An Optimal Structure of International Monetary System According to the Markowitz Portfolio Theory1

Ing. Mária Vojtková University of Economics in Bratislava Faculty of National Economy, Department of Banking and International Finance Dolnozemská cesta 1 Bratislava, 852 35 Slovakia e-mail: maria.vojtkova.nhf@euba.sk Abstract In the paper we use basic procedure of portfolio optimization introduced by Markowitz to simulate an optimal structure of international monetary system. Combination of inflation rate and share of domestic GDP on total world GDP are used as proxy variables for risk and return of single assets, in our case international currencies. Data for 26 countries used in simulations represents 97,8% of world GDP. According to our model, a multi-currency monetary system is likely to emerge in case of low inflation rate variability. The higher the change in price level and variability of inflation rate allowed the more likely we observe one or two currencies dominating the monetary system. Keywords: Markowitz portfolio theory, international monetary system JEL codes: F30, F37, G11 1. Introduction Basic dilemma of investor, payoff between risk and return, was firstly mathematically treated in a famous paper by Harry Markowitz (1952) for which he was awarded by the Nobel Prize in Economics in 1990. Since then may papers have been using idea of portfolio selection with minor or major variations.

Comprehensive overview of current state of economic research in this area is provided in the paper by Steinbach (2001). From this reason we do not provide further reference to this topic but focus more on the application of Markowitz portfolio model for modeling the structure of international monetary system. Our analysis will be partially based on a concept presented in Baggiani (2011) but extended for our purposes.

Baggiani (2011) uses Markowitz portfolio theory for illustration of Hayek’s concept of competition between various currencies in domestic economy issued by domestic commerce banks. Return of the currency is specified as expected inflation rate and as a proxy variable for a variance the volatility of inflation is used. Gold is considered to be a zero-risk currency or more precisely zero-risk asset as specified in original Markowitz portfolio theory. In the model of Baggiani (2011) existence of multi-currency system with gold as zero-risk assets is presented as a natural result coming from evolution of the system.

From the point of view of an individual investor, by managing his currency portfolio in the most efficient way he always faces a dilemma of adverse behavior between risk and return.

It is not too daring to assume that from the general point of view the monetary system should also be structured in the optimal and efficient way in order to enable its participants to act efficiently. As the current monetary system consists of various international currencies and is

1 This work was supported by the project VEGA 1/0613/12 “The intensity of the relationship between financial sector and real economy as a source of economic growth in Slovakia in the post-crisis period” (50 %) and the project of PMVP 2315026 “Theoretical and practical aspects of modeling foreign debt and economic inbalance in DSGE models.”

structured in a specific way we may generalize the problem of optimal monetary system as the problem of a portfolio choice with international currencies.

In contrast to the traditional portfolio theory where risk is calculated as a variance of asset prices and return as a simple mean of those prices, we use a broader perspective of risk and return measurement. According to our opinion, there are more concepts of risk and return related to the notion of international currency and all comes back to the theory of international currency. From this reason, we will firstly introduce basic concepts of the theory of international currency and tries to specify those economic variables that will be further on used as a measurement of risk and return of international currency.

The structure of this paper is therefore following: in the first mostly theoretical part of this paper we provide overview of the theory of international currency along with the basic mathematical description of the Markowitz portfolio model. In the second part we discuss empirical results of modeling international monetary system with the Markowitz portfolio model. Final words conclude this paper.

2. Theory of International Currency Using general definition, international currency is defined as a currency that is being used by private residents and officials for the various purposes specified in the Table 1. Three basic functions of international money can be applied for the use of currency in the international environment (Thimann, 2009.).

Currency traded internationally thus serves as a medium of exchange (the need for intervention of central banks and private use of individual market participants), unit of account (currency used in the payment system for settlement or as an anchor currency) or as a store of value (such as component of central bank foreign reserves or investment and financing currency).

Table 1 The Matrix of International Currency Usage

Source: Thimann (2009)

To what extent and in what transactions the international currency will be used is determined by several factors. In economic literature (Lim, 2006; Beckmann et al., 2001) related to the issue of currency's international status, five of the major factors (so called facilitating factors or FFS) have been specified in following way:

1. economic size and degree of openness This factor is correlated with the flow of capital and foreign trade. The larger the size and openness of the economy at the same time, the higher proportion of trade with currency issued by this economy could be expected. This arises as a result of meeting the demand and supply of the currency on foreign exchange market.

