# Free

## Covered Interest Rate Parity (IRP) – Pricing Currency Forwards

Pricing currency forward contracts – determining the appropriate future exchange rate to use – is relatively straightforward; it is based on the risk-free interest rates for the currencies involved, and the no-arbitrage condition (i.e., the forward exchange rate should make arbitrage impossible).  Because the elimination of arbitrage means that the forward exchange rate has to compensate for inequality in the risk-free interest rates – it has to restore equality, or parity – and because the parity is ensured (or covered) by the forward contract, the approach in known as covered interest rate parity (covered IRP, or CIRP).  The formula is:

$F_{PC/BC}\ =\ S_{PC/BC}\ ×\ \frac{1\ +\ r_{PC}}{1\ +\ r_{BC}}$

where:

• $F_{PC/BC}$: forward (future) exchange rate, quoted as price currency / base currency
• $S_{PC/BC}$: spot (current) exchange rate, quoted as price currency / base currency
• $r_{PC}$: risk-free rate for the price currency
• $r_{BC}$: risk-free rate for the base currency

An easy way to remember which rate goes in the numerator and which rate goes in the denominator is to think of the formula this way:

\begin{align}F_{PC/BC}\ &=\ \frac{PC_{future}}{BC_{future}}\ =\ \frac{PC_{today}\ ×\ \left(1\ +\ r_{PC}\right)}{BC_{today}\ ×\ \left(1\ +\ r_{BC}\right)}\\
\\
&= \frac{PC_{today}}{BC_{today}}\ ×\ \frac{1\ +\ r_{PC}}{1\ +\ r_{BC}}\ =\ S_{PC/BC} × \frac{1\ +\ r_{PC}}{1\ +\ r_{BC}}
\end{align}

Writing it this way shows that the future price currency is today’s price currency grown at the price currency risk-free rate (the numerators) and the future base currency is today’s base currency grown at the base currency risk-free rate (the denominators).

Note that the forward exchange rate is calculated for a certain amount of time t in the future, so the risk-free interest rates have to be the rates applicable for that amount of time.  For example, suppose you are given that the:

• spot exchange rate for US dollars and British pounds is USD/GBP 1.6453
• 180-day USD LIBOR rate is 2.4%
• 180-day GBP LIBOR rate is 3.0%

You are asked to calculate the 180-day forward USD/GBP exchange rate.  The calculation is:

\begin{align}F_{USD/GBP}\ &=\ S_{USD/GBP}\ ×\ \frac{1\ +\ r_{USD}}{1\ +\ r_{GBP}}\\
\\
&=\ 1.6453\ ×\ \frac{1\ +\ 2.4\%\left(\frac{180}{360}\right)}{1\ +\ 3.0\%\left(\frac{180}{360}\right)}\\
\\
&=\ 1.6453\ ×\ \frac{1.012}{1.015}\\
\\
&=\ 1.6404
\end{align}

Thus, the 180-day forward exchange rate is USD/GBP 1.6404.

(Note that the quoted LIBOR rates are annual, nominal rates.)

If, instead, you are given that the:

• spot exchange rate for US dollars and British pounds is USD/GBP 1.6453
• effective (annual) 180-day USD risk-free rate is 2.4%
• effective (annual) 180-day GBP risk-free rate is 3.0%

then the calculation of the 180-day forward USD/GBP exchange rate is:

\begin{align}F_{USD/GBP}\ &=\ S_{USD/GBP} × \frac{1\ +\ r_{USD}}{1\ +\ r_{GBP}}\\
\\
&=\ 1.6453\ ×\ \frac{\left(1\ +\ 2.4\%\right)^{180/365}}{\left(1\ +\ 3.0\%\right)^{180/365}}\\
\\
&=\ 1.6453\ ×\ \frac{1.011764}{1.014684}\\
\\
&=\ 1.6406
\end{align}

Thus, the 180-day forward exchange rate is USD/GBP 1.6406.

