Foreign Exchange Rate Average Noon Rate Agreement (ANR) is an agreement to buy or sell USD dollars on a future value date at a rate equal to the average rate for a specified period and adjusted by forward points agreed at the inception.
Notations used as follows.
t Valuation Date T Maturity Date
Settlement Date
Spot Exchange Rate at time
Risk-free discount rate of currency Ccy
Averaging start date K Strike price = Fixed Forward Points
Historical exchange rate at time
Forward exchange rate for time interval (t, ), where
N Notional amount
The average exchange rate, , with m historical rate averaging points and n spot rate averaging points, is computed as
if , and if .
where , and if , and if .
The average forward rate, , is then computed as
if , and if
and the forward exchange rate is computed as , where .
The actual pricing (Mark-to-Market) is done at the inventory level and is in the base currency, which is usually USD. Currently, this product can only be used by Indirect Currencies, and is normally used in CADUSD only. Therefore, this report examines indirect quote only.
The rates are quoted as . The notional currency is in USD only and the payoff currency is CAD only. The payoff at the maturity is defined as in CAD. The expected payoff is then calculated as
.
where (1 or –1) is the long / short indicator. The price of the contract in USD is obtained by dividing the expected payoff by the forward rate, , and by discounting it with USD
(1)
The price of the contract and the perturbation of the spot rate in delta calculation is in USD/CAD. Thus, the perturbation of the spot is done as and where . Thus, the USD Delta is defined by
USD Delta = (2)
ANR payoff can de decomposed into two NRCs as the following.
where the first term on the right hand side is the payoff of a NRC with strike price –K, and the second term is the payoff of a one day NRC ( ) with zero strike price . Thus, a long position of ANR is decomposed into a short of the first NRC and a long of the second NRC. The ANR pricing can be written as
(3)
where
(4)
and
(5)
The second NRC, equation (5), is actually a cash instrument and, therefore, it has zero delta value.
We examine the pricing and delta calculation with 5 test cases (for each of these test cases, ANR decomposition into 2 NRCs are also tested). The valuation date (called Spot Date in Atlas) is August 31, 2004. Actual/365 for Day Count Base, Daily Averaging frequency, and N=1,000,000 USD is used in all test cases.
It is possible that matured ANR could be in the system (not paid out to clients) and, thus, daily Mark-to-Market and Delta calculations are also done on those matured ANRs. For a matured ANR, the pricing in equations (1), (4) and (5) are modified as the following.
(6)
(7)
(8)
Here, is calculated only from historical rates and the forward rate in equation (1) and (4) becomes the spot rate in the above equations (6) and (7). Since the pricing in equations (6) and (7) involve the spot price, which varies day-to-day, there are non-zero delta values on these cases (see case 1 in table 2). Note that the 2nd NRC, equation (8), should have the maturity date same as valuation date (i.e., ). You can find other pricing models at https://finpricing.com/lib/FiBond.html
Appendix 1. Test Cases
Base Currency: USD
Underlying Currency: CAD
Principal Amount: 1,000,000 USD
Spot Rate: 1.31895 CAD/USD
Case No. Position (Buy / Sell) Start Date Maturity Date Settlement Date Strike Price 1 Sell 01-Jun-2004 30-Jun-2004 01-Jul-2004 0.0013 2 Sell 03-Aug-2004 03-Sep-2004 07-Sep-2004 -0.0075 3 Sell 18-Sep-2007 17-Oct-2007 18-Oct-2007 0.0156 4 Buy 25-Feb-2005 30-Mar-2005 31-Mar-2005 0.0105 5 Buy 10-Aug-2004 09-Sep-2004 10-Spe-2004 -0.0025
Appendix 2. Forward Foreign Exchange Rate
Number of Days from Valuation Date CAD/USD Forward Points (bps) CAD/USD Forward Outright 7 1.1500 1.319065 14 2.4500 1.319195 30 5.4500 1.319495 59 11.2000 1.320070 91 17.5000 1.320700 122 23.0000 1.321250 153 29.0000 1.321850 181 34.5000 1.322400 273 52.0000 1.324150 365 67.5000 1.325700 546 94.7500 1.328425 730 122.0000 1.331150
Appendix 3. USD Discount Factor
Number of Days from Valuation Date USD DF
7 0.99969190 14 0.99938399 30 0.99868091 61 0.99716984 91 0.99558334 181 0.99031568 273 0.98421443 365 0.97755095 730 0.94537551 1,098 0.90750025 1,462 0.86651370 1,826 0.82442859
Appendix 4. Historical Rate