This section will consider the relationships between spot and forward rates, interest rates, and maturities that exist as a result of arbitrage relationships.

Spot and Forward Rates

In the professional FX market, forward exchange rates are commonly quoted in terms of ‘points’ or ‘pips.’ These points represent the difference between the forward and spot exchange rate quotes. The scale is adjusted so that these points correspond to the last decimal place in the spot quote.

If the forward rate is higher than the spot rate, the points are positive, indicating that the base currency is trading at a forward premium. On the other hand, when the forward rate is below the spot rate, the points are negative, suggesting the base currency is trading at a forward discount. Notably, when the base currency is at a forward premium, the price currency will be at a forward discount, and vice versa.

To understand this argument, let’s look at a scenario in 2023. The spot USD/CAD exchange rate was 1.3845, and the six-month forward rate was 1.38475. This suggests that the CAD (base currency) was trading at a premium. The six-month forward points were quoted as 2.5, and this can be calculated as follows:

$$1.38475-1.3845=0.00025$$

We multiply by 10,000 to ensure the points align with the final digit of the spot exchange rate quote, which is usually the fourth decimal place. Additionally, it’s noteworthy that while points are usually quoted to at least one decimal place, the forward rate might extend to five or even more decimal places.

Among major currencies, the yen stands as an exception. Its spot rates are typically quoted to just two decimal places. Therefore, the difference between its forward and spot rates is multiplied by 100 to adjust to its two-decimal-place precision.

Forward Points and Maturity

Forward rate quotes are typically presented as the number of forward points for each specific maturity, which refers to the time interval between the spot settlement and the forward contract settlement. Often, these forward points are also termed ‘swap points’ since an FX swap encompasses both a spot and a forward transaction executed simultaneously.

To convert forward quotations into an outright forward quotation, let’s use an example with the RUB/CNY currency pair. We’ll use a table that shows maturity and forward or spot rate points.

$$ \begin{array}{l|c} \textbf{Maturity} & \textbf{Spot rate or forward points} \\ \hline \text{Spot} & 1.6459 \\ \text{One week} & -0.2 \\ \text{One month} & -0.1 \\ \text{Three months} & -5.6 \\ \text{Six months} & -12.7 \\ \text{Twelve-month} & -25.3 \\ \end{array}$$

From the table above, notice that the absolute number of points increases with maturity. This trend emerges because the number of points is directly proportional to the yield differential between the two involved countries (in this case, Russia and China), scaled by the term to maturity.

As the term to maturity extends, the absolute number of forward points increases. In the same manner, for a fixed term to maturity, a greater interest rate differential results in a higher absolute number of forward points.

Calculating Forward Exchange Rates

To calculate forward exchange rates using forward points, you divide the points by 10,000. This scales down the points to the fourth decimal place which is the last decimal place in the spot rate.(Note that for an exchange rate involving the Japanese Yen, we will divide by 100 as exchange rates involving the Japanese Yen are quoted to two decimal places). Then, you add the result to the spot exchange rate quote.

We can take the case of the six-month forward rate in the above table. Here we have:

$$1.6459+ \frac{-12.7}{10000}=1.6459-0.00127=1.64463$$

Occasionally, the forward rate points are represented as a percentage of the spot rate as opposed to an absolute number of points. Consider the illustration below that converts the above six-month forward rate from absolute points to a percentage.

$$\frac{1.6459-0.00127}{1.6459}-1=-0.077\%$$

To convert this percentage into a forward rate, we simply need to multiply the spot rate by one plus the percentage forward premium or discount:

$$1.6459 \times\left(1 +\left(-0.00077\right)\right)=1.6459\times 0.99923 =1.64463$$

Arbitrage Relationships between Spot and Forward Rates

To understand the arbitrage relations between spot and forward exchange rates, we need to consider interactions between spot rates, forward rates, and interest rates.

Note that forward exchange rates are derived from an arbitrage principle that ensures equal investment returns on two alternative yet equivalent investment opportunities.

Using a single-period analogy, an investor who has funds to invest in treasury securities has two alternatives:

1. Invest at the domestic risk-free rate (r_d).
2. Invest at the foreign risk-free rate (r_f).

Option i:

If the investor takes the first option, the fund held at the end of the period would be (1+r_d).

