# Spot Rate, Forward Rate, and Forward Premium/Discount

A spot exchange rate is the general price level in the market used to directly trade one currency for another, with the exchange occurring at the earliest possible time. The standard delivery time for spot currency transactions is no longer than T+2 (days), after which it will be deemed a forward contract.

A forward exchange rate is the price at which one currency is traded against another at some specified time in the future. The forward exchange rate must respect the arbitrage relationship, which states that the returns from two alternative but equivalent investments must be equal. We will derive the relationship between the spot and forward exchange rates from this fact.

While ignoring the bid-offer spread and the effect of market instruments, consider an investment of one unit of domestic currency for one year with the following alternatives:

• Alternative 1: A cash investment for one year at a risk-free domestic rate $$({i}_{d})$$. The investment will be worth $$(1+{i}_{d})$$ at the end of one year.
• Alternative 2: Converting domestic currency into foreign currency at the spot rate $${S}_{{f}/{d}}$$, then investing the proceeds for one year at a risk-free foreign rate of interest of $${i}_{f}$$. At the end of the investment period, the investment will be worth $${S}_{{f}/{d}}({1}+{i}_{f})$$ units of foreign currency which must be converted back to domestic currency by a forward rate $$F_{f/d}$$. Therefore, $$\frac{1}{F_{f/d}}$$ units of domestic currency would be obtained for each unit of foreign currency sold forward. In terms of the domestic currency, therefore, the investment will be worth $${S}_{{f}/{d}}({1}+{i}_{f})\frac{1}{F_{f/d}}$$.

It is important to note that the notation $$(f/d)$$ denotes “foreign/domestic currency,” where the domestic currency is assumed to be the base currency.

Back to our discussion, investments 1 and 2 are risk-free and, therefore, should give a similar return. That is, there is no chance of arbitrage opportunities. If this is true, equating the gains of the alternative investments leads us to the following formula:

$$\left({1}+{i}_{d}\right)={S}_{{f}/{d}}({1}+{i}_{f})\frac{1}{F_{f/d}}$$

We made things simple in our derivation by assuming a time horizon of one year. However, the argument holds for an investment horizon of any length. The risk-free assets used in this arbitrage relationship are typically bank deposits quoted using the reference rate (Libor until 2021, then SOFR, SONIA, etc.) for each currency involved. The day count convention for almost all deposits is Actual/360. This notation means that interest is calculated as if there were 360 days in a year.

Now, if we include the London Interbank Offered Rate (Libor) day count convention of $$\frac{\text{Actual}}{360}$$, our formula will transform into:

$$\left({1}+{i}_{d}\left[\frac{\text{Actual}}{360}\right]\right)={S}_{{f}/{d}}\left({1}+{i}_{f}\left[\frac{\text{Actual}}{360}\right]\right)\frac{1}{F_{f/d}}$$

By simple rearrangement, we can make the forward rate $$(F_{f/d})$$ the subject:

$$F_{f/d}=S_{f/d}\left(\frac{1+i_f\left[\frac{\text{Actual}}{360}\right]}{1+i_d\left[\frac{\text{Actual}}{360}\right]}\right)\ldots\ldots\ldots(i)$$

Equation (i) is a description of covered interest rate parity as discussed in Level I. It can be rearranged to give an equation for the forward premium or discount. That is:

$$F_{f/d}-S_{f/d}=S_{f/d}\left(\frac{\left[\frac{\text{Actual}}{360}\right]}{1+i_d\left[\frac{\text{Actual}}{360}\right]}\right) \left(i_f-i_d\right)$$

When $${F}_{{f}/{d}}>{S}_{{f}/{d}}$$, the domestic currency is trading at a forward premium. This will happen only if $${i}_{f}>{i}_{d}$$. Otherwise, the domestic currency is said to trade at a forward discount.

We have been using the $$(f/d)$$ notation all through. Note that we have a free hand to also switch to the $$P/B$$ (Price/Base) conventional notation and substitute it accordingly. For instance, the forward rate would be:

$$F_{P/B}={S}_{{P}/{B}}\left(\frac{{1}+{i}_{P}\left[\frac{\text{Actual}}{360}\right]}{{1}+{i}_{B}\left[\frac{\text{Actual}}{360}\right]}\right)$$

## Question

Assume that the spot (USD/CAD) is 1.0146, the 200-day Libor for USD is 1.5%, and the 200-day Libor for CAD is 5.21%. The forward premium (discount) for a 200-day forward contract for USD/CAD is closest to:

1. 0.02032.
2. -0.02032.
3. -0.02532.

#### Solution

The forward premium (discount) is given by:

$$F_{P/B}-S_{P/B}=S_{P/B}\left(\frac{\left[\frac{\text{Actual}}{360}\right]}{1+i_B\left[\frac{\text{Actual}}{360}\right]}\right) \left(i_P-i_B\right)$$

Noting that the CAD is the base currency, then:

\begin{align*} F_{USD/CAD}-S_{USD/CAD} & =1.0146\left(\frac{\left[\frac{200}{360}\right]}{1+0.0521\left[\frac{200}{360}\right]}\right) \left(0.015-0.0521\right) \\ & =-0.02032 \end{align*}

Reading 8: Currency Exchange Rates: Understanding Equilibrium Value

LOS 8 (c) Explain spot and forward rates and calculate the forward premium/ discount for a given currency.

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