12.1.3 No Arbitrage
No Arbitrage (forward) Exchange Rate Forecasting is similar to the Parity method, but it relies only on the interest rates in effect on the As-of-Date for each respective currency. Based on the relative interest rates in each country, the No Arbitrage method tells the user what the forward exchange rates must be to maintain no-arbitrage between the two currencies. Interest rates are converted to equal formats of accrual basis and compounding basis. This is achieved by converting the rates to a discount factor. (For complete details on conversion to a discount factor, see the Rate Conversion section) As a simple example, let's use annual compounding; the basic formula for Forward exchange rates would be:
Table 12-8 Basic Formula for Forward Exchange Rates
| Variable | Definition |
|---|---|
|
bat |
Exchange rate from currency b (the selected currency) to currency a (the reporting currency) at time t |
|
ba0 |
Exchange rate from currency b to currency a at As-of-Date |
|
t |
Time from As-of-Date to Start Date of the bucket |
|
rat |
The reference interest rate in currency a for term t, at As-of-Date |
|
rbt |
The reference interest rate in currency b for term t, at As-of-Date |
|
m |
The portion of year equivalent to t |
Figure 12-2 No Arbitrage Calculation
Description of the formula of No Arbitrage calculation follows:
Calculations then loop through all modeling buckets.
For example, consider the following modeling bucket configuration (and related variables):
Table 12-9 Modeling Bucket Configuration Variables
| Bucket | Term | Start Date | >End Date | t = Start Date - As-of-Date | m = t / 365 |
|---|---|---|---|---|---|
|
1 |
1 Month |
1/1/2011 |
1/31/2011 |
1 Day |
0.00274 |
|
2 |
1 Month |
2/1/2011 |
2/28/2011 |
32 Days |
0.087671 |
|
3 |
3 Month |
3/1/2011 |
5/31/2011 |
60 Days |
0.164384 |
|
4 |
6 Month |
6/1/2011 |
11/30/2011 |
152 Days |
0.416438 |
Further, the exchange rate on the As-of-Date (ba0) is 3.8, and interest rates on that date are as follows:
Table 12-10 Example of Exchange Rate on the As-of-Date and Interest Rates
| Terme/p> | rbt | rat |
|---|---|---|
|
1 Day |
3.625 |
2.1 |
|
32 Days |
4.5 |
2.625 |
|
60 Days |
4.875 |
2.75 |
|
152 Days |
5.9 |
3.5 |
Therefore, exchange rates would be:
Table 12-11 Example of Exchange Rates
| Bucket | |
|---|---|
|
1 |
3.8 * 1.000041 = 3.800154 |
|
2 |
3.8 * 1.001589 = 3.806037 |
|
3 |
3.8 * 1.003371 = 3.812808 |
|
4 |
3.8 * 1.009592 = 3.836450 |
Description of the formula of Exchange Rate Calculation follows:
Figure 12-3 Exchange Rate Calculation Formula
