Composition of Optimal Currency Portfolios: An Overview of International Currencies

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Composition of Optimal Currency Portfolios

An Overview of International Currencies

University of Amsterdam

Amsterdam Business School

BSc Economics & Business Economics - Thesis

Author:

B. Makara

Student number:

11752424

Thesis supervisor: dr. J.J.G. Lemmen

Finish date:

June 30

th

_{, 2020}

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ABSTRACT

This thesis aimed to discern the theoretical composition of an optimal currency portfolio consisting of the 8 largest currencies available to be held by both private and public agents. The portfolio has been constructed by considering the risk-return profiles of each individual currency as per the mean-variance optimization framework proposed by Markowitz. The results suggest that the global minimum-variance portfolio computed strongly resembles the actual foreign exchange market turnover attained by each currency. This implies that in aggregate, foreign exchange market participants are mean-variance optimizers and that the current average global weights assigned to each currency are theoretically optimal.

Keywords: International Currencies, Portfolio Optimization, Mean-Variance Analysis, Optimal Currency Portfolio, Contemporary Currencies and the Functions of Money

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LIST OF TABLES

Table 1: Domestic and International Functions of Money

[5]

Table 2: Currency Distribution of Global Foreign Exchange Turnover

[20]

Table 3: Exchange Rate Quotations

[22]

Table 4: Descriptive Statistics of the Acquired Dataset

[25]

Table 5: The COFER Database, 2019

[26]

Table 6: Variance-Covariance Matrix

[27]

Table 7: Correlation Matrix

[28]

Table 8: Mean-Variance Portfolio Optimization Results

[29]

Table 9: Global Minimum-Variance, COFER and BIS Foreign Exchange Turnover

[32]

LIST OF FIGURES

Figure 1: Currency Composition of Foreign Exchange Reserves, a historical overview

[7]

Figure 2: The Opportunity Set and Efficient Frontier

[17]

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CHAPTER 1: Introduction

1.1 Topic Introduction and Relevance

On a domestic scale, intra-country economic activity is to a large extent denominated in a local currency. While this differs between countries who constitute exclusive proponents of one currency and those integrated within a larger monetary union comprised of multiple individual economies, the underlying premise of using a single form of money as the basis for most internal transactions is shared. The rationale behind this has attained considerable attention in monetary economics throughout history, where the general consistency of arguments have emanated from the inefficiency of the opposing case: a barter economy. In such an environment, economic activity would be inconsistent and inefficient as the market would be unable to ensure that demand for any arbitrary product will be immediately met by the supply of another, equally demanded product. This phenomenon has been contemporarily coined the “double coincidence of wants” (Mankiw, 2019, p.127), while the idea of a natural development of a single form of money as a result of the division of labour dates back to Adam Smith (1776), an idea later corroborated by Kiyotaki and Wright (1989, p.927). A single and widely accepted currency serves three generic functions on a domestic scale: a medium of exchange, a store of value and a unit of account. Once all three are satisfied, in addition to the currency being accepted as reliable and valuable by the population, it becomes a viable representative of universal money in that environment. By extension, transactions between residents are limited to domestic currency denominations as a result of the often absolute control over the supply of what is to be widely accepted as money by a governmental body (Hartmann, 2003, p.13).

On an international scale, this topic becomes considerably more difficult. If this analysis is extended to more than one country, provided weak or non-existent capital controls on transactions, competition for currencies ensues. While the strong position of the government in the supply of a domestic currency preserves that “most transactions between residents are conducted in the domestic

currency” (Hartmann, 2003, p.13), inter-country trade has to nevertheless be denominated in one

currency, domestic or foreign. Due to increased globalisation and liberalization of economies and financial markets, this topic necessitates considerable attention. Over the years, several currencies have dominated international trade, attaining the status of an “international currency”, which is “one that is

used instead of the national currencies of the parties directly involved in an international transaction”

(Kenen, 2011, p.9). In the 19th_{century, Pound Sterling was the dominating international currency, used}

in the relative majority of inter-country transactions (Eichengreen, 2014, p.1). After the traditional gold standard was dropped early in the 20th_{century, and exchange rates were left to float in most of the largest}

economies in the world at the time, the USD attained considerably higher amounts of attention for transactions than before, albeit almost effortlessly overthrowing the pound and has since continued

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attaining global dominance (Chitu, Eichengreen & Mehl, 2012, p. 3; see also Faudot & Ponsot, 2016, p. 42; Tavlas, 1998, p. 46). However, due to contemporary global developments such as the establishment of the Eurozone, the proliferated influence of the Chinese economy on the global market coupled by the recently poor performance of US Federal Reserve (McKinnon & Schnabl, 2014, p. 33), the USD may become seriously contested as the global currency in the future.

For any currency to overthrow the USD in relative global importance, that currency must be superior in its ability to deliver the three functions of money. However, as a result of the added dimension of inter-country trade, it is necessary to extend the definitions to account for both domestic and international uses of a currency.

1.2 Literature Overview and Research Question Formulation

Vast amounts of past research attempted to discern the idea of the decaying importance of the USD in global markets, and the consequently increased importance of other currencies. Most papers concentrate on the dynamic between the USD relative to one of the two largest contenders for the coveted global currency spot: Euro and the Renminbi. Chinn and Frankel (2008, p. 19) argue that the Euro could potentially dethrone the USD over the next few years, and Lim concludes that there is “no a priori

reason to assume the euro’s role as an international store of value would not advance further” (2006,

p. 33), but that depends largely on the development of the EMU and the effects of a persistent US current account deficit on the value of the USD. Gao and Yu argue that the adoption of the Renminbi as a greater international currency would be “a balancing factor in global financial stability” (2011, p. 122), helping adjust for the prevalently imbalanced USD dominance in international trade. On the other hand, McKinnon and Schnabl argue that China is unable to “liberalize its own financial markets sufficiently

for the Renminbi to become a contender as the world’s key currency” (2014, p. 33), due to low interest

rates in the US. Others, such as Fratzscher and Mehl, conclude that the contemporary international trade patterns suggest a continued dominance of the USD, while a continuously increasing importance for the Euro and Renminbi particularly in intra-regional trade (2013, p. 1368). As such, given frequent global developments, outlooks and research methodologies, conclusions on the future of international currencies contain stark differences in current academic literature. Therefore, this thesis aims to contribute to the ongoing debate by providing an overview of the current environment of international currencies, by considering the risk-return profile provided by several large currencies. In particular, this thesis will account for the US Dollar (USD), Euro (EUR), Japanese Yen (JPY), British Pound (GBP), Chinese Renminbi (CNY), Canadian Dollar (CAD), Australian Dollar (AUD) and the Swiss Franc (CHF). Using various quantitative methods, this thesis aims to answer the following research question:

How does the risk return trade off among different currencies affect the way investors balance their portfolio? Relying primarily on the method of mean-variance portfolio optimization proposed by

Markowitz (1952), this paper will outline the relative weights that should be theoretically attained by large currencies used in international trade, and compare those weights with the actual proportion of

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each respective currency as stipulated in the Currency Composition of Official Foreign Exchange Reserves (COFER) as well as the actual proportion attained by each with respect to foreign exchange market turnover (BIS, 2019).