An Approximation

As a quick and dirty check, the percentage increase/decrease of the forward rate over the spot rate is approximately equal to the difference between the price currency’s and base currency’s risk-free rates:

$\frac{F_{PC/BC}}{S_{PC/BC}}\ -\ 1\ ≈\ r_{PC}\ -\ r_{BC}$

To illustrate this, consider the first example, above:

$\frac{1.6406}{1.6453}\ -\ 1\ =\ -0.286\% ≈\ -0.300\%\ =\ \ 2.4\%\left(\frac{180}{360}\right)\ -\ 3.0\%\left(\frac{180}{360}\right)$

Why?  In a Word: Arbitrage

The reason that the forward exchange rate must satisfy the formula given above is that any other forward rate will create an arbitrage opportunity.  The arbitrage transaction is fairly simple:

• Borrow currency A at its risk-free rate
• Convert currency A to currency B at the spot exchange rate
• Enter into a forward contract to exchange currency B for currency A at the forward exchange rate (note: the amount in the forward contract will be current amount of currency B plus the interest you will earn on currency B)
• Invest currency B at its risk-free rate
• Wait (until the forward contract matures)
• Convert currency B (principal plus interest) to currency A at the agreed forward rate
• Pay off the currency A loan (principal and interest)
• Enjoy the profit

Note that this series of transactions is risk-free (hence, arbitrage): the money is borrowed (so the arbitrageur has no money at risk), the interest rates are risk-free, and the future exchange rate is locked in at the outset by the forward contract.

To see how this works, imagine that in the first example (LIBOR rates), the quoted forward exchange rate were USD/GBP 1.6420 (higher than the required 1.6406).  In that case, the future GBP would be exchanged for more future USD than the formula specifies; this tells us that we need to start the transaction by borrowing USD (because we’ll end up with more USD than we should: we’re earning a higher interest rate on USD than we should).  If we borrow USD1,000,000, the transactions would look like this:

• Borrow USD1,000,000 at 2.4% USD LIBOR for 180 days
• Convert USD1,000,000 to GBP607,792 at USD/GBP 1.6453
• Enter into a forward contract to convert GBP616,909 (= GBP607,792 × (1 + 3%(180/360))) at USD/GBP 1.6420 in 180 days; the amount is GBP607,792 plus interest
• Invest GBP607,792 at 3% GBP LIBOR  for 180 days
• Wait 180 days; the GBP607,792 investment grows to GBP616,909
• Convert GBP616,909 to USD1,012,964 at USD/GBP 1.6420
• Pay off the USD loan for USD1,012,000 (= USD1,000,000 × (1 + 2.4%(180/360)))
• Enjoy a profit of USD964

If, instead, the quoted forward exchange rate were USD/GBP 1.6391 (lower than the required 1.6406), then the future GBP would be exchanged for fewer USD than the formula specifies; this tells us that we need to start the transaction by borrowing GBP (the reverse of the previous case: if we borrowed USD we’d end up with fewer USD than we should, earning a lower interest rate on USD than we should).  If we borrow GBP1,000,000, the transactions would look like this:

• Borrow GBP1,000,000 at 3.0% GBP LIBOR for 180 days
• Convert GBP1,000,000 to USD1,645,300 at USD/GBP 1.6453
• Enter into a forward contract to convert USD1,665,044 (= USD1,645,300 × (1 + 2.4%(180/360))) at USD/GBP 1.6391 in 180 days; the amount is USD1,645,300 plus interest
• Invest USD1,645,300 at 2.4% USD LIBOR for 180 days
• Wait 180 days; the USD1,645,300 investment grows to USD1,665,044
• Convert USD1,665,044 to GBP1,015,828 at USD/GBP 1.6391
• Pay off the GBP loan for GBP1,015,000 (=GBP1,000,000 × (1 + 3%(180/360)))
• Enjoy a profit of GBP828

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