hence, in the above example, it should be the case that

$\begin{eqnarray*} \frac{1}{(\mathsf{USDZAR \; 14})} \times \mathbb{E}_t \big[ S_{t+1} \big] = \big[ \mathsf{1} + \mathsf{0.12} \big] \\ \end{eqnarray*}$

such that

$\begin{eqnarray*} \mathbb{E}_t \big[ S_{t+1} \big] = \frac{\mathsf{1.12}}{\mathsf{1.04}} \times (\mathsf{USDZAR \; 14}) = (\mathsf{USDZAR \; 15.08}) \end{eqnarray*}$

## 4.2 Unbiasedness Hypothesis

When the forward rate equals the expected future spot rate, the forward rate is said to be an unbiased predictor of the future spot rate. This equality is summarized by the unbiasedness hypothesis:

$\begin{eqnarray} \mathbb{E}_t \left[ S_{t+1} \right] & = & F_t \tag{4.1} \end{eqnarray}$

This condition is derived from a combination of the covered interest rate parity and uncovered interest rate parity conditions. Note firstly that uncovered interest rate parity condition could be written as

$\begin{eqnarray*} \mathbb{E}_t \left[ \frac{ S_{t+1}}{S_t} \left( 1 + i^{\star}_t \right) \right] = \left( 1 + i_t \right) \end{eqnarray*}$

While the covered interest parity condition could be written as,

$\begin{eqnarray*} \frac{ F_{t}}{S_t} \left( 1 + i^{\star}_t \right) = \left( 1 + i_t \right) \end{eqnarray*}$

After setting the left-hand sides equal to one another we would have

$\begin{eqnarray*} \mathbb{E}_t \left[ \frac{ S_{t+1}}{S_t} \left( 1 + i^{\star}_t \right) \right] = \frac{ F_{t}}{S_t} \left( 1 + i^{\star}_t \right) \end{eqnarray*}$

By eliminating $$S_t$$ and $$[1 + i^{\star}_t]$$ from both sides of the exterior equations, we recover the unbiasedness hypothesis that is provided in (4.1).

## 4.3 Making this theory testable

Please take note that the inclusion of the expectations operator would ensure that this expression does not imply that the forward rate is equal to the future spot rate at each and every point in time.

Another way of making this important point is to be conscious of the fact that the estimate of the future spot rate would incorporate a forecast error, which is the difference between the realised future spot exchange rate and its forecast. These may be summarised by taking the sum of the squared forecast errors.5 It reasonable to expect exchange rate forecasts would be characterized with large variability as they behave similar to other asset prices. The extent of the forecast errors would provide us with an idea of the extent of the potential expectation errors and as such, there may be quite a large difference between the forward rate and the realized future spot rate.

Despite the fact that we may observe relatively large errors, this would not necessarily imply that the forecasts are biased, since an unbiased predictor would imply that the expected mean forecast error is zero. In our setting, we forecast the future spot rate using the forward rate so that the forecast error is the difference between the two (i.e. $$S_{t+1} - F_t$$). The unbiasedness hypothesis states nothing about the magnitude of the forecast errors, which can be large or small and can vary over time.6

The unbiasedness hypothesis in (4.1) is often identified with market efficiency. In efficient capital markets, investors cannot expect to earn profits over and above what the market supplies as compensation for bearing risk. An inefficient market is one in which profits from trading are not associated with bearing risks and are therefore considered extraordinary. The definition of market efficiency incorporates the hypotheses that people process information rationally and that they have common information on relevant variables that may help predict exchange rates. Together, these assumptions ensure that people have common expectations of the future.

# 5 Risk Premiums in the Foreign Exchange Market

The risk premium on an asset is the expected return on the asset in excess of the return on a risk-free asset. In this case, the excess return can be thought of as the uncovered foreign money market return, which we called $$exr_{t+1}$$. Denoting the foreign exchange risk premium by $$rp$$, we have $$rp_t = \mathbb{E}_t [ exr_{t+1}]$$. Different assets can have different risks, and assets that are riskier must offer higher expected returns in order to induce investors to hold them.