Based on time series estimates, Eichengreen and Frankel (1996) found that every one percent of GDP in the world total leads to 1.33 percent of central bank reserve holdings in the corresponding currency.

In addition, study of Chinn and Frankel (2005) has confirmed that there is a positive relationship between share of single country GDP to the total world GDP and the share of

central bank reserve holdings in the corresponding currency, although the relationship is likely to be nonlinear.

2. well-developed financial system Integrated and high developed financial markets provide liquidity and reduce transaction costs, uncertainty and risk. The above mentioned factors together with the existence of other factors affect the intensity of use of monetary unit in the field of saving, investment and loans The attractiveness of currency is supported at least due to the following factors:

well-developed financial markets allow participants to quickly build up and liquidate their positions in the foreign currency. As they are usually not interested in holding currency in their physical form but more in buying and selling interest-bearing securities denominated in international currency an existence of full functioning security market is advantageous.

secondly, the developed financial system is more likely to attract business from abroad where financial markets may be less developed or barriers to efficiency exists.

3. confidence in currency value To be considered a reserve currency, the international currency has to be viewed as a healthy currency without any tendency to depreciate in the long run. Even if a key currency were used only as a unit of account, a necessary qualification would be that its value does not fluctuate erratically.

Additionally, key currencies are also used as a form in which to hold assets (firms hold working balances to the currencies in which they invoice investors hold bonds issued internationally and central banks hold currency reserves). Confidence that the currency will be stable and particularly that its value will not be inflated away in the future is critical (Eichengreen, 1996).

4. political stability The historical experience shows us that there is no monetary stability without political equilibrium. As economist R. Mundell, recipient of the Nobel Prize in Economics in 1999, once stated: “when a state collapses, the currency goes up in smoke." From the historical perspective, such an international monetary union where monetary integration is not connected to political union has usually failed (Lim, 2006).

The status of international currency is not determined strictly by economic factors. Direct or indirect political support often plays a key role in promoting a currency in the international monetary relations. So called political theory of international currency has firstly been proposed and discussed in work of Strange (1971).

Later, Helleiner (2008) created characterization of selected political factors that may have a significant influence on status of international currency. According to his classification two channels of transmission may be distinguished: indirect (basic economic determinants such as liquidity, truest and transaction network) and indirect (governmental and state decisions support status of the international currency).

5. network externalities If the currency is used internationally, it will become difficult to shake its position in international monetary system. Moreover, in the case of market dominance it is not only hard to jeopardize its position, sometimes it could be almost impossible to replace it because of the high cost of such a replacement especially in short run. Individual trader has the minimum interest to leave the network which has been already built; he will consider the change only when other participants are about to decide to do the same.

The paper by Yehoue (2004) argues that trade network externalities help explain the gradual expansion of currency blocs. Countries join a currency union the sooner the more they trade with the bloc member countries and each additional member serves in a dynamic way to

attract additional members into the bloc. These arise because the payoffs of using a particular currency increase with the amount of trade with countries that use the same currency.

The empirical evidence presented in Meissner and Oomes (2003) also proves that the size of each currency bloc matters quite a lot for decision to anchor foreign currency to bloc currency. If there were an increase in trade with foreign countries pegged to a single currency e.g. the US dollar, it would be individually optimal for home country to switch to the dollar anchor too.

3. Markowitz Model for N-assets In the standard Markowitz model described in the next section there will be no zero-risk asset included. Secondly, there is no condition imposed on non-negative value of weights of single assets included into portfolio. The implications coming from these assumptions will be discussed below.

Let us now assume that investor portfolio consists of assets, where stands for asset and holds. Every asset is characterized by gross return in time , or respectively, with mean return and variance as a measurement of risk. Mathematically, gross return of an asset in time is computed as following:

[1]

where represents market price of an asset in time . Portfolio weights are computed in following way where sum of the weights in portfolio must equal one:

[2]

In vector notation the basic characteristics of portfolio optimization problem are written in following way:

[3]

Based on the previous derivations expected portfolio return may be derived in following way for assets:

[4]

Expected portfolio return is therefore computed as weighted sum of expected returns of single assets and their weights in portfolio. Here we assume that at time when portfolio characteristics are computed all characteristics of single asset are know at this time.

[5]

Variance, as the measurement of portfolio risk, is computed with help of covariance matrix and vector of asset weights for assets. In matrix form the variance is computed in following way:

[6]

where denotes covariance matrix whose values on main diagonal represents variance of

asset and covariance matrix is symmetric over main diagonal.

