1. Fi8000 Valuation of Financial Assets
Fall Semester 2009
Dr. Isabel Tkatch
Assistant Professor of Finance

2. Currency Exchange Rate (Spot)
A spot currency transaction is an exchange of one currency for another.
The currency exchange rate is a simple conversion factor:
The direct exchange rate is the number of $US to be paid for 1 unit of foreign currency (usually for the £UK and the Euro);
The indirect exchange rate is the number of foreign currency units paid for 1 $US (usually for the Swiss Franc and Japanese Yen).

3. Currency Exchange Rate
Numeric Example:
The exchange rate between the $US and £UK is $US / £UK - i.e. one has to pay $ for £1 (direct).
The same exchange rate can be presented as 1/ = £UK / $US - i.e. one has to pay £ for $1 (indirect).

4. Currency Exchange Rate
Example continued:
The exchange rate between the $US and £UK is $US / £UK.
The exchange rate between the $US and J¥ is $US / J¥.
What should be the exchange rate between the £UK and the J¥?

5. Currency Arbitrage There are at least two ways to convert £UK to J¥:
Direct conversion of £UK to J¥
Conversion using an intermediary currency:
Convert £UK to $US
Convert $US to J¥
If there is no opportunity to make arbitrage profits, both conversion methods must imply the same £UK to J¥ exchange rate.

6. Currency Exchange Rate
Example (data):
$US/£UK or £UK/$US.
$US/J¥ or J¥/$US.
We will use the no-arbitrage argument to calculate the no-arbitrage £UK/J¥ exchange rate.

7. Currency Exchange Rate
J¥/£UK = ?
£UK/J¥ = ?
£UK/$US
$US/£UK $
$US/J¥
J¥/$US

8. Currency Exchange Rate
Conversion using an intermediary currency:
Convert £UK to $US:
the price of 1 $US is £UK
Convert $US to J¥:
the price of 1 J¥ is $US
The £UK price of 1 J¥:
£UK/$US * $US/J¥ =
£UK/J¥

9. Currency Exchange Rate
The £UK/J¥ no-arbitrage exchange rate:
The £UK/J¥ exchange rate is ,
i.e. the price of 1 J¥ is £UK.
The J¥/£UK exchange rate is
1/ = ,
i.e. the price of 1 £UK is J¥.

10. Currency Exchange Arbitrage
Example continued:
The $US/£UK exchange rate is
The $US/J¥ exchange rate is
Is there an arbitrage opportunity if the market £UK/J¥ exchange rate is ?
Yes! The £UK/J¥ exchange rate in the market is different from the no-arbitrage rate (two-stage exchange rate):
Market: £UK/J¥ > £UK/J¥ :No-arbitrage
How can we make arbitrage profits?

11. Currency Exchange Arbitrage
Cross currency (triangle) arbitrage strategy:
Sell the expensive J¥: convert J¥ to £UK in one step:
1. Sell J¥ for £UK (convert J¥ to £UK)
Buy the cheap J¥: convert £UK to J¥ in two steps, using the $US as an intermediary:
2. Buy $US with £UK (convert £UK to $US)
3. Buy J¥ with $US (convert $US to J¥)
Note: this is a round trip transaction. You start with J¥ (before step 1) and you end up with J¥ (after step 3).

12. Currency Exchange Arbitrage
Cross currency (triangle) arbitrage strategy:
Sell the expensive J¥ - direct £UK to J¥ exchange rate:
1. Convert 1 J¥ to £UK.
Buy the cheap J¥ - two stages, using the $US as an intermediary:
2. Convert £UK to $US. You will get
£UK * $US/£UK = $US.
3. Convert $US to J¥. You will get
$US * J¥/$US = J¥.
Arbitrage profit: you started with 1 J¥ and ended up with J¥.

13. Currency Exchange Rate
£UK * $US/£UK
= $US
1 J¥ * £UK/J¥
= £UK $
$US * J¥/$US
= J¥

14. Currency Exchange Arbitrage
Cross currency arbitrage strategy (end up with $US):
2. Convert J¥ to £UK. You will get
J¥ * £UK/J¥ = £UK.
3. Convert £UK to $US. You will get
£UK * $US/£UK = $US.
1. Convert 1 $US to J¥. You will get
1 $US * J¥/$US = J¥.
Arbitrage profit: you started with 1 $US and ended up with $US: an arbitrage profit of $US.

15. Currency Exchange Rate
£UK * $US/£UK
= $US
J¥ * £UK/J¥
= £UK $
Start Here:
1 $US * J¥/$US
= J¥

16. Currency Exchange Rate (Forward)
Forward or Futures Contracts
An agreement between a buyer and a seller, to trade at a specific date in the future, a specific quantity of a specific currency for an agreed exchange rate.
Forward – tailored OTC market contracts for creditworthy traders and large trades.
Futures – formal markets of standardized contracts (International Monetary Market in Chicago, London International Financial Futures Exchange).