In summary, this thesis has concluded that the proposed optimal currency portfolio is considerably different from the COFER, while relatively similar to the actual foreign exchange market turnover. This implies that, on average, foreign exchange market participants are mean-variance optimizers who strive for the portfolio with the lowest possible risk coupled with the highest returns. This conclusion will be presented and justified throughout the thesis.

The rest of this thesis will be structured as follows: First, a thorough literature review will be presented, building on the ideas presented in the introduction and further outlining past research. Secondly, a quantitative analysis of the methodology utilized in this thesis will be disclosed. Thirdly, an overview of the dataset generated will be showcased. Fourthly, the attained results will be outlined alongside an analysis and discussion regarding their implications. Lastly, a conclusion will be presented summarizing the justification and relevance of this thesis and its main conclusions, closing with remarks and recommendations for future research.

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CHAPTER 2: Literature Review

2.1 The Theory of Money

Money is a versatile and convenient concept, encapsulating and formalizing the idea of “value” in an abundance of circumstances. Economic theory has delineated this notion as a tool achieving three core functions: a medium of exchange, a store of value and a unit of account (Melvin & Norrbin, 2013 p.37; Makin, 2017, p. 128; Hartmann, 2003, p.1). In tandem, all three functions strengthen and support one another for this concept to be able to enact its intended purpose. However, the distinction between these functions necessitates further attention.

The medium of exchange function posits that money represents a widely adopted instrument for which goods and services can be exchanged (Mankiw, 2019, p.127). In essence, it provides the universal and reliable means by which any agent may attain a particular product. It should be noted that any embodiment of money has to be widely established and supported as such, especially in the case of its contemporary portrayal in fiat money with no intrinsic value (König, 2001, p.8), or else it cannot reliably serve this function.

The store of value function, according to Mankiw, implies that money must also be “a way to

transfer purchasing power from the present to the future” (2019, p.127). This entails ensuring that the

purchasing power a particular agent would attain right now should be roughly the same in the future. While the current definition of money cannot uniformly conform to this specification, as inflationary pressures or other exogenous factors constantly effect real purchasing power attained in the future, which is of greater practical importance than nominal purchasing power (Friedman, 1971, p.194), it should relatively reliably warrant the ability not to burden future needs with present volumes of capital.

The unit of account function stipulates that money allows goods and services to be expressed in a common and acknowledged definition of “value” (Mankiw, 2019, p.127). While this currently differs for political and geographical reasons, as the world economy accommodates multiple currencies and by extension multiple domestic definitions of money, each ensures that agents can accurately discern the inherent worth of objects.

While most currencies correspond to these purposes on a domestic scale, the idea behind “international” money is different. According to Hartmann, a truly international currency is one which “fulfils one or several of the classical money functions for non-nationals or non-residents of the issuing

country” (2003, p.1). Thus, for a particular currency to be widely adopted as a means for consistent

international exchange, its functionality has to transcend these domestic definitions.

2.2 The International Theory of Money

A specification of what exactly an exemplar of money on an international scale must provide global markets has been devised by Cohen (1971), and later extended by Kenen (1983) and is illustrated in

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Table 1. This table outlines that besides fulfilling the three functions of money on a domestic scale, an international currency must provide additional services to various agents on an international scale. Literature limits this to two groups of agents: private individuals or organizations, and official state-owned institutions, both of which require further elaboration.

Table 1: Domestic and International Functions of Money Functions of Money

Domestic Function International Private Use International Official Use Medium of Exchange Vehicle Currency Intervention Currency

Unit of Account Quotation Currency Pegging Currency

Store of Value Investment and Financing Currency

Reserve Currency

The medium of exchange function in the context of international currencies extends to incorporate the “vehicle currency” feature for private use, and the “intervention currency” for official use. According to Goldberg and Tille, a vehicle currency is one which is used in particular transactions between countries while not being the official domestic representative of money in either one (2005, p.3). That is, given the choice of using the currency of either country in the transaction, the agents opt for a third, seemingly unrelated vehicle option. According to Melvin and Norrbin, this stems from a consideration of transaction costs (2013, p.36). In essence, when the costs associated with performing the trade using the official domestic currencies of the countries in question exceed the costs of doing so with a third, unrelated currency, the agents will choose the latter option. Hartmann argues that currencies exhibiting such characteristics require “high trading volumes and nevertheless low exchange rate

volatilities” (2003, p.55), and by extension provide relatively small transaction costs. Similarly, this

notion extends to the official use of a third-party currency by public organizations and central banks to affect exchange rate fluctuations, thereby serving as a currency for foreign exchange interventions (Melvin & Norrbin, 2013, p.36).

The unit of account function in the context of international currencies includes the “quotation currency” characteristic for private use, and the “pegging currency” for official use. While not thoroughly researched compared to the other functions (Hartmann, 2003, p.27), a quotation currency can be defined as one which serves as a globally universal measure of value for particular products (Melvin & Norrbin, 2013, p.35). That is, a currency which is used to denominate the value of products in any geographical jurisdiction or is “used to invoice merchandise trade [and] to denominate financial

transactions” (Tavlas, 1998, p.46). Melvin and Norrbin argue that this arises due to information costs

(2013, p.35), where a common currency denomination ensures that information regarding the value of particular homogeneous goods is transmitted faster in global markets. Due to this, the dominating

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international currency exhibiting this function is often used as an official pegging currency by individual economies.

The store of value function in the context of international currencies expands to encompass the “investment and financing currency” attribute for private use, and the “reserve currency” for official use. Melvin and Norrbin argue that this is generated by value stability (2013, p.36), in that agents can comfortably store purchasing power in a country if its currency has exhibited stable movements over time. That is, private individuals may be inclined to store value into a country with a stable currency, establishing it as a “dominant reserve currency in banking” (Melvin & Norrbin, 2013, p.36), and as a result central banks will be obliged to shift holdings of foreign exchange reserve in favour of it. Consequently, involatile currencies minimize the risk of losing purchasing power, and constitute prime candidates for becoming an international currency. However, since foreign exchange volatilities are a function of both global developments and the domestic economic environment, currencies abiding by this notion may inevitably experience adverse developments.