Option ii:

Alternatively, the investor could convert the domestic currency to be invested in a foreign currency using the spot rate \(S_{f/d}\). It is important to note that (f/d) is the currency quoting convention that expresses the number of foreign units per single domestic unit.

At the end of the investment period, \(S_{f/d}\ (1+r_f)\) units of foreign currency would be held by the investor. Then, the funds would have to be converted back into the domestic currency using the initial (pre-agreed) forward rate.

Note that the two investment alternatives are risk-free because they are invested in risk-free assets.

Since these investment alternatives are equal by considering the risk characteristics, the returns must also be equal. As such, we have the following relationship:

$$1+r_d=S_{f/d}\left(1+r_f\right)\left(\frac{1}{F_{f/d}}\right)$$

This formula above outlines two alternative investments (represented on either side of an equation) expected to yield equal returns. Should the returns differ, a risk-free arbitrage opportunity arises. An investor can capitalize on this by short-selling the lower-yield investment and directing those funds into the higher-yield one.

The relationship above can be rearranged to get the formula for a forward exchange rate as illustrated below:

$$F_{f/d}=\ S_{f/d}\left(\frac{1+r_f}{1+r_d}\right)$$

Where:

\(S_{f/d}\) = Current spot exchange rate.

\(F_{f/d}\) = Current forward exchange rate.

\(r_d\) = Domestic risk-free rate.

\(r_f\) = Foreign risk-free rate.

Example: Calculating the Forward Exchange Rate

Given that the spot exchange \(S_{f/d}\) is 1.502, the domestic risk-free rate for 12 months is 4%, and the 12-month foreign risk-free rate is 6.2%, the forward rate \(F_{f/d}\) is:

$$\begin{align}F_{f/d}&=S_{f/d}\left(\frac{1+r_f}{1+r_d}\right)\\ &=1.502\left(\frac{1+0.062}{1+0.04}\right)=1.5338\end{align}$$

Forward Rate as a Percentage of the Spot Rate

We can rearrange the no-arbitrage equation between the spot and exchange rates as follows:

$$\frac{F_{f/d}}{S_{f/d}}=\frac{1+r_f}{1+r_d}$$

Intuitively, from the above equation, under an f/d quoting convention, if foreign interest rates exceed domestic rates, the forward rate will be priced at a premium relative to the spot rate.

Generally speaking, without considering the quoting convention, the currency with the higher (lower) interest rate will always trade at a discount (premium) in the forward market.

We can interpret the forward exchange rate as the expected future spot rates. If we let \(F_t=F_{f/d}\), \(S_t=S_{f/d}\) and finally \(F_t=S_{t+1}\) then the above equation can be written as:

$$\begin{align} \frac{S_{t+1}}{S_t} &= \frac{1+r_f}{1+r_d}\\ \Rightarrow \%\Delta S_{t+1} &= \frac{S_{t+1}}{S_t}-1 = \left(\frac{r_f-r_d}{1+r_d}\right) \end{align}$$

As such, if we interpret forward rates as expected future spot rates, the expected percentage change in the spot rate is proportional to the interest rate differential (\(r_f-r_d\)).

The two setbacks to viewing forward rates as future spot rates are:

1. Historical data shows that forward rates are not accurate predictors of future spot rates.
2. An unconventional change in the direction of the spot rates, i.e., the above equation suggests that an increase in the domestic interest rate will cause a slower expected appreciation/ greater expected depreciation of the domestic currency, all else being equal. However, we already know that an increase in interest rates should increase the value of the domestic currency.

Forward Discount Premiums based on Spot and Interest Rates

Recall that a forward discount is when the forward exchange rate is lower than the spot exchange rate. On the other hand, A forward premium is a situation when the forward exchange rate is higher than the spot exchange rate.

Forward discounts/premiums guide one to know the currencies that will appreciate/depreciate in the near future.

Calculation

Recall the arbitrage formula:

$$F_{f/d}=S_{f/d}\left(\frac{1+i_f}{1+i_d}\right)$$

Where:

\(S_{f/d}\) = Current spot exchange rate.