Covariance matrix may be further decomposed as a product of correlation matrix and matrices with standard deviations of single assets on their diagonals:

[7]

where represents Pearson correlation coefficient. This coefficient may range

between -1 and 1 and determines strength of linear dependence between asset and .

Portfolio return may be specified in matrix form: [8]

In order to get optimal portfolio structure we solve basic optimization problem for assets, where represents unit vector with dimensions and represents portfolio return required by investor for which we minimize total portfolio risk:

[9]

By assumption,2 optimization problem stated above has a nice solution in following form:

[10]

In the optimal portfolio structure portfolio risk is determined by following function:

[11]

where . [12]

Structure and portfolio risk are dynamically dependent on requested portfolio return . Among other factors that may influence value of optimal weights are asset returns and variance matrix. In case that there is no change in number of assets included in portfolio the variance matrix and asset returns remain constant for every required risk-return combination.

Optimization problem stated in [9] may be reformulated in a mirror form when an investor maximizes his return given required level of portfolio risk. The below specified optimization problem may be solved by Lagrangian. Hence:

2 In order to get a plausible solution to the optimization problem stated vector needs to be parallel to the unit

vector . This condition is met when . Otherwise, the problem does not have a

solution. Additionally, covariance matrix is to be invertible and strictly positive semi-definite.

[13]

The standard Markowitz model of optimal portfolio assumes that there is possibility to go short on different assets included in portfolio when required by optimal structure. In mathematical terms, the following condition is omitted in the standard model described above:

[14]

However, this condition is necessary to be included into the model that will be able to model optimal structure of international monetary system pictured here as a portfolio consisting of various international currencies. Economically speaking, it is not plausible to assume that there is possibility to go short on some international currency (e.g. euro) at the level of global monetary system. Mathematical solution to the optimization problem described in [9] with new optimal conditions as in [14] belongs to the area of convex quadratic programming problems and can by solved numerically by number of efficient algorithms.

In this paper we consider the easiest way how to compute optimal weights of international currencies in international monetary system as we use built-in algorithm for quadratic optimization in MS excel called Solver.

3.1. Application of the Markowitz Model to Problem of Optimal Structure of International Monetary System

In the standard Markowitz model all assets are characterized by two variables – return and risk, or expected return and variance of returns respectively. Let us assume now that each international currency may be also characterized by those two variables described above. However, in case of international currency expected return and risk cannot be easily computed as mean or variance of prices because there is no single price available3 in international monetary system.

Luckily, basic characteristics of international currency are easily to be specified once we look at the macro level. Taking into consideration five basic facilitating factors (FFs) described above that have strong influence on a position of international currency we may use the Markowitz portfolio theory with those 5 FFs as measurement of risk and return:

- return variable – network externalities (share of international currency on total daily turnover in exchange rate markets, relative economic strength of issuing economy on total world GDP, value of shares or bonds denominated in international currency on total value of shares or bonds issued etc.). In this case we assume that higher share of international currency reflects higher returns coming from network externalities, hence higher return of international currency is favourable.

- risk variable – as a proxy of risk variable all the factors that may negatively influence status of international currency in international monetary system may be used (inflation rate in issuing economy, volatility of exchange rate, share of total debt on GDP, share of external debt on GDP or share of balance of payment deficit or surplus on GDP etc.).

3 Exchange rate is considered to be price of domestic currency in terms of units of foreign currency. However, because of the bilateral notation of exchange rate it is problematic to analyze whether change in exchange rate has been caused by change of fundamentals relevant for domestic of foreign currency. Although we may take into consideration a change of price of one international currency relative to other international currency we are not able to get a change of overall price of international currency in general. This problem is likely to be solved using concept of currency invariant index proposed by Hovanov et al. (2004). However, we will not use this concept in our paper as we deal with this issue in different manner.

4. Methodology Basic principles of Markowitz optimization theory have been applied for different combinations of return-risk characteristics. Data used in the simulation were mostly drawn from International Monetary Fund database on quarterly basis for 26 countries.4 Data for Chinese GDP needed to be taken from Chinese statistical yearbook published by National Bureau of Statistics of China. Chinese GDP was originally denominated in Chinese renmimbi, therefore recalculated with period average exchange rate between USD and renmimbi as published by International Monetary Fund. All the time series start from 1Q1999 as a datum relevant for creation of the Euro Area and were seasonally adjusted by International Monetary Fund. Data for Chinese GDP were seasonally adjusted by Census X12 filter built in Eviews software.