It follows that currencies which reliably exhibit these functions naturally take on the role of the global currencies dominating most inter-country transactions. Mundell has outlined that such currencies and their underlying economies must contain a large transaction area, a stable monetary policy, an absence of controls, a strong central state, have been initially backed by gold or silver, exhibit a sense of permanence, and contain low interest rates (1999, pp.441-442). That is, only strong, large, stable, and globally relevant economies may contest each other for the coveted spot of the global currency (see also Dwyer & Lothian, 2002, p.1). However, frequent, and significant economic developments slowly deduct from the capability of a reigning global currency to sustain its function well enough relative to other, increasingly admissible currencies.

2.3 The Global Currency: An Overview

Numerous representatives of money have served the role of a global currency throughout history (Dwyer & Lothian, 2002; Eichengreen, 2014). While the reasons for the dissolution of each dominating currency differ, history proves that the spot is not absolute. From the middle ages to contemporary society, any global representative of money has eventually been replaced by a more suitable successor. Current academic and practical consensus postulates that the USD is the reigning international currency used in most inter-country transactions. The USD overthrew the previously international Pound Sterling after the first world war (Eichengreen, 2010, p.723), and formally affirmed its spot as the global currency after the establishment of the Bretton Woods system in 1944. It managed to keep this status ever since (Krugman, 1984, p.261; Bracke & Bunda, 2011, p.6). This effectively implies that the USD conforms most optimally to the international adaptation of the three functions of money relative to other currencies available. However, it has experienced slight volatilities in recent years, according to the COFER dataset compiled by the IMF, as illustrated in Figure 1.

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Figure 1: Currency Composition of Foreign Exchange Reserves, a historical overview1

According to this metric, the trend after the introduction of the Euro in 1999 progressed unfavourably towards the position of the USD in international trade. In particular, the Euro has seized a considerable portion of the worlds reserve holdings, and the Chinese Renbimbi has recently also began attaining traction. These developments have sparked numerous debates among academics whether the time has come for the incumbent USD to be seriously challenged by these increasingly prominent alternatives.

Since its inception, the Euro has steadily maintained and occasionally even strengthened its role as the second most dominant international currency. According to the ECB, the Euro has shown slight signs of its rising global influence as an international currency over the past couple of years (ECB, 2019, p.3). In particular, the share of Euros in foreign exchange reserves increased at the expense of a decline in holdings of the USD, as several emerging markets used USD holdings for currency stabilization purposes following financial volatilities (ECB, 2019, p.8). However, the Euro has lost attractiveness as an investment currency as the euro area capital markets have experienced a decline in fund inflows (ECB, 2019, p.11), and its standing as a pegging currency has remained realtively stable (ECB, 2019, p.17). In a similar light, Bénassy-Quéré argues that while the Euro has attained dominance on a regional scale, it is still far from being considered a truly international currency. According to the author, the Euro has largely developed “as a store-of-value, for both the official and the private sectors” (2015, p.3), while it fails to serve as both a dominating vehicle and an invoicing currency, where most pegs towards the Euro constitute “neighbouring countries and former African colonies” (2015, p.4). The view

1_{The COFER dataset changes continuously based on reports from included countries. This figure presents the}

developments of its composition for the past 20 years.

0.00 20.00 40.00 60.00 80.00 100.00 120.00 Sh a re , % Year

COFER, historical overview

Shares of U.S. dollars Shares of euro Shares of Chinese renminbi Shares of Japanese yen Shares of pounds sterling Shares of Australian dollars Shares of Canadian dollars Shares of Swiss francs Shares of other currencies

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that the Euro constitutes a largely international store of value and a considerably weaker unit of account or medium of exchange is corroborated by multiple authors. Viceira and Gimeno illustrate that the Euro serves as an effective interest-risk hedging tool for any investor (2010, p.28). Lim stipulates that the role of the Euro as an international store of value has no reason to stagger, while the pace at which it develops may be impacted by “structural reforms in the EMU and impact of U.S. current account deficits on the

dollar’s value” (2006, p.33). Additionally, unless the value of the dollar depreciates, the future of the

Euro as a vehicle currency is bleak (Lim, 2006, p.34). Thus, in aggregate the Euro conforms relatively less to the requirements of a truly international currency compared to the USD. While it does represent a strong candidate for an international store of value, it currently lacks in other features to be considered an international medium of exchange and a unit of account. This, however, does not dismiss the chance that global developments and particularly US developments may play in favour of the Euro.

The Chinese economy has experienced steady levels of high GDP growth over the past two decades. According to the World Bank (2018), the annual growth rate in GDP in 2018 was 6.567%, which is more than twice that of the US at 2.927% and the EU at 2.127%. Despite current declines in global GDP growth as a result of the Covid-19 pandemic, due to the perpetually growing presence of the Chinese economy on the international scale, academics began exploring the possibility of the Renminbi to overtake the USD as the new global currency. McKinnon and Schnabl outline that east Asian economies, including China, rely on the USD as the primary intervention currency as well as the denominator for the majority of intra-region transactions (2014, p.1). This illustrates that contrary to the Euro, which constitutes the dominating invoicing currency between EU countries (Langedijk, Karagiannis & Papanagiotou, 2016, p.11), the Renminbi has not yet penetrated its regional market in this regard. Similarly, due to a relatively undeveloped financial market, the Renminbi cannot yet compete with the USD as an international vehicle currency (Gao & Yu, 2011, p.119). Furthermore, Bowles and Wang stipulate that the Renminbi currently plays a relatively small role in global reserve holdings, as a result of it being “a non-fully convertible currency” (2013, p.1365), nor is it used as a pegging currency by any country (2013, p.1376). That is, scholars are consistent with the still relatively insignificant position of the Renminbi in the COFER database, pertaining to its currently weak ability to conform to the three functions of money on an international scale. Despite this, given the rapid growth in importance of the Chinese economy and plans to promote its financial markets, Eichengreen argues that the Chinese officials aspire to internationalize the Renminbi. However, due to the current structure of its economy and financial markets, such a feat would require “creating a deep and liquid financial

market open to foreign investors” (2010, p.727), increase the domestic debt market and an overhaul of

the Chinese development model to generate global demand for the Renminbi (2010, p.728). Given the right strategy and execution, the Chinese economy is becoming too large and influential for the Renminbi not to pose a significant threat to the USD. Current literature supports the USD as the optimal international currency available. However, proper developments in its largest contenders, the EU and

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China, coupled with adverse developments in the US economy, may shift the role of the USD in the future.

As a result of the already established academic debate and theory regarding international currencies, this thesis attempts to answer three hypotheses. The first hypothesis to be explained is that

the theoretically optimal weights to be allocated to each currency will differ from the composition as stipulated by the COFER, as a direct result of the fact that a truly international currency should conform

to more than the function of an official reserve currency, which is captured within the database. Thus, it is expected that the computed portfolio composition will not be identical to that of the COFER. The second hypothesis to be explained is that the proposed portfolio with the lowest risk and highest return

will most resemble the COFER composition, as past research implies that the value of an international

currency has to be relatively invariant. The third hypothesis to be explained is that not all considered

currencies will receive significant weights in the optimal portfolio, making them relatively unnecessary in the context of international trade, the logic behind which stems from the inability of some of the

aforementioned currencies to efficiently conform to the theory behind a truly international currency. For instance, Switzerland does not have the size or global economic impact for the CHF to be theoretically considered globally relevant.

