University of Southern California Dissertations and Theses

A Portfolio Approach To Domestic And Foreign Investment

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A Portfolio Approach To Domestic And Foreign Investment

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Content I I 72-17 ,495 NOREIKO, Gary Victor, 1944- A PORTFOLIO APPROACH TO DOMESTIC AND FOREIGN INVESTMENT. University of Southern California, Ph.D., 1972 Economics, theory University Microfilms, A X ER O X Com pany, Ann Arbor, Michigan © 1972 Gary Victor Noreiko ALL RIGHTS RESERVED THIS DISSERTATION HAS BEEN MICROFILMED EXACTLY AS RECEIVED A PORTFOLIO APPROACH TO DOMESTIC AND FOREIGN INVESTMENT by Gary Victor Noreiko A Dissertation Presented to the FACULTY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (Economics) February 1972 UNIVERSITY O F SO U T H E R N CALIFO RNIA TH E GRADUATE SC H O O L UN IV ERSITY PARK LOS ANG ELES, C A LIFO R N IA 9 0 0 0 7 This dissertation, written by Gary Victor Noreiko under the direction of h'ks . . . . Dissertation C om m ittee, and approved by all its m em bers, has been presented to and accepted by T h e G radu ate School, in partial fulfillm ent of require ments of the degree of D O C T O R O F P H I L O S O P H Y Dean n „ ,, February, 1972 PLEASE NOTE: Some pages may have indistinct print. Filmed as received. University Microfilms, A Xerox Education Company ACKNOWLEDGMENTS For his great help and encouragement in writing this dissertation, I would like to express my thanks and appreciation to Professor Gerhard Tintner. I am indebted to Professor Elliott for his helpful supervision and guidance, and would like to thank Professor Maass for his encouragement along with assistance in the financial aspects of the analysis developed in this dissertation. In addition, appreciation is expressed to the Computer Science Laboratory of the University of Southern Califor nia for the use of their computer facility. - ii - TABLE OP CONTENTS Chapter Page ACKNOWLEDGMENTS ....................................... ii LIST OP TABLES....................................... v LIST OF ILLUSTRATIONS...................................vi I. INTRODUCTION ................................... 1 The Problem..................................... 1 Hypotheses ..................................... 2 Methodology ..................................... 5 Sources ......................................... 6 Organization of the Dissertation ............. 10 II. REVIEW OP THE LITERATURE.........................14 International Capital Plow Theory ............. 14 Portfolio Analysis ............................ 21 The Application of Portfolio Analysis to International Investment ................... 30 III. THE MODEL...................... 36 Objective Function ............................ 36 Investment Search .............................. 58 Taxes..............................................64 The Complete M o d e l ...............................75 - iii - iv Chapter Page IV. ECONOMETRIC ANALYSIS .......................... 81 Relative Dispersion ............................ 81 Indirect Portfolio Analysis .................... 85 An Autocorrelation Model ...................... 89 An Econometric Model ........................... 93 Application of the Logarithmic Normal Diffusion Process ............................. 102 V. EVALUATION OF THE PORTFOLIO MODEL...............1H- The Markowitz Portfolio Model .................. Ill The Sophisticated Portfolio Model .............. 137 Predictions of the Sophisticated Portfolio Model....................................... . 144 VI. CONCLUSION.................................. . 154 Summary...........................................154 Results from Testing the Hypothesis ............ 156 Revaluation of the Basic Hypothesis ............ 158 APPENDIX............................................... 162 BIBLIOGRAPHY 165 LIST OP TABLES Table Page I. The Complete Model............................76 II. Relative Variation ............................. 84 III. Rate of Return Means and Variances..........112 IV. Area Correlation M a t r i x .....................114 V. Industry and Area Correlation Matrix........114 VI. The Dimensionless Third Moment ................ 138 VII. Estimated Range of Investment Proportions . . 142 VIII. Input Parameters for the 1969 Prediction . . . 146 - v - LIST OF ILLUSTRATIONS Figure Page 1. The Efficient S e t ............................... 26 2. The Profit of an Investment Suitable for Limited Uncertainty .......................... 4l 3. The Profit of an Investment Suitable for Great Uncertainty............................ 43 4. A Positively Skewed Distribution of Rates of Return..................................... 54 5. A Three Dimensional Efficient S e t............. 56 6. A Two Dimensional Representation of the Efficient Set when Skewness is Present . . . 57 7. A Comparison of Relative Variation and Investment Search ............................ 60 8. The Impact of Differential Tax Policies for Developed and Less Developed Countries . . . 68 9. The Major Forms of Corporate Structure for an Internationally Oriented Firm ........... 