You may think that the riskiness of an investment in an asset is determined by the uncertainty associated with the asset’s payoff. For example, the risk premium on currency speculation must be linked to the variability of exchange rate changes. After all, the conditional distribution of the future exchange rate will be wider the more variable such changes are.

However, this is not the case. The reason is that investors care about the expected return and risk of their whole portfolio of assets, not necessarily about the risk of an individual asset viewed in isolation. Note that the uncertainty of a return can always be decomposed into a part that is systematic and a part that is unsystematic. Hence,

$\begin{eqnarray*} \text{Individual asset return uncertainty} = \text{Systematic risk} + \text{Unsystematic risk} \end{eqnarray*}$

Portfolio theory dictates that assets whose returns contain only unsystematic risk do not contribute to the variance of the portfolio and, consequently, do not command any risk premium as it would be possible to use diversification to reduce this risk to zero. Similarly, as systematic risk measures how much an asset return moves with the market, it cannot be diversified away, and the risk involved commands a risk premium.

The capital asset pricing model (CAPM) suggests that it is the covariance of an asset’s return with the return on the market portfolio that determines the asset’s systematic risk and hence its risk premium. The model also provides an easy-to-implement procedure to put an actual number on the risk premium, where the systematic risk of an individual asset is fully described by its beta with respect to the market portfolio, which is measured as

$\begin{eqnarray*} \text{Beta on asset } i = \frac{\text{Covariance (Asset return) } i \text{, Market portfolio return)}}{\text{Variance (Market portfolio return)}} \end{eqnarray*}$

Higher betas indicate higher systematic risk, and the CAPM postulates that

$\begin{eqnarray*} \text{Risk premium on asset } i = (\text{Beta on asset } i ) \times ({\text{Risk premium on market portfolio)}} \end{eqnarray*}$

Hence, the investor requires compensation in excess of the risk-free rate (that is, a risk premium) if the beta of the asset is positive, as are the betas of most equity investments. In addition, as a forward contract is an asset, there is potential for a risk premium. Taking a position in a forward contract involves no investment of funds at the point in time when the contract is set, and it is not necessary to compensate the investor for the time value of money. However, the dollar profits and losses on the forward contract can still covary systematically with the return on a portfolio of currency assets.

Hence, if the profitability of Mr. Buy’s purchase of foreign currency at the forward exchange rate covaries positively with the dollar return on the market portfolio, Mr. Buy will view the forward contract as risky and will demand an expected profit. As noted previously, though, Ms. Sell’s profits and losses on the forward contract are the opposite of Mr. Buy’s profits and losses. Hence, if Mr. Buy’s dollar profit is positively correlated with the dollar return on the market portfolio, the covariance of the dollar profit on Ms. Sell’s side of the forward contract is negatively correlated with the dollar return on the market portfolio.

In this case, when Ms. Sell enters into the contract, she obtains an asset that reduces the variability of her overall portfolio. She consequently willingly holds this contract at an expected loss. Again, this is like portfolio insurance. From Ms. Sell’s perspective, the expected loss is balanced by the fact that the forward contract performs well when the rest of her portfolio does poorly. Consequently, there can be a risk premium that causes the forward rate to be a biased predictor of the future spot rate.

According to the CAPM, such a risk premium should depend on the beta of the (excess) return to currency speculation.

# 6 Empirical applications of uncovered interest rate parity and unbiasedness hypothesis

The first potential problem that we would encounter when trying to test the hypothesis, $$\mathbb{E}_t \left[ S_{t+1} \right] = F_t$$ is that spot rates and forward rates move together over time in a persistent manner. Hence, a test on the levels of the variables would almost always fail to reject the unbiasedness hypothesis, even when the hypothesis was false. Therefore, it is necessary to transform the variables to ensure that they take on functional forms that are stationary.