17. Pricing Currency Forwards
There are at least two ways to invest money in a risk-free asset for one year:
Domestic risk-free investment
Buy US Treasury Bills
Foreign risk-free investment
Convert $US for foreign currency
Buy foreign risk-free bonds for 1 year
Convert the foreign currency back to $US (forward contract)
If there is no opportunity to make arbitrage profits, both investment strategies should have the same dollar denominated risk-free return.

18. Covered Interest Arbitrage
Numeric Example:
Suppose you would like to invest $100,000 in a risk-free security.
In the US the annual risk free rate is 5.00%, while in the UK the annual risk free rate is 5.20%.
Is there an arbitrage opportunity? – Find a way to compare the domestic and foreign investment strategies.

19. Covered Interest Arbitrage
Numeric Example Continued:
We need the spot and forward (one year) $US/£UK exchange rates to answer that question.
Note that if we do not use a forward contract to “lock in” the exchange rate, the foreign alternative becomes a risky rather than risk-free investment strategy (exchange rate risk).
Is there an opportunity to make arbitrage profits, if the spot exchange rate is $US/£UK and the (one year) forward rate is $US/£UK?

20. Comparing the Two Strategies
1. Domestic risk-free investment:
1a. Buy US Treasury Bills
Strategy
t = 0
t = 1
CF ($US)
CF (£UK)
Buy T-Bills
(5.00%)
-100,000
+105,000
Total

21. Comparing the Two Strategies
2. Foreign risk-free investment:
2a. Convert $US for the foreign currency (£UK)
2b. Buy foreign (£UK denominated) risk-free bonds
2c. Convert the foreign currency (£UK) back to $US (forward rate)
Strategy
t = 0
t = 1
CF ($US)
CF (£UK)
Convert $US to £UK (spot rate )
-100,000
+59,701
Buy UK risk-free bonds
(5.20%)
-59,701
+62,806
Convert £UK to $US
(forward rate )
+103,630
-62,806
Total

22. Arbitrage Strategy Buy Cheap: Domestic risk-free investment
Buy US Treasury Bills
 get 5% dollar denominated risk free rate
Sell Expensive: Foreign risk-free investment
Convert £UK to $US
Short sell UK risk-free bonds for 1 year
Convert $US back to £UK (forward contract)
 pay 3.63% dollar denominated risk free rate

23. Covered Interest Arbitrage
Strategy
t = 0
t = 1
CF ($US)
CF (£UK)
Buy US T-Bills
(5.00%)
-100,000
+105,000
Convert £UK to $US (spot rate )
+100,000
-59,701
Sell UK risk-free bonds
(5.20%)
+59,701
-62,806
Convert $US to £UK
(forward rate )
-105,000
+63,636
Total
+803

24. No-Arbitrage Forward Exchange Rate
( F0($US/£UK) = )
Strategy
t = 0
t = 1
CF ($US)
CF (£UK)
Buy US T-Bills
(5.00%)
-100,000
+105,000
Convert £UK to $US (spot rate )
+100,000
-59,701
Sell UK risk-free bonds
(5.20%)
+59,701
-62,806
Convert $US to £UK
(forward rate F0)
-62,806 * F0 = -105,000
+62,806
Total =0
=0 +803

25. No-Arbitrage UK Risk Free Rate
( rUK = % )
Strategy
t = 0
t = 1
CF ($US)
CF (£UK)
Buy US T-Bills
(5.00%)
-100,000
+105,000
Convert £UK to $US (spot rate )
+100,000
-59,701
Sell UK risk-free bonds
( ruk )
+59,701
-59,701(1+r)
Convert $US to £UK
(forward rate )
-105,000
+63,636
Total
=0 +803

26. Interest Rate Parity (Covered Interest Arbitrage)
Intuition:
If two investments are risk-free they must have the same rate of return. Therefore, any difference in the domestic and foreign risk-free rates must be offset by a difference in the spot and forward exchange rates.

27. Interest Rate Parity (Covered Interest Arbitrage)
Notation:
E0 = spot exchange rate ($US/£UK) or (£UK/$US)
F0 = forward exchange rate ($US/£UK) or (£UK/$US)
* Note that if you use the £UK/$US (indirect) exchange rate you will also have to reverse the ratio of interest rates.

28. Practice Problems Practice Problem #1
The annual risk-free rate in the US is 5.00% while in Japan it is 3.20%.
What should be the spot J¥/$US exchange rate, if the (one year) forward J¥/$US exchange rate is ?
Answer: E0(J¥/$US) =

29. Practice Problems Practice Problem #2
The annual risk-free rate in the US is 4.60% while in Japan it is 3.50%.
The spot J¥/£UK exchange rate is
The spot $US/£UK exchange rate is
The (one year) forward J¥/£UK exchange rate is
The (one year) forward $US/£UK exchange rate is
Describe an arbitrage transaction: write down the strategy in the table format presented in the lecture notes.

30. Practice Problems BKM Ch. 23: 7th Ed.: 10, 12-14.
8th Ed.: 11-12, CFA: 2,3.
Practice problems:
Forward and futures contracts 1-5;
Currency exchange rates 6-9.