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CHAPTER 3: Methodology

3.1 Markowitz Mean-Variance Portfolio Optimization

To compute the theoretically optimal composition of currencies to be held in portfolios by either private individuals or public organizations, a frequently applied method is the mean-variance portfolio optimization procedure. First introduced by Harry Markowitz in 1952, its general premise is to construct the weights of held securities such that the overall portfolio risk-return relationship is optimized in a way that suits the investor in question. This hinges on the “expected returns-variance of returns” maxim (Markowitz, 1952, p.77), which posits that investors wish to maximize discounted expected returns and simultaneously minimize risk. In other words, the method assumes investors seek higher returns and are risk averse. The underlying idea of the model is that by holding more than one security, provided non-perfect correlations, the investor benefits from an increase in the risk-return characteristic of the newly constructed portfolio relative to the base case of holding a single asset, even if all the securities under consideration have the same standalone risk-return profiles (Markowitz, 1952, p.78). The reason for this stems from diversification benefits associated with combining securities with non-perfect correlations, an idea which will be later disclosed in greater detail. In cases of unequal standalone risk-return profiles of each considered asset 𝑖, the proportions to be held in any such asset in the combined portfolio also affect its risk-return profile. Certain combinations of proportions may provide the investor with optimal risk-return characteristics out of all possible proportion combinations, conditional on the characteristics of the securities under analysis. The mean-variance procedure proposed by Markowitz aims to compute this optimal weight combination while considering diversification benefits and the standalone profiles of each available asset.

The weights 𝑤𝑖 to be allocated to each individual currency 𝑖 can be simplified into a single formula if the number of currencies under analysis, 𝑛, equals 2 (see Bodie et al., 2018, or any other textbook about portfolio theory). However, in case 𝑛 > 2, formally matrix algebra and practically data manipulation software are utilized to compute the optimal weights for each currency in the overall portfolio as a single formula is impossible to be derived. The logic behind the former will be illustrated in this section, while the actual results of this thesis will be computed using the latter.

3.1.1 Input Variable Definitions

In order to formalize the methodology behind this procedure, certain variables have to be defined in various forms:

1. (Expected) Return of an Individual Security

Denoted 𝐸(𝑟𝑖), it constitutes the first central moment of a distribution and it outlines the expected return to be attained by any discrete random variable 𝑖. Provided an agent “starts with observation and

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experience and ends with beliefs about the future performances of available securities” (Markowitz,

1952, p.77) which are held with a high degree of certainty, the expected return can be computed as a weighted average of those beliefs, weighted by the respective probability assigned to each. While such security analysis is out of the scope of this thesis, in practice the future yield distribution of a security is often unpredictable and cannot be reliably estimated. In fact, evidence suggests that even large institutional investors who strive off of such predictions often fail at properly anticipating future security price movements (see Carhart, 1997). For this reason, the expected future return will be computed as the arithmetic mean of past realized returns:

𝐸(𝑟𝑖) = 𝜇𝑖 = 1 𝑁∑ 𝑌𝑖

𝑁

𝑖=1

That is, the expected future returns correspond to the average of past realized returns 𝑌𝑖 for a sample period of 𝑁 years. The choice of the arithmetic mean over other return measures stems from its capacity to serve as the optimal predictor for future returns (Bodie et al., 2018, p.129), which is a desirable feature for the purpose of mean-variance analysis in the absence of security analysis.

2. (Sample) Standard Deviation of an Individual Security

Denoted 𝜎𝑖, it constitutes the second central moment of a distribution and it corresponds to the most frequently adopted representative of risk in financial analyses. Similarly to the rationale behind the expected return, because the future yield distribution of any random variable 𝑖 is often unpredictable and cannot be reliably estimated, the future expected standard deviation is approximated with past realized returns. In effect, it measures the dispersion of squared deviations from the mean. To compute it, it is first necessary to find the variance, and take its squared root to arrive at a more readable metric:

𝜎_𝑖2= 1 𝑁 − 1∑(𝑌𝑖− 𝜇) 2 𝑁 𝑖=1 𝜎𝑖 = √𝜎𝑖2

Note that since the data considered correspond to a sample drawn from the actual population distribution, a correction of 𝑁 − 1 is appropriate to compute the sample variance instead of the population variance. However, since the sample itself is relatively extensive, the correction is a mere technicality. It should be noted that the standard deviation is an appropriate measure of risk provided an approximately normally distributed dataset. Absent this feature, it cannot capture the added adverse feature of a higher probability of negative returns, occurring in cases of negative skewness and high kurtosis (Bodie et al., 2018, p.138). While a higher probability of positive return realizations is advantageous, a negative skewness and high kurtosis can be detrimental for investors if overlooked. In such cases, other risk measures accompanied by the standard deviation may better represent the inherent riskiness of the returns of any random variable. To name some examples, the Value at Risk (VAR),

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expected shortfall (ES), Lower Partial Standard Deviation (LPSD) or the Generalized Autoregressive Conditional Heteroskedasticity (GARCH) measures should be employed, depending on the severity and specificity of the distribution.

3. (Sample) Covariance Between Two Securities

The Markowitz procedure culminates in one optimal final portfolio 𝑃 of all available assets given assumed constraints and optimization goals. As will be outlined later on, it will also be necessary to compute the expected return 𝐸(𝑟𝑃) and standard deviation 𝜎𝑝 of this final portfolio. Consequently, 𝜎𝑝 requires as an input the various combinations of covariances between all individual assets in that portfolio. The reason for this will be illustrated in point 6 below, which delves into the computation of 𝜎𝑝 and illustrates why covariances are central to the ideas proposed by Markowitz. As such, it is necessary to define the computation of covariance between two discrete random variables 𝑋 and 𝑌 (Markowitz, 1952, p.80): 𝐶𝑜𝑣(𝑋, 𝑌) = 𝜎𝑋𝑌 = 1 𝑁 − 1∑(𝑋𝑖− 𝐸(𝑟𝑋))(𝑌𝑖− 𝐸(𝑟𝑌)) 𝑁 𝑖=1

Similarly to the standard deviation of one random asset, a correction of 𝑁 − 1 applies here for the same reason. It follows that a positive covariance term implies that the two random variables tended to move together, whereas a negative covariance term implies the opposite.