70 10. An Illustration of the Solution to the Complete Portfolio Model ................... 78 11. A Comparison of the Results of Various Models for Domestic Investment ............. 108 12. A Comparison of the Results of Various Models for Foreign Investment ............... 110 13. The Impact of Changes in A on the Invest ment Proportions...............................119 14. The Efficient Set Generated from the Objective Inputs ............................ 122 - vi - vii Figure Page 15. The Influence of Shifts in A as Foreign Variance Varies on the Foreign Invest ment Proportion.................................126 16. The Impact on the Foreign Investment Proportion of Shifts in A as y2 Varies . . . 128 17. The Three Dimensional Surface for a Single A Parameter.............................130 18. The Influence of Negative Variations in P12 and Variations in A on the Dependent Variables x2 ...................... 132 19. The Influence of Positive Variations in Pip and Variations in A on the Dependent Variables x2 ................................... 133 20. Efficient Sets with Various Values of Correlation..................................... 135 21. The Importance of Skewness for Two Distri butions of Rates of Return G and H ........... 160 CHAPTER I INTRODUCTION The Problem Risks in the formulation of expectations about future foreign exchange rates, levels of sales and pro fits in both domestic and foreign markets, and cost differ entials between world capital markets are some of the factors which firms consider in initiating or extending foreign operations. Adding to these considerations are uncertainties due to wars, silent revolutions that change modes of thought, and violent political revolutions whose impact is most difficult to anticipate. Moreover, the modern corporation with domestic and foreign interests is confronted with the evaluation of information on rate of return, risk, and uncertainty in varied parts of the world with different cultures, laws, and customs. Confronted with the complexity of both economic and noneconomic events on the problem of analyzing world wide investments, only limited attempts have been made concerning analysis of a firm's domestic and foreign - 1 - investment. This dissertation will deal with the problem of analyzing worldwide investments through a portfolio allocation model and the evaluation of variables that influence the rate of return of an investment. Hypotheses In the context of analyzing the complex environment of domestic and foreign investment from the corporate viewpoint two general levels of analysis are pursued. On the first level hypotheses related to worldwide invest ment are evaluated. The second level of analysis is higher than the first as it evaluates hypotheses closely asso ciated with and extensions of Markowitz portfolio analysis. Primary emphasis on the first level of analysis is directed towards analyzing rate of return. The esti mation of rates of return as inputs in the portfolio model represents a necessary requirement for constructing a sound portfolio of investments. The analysis of hypotheses related to rates of return is conducted in Chapter IV. As the analysis proceeds the degree of complexity of models applied increases. One hypothesis evaluated is that foreign rates of return have greater relative variation than American rates of return. To test the information con tent of previous rates of return the hypothesis that pre vious rates of return from stable areas have greater information content than those of unstable areas is evaluated. After this point in the analysis the degree of complexity of the models employed greatly increases. One approach is to build an econometric model that will estimate the rate of return as a function of important independent variables. The independent variables terms of trade and the growth rate of national product are evaluated for their impact on the rate of return for the United States, Canada, Europe, and Latin America. A hypothesis tested is that the terms of trade is more significant for less developed regions and that growth rates are more sig nificant for developed regions. An alternative approach is to argue that rates of return are determined by such a large number of simultaneous interdependent variables that the more exact econometric model building is inappro priate. In this event a stochastic process that accounts for randomness can best explain rates of return. The logarithmic normal diffusion process is the stochastic process specifically applied. A hypothesis evaluated tests whether a regression model or this stochastic pro cess is more appropriate for explaining rates of return. On the second level of analysis hypotheses con cerned with portfolio analysis per se are tested. An implicit hypothesis is that Markowitz portfolio analysis serves as a foundation for analyzing domestic and foreign investment. The portfolio model constructed in Chapter III shows that the implicit hypothesis builds on, modi fies, and extends the basic Markowitz portfolio model. For instance, risk in the simple Markowitz portfolio model is solely concerned with the variance of the rate of re turn distribution. The portfolio model