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

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).

Currency Exchange Rate Numeric Example: The exchange rate between the $US and £UK is 1.6757 $US / £UK - i.e. one has to pay $1.6757 for £1 (direct). The same exchange rate can be presented as 1/1.6757 = 0.5968 £UK / $US - i.e. one has to pay £0.5968 for $1 (indirect).

Currency Exchange Rate Example continued: The exchange rate between the $US and £UK is 1.6757 $US / £UK. The exchange rate between the $US and J¥ is 0.007331 $US / J¥. What should be the exchange rate between the £UK and the J¥?

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.

Currency Exchange Rate Example (data): 1.6757 $US/£UK or 0.5968 £UK/$US. 0.007331 $US/J¥ or 136.40 J¥/$US. We will use the no-arbitrage argument to calculate the no-arbitrage £UK/J¥ exchange rate.

Currency Exchange Rate £ J¥/£UK = ? £UK/J¥ = ? 0.5968 £UK/$US 1.6757 $US/£UK ¥ $ 0.007331 $US/J¥ 136.40 J¥/$US

Currency Exchange Rate Conversion using an intermediary currency: Convert £UK to $US: the price of 1 $US is 0.5968 £UK Convert $US to J¥: the price of 1 J¥ is 0.007331 $US The £UK price of 1 J¥: 0.5968 £UK/$US * 0.007331 $US/J¥ = 0.004375 £UK/J¥

Currency Exchange Rate The £UK/J¥ no-arbitrage exchange rate: The £UK/J¥ exchange rate is 0.004375, i.e. the price of 1 J¥ is 0.004375 £UK. The J¥/£UK exchange rate is 1/0.004375 = 228.5641, i.e. the price of 1 £UK is 228.5641 J¥.

Currency Exchange Arbitrage Example continued: The $US/£UK exchange rate is 1.6757. The $US/J¥ exchange rate is 0.007331. Is there an arbitrage opportunity if the market £UK/J¥ exchange rate is 0.004494? Yes! The £UK/J¥ exchange rate in the market is different from the no-arbitrage rate (two-stage exchange rate): Market: 0.004494 £UK/J¥ > 0.004375 £UK/J¥ :No-arbitrage How can we make arbitrage profits?

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).

Currency Exchange Arbitrage Cross currency (triangle) arbitrage strategy: Sell the expensive J¥ - direct £UK to J¥ exchange rate: 1. Convert 1 J¥ to 0.004494 £UK. Buy the cheap J¥ - two stages, using the $US as an intermediary: 2. Convert 0.004494 £UK to $US. You will get 0.004494 £UK * 1.6757 $US/£UK = 0.00753 $US. 3. Convert 0.00753 $US to J¥. You will get 0.00753 $US * 136.40 J¥/$US = 1.02717 J¥. Arbitrage profit: you started with 1 J¥ and ended up with 1.02717 J¥.

Currency Exchange Rate £ 0.004494 £UK * 1.6757 $US/£UK = 0.00753 $US 1 J¥ * 0.004494 £UK/J¥ = 0.004494 £UK ¥ $ 0.00753 $US * 136.40 J¥/$US = 1.02717 J¥

Currency Exchange Arbitrage Cross currency arbitrage strategy (end up with $US): 2. Convert 136.40 J¥ to £UK. You will get 136.40 J¥ * 0.004494 £UK/J¥ = 0.6130 £UK. 3. Convert 0.6130 £UK to $US. You will get 0.6130 £UK * 1.6757 $US/£UK = 1.02717 $US. 1. Convert 1 $US to J¥. You will get 1 $US * 136.40 J¥/$US = 136.40 J¥. Arbitrage profit: you started with 1 $US and ended up with 1.02717 $US: an arbitrage profit of 0.02717 $US.

Currency Exchange Rate £ 0.6130 £UK * 1.6757 $US/£UK = 1.02717 $US 136.40 J¥ * 0.004494 £UK/J¥ = 0.6130 £UK ¥ $ Start Here: 1 $US * 136.40 J¥/$US = 136.40 J¥

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).

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.

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.

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 1.6750 $US/£UK and the (one year) forward rate is 1.6500 $US/£UK?

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

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 1.6750) -100,000 +59,701 Buy UK risk-free bonds (5.20%) -59,701 +62,806 Convert £UK to $US (forward rate 1.6500) +103,630 -62,806 Total

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

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 1.6750) +100,000 -59,701 Sell UK risk-free bonds (5.20%) +59,701 -62,806 Convert $US to £UK (forward rate 1.6500) -105,000 +63,636 Total +803

No-Arbitrage Forward Exchange Rate ( F0($US/£UK) = 1.6718 ) Strategy t = 0 t = 1 CF ($US) CF (£UK) Buy US T-Bills (5.00%) -100,000 +105,000 Convert £UK to $US (spot rate 1.6750) +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

No-Arbitrage UK Risk Free Rate ( rUK = 6.5909% ) 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 1.6750) +100,000 -59,701 Sell UK risk-free bonds ( ruk ) +59,701 -59,701(1+r) Convert $US to £UK (forward rate 1.6500) -105,000 +63,636 Total =0 +803

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.

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.

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 107.875? Answer: E0(J¥/$US) = 109.7565

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 205.00 The spot $US/£UK exchange rate is 1.8825 The (one year) forward J¥/£UK exchange rate is 204.00 The (one year) forward $US/£UK exchange rate is 1.8900 Describe an arbitrage transaction: write down the strategy in the table format presented in the lecture notes.

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.