Variable share of GDP on total world GDP has been computed from a total GDP produced by selected 26 countries, not the total world GDP. This is a plausible computation as the share of total GDP produced by 26 countries5 in our sample represents 97,8% of world GDP, therefore conclusions drawn from our simulation are relevant also for global monetary system.

For the purpose of this paper we decided to test relationship between variability of inflation rate in countries issuing international currency and their respective share on total GDP produced by all countries included in our sample. At this place, we would like to point out that there are many other combinations available for analyzing optimal structure of international monetary system. Some of them are proposed in Section 1.4. of this paper. As reporting results for such an analysis is way behind the scope of this paper throughout the rest of the paper we will focus only on the combination of variability of inflation rate vs. share of GDP on total GDP. Thus we analyze four plausible combination specified as following:

1. Profit maximization given level of risk a) Share of GDP vs. Variability of inflation rate – model I b) Share of GDP vs. Change in price level – model II

2. Risk minimization given expected return a) Share of GDP vs. Variability of inflation rate – model I b) Share of GDP vs. Change in price level – model II

4.1. Model with variability of inflation rate As derived in the section 1.3. of this paper standard approach considers variance as a measurement of risk related to a single asset. Using inflation rates of issuing countries as a variable and their variances as measurement of risk related to the usage of a currency, in economic terms we assume that not the absolute level of an inflation rate is perceived as a risk but variation in inflation rate represent negative aspects of international currencies that may undermine their role in international monetary system. Thus, not the absolute level of inflation rate but variations in inflation rates are considered as a risk here. This approach was previously used in Baggiani (2011).

At this place it is necessary to spend some time by discussing interpretation of empirical results obtained by testing this model. Expected portfolio return in the Markowitz environment with a GDP share as a proxy variable for return of asset may be interpreted as a sum of GDP shares weighted by their respective optimal weights in the optimal structure of monetary system. Similarly, expected portfolio risk represents sum of inflation rate variability

4 Australia, Canada, Chile, Czech Republic, Denmark, Hungary, Iceland, Israel, Japan, South Korea, Mexico, New Zealand, Norway, Poland, Sweden, Switzerland, Turkey, United Kingdom, United States, Euro Area (17 member states), Brazil, India, Indonesia, Russian Federation, South Africa, China. 5 We talk about countries as we consider Euro Area as one homogeneous country although it includes 17 member countries. Correctly, we include to our sample 25+17=42 sovereign countries.

of countries in the sample along with interaction between them weighted by their respective optimal weights calculated by optimization procedure.

4.2. Model with change in price level On contrary to Baggiani (2011), we would like to test a model where not the variation in inflation rate but an absolute change in price level is considered to be crucial for a stable and optimal structure of international monetary system. From this reason, we decided to follow following steps:

1. absolute values of the CPI index (consumer price index) are used as a variable by which the change in price level is derived,

2. on contrary to the standard calculation of variance and covariance for single asset and covariance matrix respectively, as presented in the Section 1.3. of this paper, we use following equation for variances calculated and used in our model:

where represents ordering of currencies included in our sample and represents realization of CPI for each currency. On contrary to standard calculation of variance we do assume that mean of the CPI is fixed for each currency at a level 100. The higher the deviation of the current CPI from a fixed value 100 the higher the variation in a price level in country, thus higher the risk linked to currency. As the fixed level 100 for the CPI is standardized for each country in our sample (year 2005) we may use this fixed point for calculation of variances throughout our sample.

3. The same reasoning is used for calculation of covariances. All variances and covariances are used for building standard covariance matrix as specified in the Section 1.3. of this article.

4. Optimal weights are calculated with help of a build-in tool in MS Excel called Solver. As in the previous model I, interpretation of results in model II is quiet similar. Expected portfolio return in the Markowitz environment with a GDP share as a proxy variable for return of asset may be interpreted as a sum of GDP shares weighted by their respective optimal weights in the optimal structure of monetary system. However, in this case expected portfolio risk represents sum of change in price level of countries in the sample and interaction between them weighted by their respective optimal weights in the optimal structure of monetary system, not the inflation rate variability.

5. Empirical Results In the following text we would like to discuss results of our empirical analysis based on a procedure described in the previous part of this paper. Concretely, we will focus on interpretation of results obtained by testing variability of inflation rate (model I) and change in price level (model II) in the profit maximization environment.

