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### Lecture 7: The Forward Exchange Market

### Appendix A

Determining the Appropriate Forward Exchange Quote: The Interest Rate Parity Model

China’s Foreign Exchange Trade System

- China’s Foreign Exchange Trade System (CFETS) was founded in April 1994 as part of China’s FX reforms. Today CFETS plays a significant role in managing the Yuan exchange rate.
- CFETS is a sub-institution of the People\'s Bank of China (PBC). Its main foreign exchange functions include: providing a system for foreign exchange trading; organizing interbank FX trading, providing information on the FX, market; and engaging in other businesses authorized by the PBC.
- CFETS is headquartered in Shanghai.

How do Market Makers Determine the Forward Exchange Rate?

- The quoted forward rate is not a reflection of where market makers think the spot exchange rate will be on that forward date .
- Lloyds Bank, UK (Corporate Banking and Treasury Training Publication) : “Forward rates .. are not the dealer\'s [i.e., market maker bank’s] opinion of where the spot rate will be at the end of the period quoted.”
- So what determines the forward rate?
- Quick answer: Interest rate differentials between currencies being quoted, or the Interest Rate Parity Model.
- To develop this concept, and the Interest Rate Parity Model, we will work through the following example.

Consider Cross Border Investing

- Assume a U.S. investor has $1 million to invest for 1 year and can select from either of the following 1 year investments:
- (1) Invest in a U.S. government bond and earn 2.0% p.a.
- (2) Invest in an Australian government bond and earn 5.5% p.a.
- If the U.S. investor invests in Australian government bonds, he/she will receive a known amount of Australian dollars in 1 year when the bond matures.
- Principal repayment and interest payment both in AUD.

Risk of Investing Cross Border

- Question: Using the previous example, what is the risk for the U.S. investor if he/she buys the 1 year Australian government bond?
- Answer: The risk associated with foreign exchange exposure in AUD (open position).
- The U.S. investor will be paid a specified amount of Australian dollars 1 year from now:
- The risk is the uncertainty about the Australian dollar spot rate 1year from now.
- If the Australian dollar (spot) weakens, the U.S. investor will receive fewer U.S. dollars at maturity:
- Example: If the Australian dollar weakened by 2% by the end of the year, this reduces the return on the Australian investment (from 5.5 % to 3.5%).

Solution to The Currency Risk for the U.S. Investor

- Question: Using the previous example, how could the U.S. investor manage the risk associated with this Australian dollar exposure?
- Solution: The US investor can cover the Australian dollar investment by selling Australian dollars 1 year forward (a short position).
- Australian dollar amount which the investor will sell forward would be equal to the principal repayment plus earned interest (Note: this was the known amount of AUD to be received in 1 year).

Calculating the U.S. Dollar Equivalent of the Maturing AUD Government Bond when Covered

- Assume:
- A 1 year Australian Government Bond with a par value of 1,000AUD (assume you purchased 100 of these at par)
- Assume an annual coupon of 5.5% (payable at the end of the year)
- Assume the following market maker bank quoted exchange rates:
- AUD/USD spot 1.0005/1.0009
- AUD/USD 1 year forward 0.9650/0.9657
- Calculate the USD covered amount when the bond matures:

______________________

Answer: U.S. Dollar Equivalent of the Maturing AUD Government Bond

- Amount of AUD to be received in 1 year from maturing bonds:
- Par value = AUD1,000 x 100 = AUD100,000
- Interest (5.5% coupon) = 100,000 x 0.055 = AUD5,500
- Total received = AUD105,500 (to be sold forward)
- Exchange rates:
- AUD/USD spot 1.0005/1.0009
- AUD/USD 1 year forward 0.9650/0.9657
- USD covered amount (to be received in 1 year) = AUD105,500 x 0.9650 = USD101,807.50

Concept of Covered Return

- The covered return (i.e., hedged return) on a cross border investment is the return after the investment’s foreign exchange risk has been covered with the appropriate forward contract.
- The forward exchange rate will determine the “covered” investment return for the U.S. investor.
- In the previous example, how would you determine the covered return (as a %) to the U.S. investor?

Calculating the Covered Return

- Answer: Calculate the yield to maturity on the investment when covered.
- Note: Yield to Maturity is the internal rate of return (IRR), or the discount rate that sets the present value of the future cash inflow to the price of the investment,
- So given:
- AUD/USD spot 1.0005/1.0009
- AUD/USD 1 year forward 0.9650/0.9657
- USD Purchase Price = AUD100,000 x 1.0009 = USD100,090
- USD Hedged Equivalent Cash Inflow in 1 year = AUD105,500 x 0.9650 = USD101,807.50
- Solve for the IRR (k): -100,090 = 101,807.50/(1+k)
- http://www.datadynamica.com/IRR.asp
- k = 1.72% (Why is this different from the 5.5%)
- Answer: Because AUD is selling at a 1 year forward discount.