One way to accomplish this objective is to divide both variables by $$S_t$$. In addition, it is also usually convenient to subtract one from both sides to make the representation easy to interpret. Hence,

$\begin{eqnarray} \nonumber \frac{F_{t}}{S_t} - \frac{S_t}{S_t} &=& \frac{\mathbb{E}_t \left[ S_{t+1} \right]}{S_t} - \frac{S_t}{S_t} \\ \nonumber \frac{F_{t}-S_t}{S_t} &=& \frac{\mathbb{E}_t \left[ S_{t+1} \right] - S_t}{S_t} \\ fp_t &=& \mathbb{E}_t \left[ s_{t+1} \right] \tag{6.1} \end{eqnarray}$

where $$fp_t$$ represents the forward premium (or discount) on the currency. The right-hand side represents the expected rate of appreciation (depreciation) of the currency. This expression states that the unbiasedness hypothesis requires the forward premium (discount) on the currency should be equal to the market participants’ expectations about the rate of appreciation (depreciation) of the currency over the time period $$t+1$$. If the hypothesis holds, the expected return from currency speculation will be exactly zero.

The next problem that we would have with testing the unbiasedness hypothesis is that it contains a variable that cannot be observed by a statistician: the conditional expectation of the rate of appreciation of one currency in terms of another. This conditional expectation is formed by market participants on the basis of their information set. Hence, in order to test the unbiasedness hypothesis, a statistician must specify how investors and speculators form their expectations.

Typically, when statisticians are confronted with an unobservable variable, they make an auxiliary assumption to develop a test of the underlying hypothesis. As in most other areas of financial economics, the most popular auxiliary assumption is that investors have rational expectations. If investors have rational expectations, they do not make systematic mistakes, and their forecasts are not systematically biased. When investors have rational expectations, we can decompose the realized (observed) rate of appreciation into its conditional expectation plus an error term that does not depend on time $$t$$ information,

$\begin{eqnarray*} \text{Realized appreciation } = \text{ Expected appreciation } + \text{ Forecast error} \end{eqnarray*}$

The error term can be viewed as news that moved the exchange rate, but the news, by definition, was unanticipated by rational market participants at time $$t$$. The use of rational expectations would imply that both the conditional mean and the unconditional mean of the forecast error term are zero. Hence, (6.1) could then be written as,

$\begin{eqnarray} s_{t+1} = fp_t + \varepsilon_{t+1} \tag{6.2} \end{eqnarray}$

where $$\varepsilon_{t+1}$$ represents the forecast error, while the other two variables, $$s_{t+1}$$ and $$fp_t$$, are observed.

## 6.1 Mean forecast error

Possibly the first thing to investigate is whether the forecast error in (6.2) is indeed significantly different from zero. This can be achieved by testing the null hypothesis that the unconditional mean of the realized rate of appreciation is equal to the unconditional mean of the forward premium. These statistics are reported in Table 1, where we can conclude that the mean of a time series is significantly different from zero at the 95% confidence level if the sample mean is more than 1.96 standard errors from zero. In this case, the standard error of the sample mean for a time series can be calculated as $$\sigma / \sqrt{T}$$, where $$\sigma^2 = \sum^T_{t=1} (x_t - \hat{\mu})^2 / T$$ denotes the sample variance of the series, and $$\hat{\mu}$$ denotes the sample mean of the series. The confidence level of the test that the mean is zero is provided below the standard error in the table.

These results suggest that not a single mean rate of depreciation is sufficiently large relative to its standard error that we can be more than 90% confident that it is significantly different from zero (which are shown in the first column). In contrast with these the results in the second column, which show the mean forward premium for the currencies, suggest that we can be quite confident that all the unconditional means of the forward premiums are significantly different from zero.