The Markowitz procedure requires the computation of covariances between all 𝑁 assets under analysis, which can be illustrated in a so-called variance-covariance matrix 𝛴 (Benninga, 2014, p.207):

𝛴 = [ 𝜎11 𝜎12 𝜎13 ⋯ 𝜎1𝑁 𝜎21 𝜎22 𝜎23 ⋯ 𝜎2𝑁 𝜎31 𝜎32 𝜎33 ⋯ 𝜎3𝑁 ⋮ ⋮ ⋮ ⋯ ⋮ 𝜎𝑁1 𝜎𝑁2 𝜎𝑁3 ⋯ 𝜎𝑁𝑁]

Because the covariance of a random variable with itself is equivalent to its variance, the above matrix has been coined the variance-covariance matrix as it contains both covariance terms between different assets as well as variance terms of individual assets along the main diagonal.

4. Correlation Between Two Securities

Despite the covariance representing an excellent measure for the co-movement of two random variables, it is nevertheless practically uninterpretable. According to Stock and Watson, “because the covariance

is the product of X and Y, deviated from their means, its units are, awkwardly, the units of X multiplied by the units of Y” (2020, p.71), such that the covariance is problematic to understand or compare as the

units in which it is presented in have no real understandable basis. The solution for this is to transform the covariance statistic into one which conveys the exact same information in a form that accommodates

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fast understanding and interpretability of the units of denomination. The descriptive statistic which conforms to these specifications is the correlation, denoted 𝐶𝑜𝑟𝑟(𝑋, 𝑌) = 𝜌𝑋𝑌, which is computed as follows:

𝜌𝑋𝑌 = 𝜎𝑋𝑌 𝜎𝑋𝜎𝑌

It corrects for the “units” problem by dividing the covariance term by the product of the individual standard deviations of the two random variables, the units of which are equal to that of the covariance term. As such, “the units cancel, and the correlation is unit free” (Stock & Watson, 2020, p.71). As a result of this transformation, the correlation provides the exact same information as the covariance but limits the set of values it can attain to (−1; 1), making it relatively simple to understand and compare.

While not pivotal for the mean-variance framework itself, a correlation matrix will be constructed to supplement the theoretical explanations behind the implications of the methodology and its application to the chosen dataset in this thesis.

5. Expected Return of the Portfolio

Given any arbitrary assignment of weights 𝑤𝑖 to any asset 𝑖 out of the total of 𝑁 assets under consideration, it is of utmost importance to compute the expected return of the entire portfolio, denoted 𝐸(𝑟𝑃), for comparison and reference. It is computed as follows:

𝐸(𝑟𝑃) = ∑ 𝑤𝑖𝐸(𝑟𝑖) 𝑁

𝑖=1

As such, it constitutes a simple weighted average return of the expected returns of its underlying individual assets, 𝐸(𝑟𝑖), weighted by the proportion of total funds invested in each, 𝑤𝑖 (Markowitz, 1952, p.80).

However, since any Markowitz model with 𝑁 > 2 cannot be condensed into a simple formula, the expected return of the portfolio has to be transformed into matrix notation for further use. For this purpose, assume a 1𝑥𝑁 column matrix 𝑅 representing the expected returns of all random variables under question, and a 1𝑥𝑁 column matrix 𝑊 representing the allocated weights for each asset 𝑖 in the final portfolio (Benninga, 2014, p.206): 𝑅 = [ 𝐸(𝑟1) 𝐸(𝑟2) 𝐸(𝑟3) ⋮ 𝐸(𝑟𝑁)] 𝑊 = [ 𝑤1 𝑤2 𝑤3 ⋮ 𝑤𝑁]

Each 𝐸(𝑟𝑖) in 𝑅 represents a random variable, such that 𝑅 itself is constructed as a random variable by extension. On the other hand, each 𝑤1 is not random but rather chosen by the agent in question for the

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purpose of portfolio construction. As such, 𝑊 is a matrix representing a collection of constructed variables.

If the transpose of 𝑊 is denoted as 𝑊𝑇_{, the expected return of the final portfolio can be} illustrated in matrix notation as follows (Benninga, 2014, p.206):

𝐸(𝑟𝑃) = 𝑊𝑇𝑅 = [𝑤1 𝑤2 𝑤3 ⋯ 𝑤𝑁] [ 𝐸(𝑟1) 𝐸(𝑟2) 𝐸(𝑟3) ⋮ 𝐸(𝑟𝑁)] = 𝑤1𝐸(𝑟1) + 𝑤2𝐸(𝑟2) + 𝑤3𝐸(𝑟3) + ⋯ + 𝑤𝑁𝐸(𝑟𝑁) = ∑ 𝑤𝑖𝐸(𝑟𝑖) 𝑁 𝑖=1

Which simplifies to the base case representation illustrated earlier. Such notation will be crucial in subsequent computations.

6. (Expected) Standard Deviation of the Portfolio

It then follows

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that the more said deviations oppose each other, the lower will be the effective variability of returns in the portfolio

𝑃.

That is, the logic behind such a setting is that of attainable diversification from holding multiple assets with correlation coefficients smaller than 1. This is mathematically illustrated in the equation for

𝜎

_𝑃2 above, where the covariance term, 𝜎𝑖𝑗, effectively decreases the contribution to total portfolio variance of any two random variables provided the correlation between them is smaller than 1. However, this idea not only decreases the total portfolio variance if applied correctly, it can do so without hampering expected portfolio returns. Assume a situation with 𝑁 identical assets in both 𝐸(𝑟𝑖) and

𝜎

_𝑖2,but with 𝜎𝑖𝑗 < 1 for all 𝑖, 𝑗. It can be shown that by proper combinations of weights attached to each asset 𝑖, total portfolio variance will be smaller than the weighted average of the variances of the individual assets, while simultaneously the expected return will remain the weighted average of the expected returns of the individual assets (Markowitz, 1952, p.89). That is, any combination of weights assigned to each asset 𝑖 will be strictly preferred by any investor over holding a single one of the assets considered, purely as a result of added diversification benefits. While it may occur that a single asset provides the optimal risk-return profile when compared to various combinations of portfolios of multiple assets (Markowitz, 1952, p.89), the logic remains for most practical situations.

Contrary to the method of solving for the expected portfolio return, the computation of the variance of the final portfolio becomes increasingly more difficult as 𝑁 increases. Due to time efficiency and error minimization, the portfolio standard deviation can, similarly to portfolio expected return, be transformed into matrix algebra. Given the previously defined matrices, the variance of the final portfolio is computed as follows (Benninga, 2014, p.207):

𝐸(

𝜎

𝑝2)

=

𝑊𝑇𝛴𝑊

= [

𝑤1 𝑤2 𝑤3 ⋯ 𝑤𝑁

]

[

𝜎11 𝜎12 𝜎13 ⋯ 𝜎1𝑁 𝜎21 𝜎22 𝜎23 ⋯ 𝜎2𝑁 𝜎31 𝜎32 𝜎33 ⋯ 𝜎3𝑁 ⋮ ⋮ ⋮ ⋯ ⋮ 𝜎_𝑁1 𝜎_𝑁2 𝜎_𝑁3 ⋯ 𝜎_𝑁𝑁

]

[

𝑤1 𝑤2 𝑤₃ ⋮ 𝑤_𝑁

]

= ∑ ∑

𝑤𝑖𝑤𝑗𝜎𝑖𝑗 𝑗∈{𝑁} 𝑖∈{𝑁}

Which, when solved, simplifies to the base case presented earlier. From this, the standard deviation can be extracted by taking the square root of the variance.