Another Example of a Covered Return

- Assume the following:
- A 1 year Japanese Government Bond with a coupon of 1%.
- Par value of 100,000 yen and selling at par.
- Exchange Rates:
- USD/JPY spot: 76.61/76.65
- 1 year forward: 73.50/73.55
- Calculate the covered return for a U.S. investor on the above JGB

Answer to JGB Covered Return

- Step 1: Calculate the USD purchasing price of the JGB:
- 100,000/76.61 (note this is spot bid) = 1305.31
- Step 2: Calculate the yen inflow expected in 1 year:
- 100,000 x 1.01 = 101,000 (note: coupon rate is 1%)
- Step 3: Calculate the USD equivalent of the 1 year yen inflow using a forward contract.
- 101,000/73.55 = 1373.22 (note this is 1 year ask)
- Ask is the price at which the bank will sell you dollars.
- Step 4: Calculate the IRR (using the web site)
- -1305.31= 1373.22/(1+k); k = 5.21% (Why is this different from the 1%)

Covered Interest Arbitrage

- Covered interest “arbitrage” is a situation that occurs when a covered return offers a higher return than that in the investor’s home market.
- As an example assume:
- 1 year interest rate in U.S. is 4%
- 1 year interest rate in Australia is 7%
- AUD 1 year forward rate is quoted at a discount of 2%.
- In this case, a U.S. investor could invest in Australia and
- Cover (sell Australian dollars forward) and earn a covered return of 5% (7% - 2%) which is 100 basis points greater than the U.S. return
- This is covered interest arbitrage: earning more (when covering) than the rate at home.

Explanation for Covered Interest Arbitrage Opportunities

- Covered interest arbitrage will exist whenever the quoted forward exchange rate is not priced correctly.
- If the forward rate is priced correctly, covered interest arbitrage should not exist.
- Going back to our original example:
- (1) Invest in a U.S. government bond and earn 2.0%.
- (2) Invest in an Australian government bond and earn 5.5%
- If the AUD 1 year forward were quoted at a discount of 3.5%, then the covered return (2%) and the home return (2%) would be equal.

The Appropriate Forward Exchange Rate and the Interest Rate Parity Model

- The Interest Rate Parity Model (IRP) offers an explanation of the market’s correctly priced (i.e., “equilibrium”) forward exchange rate.
- This equilibrium rate is the forward rate that precludes covered interest arbitrage
- The Interest Rate Parity Model states:
- “That in equilibrium the forward rate on a currency will be equal to, but opposite in sign to, the differencein the interest rates associated with the two currencies in the forward transaction.”
- Thus, the equilibriumforward rate is whatever forward exchange rate willinsure that the two cross border investments will yield similar returns when covered.

How is the Forward Rate Calculated?

- Market maker banks calculate their quoted forward rate is calculated from three observable numbers:
- The (current) spot rate.
- A foreign currency interest rate.
- A home currency interest rate (assume to be the U.S.).
- Note: The maturities of the interest rates used should be approximately equal to the calculated forward rate period (i.e., maturity of the forward contract).
- What interest rates are used?
- Interbank market (wholesale) interest rates for currencies (sometimes called euro-deposit rates). Large global banks quote each other and clients market interest rates in a range of currencies.

Example: October 11, 2012

- http://www.forexpros.com/rates-bonds/forward-rates

Forward Rate Formula for European Terms Quote Currencies

- The formula for the calculation of the equilibrium European terms forward foreign exchange rate is as follows:
- FTet = Set x [(1 + INTf) / (1 + INTus)]
- Where:
- FTet = forward foreign exchange rate at time period T, expressed as units of foreign currency per 1 U.S. dollar; thus European terms, i.e., “et”
- Set = today\'s European terms spot foreign exchange rate,
- INTf = foreign interest rate for a maturity of time period T (expressed as a percent, e.g., 1% = 0.01)
- INTus = U.S. interest rate for a maturity of time period T

Example: Solving for the Forward European Terms Exchange Rate

- Assume the following data:
- USD/JPY spot = ¥120.00
- Japanese yen 1 year interest rate = 1%
- US dollar 1 year interest rate = 4%
- Calculate the 1 year yen forward exchange rate:
- Set up the formula and insert data.
- FTet = Set x [(1 + INTf) / (1 + INTus)]