constructed in this dissertation extends the concept of risk to include not only variance, but also the skewness of the distribu tion of rates of return. A basic hypothesis to the impli cit hypothesis is that a sophisticated three dimensional objective function of rate of return, variance and skew ness represents an improvement over the two dimensional objective function of the simple portfolio model that only considers rate of return and variance. In this manner a more sophisticated portfolio model is constructed that derives its foundations from general portfolio analysis. Hypotheses concerned with the input parameters of the portfolio model are also tested. These hypotheses are (1) greater foreign than domestic rates of return; (2) greater foreign variance of rates of return; (3) the exis tence of positive correlation between rates of return of worldwide investments; and (4) the prevalence of positive skewness for rates of return distributions. Chapter V applies the simple and sophisitcated portfolio models to worldwide investment as well as the evaluation of the 5 input parameter hypotheses. The analysis of Chapter V shows how the addition of sophistication allows portfolio analysis a greater capacity for analyzing world investment situations. Methodology Mathematical and econometric techniques are ap plied to the problem of analyzing worldwide investment. The building of a more complex model based on a simple portfolio model is the primary mathematical method for revealing variables significant for domestic and foreign corporate investment. A general n sector model is con structed to reflect aspects discussed in the literature. For empirical applications a two sector model is generally applied. From the viewpoint of a United States corporation subscripts i or j equal to one defines domestic investment and subscripts i or j equal to two defines foreign invest ment. The application of a two sector portfolio model allows the analysis to focus on significant properties of the rate of return distribution. An econometric approach is the method applied to the analysis of rates of return. Here, a large number of models are applied to the time series rate of return data. Thus, a mathe matical approach that relies on model building and econometric analysis are the methods followed. 6 Sources Sources of data depend on the type of model con structed and the definition of the model's variables. Thus, interest is focused on aggregate industrial rates of return and measures of investment proportions. The sources of data for the most important variables will be described, and the data sources for less important variables will be stated in their particular sections. Applications toward worldwide investment of a firm increases difficulties of acquiring data. Unlike the relative simplicity of determining the actual rate of return from a security as dividends plus the price change divided by the average share price, the great complexity of a firm's hierarchy allows several possibilities for calculating the rate of return. For example, the actual rate of return could be measured as the ratios of earnings to total assets, earnings to stockholders equity, or earnings to assets of plant and equipment. The measure of rate of return generally applied for domestic invest ment in this dissertation is the ratio of earnings to stockholders equity. A data source for this measure of rate of return is the Quarterly Financial Report for 7 Manufacturing Corporations.'1 ' In a similar manner foreign rates of return will be defined in terms of equity as the ratio of earnings to the value of direct investment. The Department of Commerce's Survey of Current Business pro- 2 vldes data on this measure of rate of return. The data taken from the previous sources to form the basic 1950- 1966 time series and the succeeding years utilized for predictions should be considered an extremely good measure of rate of return due to limited errors of observation. The previous data is employed for the portfolio model and utilized for most of the econometric analysis. In con trast, the rate of return calculated as earnings to total assets derived from "Fortune" should be considered a rough measure of rate of return.-^ This cross section data for 1963, 1964, and 1965 has errors of observation due to the different accounting procedures applied in different countries. This data is primarily applied to regressions 1Federal Trade Commission and Securities and Ex change Commission, Quarterly Financial Report for Manu facturing Corporations. (Washington, D.C.: Government Printing Office, 1950-1970). 2 Department of Commerce, Survey of Current Business. (Washington, D.C.: Office of Business Economics, U. S. Department of Commerce, 1951-1970), XXX-L. ^Fortune, "Fortune's 500 and 200," (Chicago: Time Incorporated, July, August, 1964; July, August, 1965; and July, August, 1966). 