I. Conceptual Questions
- What is the law of one price?
Answer: The law of one price states identical assets must have the same price wherever they are bought or sold. The law of one price is enforced by arbitrage activity between identical assets. In a perfect market without transaction costs, the law of one price must hold for there to be no arbitrage opportunities.
- What is an arbitrage profit?
Answer: Arbitrage profit is a profit obtained through the simultaneous purchase and sale of the same or equivalent securities such that there is no net investment or risk. Arbitrage will drive the prices of identical assets into equilibrium and enforce the law of one price.
- What is the difference between locational, triangular, and covered interest arbitrage?
Answer: Locational arbitrage is conducted between two physical locations, such as between currency prices at two different banks (such that ASf/dBSd/f ¹ 1 for banks A and B and currencies d and f). Triangular arbitrage is conducted across three different cross exchange rates (such that Sd/e Se/fSf/d ¹ 1 for currencies d, e, and f). Covered interest arbitrage takes advantage of a disequilibrium in the interest rate parity condition [(Ftd/ f) / (S0d/ f)] ¹ (1 + id) / (1 + i f)]t between currency and Eurocurrency markets
II. True/False
- No-arbitrage condition is also the condition for the law of one price.
True
- Purchasing power parity is the law of one price
True
- According to Purchasing power parity, exchange rates are not determined by market demand and supply, but determined by nominal prices in each country.
False. The nominal prices reflect the market demand and supply.
- The exchange rate equilibrium does not prohibit you to make profit just from buying and selling different currencies under the exact same market condition.
False. Exchange rate equilibrium eliminates arbitrage opportunities.
- The exchange rate equilibrium is the state when there is no arbitrage opportunity and the law of one price prevails.
True.
- Interest rate parity (covered interest parity) shows how the price of a currency forward is determined.
True.
III. Problems
1. Expected inflation over the next year is E[p] = 10%. What nominal interest rate i should investors charge on the following assets?
a. Investors require a real rate of return of ι = 2 percent on a one-year corporate bond.
i=(1+ι)(1+E[p])-1=1.02*1.10-1=12.2%
b. Investors require a real return of ι = 6 percent on a portfolio of stocks.
i=(1+ι)(1+E[p])-1=1.06*1.10-1=16.6%
c. Investors require a real return of ι = 10 percent on an investment in an oil field.
i=(1+ι)(1+E[p])-1=1.10*1.10-1=21.0%
2. Calculate the following cross exchange rates:
a. If exchange rates are 200 yen per dollar and 50 US cents per Swiss franc, what is the exchange rate of yen per franc?
S¥/$*S$/SFr*SSFr/¥=1
SSFr/¥=1/(200*0.50)=0.01
S ¥/SFr=1/0.01=100
b. The dollar is trading at ¥100/$ and at SFr1.60/$. What is the yen per franc rate?
S$/SFr=1/ SSFr/$=1/1.60=0.625$/SFr
S¥/$*S$/SFr*SSFr/¥=1
SSFr/¥=1/(100*0.625)= SFr 0.016/¥
S ¥/SFr=1/0.016=¥62.5/SFr
3. As a percentage of an arbitrary starting amount, about how large would transaction costs have to be to make arbitrage between the exchange rates S0SFr/$=SFr1.7223/$, S0$/¥=$0.009711/¥, and S0¥/SFr=61.740 unprofitable?
S0SFr/$S0$/¥S0¥/SFr=1.7223*0.009711*61.740=1.0326>1
By selling SFr with ¥, selling ¥ with $ and selling $ with SFr, there can be an arbitrage of (1.0326/1-1)=3.26%.
Therefore, the total transaction cost should be 3.26% through the transactions of selling SFr with ¥, selling ¥ with $ and selling $ with SFr.
4. The Mexican peso is quoted in direct terms at ‘¥28.74/MXN BID and ¥28.77/MXN ASK’ in Tokyo. The yen is quoted in direct terms in Mexico City at ‘MXN0.0341600/¥ BID and MXN0.03420/¥ ASK.’
a. Calculate the bid-ask spread as a percentage of the bid price from the Japanese and from the Mexican perspective.
Japanese perspective:
Direct quote: (28.77-28.74)/28.74=0.104% or 10.4 basis points
Mexican perspective:
Direct quote: (0.03420-0.03416)/0.03416=0.117% or 11.7 basis points
b. Is there an opportunity for profitable arbitrage? If so, describe the necessary transactions using a ¥1 million starting amount. Take your profit in yen.
Japanese indirect quotes:
BID: 1/28.74=MSN0.03479/¥ ASK: 1/28.77=MSN0.03475¥
Mexican indirect quotes:
BID: 1/0.03416=¥29.274/MSN ASK: 1/0.03420=¥29.240/MSN
Arbitrage opportunities: S¥/MSNSMSN/¥=1?
Buy denominating currency: S¥/MSNSMSN/¥=28.77*0.03420=0.984<1
Profit ratio: 1/0.984-1=0.016
Profit: ¥1m*0.016=¥16,000
Sell denominating currency: S¥/MSNSMSN/¥=29.240*0.03475=1.016>1
Profit ratio: 1.016/1-1=0.016
Profit: ¥1m*0.016=¥16,000
Either buy or sell the denominating currency, you can get ¥16,000 of arbitrage profit.
5. Quotes for the US dollar and Thai baht (Bt) are as follows:
Spot contract midpoint S0Bt/$=Bt24.96/$
1-year forward contract midpoint F1Bt/$=Bt25.64/$
1-year Eurodollar interest rate i$=6.125% per year
- Your newspaper does not quote 1-year Eurocurrency interest rates on Thai baht. Make your own estimate of iBt.
F1Bt/$/ S0Bt/$=(1+ iBt)/(1+ i$)
iBt= F1Bt/$(1+ i$)/ S0Bt/$-1=9.016%
- Suppose that you can trade at S0Bt/$, F1Bt/$, and i$ and that you also can either borrow or lend at a Thai Eurocurrency interest rate of iBt=10% per year. Base on $1 million initial amount, how much profit can you generate through covered interest arbitrage?
F1Bt/$/ S0Bt/$=1.02724
(1+ iBt)/(1+ i$)=1.03651
F1Bt/$/ S0Bt/$<(1+ iBt)/(1+ i$)
Therefore Buy $ forward, Sell $ spot, Invest Bt, and Borrow $. To be more specific:
Step 1: Borrow $1 million at 6.125%, which means to repay interest of $1m*6.125%=$61,250 in a year.
Step 2: Sell $1 million spot for $1m*Bt24.96/$=Bt24.96m
Step 3: Invest Bt24.96 million at 10% to expect Bt24.96m*(1+10%)=Bt27.456m in a year
Step 4: Buy $ forward using Bt27.456 million to get 27.456m/25.64=$1.07082m
Repay interest $61,250, covered interest arbitrage profit is $70,820-$61,250=$9570