Mean
Rate of Appreciation
Difference
(S.E.)
(S.E.)
(S.E.)
Currency $$\hspace{1cm}$$ Conf. Conf. Conf.
EURUSD 2.04 1.45 0.58
(-1.98) (-0.25) (-2.02)
0.7 1 0.23
GBPUSD -0.41 -2.24 1.62
(-1.93) (-0.23) (-1.99)
0.17 1 0.59
USDJPY -2.82 -3.31 0.48
(-2.05) (-0.23) (-2.13)
0.83 1 0.18
EURJPY -1.44 -1.86 0.42
(-2.03) (-0.18) (-2.07)
0.52 1 0.16
EURGBP 2.81 3.5 -0.69
(-1.64) (-0.25) (-1.67)
0.91 1 0.32
GBPJPY -3.7 -5.34 1.65
(-2.29) (-0.19) (-2.34)
0.89 1 0.52

Table 1: Means of Monthly Rates of Appreciation, Forward Premiums, and their Differences

The third column in Table 1 tests the hypotheses that the means of the 1-month forward premiums are equal to the means of the 1-month rates of appreciation. In no case is there sufficient evidence to reject the null hypothesis with 90% confidence that the difference in the mean values for $$s_{t+1} - fp_t$$ is significantly different from zero. The largest confidence level is only 0.59 for the dollar value of the pound. Hence, we are unable to reject the null hypothesis that the unconditional mean of the realized rate of appreciation is equal to the unconditional mean of the forward premium.

From the conditions from interest rate parity the forward premium on a foreign currency is equal to the interest differential between the domestic currency and the foreign currency. Hence, countries with high nominal interest rates have currencies that should depreciate in value over time relative to the currencies of countries that have low nominal interest rates. Hence, failure to reject the unbiasedness hypothesis with the test of unconditional means supports the proposed fact quite strongly. For example, it has been noted that the euro interest rates were approximately 1.86% higher than the interest rates in Japan. Table 1 suggests that this is fairly similar to the average depreciation of the euro relative to the yen, which was 1.44% over the sample period.

## 6.2 Regression Tests of the Unbiasedness of Forward Rates

We can also use a simple linear regression model to test the unbiasedness hypothesis. Such a model may be specified as,

$\begin{eqnarray} s_{t+1} = \phi_1 + \phi_2 fp_t + \varepsilon_{t+1} \tag{6.3} \end{eqnarray}$

where the unbiasedness hypothesis would hold when $$\phi_1 =$$ 0 and $$\phi_2 =$$ 1. The regression tests of the unbiasedness hypothesis are presented in Table 2, which presents the estimated parameters and their standard errors for regressions using the six exchange rates as in Table 1. The standard errors are presented in parentheses below the estimated coefficients. The confidence levels of the tests that $$\phi_1 =$$ 0 and $$\phi_2 =$$ 1 are presented below the standard errors. Values of the confidence level that are above 0.90 indicate that we can reject the null hypothesis with 90% confidence.

Coefficients on Regressors
Const
(S.E.)
(S.E.)
Currency $$\hspace{1cm}$$ Conf. ($$\phi_1$$ = 0) Conf. ($$\phi_2$$ = 1) $$R^2$$
EURUSD 3.26 -0.84 0.004
(-2.31) (-0.81)
0.84 0.98
GBPUSD -3.84 -1.68 0.016
(-2.24) (-0.82)
0.91 1
USDJPY -10.03 -2.18 0.023
(-2.67) (-0.64)
1 1
EURJPY -4.46 -1.62 0.008
(-2.3) (-0.87)
0.95 1
EURGBP 4.7 -0.54 0.003
(-2.56) (-0.65)
0.93 0.98
GBPJPY -17.17 -2.52 0.020
(-5.34) (-0.84)
1 1

Table 2: Regression Tests of the Unbiasedness Hypothesis

Note that all six of the estimated values of $$\phi_2$$ are significantly different from unity. Perhaps more surprisingly, notice that all the estimated slope coefficients are negative. The estimated values of $$\phi_2$$ range from -2.52 for the yen value of the pound to -0.54 for the pound value of the euro. Consequently, the regressions suggest the existence of a forward rate bias (i.e. the forward rate does not equal the expected future spot rate). The regression evidence thus qualifies the use of the unbiasedness hypothesis. Treasurers in MNCs and global portfolio managers must realize that there is a potential cost to hedging foreign currency risk because the forward rate is not necessarily the best forecast of the future exchange rate.