8 of limited importance. For both cases of time series and cross section data the rate of return must be calculated due to a lack of established data. There are sources of data problems in focusing the analysis in terms of the investment proportion of a firm, which in contrast, are not present in stock market appli cations. The measure of aggregate domestic and foreign investment should adequately reflect reality through the investment proportion and be readily obtainable from avail able sources. Defining foreign investment as direct investment emphasizes tangible and intangible control over an existing firm or the establishment of a new firm in a foreign country. However, the utilization of this definition presents the problem that a clearly defined domestic equivalent to direct investment is difficult to find. Statistics on the changes of influence or control of domestic firms are generally not available. Thus, direct investment and a measure of domestic investment are not suitable for establishing an investment proportion. To overcome the previous difficulties foreign and domes tic investment are defined as foreign plant and equipment expenditures and domestic plant and equipment expenditures, respectively. The major source for both domestic and foreign plant and equipment expenditures is the Survey of 9 Current Business.** This definition allows the parent corporation to exercise policies of creating a foreign affiliate, initiating joint ventures, and purchasing con trol of an existing foreign firm as it adjusts its compo sition of investment to reflect changes of the profita bility distribution. Thus, the measure of rate of return generally applied in this dissertation is an equity measure of earnings divided by stockholders equity for domestic in vestment and earnings divided by the book value of direct investment for foreign investment. The period covered for general analysis is 1950-1966. The measure of rate of return for the period considered is a good measure as it adequately reflects changes in rates of return. To establish a measure of the investment proportions an im portant component is total plant and equipment expenditures that includes the sum of both domestic and foreign expen ditures of American firms. The foreign investment propor tion is defined as foreign plant and equipment expenditures divided by total plant and equipment expenditures, and the domestic investment proportion is similarly defined as domestic plant and equipment expenditures divided by total plant and equipment expenditures. The previous discussion ^Department of Commerce, op. cit. 10 summarizes measures of variables important to the analysis. Organization of the Dissertation Chapter II reviews the literature from three perspectives. First, traditional capital flow theories are surveyed. These theories are primarily concerned with portfolio Investments in bonds and stocks where em phasis is placed on yield differentials to the general neglect of risk. In contrast, corporate investment em bodies technology, management skill, and physical capital of plant and equipment to achieve the goals of a firm In the face of a most complex probability distribution that is difficult to characterize. Second, Important properties of the Markowitz portfolio model are described. One of the useful outputs of the model is the efficient set concept that shows for different compositions of investment the greatest return at each level of risk. The efficient set and properties of the model have been applied directly to analyze stock market investments. Third, the direct application of the Markowitz portfolio model, which in trinsically applies the relations and properties of the model, Is described for international stock market invest ments. This dissertation will primarily utilize and expand the portfolio model in a direct manner to analyze worldwide corporate investment. In addition, for purposes of depth 11 and completeness, a critique is developed of a regression technique that only in a superficial, indirect manner can be viewed as portfolio analysis. A model to analyze worldwide investments confronted by a multinational corporation is constructed in Chapter III. The simple Markowitz portfolio model is the basic building block on which a sophisticated model is constructed to take fuller account of the properties of the rate of return distribution. The objective function of the model constructed explicitly considers both variance and skewness as a measure of the risk of the distribution of rates of return and uncertainty reflected in the rate of return. The importance of investment search and the impact of taxes on the efficient set are also considered. Further refinements are introduced by the addition of constraints that confront a worldwide corporation. Then, the complete sophisticated portfolio model is presented graphically to show the full impact of a three dimensional objective function that measures the rate of return, variance and skewness along with the added constraints. Chapter IV applies econometric techniques to analyze important hypotheses related to the rate of return input for portfolio analysis. For nine foreign and domes tic industries based on the standard industrial classifi cation a hypothesis concerning relative variation of rates 12 of return is tested. Some indirect portfolio analysis is also shown. The hypothesis concerning investment search formulated in the previous chapter is