Lecture 02 - International Financial Modelling PDF

International Financial Modelling Cosimo Munari Department of Economics University of Verona Lecture 2 8 October 2024 1 / 18  Direct and indirect exchange rates We fix two currencies referred to...

International Financial Modelling Cosimo Munari Department of Economics University of Verona Lecture 2 8 October 2024 1 / 18  Direct and indirect exchange rates We fix two currencies referred to as currency 1 and currency 2. In the following we take the perspective of currency 1 (typically our domestic currency). In other words, currency 1 is our base currency. The direct exchange rate or spot rate or spot price between currency 1 and 2 relative to currency 1 is the number of units of currency 1, denoted by Sdir , that can be exchanged into one unit of currency 2: currency 1 ⇐⇒ currency 2 Sdir 1 The indirect exchange rate or spot rate or spot price between currency 1 and 2 relative to currency 1 is the number of units of currency 2, denoted by Sind , that can be exchanged into one unit of currency 1: currency 1 ⇐⇒ currency 2 1 Sind 2 / 18  Direct and indirect exchange rates “Direct” and “indirect” have to be always understood from the perspective of the base currency, in our case currency 1: The direct exchange rate between currency 1 and 2 relative to currency 1 is expressed in units of currency 1. Equivalently, it represents the price of one unit of currency 2 in terms of currency 1 (hence direct). The indirect exchange rate between currency 1 and 2 relative to currency 1 is expressed in units of currency 2. Equivalently, it represents the price of one unit of currency 1 in terms of currency 2 (hence indirect). 3 / 18  Direct and indirect exchange rates “Direct” and “indirect” always depend on the base currency: The direct exchange rate between currency 1 and 2 relative to currency 1 coincides with the indirect exchange rate between currency 1 and 2 relative to currency 2. The indirect exchange rate between currency 1 and 2 relative to currency 1 coincides with the direct exchange rate between currency 1 and 2 relative to currency 2. 4 / 18  Example of direct and indirect exchange rates Example. The table below displays direct and indirect exchange rates between EUR and some foreign currencies on 01.10.2024 as retrieved from the website of the European Central Bank. EUR is the base currency. Foreign Currency Direct Exchange Rate Indirect Exchange Rate USD 0.902 1.108 GBP 1.203 0.831 CHF 1.064 0.939 As said before, the values under the column “Direct Exchange Rate” are all expressed in EUR, while the values under the column “Indirect Exchange Rate” are expressed in the various foreign currencies. 5 / 18  Direct and indirect exchange rates: Who uses what? It is standard to use one’s domestic currency as the base currency. This means, e.g., that the base currency adopted in Italy is always EUR. Most currency quotes provided in newspapers and websites are direct, but there are two notable exceptions where indirect exchange rates are used: Exchange rates between GBP and foreign currencies are often indirect. This is probably because the GBP-based monetary system was highly nondecimal (1 pound = 20 shilling = 240 pence) and it was therefore easier to express one unit of GBP in a foreign currency. Exchange rates between EUR and foreign currencies are often indirect. This is probably because EUR is younger compared to the main reference currencies around the world and it was therefore natural to express one unit of the newer currency in terms of the existing ones. 6 / 18  Relationship between direct and indirect exchange rates Let Sdir and Sind be the direct and indirect exchange rates between currency 1 and 2 relative to currency 1. To avoid price inconsistencies, we must have Sdir Sind = 1. This shows that we can easily switch from direct to indirect exchange rates and viceversa. From now on, we focus on direct exchange rates and drop both the term and the subscript “direct” for convenience. 7 / 18  Relationship between direct and indirect exchange rates To justify the relationship between direct and indirect exchange rates, note that we can make the following transactions: currency 1 currency 2 Sdir =⇒ 1 1 =⇒ Sind Multiplying the second line by Sdir shows that we can also make the following: currency 1 currency 2 Sdir =⇒ 1 Sdir =⇒ Sdir Sind In words, we can exchange Sdir units of currency 1 for either 1 unit or Sdir Sind units of currency 2, forcing Sdir Sind = 1. 