Foreign Exchange Forwards
Foreign Exchange forward contracts are transactions in which counterparties agree to exchange a specified amount of
different currencies at some future date, with the exchange rate being set at the time the contract is entered into.
The user is protected from adverse movements in future FX rates, but he also does not benefit from favourable movements.
Foreign Exchange forwards remove uncertainty and are therefore valid instruments for users to mitigate the downside foreign exchange
risk for future transactions denominated in a foreign currency.
Foreign Exchange forward rates, Foreign Exchange spot rates, and interest rates are interrelated by the interest rate parity (IRP) principle.
This principle is based on the notion that there should be no arbitrage opportunity between the
Foreign Exchange spot market, Foreign Exchange forward market, and the term structure of interest rates in the two countries.
Typical Pricing Function
Variable | Description |
Fx_{fwd} | fair forward FX rate (quoted in units of domestic currency per unit of foreign) |
Fx_{spot} | spot FX rate (quoted in units of domestic currency per unit of foreign) |
r_{d} | domestic interest rate (for term of forward) quoted on a simple interest basis |
r_{f} | foreign interest rate (for term of forward) quoted on a simple interest basis |
AF_{d} | domestic accrual factor |
AF_{f} | foreign accrual factor |
Example
Given the following spot FX spot and money market rates, what should be the theoretical 90 day forward FX rate?
spot rate = 1.3500 CAD/USD
Canadian 90-day Libor = 4.50%
US 90-day Libor = 3.80%
answer:
Fx_{fwd} = 1.35 X (1 + ((90/365) * 0.045)) = 1.3521
(1 + ((90/360) * 0.038))
Argument | Description | Sample Data | Switch |
FX_spot | spot price of underlying currency (domestic per 1 unit of foreign) | 1.35 | |
rate_dom | domestic rate (for term of forward) | 0.045 | |
rate_for | foreign rate (for term of forward) | 0.038 | |
d_s | settlement date | 23-Sep-03 | |
d_del | future delivery date | 22-Dec-03 | |
acc_dom | accrual method for domestic rate | 1 | actual / 365 |
acc_for | accrual method for foreign rate | 2 | actual / 360 |
Result
The fair forward FX rate is $1.3521 CAD/USD.
The fair basis is $0.0021, i.e., 21 basis points
Interest Rate Parity
Using the above example to
illustrate the principles of IRP, if you borrowed $100 CAD at 4.5%, bought USD on
the spot market, invested the proceeds in the US at 3.80 %, and sold forward the
same amount, you should not be able to extract any arbitrage proceeds from the
process.
1. Convert the CAD today | (100/1.3500) = $74.074 USD |
At the end of 90 days: | |
2. pay off loan | 100 * (1+(0.045*90/365)) =$101.11 CAD outflow |
3. USD investment | 74.07 * (1+(0.0380*90/360)) = $74.778 USD inflow |
The forward rate should be (101.11/74.778) = 1.3521 CAD/USD
If the forward rate is higher than
this value, then you could make riskless profit by
following the above strategy.Suppose the forward rate was 1.3600 CAD/USD.
The profit you would make would be
(74.778 * 1.3600) - (101.11) = $0.588 CAD for every $100 CAD you borrowed.By using $1 million, your riskless profit would be $5,880.00.
This cannot persist very long in real markets.
If the forward rate was lower than 1.3521, then the reverse strategy can be used.
Suppose the forward rate was 1.3500.
You would then borrow $100 USD, buy CAD on the spot market, and invest in the Canadian money market,
and buy the USD forward to pay off the loan.
This is shown as follows:
Convert to CAD today | 100 * 1.3500 = $135.00 CAD |
90 days from today | |
pay off loan | 100 * (1 + (0.0380 * 90/360)) = 100.95 USD outflow |
CAD investment | 135 * (1 + (0.0450 * 90/365)) = 136.50 CAD inflow |
purchase USD forward | 136.50 / 1.3500 = 101.11 USD |
net profit (per $100 USD) | 101.11 - 100.95 = $0.16 USD |
Quoting Conventions
The previous example was somewhat oversimplified.
There was no bid-ask spread in the spot FX rate and the loan and investment interest rates were assumed equal.
In reality, to calculate valid forward prices one needs to have the valid bid-ask prices of the
spot rates, and separate loan and investment interest rates.
Forward rates can be quoted in two ways,
as an "outright" quote,
or as forward points (also called a swap rate).
The outright quote is simply a bid-ask price same as the spot market quotes.
The forward points are the amount that needs to be added to or subtracted from the spot rates.
The following illustrates the two quoting methods:
spot rates | 1.3500 - 1.3505 CAD/USD |
forward points | 90 - 95 |
outright forward rates | 1.3590 - 1.3600 CAD/USD |
Premiums and Discounts
As previously discussed, the forward rates are closely related to the spot rates and interest rates of the two countries.