Example: Solving for the Forward European Terms Exchange Rate

- Assume the following data:
- USD/JPY spot = ¥120.00
- Japanese yen 1 year interest rate = 1%
- US dollar 1 year interest rate = 4%
- Calculate the 1 year yen forward exchange rate:
- FTet = Set x [(1 + INTf) / (1 + INTus)]
- FTet = ¥120 x [(1 + .01) / (1 + .04)]
- FTet = ¥120 x .971153846
- FTet = ¥116.5384615

Forward Rate Formula for American Terms Quote Currencies

- The formula for the calculation of the equilibrium American terms forward foreign exchange rate is as follows:
- FTat = Sat x [(1 + INTus) / (1 + INTf)]
- Where:
- FTat = forward foreign exchange rate at time period T, expressed as the amount of 1 U.S. dollars per 1 unit of the foreign currency; thus American terms, or at)
- Sat = today\'s American terms spot foreign exchange rate.
- INTus = U.S. interest rate for a maturity of time period T (expressed as a percent, e.g., 4% = 0.04)
- INTf = Foreign interest rate for a maturity of time period T

Example: Solving for the American Terms Forward Exchange Rate

- Assume the following data:
- GPB/USD spot = $1.9800
- UK 1 year interest rate = 6%
- US dollar 1 year interest rate = 4%
- Calculate the 1 year pound forward exchange rate:
- Set up the formula and insert data:
- FTat = Sat x [(1 + INTus) / (1 + INTf)]

Example: Solving for the American Terms Forward Exchange Rate

- Assume the following data:
- GPB/USD spot = $1.9800
- UK 1 year interest rate = 6%
- US dollar 1 year interest rate = 4%
- Calculate the 1 year pound forward exchange rate:
- FTat = Sat x [(1 + INTus) / (1 + INTf)]
- FTat = $1.9800 x [(1 + .04) / (1 + .06)]
- FTat= $1.9800 x .9811
- FTat = $1.9426

Calculating the forward rate for periods less than and greater than one year

Formulas and Interest Rates

- The formulas used in the previous slides show you how to calculate the forward exchange rate 1 year forward.
- The following slides illustrate how to adjust the forward rate formula for periods other than 1 year.
- Important:
- All interest rates quoted in financial markets are on an annual basis, thus and adjustment must be made to allow for other than annual interest periods.

Forwards Less Than 1 Year: European Terms

- FTet = Set x [(1 + ((INTf) x n/360)) / (1 + ((INTus) x n/360))]
- Where:
- FT = forward foreign exchange rate at time period T, expressed as units of foreign currency per 1 U.S. dollar;
- Set = today\'s European terms spot foreign exchange rate.
- INTf = foreign interest rate for a maturity of time period T
- INTus = U.S. interest rate for a maturity of time period T
- n = number of days in the forward contract (note: we use a 360 day year in this formula).
- Note: What we have added to the original formula is an adjustment for the time period (n/360)

European Terms Example: Less than 1 year

- Assume:

USD/JPY spot = 82.00

6 month Japanese interest rate = 0.12%*

6 month U.S. interest interest rate= 0.17%*

*These are interest rates expressed on an annual basis.

- Calculate the 6 month forward yen
- FTet = Set x [(1 + ((INTf) x n/360))/ (1 + ((INTus) x n/360))]

Ftet = 82.00 x [(1 + ((0.0012 x 180/360))/((1 + ((0.0017 x 180/360))]

FTet = 82.00 x (1.0006/1.00085)

FTet = 82.00 x .9997

FTet= 81.9795

Forwards More Than 1 Year: American Terms

- FTat = Sat x [(1 + (INTus)n / (1 + (INTf)n]
- Where:
- FT = forward foreign exchange rate at time period T, expressed as the amount of 1 U.S. dollars per 1 unit of the foreign currency.
- Sat = today\'s American terms spot foreign exchange rate.
- INTus = U.S. interest rate for a maturity of time period T
- INTf = Foreign interest rate for a maturity of time period T
- n = number of years in the forward contract.

American Terms Example: More than 1 Year

- Assume:

GBP/USD spot = 1.5800

5 year United Kingdom interest rate = 1.05%*

5 year United States interest rate = 1.07%*

*These are interest rates expressed on an annual basis.

- Calculate the 5 year forward pound:

FTat = Sat x ((1 + INTus)n/(1 + INTf)n)

FTat = 1.5800 x ((1 + 0.0107)5/(1 + 0.0105)5)

FTat = 1.5800 x (1.05466/1.05361)

FTat = 1.5800 x 1.001

FTat = 1.5816 (Note: This is the forward 5 year rate)

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