8 / 18  Exercise with direct and indirect exchange rates Exercise. The table below displays some direct and indirect exchange rates between EUR and a number of foreign currencies. The base currency is EUR. Foreign Currency Direct Exchange Rate Indirect Exchange Rate USD ? 1.093 GBP ? 0.859 CHF 1.043 ? JPY ? 159.040 INR 0.011 ? BRL 0.184 ? Question. Fill in the spots with the question mark. 9 / 18  Exercise with direct and indirect exchange rates: Solution Solution. It suffices to invert the given exchange rates: Foreign Currency Direct Exchange Rate Indirect Exchange Rate 1 USD 1.093 = 0.915 1.093 1 GBP 0.859 = 1.164 0.859 1 CHF 1.043 1.043 = 0.959 1 JPY 159.040 = 0.006 159.040 1 INR 0.011 0.011 = 90.909 1 BRL 0.184 0.184 = 5.435 10 / 18  Beware of rounding! In our calculations we will often need to round numbers to a certain decimal digit to avoid excessively long expressions. This occurs when dealing with: rational numbers whose sequence of digits is very long as in 1022 = 1.114503817. 917 rational numbers whose sequence of digits is infinite as in 7 = 2.3333333333333333333333... 3 irrational numbers, whose sequence of digits is always infinite as in √ 2 = 1.4142135623730950488016... In any of these cases we round the numbers, i.e., we stop the sequence at some pre-specified decimal digit potentially adjusting the last digit to account for what has been deleted. This procedure may lead to tiny inconsistencies! 11 / 18  Rounding convention In this course we follow this convention when rounding at the kth digit: Delete all digits after the kth one. If the (k + 1)th digit is greater or equal to 5, then add 1 to the kth digit. If the (k + 1)th digit is less or equal to 4, then leave the kth digit as it is. Example. Consider rounding at the third decimal digit the numbers n1 = 0.91491308, n2 = 18.8364888, n3 = 9.78950195. The rounded numbers become n1 = 0.915, n2 = 18.836, n3 = 9.790. 12 / 18  Issues with rounding Example. Consider the first line in the table on slide 10: Foreign Currency Direct Exchange Rate Indirect Exchange Rate 1 USD 1.093 = 0.915 1.093 To get the direct exchange rate between EUR and USD relative to EUR we rounded at the third decimal digit the number 1 = 0.9149130833. 1.093 We know that the product between the direct and indirect rates is equal to 1. However, rounding makes this relationship fail because 0.915 · 1.093 = 1.000095 > 1. However, note that the right relationship continues to hold in rounded terms! 13 / 18  How to express exchange rates: Our convention Example. Consider exchanging EUR and USD. In the sequel we will write “the exchange rate is 0.915 EUR/USD”. By this we mean that 0.915 is the (direct) exchange rate between EUR and USD relative to EUR. It is sometimes convenient to assign a symbol to the exchange rate. If we denote by S the (direct) exchange rate between EUR and USD relative to EUR, then we write S = 0.915 EUR/USD. The expression “EUR/USD” should be read “EUR per USD”. 14 / 18  Calculations with exchange rates Example. Consider exchanging EUR and USD at the exchange rate S = 0.915 EUR/USD. Questions. (1) How many EUR does one need to get 500 USD? (2) How many USD does one need to get 500 EUR? Answers. (1) One needs 0.915 · 500 = 457.50 EUR to get 500 USD. 1 (2) One needs 0.915 · 500 = 546.45 USD to get 500 EUR. 15 / 18  Calculations with exchange rates Example. Consider exchanging EUR and CHF at the exchange rate S = 1.043 EUR/CHF. Questions. (1) How many EUR does one get in exchange for 200 CHF? (2) How many CHF does one get in exchange for 200 EUR? Answers. (1) One gets 1.043 · 200 = 208.60 EUR in exchange for 200 CHF. 1 (2) One gets 1.043 · 200 = 191.75 CHF in exchange for 200 EUR. 16 / 18  Exercises with exchange rates Exercise. Consider exchanging EUR and GBP at the exchange rate S = 0.859 GBP/EUR. Questions. (1) How many EUR does one need to buy 1000 GBP? (2) How many GBP does one need to buy 1000 EUR? (3) How many EUR can one buy with 500 GBP? (4) How many GBP can one buy with 500 EUR? 17 / 18  Exercises with exchange rates: Solution Solution. 1 (1) One needs 0.859 · 1000 = 1164.14 EUR to buy 1000 GBP. (2) One needs 0.859 · 1000 = 859 GBP to buy 1000 EUR. 1 (3) One buys 0.859 · 500 = 582.07 EUR with 500 GBP. (4) One buys 0.859 · 500 = 429.50 GBP with 500 EUR. 18 / 18