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# Calculate and Interpret a Forward Discount or Premium

## Forward Discount

A forward discount is when the domestic current spot exchange rate is traded at a higher level than the current domestic future spot rates. The analysis of the expectations from the market depends mostly on discounts and premiums. Also, they enable one to know the currencies that should appreciate and those that will depreciate in the near future.

A forward premium is a situation when the forward exchange rate is higher than the spot exchange rate. Conversely, a forward discount is when the forward exchange rate is lower than the spot exchange rate.

Irrespective of the quoting convention, the currency with the higher (lower) interest rate will always trade at a discount (premium) in the forward market.

## Calculation

Interest parity states that both the spot and forward exchange rates between two currencies must be at equilibrium with the two nation’s interest rates. The formula includes four variables:

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

Where

$$S_{f/d}$$ = current spot exchange rate

$$F_{f/d}$$ = current forward exchange rate

$$i_d$$ = domestic interest rate

$$i_f$$ = foreign interest rate

We can rewrite the above equation to show the forward rate as a percentage of the spot rate:

$$\frac{F_{f/d}}{S_{f/d}} = \frac{1 + i_f}{1 + i_d}$$

It is easy to see that, given the foreign/domestic convention (f/d), the forward rate will be at a premium to the spot rate if the foreign interest rates are higher than the domestic interest rate.

Let’s say the current Kenyan Shilling to Ugandan Shilling exchange rate is Sh 100. The domestic interest rate in Kenya is 5%, and the foreign interest rate is 4.75%, causing the resulting equation to be:

$$F = \text{Ksh} 100 (\frac{1.0475}{1.05}) = \text 99.7619$$

The forward rate relates to the spot rate by a premium or discount, which is proved in the following relationship:

$$F = S (1 + x)$$

Where F is the current premium or discount.

Spot exchange rates differ from the forward currency exchange rates. When the forward currency exchange rate happens to be higher than the spot rate, then the currency is said to be at a premium. Conversely, discounts occur when the spot rates are higher than the forward exchange rates. Hence, a negative premium is equal to a discount.

A simpler way to look at it is when $$F/S – 1 > 0$$, the denominator is at a discount. When $$F/S – 1 < 0$$, the denominator is at a premium, and the numerator is at a discount. For instance, if we want to ascertain the premium of a forward trading rate, we can also do that using the formula below:

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

Where (1 + i_f τ) is the interest earned in the foreign market and (1 + i_d τ) is the interest earned in the domestic market. Where τ is the fraction of the investment horizon at which the interest rates are quoted.

### Example: Forward Discount or Premium

If we want to know the 31-day forward exchange rate from a 31-day domestic risk-free interest rate of 2.5% per year, given that the foreign 31-day risk-free interest rate is 3.5% with a spot exchange rate, $$S_{f/d}$$, of 1.5630, then we simply have to substitute these values into the forward rate equation:

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

$$F_{f/d}=1.5630 (\frac{1+0.035×\frac{31}{360}}{1+0.025×\frac{31}{360}})=1.5643$$

$$F_{f/d} – S_{f/d} = 1.5643 – 1.5630 = 0.0013$$

Since forward premiums or discounts are usually quoted in pips or points (1/100 of 1%), multiplying the result by 10,000 will give us $$0.0013 \times 10,000 = 13$$ pips, which is the forward trading premium quoted in pips or points.

### Question

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

A. When the forward rate expressed in the domestic currency is below the spot rate.

B. When the forward rate expressed in the domestic currency is above the spot rate.

C. When the forward rate expressed in the foreign/domestic currency is at equilibrium.

Solution

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

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