3.1.2 Finding the Optimal Weights

While the Markowitz procedure can have multiple constraints in reality, most of them are optional. The only constraint that is required in any optimization of this nature is that the sum of the portfolio weights assigned to each 𝑖𝑡ℎ_{security of the 𝑁 securities under analysis has to equal 1. In mathematical notation:}

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16 ∑ 𝑤𝑖

𝑁

𝑖=1

= 1 Or equivalently in matrix notation:

𝑊𝑇1 = [𝑤1 𝑤2 𝑤3 ⋯ 𝑤𝑁] [ 1 1 1 ⋮ 1]

This makes sense, as the weights constitute the proportion of our total fixed funds to be invested into each asset 𝑖. The sum of all assigned weights has to be such that all considered funds to be invested are accounted for. Given this as the sole mandatory constraint for the problem, the Markowitz procedure transforms into a constrained optimization problem. This thesis primarily aims to compute the minimum-variance portfolio composition of the aforementioned 𝑁 = 8 currencies, which implies that the optimal currency portfolio to be held by both private and public agents should be the one with the lowest possible variance with the highest possible expected return. Such a specification best resembles the characteristics of international currencies presented in Chapter 2. As such, the principal constrained optimization problem for this thesis corresponds to minimizing the expected portfolio variance, 𝜎𝑃2, subject to the constraint ∑𝑁𝑖=1𝑤𝑖 = 1 (Benninga, 2014, p.210):

min ∑ ∑

𝑤

𝑖

𝑤

𝑗𝜎𝑖𝑗 𝑗∈{𝑁} 𝑖∈{𝑁} 𝑠𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜 ∑ 𝑤𝑖 𝑁 𝑖=1 = 1

Such a constrained optimization problem would require the use of the Lagrange multiplier coupled with partial differential calculus to obtain 𝑁 first-order conditions together with 𝑁 unknown weight variables. Consequently, matrix algebra would have to be utilized to solve a system of 𝑁 linear equations with 𝑁 unknowns. For the interested reader, a detailed derivation of the conclusion to this problem can be found in multiple sources (see Abonongo et al., 2017; Back, 2017, Ch.5).

It is crucial to point out that the Markowitz procedure, as presented above, is not limited in what should be optimized nor in the specificities of the constraints conceived in the optimization. While the default global minimum variance portfolio (GMVP) is computed as stipulated above, it may be that the investor in question is concerned rather with maximizing returns or optimizing the risk-return trade-off. Similarly, while the constraint ∑𝑁𝑖=1𝑤𝑖= 1 must always hold, investors can add multiple auxiliary constraints such as the inability for any 𝑤𝑖 < 0, thereby eliminating the possibility of short selling. This makes the method flexible and potentially applicable in a wide range of investment situations.

3.1.3 Graphical Logic Behind the Markowitz Procedure

To illustrate the logic behind the minimum-variance portfolio found by completing the optimization problem presented above, it is beneficial to graph the risk-return characteristics of any portfolio

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construction from the given set of investable assets 𝑁. For the purpose of illustration, specific data is irrelevant, as the same idea applies regardless of the magnitude of 𝑁 or the characteristics of the assets under analysis. That is, assume a random collection of 𝑁 investable assets, each with random estimates of return, standard deviation, and covariance between each other. For any such selection, it is possible to construct a graph which plots portfolios with a desired level of expected return and its corresponding point of minimal variance. Such a graph is called the minimum-variance frontier of risky assets, as is illustrated in Figure 2.

Figure 2: The Opportunity Set and Efficient Frontier2

Source: Bodie, Kane & Marcus, 2018, p.209

It follows that investors strictly prefer to attain any arbitrary level of expected return by holding the minimum variance portfolio corresponding to that level of expected return over any other portfolio attaining the same expected return, as the minimum-variance portfolio will have by definition the smallest standard deviation of all possible portfolios. Additionally, any investor will strictly prefer only those minimum-variance portfolios which are located at or above the GMVP, as those provide a higher level of expected return for the same level of risk than portfolios below the GMVP. As such, the only truly investable and mean-variance efficient portfolios available are illustrated by the efficient frontier as indicated in Figure 2. From this set of efficient portfolios, the investor may choose one which corresponds to any possible level of expected return, at the expense of higher risk. This implies that the GMVP, attained through the constrained minimization problem presented above, may not be the optimal portfolio for every investor (Markowitz, 1952, p.79). However, it still follows that the GMVP is one which provides the investor with the highest level of expected return for the lowest level of risk.

2_{The opportunity set outlines the various possible minimum-variance portfolios attainable for a given desired}

level of expected return. The efficient frontier is the portion of the opportunity set strictly preferred by any agent over any other portfolio combination at that given level of expected return

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3.1.4 Data Manipulation Software as the Tool Used for Attained Results

Actually solving the constrained optimization problem for various combinations of weights and differing constraints requires an immense amount of computations, the disclosure of which is impractical in this thesis. Thus, the actual results shall be attained and communicated through the use of data manipulation software. In particular, Microsoft Excel will serve as the primary means for computing the theoretically optimal weights assigned to each currency under various assumptions. This software contains built-in functions for computing all required inputs, as well as a customizable tool called the “Solver” for the optimization procedure itself. In addition, the statistical software Stata will also be utilized for auxiliary computations as well as graphical depictions of various key characteristics of the return series considered.