For our purposes, interpretation of model with profit maximization problem has more significant economic meaning than the risk minimization problem (or in this case maximization of a sum of weighted GDP shares) as the former one allows us to observe change in optimal structure by changing levels of inflation rate or changes in price level, respectively. Yet, in order not to miss some piece of important information we will report both of them.

5.1. Profit maximization problem This model allows us to maximize sum of weighted GDP shares for a given level of a risk, in our case for a given level of either inflation rate volatility or deviation of inflation rate from its zero value (level 100 in case of CPI index).

Graph 1 Relationship between optimal weights in portfolio (y axis) of various currencies and required portfolio risk (x axis) – model I

As we may observe in the Graph 1 and 2 as soon as we allow for a higher level of inflation rate in global environment, the US dollar becomes dominant currency in the international monetary level. Additionally, in environment of high inflation rate Russian rubel surpasses euro or Japanese yen respectively.

The difference between model I (with a variability of inflation rate) and model II (with an absolute level of inflation rate) lies in underlining Japanese yen as a significant player in the international monetary system and a diminishing role of a common European currency. Explanation for different structure of international monetary system is easy to find: while in case of an absolute level of inflation rate Japanese yet dominates euro because of period of deflation in Japan, euro uses its advantage as a currency with relatively stable levels of inflation rate in model I.

Graph 2 Relationship between optimal weights in portfolio (y axis) of various currencies and required portfolio risk (x axis) – model II

Empirical results presented here deliver also one very interesting observation: for achieving environment with a low level of inflation rate variability (not necessary low level of inflation rate in absolute value) the optimal structure of international monetary system needs to consist of multiple currencies. The higher the level of inflation rate and inflation rate variability the more the monetary system becomes dominated by one or maximum two currencies.

5.2. Risk minimization problem Results presented here represent a mirror image of a profit maximization problem. When we allow for low levels of a weighted sum of a GDP shares the structure of international monetary system becomes more diverse. As we have seen in the previous section the more diverse structure of a monetary system comes hand in hand with a low inflation rate variability environment. Yet, this is not a case for a low inflation environment as that comes only in the case of high dominance of one or two currencies.

Logically, by increasing required level of portfolio return the currencies with a high GDP share becomes more dominant as the likelihood of high inflation rate environment with high inflation rate variability is increasing.

Second observation may be attributed to the fact that some of the currencies are likely to never achieve significant position at international level. This is mostly due to their low shares of GDP on total world GDP, the fact that does not allow them to become dominant even in the case of low inflation rate in absolute or relative terms attributed to them.

Graph 1 Relationship between optimal weights in portfolio (y axis) of various currencies and required portfolio return (x axis) – model I

Thirdly, common currency of eurozone area does not play a significant role in case of a change in price level variable. The important feature of a eurozone lies in stability of the price level thanks to the active role of the European central bank who has committed itself to maintain stable inflation rate at two percent. Thus, euro is likely to play an important role in international monetary system that aims at keeping changes of price level stable. Yet, the US dollar does still holds its dominant position when we take into consideration strength of the US economy.

Graph 2 Relationship between optimal weights in portfolio (y axis) of various currencies and required portfolio return (x axis) – model II

6. Concluding Remarks In this paper we use basic principles of Markowitz portfolio theory to derive optimal structure of international monetary system. As a proxy variable for expected return and risk we use GDP share of issuing country on total GDP for the former and variability of inflation rate and change in price level for the latter one.

According to our model, a multi-currency monetary system is likely to emerge in case of low inflation rate variability. The higher the change in price level and variability of inflation rate allowed the more likely we observe one or two currencies dominating the monetary system.

Second important observation is attributed to the fact, that some of the currencies are unlikely to ever achieve international significance. Even in the multi-currency model of monetary system, only selected currencies will dominant the international relations. This is due to their strong economic position measured by level of GDP even in a presence of higher inflation rate or inflation rate volatility.

Thirdly, common European currency euro may take advantage of policy of inflation rate targeting in case that a factor or a variability of inflation rate is important for maintaining the position of international currency. In this case, position of euro is supported by economic strength of the Eurozone, thus euro might become a significant player in the international monetary system.

We are aware of the fact that an analysis of a structure of international monetary system is a very complex problem. By using Markowitz portfolio theory we constrain our analysis on interaction of two factors that are important for determining the structure of a portfolio – risk and return. While in case of a portfolio theory this might be enough, in case of an optimal structure of international monetary system model with more variables included would be more appropriate. One possible way how to partially solve this problem is to test other combination of risk and return duality in environment of Markowitz theory as suggested in theoretical part of this paper. This we leave for a future research.

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