Since negative values for $$\phi_2$$ are found in the cross-rate regressions as well, the explanation of this phenomenon for the dollar exchange rates should not be sought in a story about common movements of the dollar relative to other currencies, nor could it be due strictly to U.S. policy. Apparently, the explanation must encompass the behaviour of all major foreign exchange markets.

Note also that the explanatory power of the regressions, which is measured by the $$R^2$$ values, is quite low. The largest $$R^2$$ is 2.3%. The appropriate way to interpret this finding is that there is some ability of the forward premium to predict the rate of appreciation, but the unanticipated component in the rate of appreciation is large relative to its predictable component.

This regression for the unbiasedness hypothesis could also be used to generate a forecast for the future changes in spot exchange rates, where we could use the estimated values of $$\hat{\phi}_1$$ and $$\hat{\phi}_2$$ to generate values for $$\mathbb{E}_t [s_{t+1}]$$. In addition, we could also make use of this regression to calculate the forward market return, which is given as $$s_{t+1} - fp_t$$ and would represent the return from holding a long forward position in the foreign currency. Hence,

$\begin{eqnarray} \nonumber \mathbb{E}_t [s_{t+1}] &=& \hat{\phi}_1 + \hat{\phi}_2 fp_t \\ \text{or } \mathbb{E}_t \big[s_{t+1} - fp_t \big] &=& \hat{\phi}_1 + (\hat{\phi}_2 -1) fp_t \tag{6.4} \end{eqnarray}$

People familiar with the results of the unbiasedness regressions just presented often argue that the negative slope coefficients imply that currencies trading at a forward discount will strengthen, in contrast to the prediction of the unbiasedness hypothesis, which implies that discount currencies are going to weaken. Unfortunately, this interpretation of the regression is wrong because it ignores the value of the constant term in the regression.

Table 3 shows the importance of the constant in the regression, using the USDJPY equation as an example. We consider a forward discount on the dollar of 3.31%, the sample average (see Table 1), implying that Japanese yen interest rates were on average approximately 3.31% less than U.S. dollar interest rates. On the first line of Table 3, we repeat the prediction of the theory: If the dollar is at a 3.31% discount, it should be expected to depreciate by 3.31%. If we were to use the regression and ignore the constant as in the computation on the second line, the prediction is a 7.22% appreciation of the dollar, so that the dollar indeed gives a higher yield and is expected to appreciate substantially.

However, the correct interpretation is on the third line of Table 3, which uses the regression with the estimated coefficients as in (6.4) to determine an estimate of expected dollar depreciation or appreciation. The dollar is now expected to weaken, but only by 2.82%. This is the average depreciation of the dollar over the sample period (see Table 1), and most importantly, it is lower than the depreciation the forward discount suggests.

However, the regression still implies that a speculator should buy dollars forward if he believes the prediction of the regression will be borne out. That is,

$\begin{eqnarray*} \mathbb{E}_t [fmr_{t+1} ] &=& \mathbb{E}_t [s_{t+1} - fp_t] \end{eqnarray*}$
(Expected forward market return) $$=$$ -2.82% - (-3.31%) $$=$$ 0.49%

$$fp_t$$ $$\phi_1$$ $$\phi_2$$ $$\mathbb{E}_t [ s_{t +1}]$$ $$\mathbb{E}_t [ fmr_{t}]$$
Uncovered Interest Rate Parity -3.31% 0 1 -3.31% 0%
Naive Interpretation -3.31% 0 -2.18 7.22% 10.53%
Actual Interpretation -3.31% -10.03 -2.18 2.82% 0.49%
(large discount) -5.00% -10.03 -2.18 -0.87% 5.87%

Table 3: Interpreting the Unbiasedness Regression

The expected forward market return from buying dollars forward is now positive. When the forward discount is unusually large, there can be an expected dollar appreciation, and the expected return from going long dollars increases substantially. The last line in Table 3 demonstrates this for a forward discount of 5%.