\(F_{f/d}\) = Current forward exchange rate.

\(r_d\) = Domestic risk-free rate.

\(r_f\) = Foreign risk-free rate.

Note that in the above formula, we assumed a single-period analogy. Suppose the investment term is a fraction, \(\tau\), of the period for which the interest rates are quoted. The interest earned in the domestic and foreign markets would then be \(r_d\tau\) and \(r_f\tau\) respectively.

As such, the arbitrage formula transforms into:

$$F_{f/d}=S_{f/d}\left(\frac{1+r_f\tau}{1+r_d\tau}\right)$$

Intuitively, if we wish to calculate the forward premium or discount, we find the difference between the forward and spot exchange rates:

$$\begin{align}F_{f/d}-S_{f/d}&=S_{f/d}\left(\frac{1+r_f\tau}{1+r_d\tau}\right)-S_{f/d}\\&=S_{f/d}\left[\left(\frac{1+r_f\tau}{1+r_d\tau}\right)-1\right]\\&=S_{f/d}\left(\frac{r_f-r_d}{1+r_d\tau}\right)\tau\\ \therefore F_{f/d}-S_{f/d}&=S_{f/d}\left(\frac{r_f-r_d}{1+r_d\tau}\right)\tau\end{align}$$

Example: Forward Discount or Premium

Let’s say we have a 31-day forward exchange rate. The domestic 31-day risk-free interest rate is 2.5% per year, and the foreign 31-day risk-free interest rate is 3.5%. The spot exchange rate is 1.5630. We can calculate the forward premium or discount using two methods:

Method 1:

We first calculate the forward exchange rate, and then we subtract the spot exchange rate from it:

$$\begin{align}F_{f/d}&=S_{f/d}\left(\frac{1+r_f\tau}{1+r_d\tau}\right)\\&=1.5630\left(\frac{1+0.035\times\frac{31}{360}}{1+0.025\times\frac{31}{360}}\right)=1.56434\end{align}$$

Hence, the forward trading premium is:

$$F_{f/d}-\ S_{f/d}=1.56434-1.5630=0.00134$$

Since forward premiums or discounts are usually quoted in pips or points, multiplying the result by 10,000 will give us \(0.00134\times10,000=13.4\) pips, which is the forward trading premium quoted in pips or points.

Method 2:

Inserting the data directly into the formula:

$$\begin{align}F_{f/d}-S_{f/d}&=S_{f/d}\left(\frac{r_f-r_d}{1+r_d\tau}\right)\tau\\&=1.5630\left(\frac{0.035-0.025}{1+0.025\times\frac{31}{360}}\right)\times\frac{31}{360}\\&=0.00134\approx13.4\ \text{pips} \end{align}$$

Question #1

An Italian company has secured a contract with a US client, expecting a payment of USD 40 million in 45 days. The finance manager of the Italian firm wishes to hedge the FX risk of this deal and gets the following rates from a broker:

USD/EUR spot rate: 0.9220

One-month forward points: +2.0

According to the exchange rate information provided, the finance manager could hedge the FX risk by:

1. Buying euro (selling US dollars) at a forward rate of 0.9222.
2. Buying euro (selling US dollars) at a forward rate of 0.9200.
3. Selling euro (buying US dollars) at a forward rate of 0.9200.

Solution

The correct answer is A.

The Italian company would aim to change the US dollar to its home currency, the euro (it intends to sell US dollars and buy euros), using a forward rate calculated as 0.9220 + (+2.0/10,000) = 0.9222.

By doing this, the company is able to protect itself against any unfavorable movements in the foreign exchange rate over the next 45 days (before payment is received). In other words, the company locks in the exchange rate of USD/EUR = 0.9222 now. This can be achieved using a forward contract, which allows the company to exchange currencies at a predetermined rate in the future.

Question #2

When is a foreign currency most likely trading at a forward premium?

1. When the forward rate expressed in the domestic currency is below the spot rate.
2. When the forward rate expressed in the domestic currency is above the spot rate.
3. When the forward rate expressed in the foreign/domestic currency is at equilibrium.

Solution

The correct answer is B.

A foreign currency is at a forward premium if the forward rate expressed in domestic

currency is above the spot rate.