3.2 The Sharpe Ratio

Financial literature and academia recognized that expected returns of an investment opportunity carry meagre decision-making significance if considered in solitary. Comparing two securities with varying levels of expected returns, the superior option according to this metric is not always strictly preferred. This fact emanates from the empirical basis in financial markets that higher returns are often accompanied by higher levels of risk. This idea has been properly formalized in 1966 by William F. Sharpe, in a paper investigating methods of measuring mutual fund performance. Assuming that the performance of any portfolio 𝑃 can be properly captured by its first two moments, 𝐸(𝑟𝑃) and 𝜎𝑃, coupled with the expectation that any investor is able to borrow and lend at a risk-free rate 𝑟𝑓 (1966, p.121), Sharpe argues that the aforementioned risk-return doctrine can be conveniently captured in the following ratio: 𝐸(𝑟𝑃)−𝑟𝑓

𝜎𝑃 (1966, p.122). Consequently, the optimal portfolio to be chosen is “the one for which 𝐸(𝑟𝑃)−𝑟𝑓

𝜎𝑃 is the greatest” (Sharpe, 1966, p.122), as that corresponds to attaining the highest level of expected returns in excess of the risk-free rate per unit of risk as measured by the standard deviation. Note that the risk-free rate is subtracted from the expected return, as that constitutes the riskless base investment opportunity attainable by any investor and should not be accountable for any unit of risk inherent in any random portfolio. However, the currency market provides no immediate candidate for a risk-free investment with which any currency portfolio performance ought to be compared (Melvin & Shand, 2010, p.1). Consequently, the equity market risk-free rate based on the rates offered by the US Treasury will be used as a proxy. On the one hand, as private agents usually do not buy and hold currencies due to continuously rising levels of inflation, which would diminish real purchasing power in the future, a shorter term rate may be warranted. On the other hand, public institutions need to keep some currencies on hand even for prolonged periods, such that a longer term rate may also be suitable. In practice, the majority of all daily foreign exchange market trades are executed by non-public agents (BIS, 2019), thereby a shorter term benchmark risk-free rate is more sensible. For this reason, the

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average of the 1-year maturity Treasury Bill rates over the considered sample period, computed to be 0.66% as showcased in Appendix A, shall be utilized.

Because this measure supplies a simple yet accurate way to quantify the risk-return relationship for virtually any security with computable first and second moments, it has been widely adopted as such in financial literature. Thus, the results of a mean-variance portfolio optimization problem, represented in the newly constructed portfolio expected return and standard deviation, can be conveniently compared with other portfolios or investment opportunities to objectively demarcate superiority. For this reason, the “Sharpe ratio” will be utilized in this thesis as the primary metric for quantifying the viability of attained portfolio combinations.

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CHAPTER 4: Data

4.1 Data Under Analysis

The data used in this thesis consists of two primary types. First, a time series of daily returns for all the currencies considered. This is imperative for performing the Markowitz procedure outlined in Chapter 3. With such a dataset, it is possible to compute all necessary input variables for the procedure and proceed with the optimization. The currencies featured in this time series are the US Dollar (USD or $), the Euro (EUR or €), the Japanese Yen (JPY or JP¥), the British Pound (GBP or ₤), Chinese Renminbi/Yuan (CNY or ¥), the Canadian Dollar (CAD or C$), the Australian Dollar (AUD or AU$) and the Swiss Franc (CHF or

₣

). The justification for analysing these particular currencies stems from them currently enacting the role of the 8 largest currencies in terms of global foreign exchange turnover according to the latest Triennial Central Bank Survey of Foreign Exchange Turnover compiled by the Bank of International Settlements (BIS, 2019) illustrated in Table 2. This suggests that these 8 currencies acquire the highest levels of demand on international markets, implying significant global importance relative to other existing currencies. This consideration is consistent with the aim of this thesis to detect the contemporary international currencies and their primary contenders.

Table 2: Currency Distribution of Global Foreign Exchange Turnover

Currency Distribution of Global Foreign Exchange Turnover in April, expressed as a % of average daily turnover

April 2013 April 2016 April 2019

Currency Rank Weight Rank Weight Rank Weight

USD 1 43.5% 1 43.8% 1 44.2% EUR 2 16.7% 2 15.7% 2 16.1% JPY 3 11.5% 3 10.8% 3 8.4% GBP 4 5.9% 4 6.4% 4 6.4% AUD 5 4.3% 5 3.4% 5 3.4% CAD 7 2.3% 6 2.6% 6 2.5% CHF 6 2.6% 7 2.4% 7 2.5% CNY 9 1.1% 8 2.0% 8 2.2% HKD 13 0.7% 13 0.9% 9 1.8% NZD 10 1.0% 10 1.0% 10 1.0%

Source: Bank for International Settlements, Triennial Central Bank Survey – Global foreign exchange market turnover in 2019.

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Table 2 illustrates that the USD and EUR consistently attain the first and second spots respectively, hovering around the same percentage share of average daily foreign exchange turnover. Similarly, the GBP, CAD and CHF manage to account for roughly the same share of average daily global foreign exchange turnover in each 3 year interval, with slight developments in respective rankings as a result of alterations in other currencies. In contrast, the JPY and AUD have experienced the largest steady decreases in weights from all analysed currencies, hinting their decreasing dominance on global markets. Of particular importance is the steady increase in the share attributed to the CNY, corroborating the theoretical implication of the rising importance of the Chinese economy on global markets. Despite still encompassing a relatively small share relative to other currencies, the consistently significant increases imply the CNY continues to ascertain a larger role in global foreign exchange markets.

Second, as already outlined in the introduction, the aim of this thesis is to compare the theoretically optimal weights attained using the Markowitz portfolio optimization theory to the weights actually allocated to each currency as an optimal basket of foreign currencies represented by the COFER compiled by the IMF. As illustrated in Figure 1, the most recent COFER database (IMF, 2019) considers the 8 identical dominating currencies as found when ranking the global foreign market turnover in Table 2 to be held in an optimal basket of foreign currencies. This illustrates consistency in the assumptions presented in this section of the thesis. Thus, a quarterly-compiled time series of the COFER database is similarly required.

Both time series of daily currency returns and the quarterly COFER was generated for the time period of April 2010 until April 2019. The reason for this particular range stems from the consideration of the global financial crisis of 2008/2009, which significantly negatively impacted global trade and could have biased attained results. Additionally, the time series of daily currency returns assumes an approximate number of trading days per year to be 252 for all currencies. Finally, exchange rate information for all 8 currencies was extracted from the FactSet database3_{, while data regarding the}

COFER was acquired from the IMF.

4.1.1 SDR Denomination

The time series of daily currency returns had to be constructed such that each daily exchange rate change is expressed in the same denominating currency, or else any attained results would have been rendered uninterpretable and incomparable. To this end, the Special Drawing Rights (SDR) have been chosen as an unbiased denominator. The reason for this stems from two arguments. On the one hand, in order to compute the shares to be allocated to each of the 8 currencies in question, not one could serve as the denominating currency as it could not form a part of the optimal portfolio by construction. On the other hand, studies have shown that the choice of the denominating currency is not perfectly unbiased and changes the shares attributable to any arbitrary currency under analysis (see Papaioannou et al., 2006,

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p. 541). The SDR was chosen as it fixes both problems. According to the IMF, “the SDR is an

international reserve asset, created by the IMF in 1969 to supplement its member countries’ official reserves” (2020). In effect, an SDR is a form of reserve money that grants its holder a potential claim

on its underlying assets. While it is not technically a currency in and of itself (IMF, 2020), it represents a bundle of various important international currencies, weighted by their respective importance in foreign exchange markets. Its underlying value is computed daily, while it is rebalanced every 5 years (IMF, 2020). As illustrated in Appendix B, the current composition of the SDR consists of the USD, EUR, CHY, JPY and GBP.