# 7 Conclusion

Well over one hundred papers document, in some form or another, the *forward premium anomaly}, where future exchange rate changes do not move one-for-one with interest rate differentials across countries. The general finding is that over 1 to 6 months horizons there is consistent evidence of the {forward premium anomaly} over many currencies. In these cases, the average value of $$\hat{\phi}_2$$ is -0.88 (Froot and Thaler 1990). It has also been shown that the magnitude of $$\phi_2$$ is time varying and is also possibly regime specific. However, over the long run the average interest rate differential and currency depreciation appear to move as expected by the theory.

# 8 Appendix: Alternative implementation of the regression equation

If we also assume that investors are risk-neutral and in the absence of taxes and on capital transfers we can derive a regression equation for the unbiasedness hypothesis, $$\mathbb{E}_t [S_{t+1}] = F_t$$. For example, under the assumption of log-normality, we can rewrite this expression as,

$\begin{eqnarray*} \mathbb{E}_t \Big[ \exp \big[ \log \left( S_{t+1} \right) \big] \Big] = \mathbb{E}_t \big[ \exp \left( s_{t+1} \right) \big] &=& F_t \\ \exp \left( \mathbb{E}_t s_{t+1} + \frac{1}{2} \textrm{var}_t \left( s_{t+1} \right) \right) &=& F_t \end{eqnarray*}$

log both sides to derive,

$\begin{eqnarray*} \mathbb{E}_t s_{t+1} = f_t - \frac{1}{2} \textrm{var}_t \left( s_{t+1} \right) \end{eqnarray*}$

Now drop the variance and subtract $$s_t$$ to get a relatively neat approximation

$\begin{eqnarray*} \mathbb{E}_t \Delta s_{t+1} = f_t - s_t \approx i_t - i^{\star}_t \end{eqnarray*}$

This could then be rewritten as

$\begin{eqnarray*} \Delta s_{t+1} = \mathbb{E}_t \Delta s_{t+1} + \varepsilon_{t+1} \end{eqnarray*}$

In addition, under the assumption of {rational expectations}, $$\varepsilon_{t+1}$$ is orthogonal to $$\mathbb{E}_t \Delta s_{t+1}$$. If we substitute in the unbiasedness hypothesis condition, we would then have:

$\begin{eqnarray*} \Delta s_{t+1} = \left( f_t - s_t \right) + \varepsilon_{t+1} \end{eqnarray*}$

which may be used to formulate the regression model

$\begin{eqnarray*} \Delta s_{t+1} = \phi_1 + \phi_2 \left( f_t - s_t \right) + \varepsilon_{t+1} \end{eqnarray*}$

that can be applied to the data.

# 9 References

Akram, Farooq, Dagfinn Rime, and Lucio Sarno. 2008. “Arbitrage in the Foreign Exchange Market: Turning on the Microscope.” Journal of International Economics 76: 237–53.

Bekaert, Geert, and Robert J. Hodrick. 2012. International Financial Management. New York: Prentice Hall.

Froot, Kenneth A., and Richard H. Thaler. 1990. “Foreign Exchange.” Journal of Economic Perspectives 4: 179–92.

1. For example, if you have recently completed a research project for an American company, then depending on the respective interest rates you would be able to assess whether it is better to hold his money in the U.S. for a month and convert later, or to convert immediately and hold the money in S.A.

2. This figure has been taken from Bekaert and Hodrick (2012) and has been adjusted from the different currencies in this example.

3. Although the use of a normal distribution would not be appropriate in this case, it will assist with developing intuition. We will consider the implications of using a non-normal distribution at a later point in time.

4. You can either make use of statistical tables, the R function pnorm(z,0,1), or the Excel function NORM.S.DIST(z,TRUE) to find the probability that an event will occur for a given standard normal variable. In the above example the value for $$z$$ is -0.8846.

5. After calculating this statistic we could take its square root to find the root-mean squared-error.

6. If you knew that forecast errors were biased then you could potentially derive a more accurate forecast that could be used to derive profitable investment opportunities.