Question

Suppose the price of a British pound is \(\$ 2.00\) and the price of a German mark is \(\$ 0.40\). a) What should the price of a British pound be in terms of German marks? b) If the market was paying 6 marks for every British pound, how might an Englishman make an arbitrage profit?

Short answer

The exchange rate between British pounds and German marks should be 5 marks for every British pound. However, given the market exchange rate of 6 marks per British pound, an Englishman can make an arbitrage profit by converting £1,000 into $2,000, then into 5,000 marks, and finally back to £833.33, making a profit of £166.67 in the process.

Step-by-step solution

Calculate the Exchange Rate between British Pounds and German Marks

To find the exchange rate between British pounds and German marks, we will use the given prices of each currency in dollars. Price of 1 British pound = $2.00 Price of 1 German mark = $0.40 Exchange rate (British pound to German mark) = (Price of 1 British pound) / (Price of 1 German mark)

Calculate the Exchange Rate

Now plug in the given values and calculate the exchange rate: Exchange rate (British pound to German mark) = \(2.00 / \)0.40 Exchange rate (British pound to German mark) = 5 So the exchange rate between British pounds and German marks is 5 marks for every British pound.

Check the Market Exchange Rate for an Arbitrage Opportunity

We are given that the market exchange rate is 6 marks for every British pound, which is different from our calculated exchange rate of 5 marks per British pound. Since there is a discrepancy, there is an arbitrage opportunity.

Describe the Arbitrage Process

An Englishman could take advantage of this arbitrage opportunity in the following way: 1. Convert British pounds to dollars in the UK, where the exchange rate is 1 pound = $2.00. 2. Convert dollars to German marks in Germany, where the exchange rate is 1 mark = $0.40. 3. Exchange German marks to British pounds using the given market exchange rate of 6 marks per British pound. 4. Bring the British pounds back to the UK, making a profit in the process. Let's consider an example of exchanging £1,000.

Calculate the Arbitrage Profit

1. Convert £1,000 to dollars: £1,000 * (\(2.00/£) = \)2,000 2. Convert $2,000 to German marks: \(2,000 / (\)0.40/mark) = 5,000 marks 3. Convert 5,000 marks to British pounds using the market exchange rate: 5,000 marks * (1 pound / 6 marks) = £833.33 The Englishman started with £1,000 and now has £833.33 worth of British pounds (while essentially completing a full circle of exchanges). The difference, £166.67, represents a profit made through arbitrage.

Key concepts

Arbitrage Opportunity

In the world of finance, an arbitrage opportunity occurs when a price discrepancy allows for a risk-free profit to be made by conducting a series of transactions. Essentially, it exploits the differences in the price of a single asset or similar assets across different markets. To understand this, let's dive into our exercise example.

Using the given prices of currencies, an Englishman can make a profit by taking advantage of the difference between the local exchange rates and the market exchange rate. Exchange rates should theoretically be consistent. However, when the market exchange rate offers 6 German marks for one British pound, while the calculated exchange rate based on currency values in dollars suggests it should be 5, it creates an opportunity.

Savvy individuals can 'buy low and sell high,' in this case, by converting currencies where they are undervalued and converting them back where they are overvalued. Leveraging the arbitrage opportunity step by step as outlined by the solution, the Englishman converts currencies in such a way that he ends up with more pounds than he started with, thereby making a profit with no real investment risk, as long as the price discrepancy exists.

Currency Conversion Calculation

Understanding how to perform a currency conversion calculation is essential for identifying arbitrage opportunities. The process involves multiple conversions and can be summarized by a simple formula: Exchange rate = (Price of Currency A) / (Price of Currency B).

In our exercise, the Price of 1 British pound is given as \(2.00, and the Price of 1 German mark as \)0.40. The exchange rate is calculated by dividing the price of the pound by the price of the mark. Thus, the proper exchange rate, under the assumption of no market inefficiency, should be 5 marks for 1 British pound. If you know how much one currency is worth in terms of another, you can convert between currencies.

To validate the exchange rate or to engage in currency conversion, one must monitor the exchange rates closely. A solid grasp on these calculations could enable someone to perform quick conversions and potentially benefit from arbitrage opportunities presented by fluctuations in exchange rates.

Exchange Rate Discrepancy

The exchange rate discrepancy refers to a situation where there is a difference in the value of a currency between two (or more) markets or calculations. This is what can lead to an arbitrage opportunity. In our original problem, there's a discrepancy between the calculated exchange rate of 5 German marks per British pound and the market exchange rate of 6 marks per pound.

Such discrepancy is unusual in efficient markets due to the large number of participants and high volume of transactions which usually correct these mispricings quickly. However, they can occur due to market inefficiencies, differences in information access, or rapid changes in market conditions.

For someone looking to perform arbitrage, spotting these discrepancies is crucial. The ability to quickly do the maths and act on an exchange rate discrepancy can result in profit. It's a race against time, as other market participants may also notice the discrepancy, and their actions will eventually correct the price imbalance.