A result of the IRP
theory is that for the country with the higher interest rate, its currency is
weaker in the forward market than in the spot market. As shown in the previous
example, the Canadian interest rate was higher than the US interest rate, and
the resulting theoretical forward rate was 1.3521 CAD/USD, compared with the
spot rate of 1.3500 CAD/USD.
The terms premium and discount refer to whether the forward rates are higher or lower than the spot rates.
A premium means the forward price is higher than the spot price and a discount
means lower.
In this case the forward rate is then at a premium of 21 points.
In order to apply forward points to a spot rate to come up with an outright quote for the forward price,
one needs to know whether the forward points are premiums or discounts.
For premiums, bid forward points are added to bid spot prices and ask forward points are added to ask spot prices.
For discounts bid forward points are subtracted from ask spot prices and ask forward points are subtracted from bid spot prices.
If all this seems confusing, just remember the rule that the
bid-ask spreads of the forward price should always be greater than the spot price,
and that the sum of the bid-ask spreads of the spot price and forward points should equal the spread of the forward price.
The bigger spread in the forward market can be viewed as compensation for the increased risk the market maker takes
in the forward market relative to the spot market.
Example of a premium: | | Spread |
spot rates | 1.3500 - 1.3505 CAD/USD | 5 points |
forward points | (+90) - (+95) | 5 points |
outright forward rates | bid = 1.3500 + 0.0090 = 1.3590 | |
| ask = 1.3505 + 0.0095 = 1.3600 | |
| 1.3590 - 1.3600 CAD/USD | 10 points |
| | |
Example of a discount: | | Spread |
spot rates | 0.7405 - .7410 USD/CAD | 5 points |
forward points | (-90) -(-95) | 5 points |
outright forward rates | bid = 0.7405 - 0.0095 = 0.7310 | |
| ask = 0.7410 - 0.0090 = 0.7320 | |
| quote = 0.7310 - 0.7320 CAD/USD | 10 points |
Detailed Forward Rate Calculation
We will now provide sample
calculations of both a premium and a discount forward price using more realistic
data.Consider the following
scenario:
spot rate CAD/USD | 1.3500 - 1.3506 | |
spot rate USD/CAD | 0.7404 - 0.7407 | |
90 day Canadian interest rates | 5.98 % investment | 6.02 % loan |
90 day US interest rates | 3.92 % investment | 3.98 % loan |
Example of a forward premium calculation:
If a Canadian company needs to have USD in 90 days, it can buy USD forward, or pursue the following strategy.
Borrow CAD, convert to USD, invest in USD for 90 days.
Canadian borrowing costs | 1 + (0.0602 * (90/365)) = 1.01484 CAD |
Converting CAD to USD | 1.3506 CAD |
Total cost | 1.01484 * 1.3506 = 1.3706 CAD |
US investment income | 1 + (0.0392 * (90/360)) = 1.0098 USD |
The forward price required for the two strategies to break-even (i.e. no arbitrage) would be 1.3706 / 1.0098 = 1.3575 CAD/USD.
Note this is the forward ask price of the USD.
The inverse of this (1 / 1.3575 = 0.7366) is then the forward bid price of the CAD (i.e. sell USD / buy CAD).
CAD/USD forward points = 1.3575 - 1.3506 = +0.0069 (a premium)
USD/CAD forward points = 0.7366 - 0.7404 = -0.0038 (a discount)
Example of a forward discount calculation:
If a US company needs to have CAD in 90 days, it can buy CAD forward, or pursue the following strategy.
Borrow USD, convert to CAD, invest in CAD for 90 days.
US borrowing costs | 1 + (0.0398 * (90/360)) = 1.00995 USD |
Converting USD to CAD | 0.7407 USD |
Total cost | 1.00995 * 0.7407 = 0.7481 USD |
Canadian investment income | 1 + (0.0598 * (90/365)) = 1.01475 CAD |
The forward price required for the two strategies to break-even (i.e. no arbitrage) would be 0.7481 / 1.01475 = 0.7372 USD/CAD.
Note this is the forward ask price of the CAD.
The inverse of this (1 / 0.7372 = 1.3565) is then the forward bid price of the USD (i.e. sell CAD / buy USD).
USD/CAD Forward points = 0.7372 - 0.7407 = -0.0035 (a discount)
CAD/USD Forward points = 1.3565 - 1.3500 = +0.0065 (a premium )
To summarise the results | | Spread |
Forward rate CAD/USD | 1.3565 - 75 | 0.0010 |
Forward points CAD/USD | (+0.0065) - (+0.0069) | 0.0004 |
Forward rate USD/CAD | 0.7366 - 72 | 0.0006 |
Forward points USD/CAD | (-0.0035) - (-0.0038) | 0.0003 |
| | |
Recall the spot rates | | |
Spot rate CAD/USD | 1.3500 - 1.3506 | 0.0006 |
Spot rate USD/CAD | 0.7404 - 0.7407 | 0.0003 |
As a check on the calculation, note that the spread of the spot plus the spread of the forward points equal the spread of the outright forward price.