Before delving into the actual dataset generated, it is important to disclose the two dimensions considered with any exchange rate analysis. First, the way in which quoted exchange rates are expressed. As with any exchange rate analysis, for the purpose of readability and interpretability of attained results, it is paramount to outline what the actual exchange rate refers to: whether it outlines the ratio of a unit of a base currency to a reference currency or vice versa. In this thesis, any exchange rate refers to the number of a base currency required to attain a unit of the reference or denominating currency. To illustrate, if any agent requires $1.3 to attain ₤1, and the GBP is quoted as the denominating currency, then the proper exchange rate would be denoted as $

₤= 1.3

1 = $1.3/₤. As the SDR represents the common denominator for all quoted exchange rates, it follows that each of the 8 types of exchange rates has been compiled in accordance to the specifications illustrated in Table 3.

Table 3: Exchange Rate Quotations4

Exchange Rate Quotations

Currency Exchange Rate Quotation

USD $ 𝑆𝐷𝑅 EUR € 𝑆𝐷𝑅 JPY 𝐽𝑃¥ 𝑆𝐷𝑅 GBP ₤ 𝑆𝐷𝑅 AUD 𝐴𝑈$ 𝑆𝐷𝑅 CAD 𝐶$ 𝑆𝐷𝑅 CHF ₣ 𝑆𝐷𝑅 CNY ¥ 𝑆𝐷𝑅

4_{The format in which the exchange rates for each considered currency will be presented and used throughout}

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Second, the way in which the exchange rates themselves were compiled. While the FactSet database contained time series of exchange rates between any two currencies, the conversion to an SDR denomination had to be performed manually. For this reason, a third time series was required, namely that of the value of an SDR, which was readily available through FactSet as well. However, the value of an SDR is always quoted in terms of one USD (see Appendix B). Thus, each final exchange rate consisted of the cross-exchange rate between each respective currency denominated in USD with the value of an SDR by definition also denominated in USD. For instance, to attain the time series of exchange rates €

𝑆𝐷𝑅, a cross-exchange rate was computed by first attaining €

$ from the FactSet database for all required dates, and multiplying it with the corresponding value of an SDR for the same dates,

$

𝑆𝐷𝑅, to attain the correct and finalized exchange rate € $∗

$ 𝑆𝐷𝑅=

€

𝑆𝐷𝑅. After equivalent computations for all other currencies, the final exchange rate time series was generated. Appendix Cshowcases a fraction of the dataset, clearly outlining the logic in its generation. For access to the entire sample, any interested reader can contact the author of this thesis.

4.2 Overview and Descriptive Statistics of the Acquired Dataset

Figure 3plots the daily prices of each currency in terms of SDR as a function of time from 2000 to 2019. The sample was extended to showcase a broader time period of the historical distribution of currency prices, as well as to corroborate the justification for the chosen sample period in this thesis. In particular, the crisis of 2008/2009, as illustrated in Figure 3, depicts relatively abnormal developments in the prices of currencies.

Figure 3: SDR-Denominated Currency Prices Over-Time5

Source: Paper’s dataset, derived from the FactSet database

5_{Two vertical axes have been used in this graph as the price of JPY and CNY, in terms of SDR, are considerably}

higher than that for the other currencies

0 50 100 150 200 0 0.5 1 1.5 2 2.5 3 04/10/2000 02/09/2004 12/10/2007 10/10/2011 08/10/2015 06/10/2019 P ri ce s JP Y an d C NY P ri ce s USD, E UR , GB P , A UD , C A D an d C HF Date

Currency Prices 2000-2019

In SDR

USD/SDR EUR/SDR GBP/SDR AUD/SDR

CAD/SDR CHF/SDR JPY/SDR CNY/SDR

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Figure 3 graphically illustrates volatilities exhibited by each currency. For the sample period used in this thesis, the JPY has exhibited large volatilities in value both up and down. The AUD and CAD showcase a relatively stable and high upward trend, while the EUR similarly implies an upward trend but of a smaller magnitude. The USD and CHF exhibit slightly negative while persistent price developments. Finally, the values of the CNY and GBP are relatively stable in the given sample period. For a quantitatively informative summary, refer to Table 4 which illustrates primary descriptive statistics for the sample dataset utilized in all subsequent computations. Before delving into interpretations, a two things should be noted. First, all returns have been computed as log returns following:

𝑟𝑖 = ln ( 𝑃𝑖,𝑡=𝑖 𝑃𝑖,𝑡=𝑖−1

)

Where 𝑃 represents the price of a particular currency 𝑖 in terms of SDR. Such return computations were generated for the entire time series and form the primary time series transformation on which Table 4 is based. Second, daily means, medians, variances, minima, and maxima were all annualized by a multiplication procedure with a factor 252, whereas the standard deviation has been annualized by a multiplication with a factor √252. The CAD exhibited the highest average annual mean returns of 1.97%, followed by the AUD with 1.74% and the EUR with 1.03%. The lowest average annual mean returns emerged in the CHF at -1.56%, accompanied by CNY with -1.16% and the USD with -0.98%. Interestingly, the median returns painted a substantially different picture. The JPY possessed the highest annual median returns of 0.99%, and the CAD follows shortly after with 0.76%. Noteworthy is the lowest annual median return exhibited by the AUD at -4.057%, which was paradoxically ranked second best according to the average annual mean returns. Given the wide differences between the median and mean of most currencies, it is probable the time series are not perfectly symmetric such that the mean suffers a bias from the presence of large outliers. Furthermore, the riskiest currency was the AUD, as per the annual standard deviation of 9.09%, together with the CHF and JPY at 8.45% and 8.35% respectively. Contrarily, the safest currency was the USD with an annual standard deviation of 4.08%, coupled with the CNY and EUR with 4.39% and 5.14% respectively. These statistics imply some consistency in financial markets. The USD was rightfully among the currencies with lowest average annual returns as it is the currency with the lowest volatility, and similarly the CAD attained the highest average annual returns due to a significantly high standard deviation of 7.31%. However, the profiles of certain currencies render them relatively unattractive. For instance, the average annual returns of the CHF were the lowest of all currencies analysed, while it was simultaneously the second most volatile.

When it comes to the assumption of a normal distribution of returns exhibited by the time series of the currencies in question, the third and fourth central moments, skewness, and kurtosis respectively, can extend the debate. These additional descriptive statistics provide relatively simple and reliable rules of thumb in discerning the assumption of normality. Skewness “measures the asymmetry of a

Composition of Optimal Currency Portfolios: An Overview of International Currencies