tested. An econo metric model that also tests the importance of the terms of trade and the growth rate of national product is devel oped to explain rate of return. An alternative approach of applying the logarithmic normal diffusion process to explain rates of return is formulated. It is hypothe sized that this diffusion process represents a superior method for predicting rates of return when a strong trend is present. The direct application of a two sector portfolio model is developed in Chapter V. First, the simple port folio model’s input parameters rate of return, variance, and correlation of rates of return are presented. A sensitivity analysis is conducted to show problems for the objective application to worldwide investment of the simple portfolio model. In part the problems encountered are related to the measure of risk. Second, the objective function of the sophisticated portfolio model is shown to overcome problems encountered by the simple portfolio model. The hypothesis of the prevalent existence of posi tive skewness is also tested. Through this form of analy sis, support is given to the basic hypothesis that the sophisticated portfolio model is an improvement over the simple model. Then, predictions of foreign and domestic investment are developed from the sophisticated portfolio model by utilizing the functional formulation of Chapter III and the econometric analysis of Chapter IV to generate input rate of returns. In this manner portfolio analysis is directly applied to problems of the real world. The concluding chapter concisely restates the sub stantive elements revealed in the whole study. Significant problems encountered and the future direction for exten sions of portfolio analysis are also described. CHAPTER II REVIEW OP THE LITERATURE International Capital Flow Theory A review of the works significant for this disser tation requires penetration into the areas of international capital theory and portfolio analysis. International capital theory shows how traditional economic theory has approached the analysis of domestic and foreign investment. Portfolio analysis represents a relatively new technique of investment analysis. A discussion of the works that directly apply the intrinsic aspects of portfolio analysis to worldwide investment points in the direction of the approach followed by this dissertation. Unfortunately, the succeeding analysis of inter national capital theory shows a predominate emphasis on only one component of actual international capital flows. This component is called portfolio investment and is motivated by a desire for income. In contrast, the gener ally neglected component is direct investment that is motivated by the desire to control or influence a firm to - 14 - 15 fulfill goals of corporate strategy. Through the control of a firm, plant and equipment expenditures can be directed to achieve corporate goals. A consequence of the focus on portfolio investment is the lack of emphasis on variables important to the investment decisions of a firm. For example, risk and uncertainty are only implicit or completely neglected in models of capital flows. The succeeding discussion will show the general direction and form of international capital theory. The interest rate has a central role in classical analysis as well as more contemporary theory. Classical theory has capital attracted to the country with higher interest rates or favorable interest rate differentials. The world allocation of capital with respect to interest rates, under the assumption of pure competition, maximizes world output as the interest rate equals the marginal product of capital. This results in the flow of capital toward the area of greatest productivity. Contemporary analysis has introduced factor endowments and the inter dependence between trade and factor endowments. Briefly, Heckscher-Ohlin trade theory finds the basis for inter national trade in terms of the different relative factor ■^Peter B. Kenen, "Private International Capital Movements," International Encyclopaedia of the Social Sciences, 2nd ed., VIII, 29-30. endowments of a country. Through comparative advantage a country exports products intense in its relatively abundant factors and imports products intense in its relatively scarce endowments. The strength of the pre vious structure for capital theory is reflected as both "Ohlin and Inversen have applied the general theory of trade and factor movements to movements of capital."^ Thus, differences in relative factor endowments influence both trade and capital movements. A scarcity of capital increases the interest rate, which can cause a favorable interest rate differential, and thereby attract capital. The capital inflows due to interest rates can alter rela tive factor endowments and thus influence trade. To further characterize the direction of inter national capital theory many of the theoretical structures developed earlier for international trade theory are directly applied to international capital theory. For example, a model developed explicitly applies Meade's geometry of international trade to problems of capital 3 flows. The familiar conclusion derived from the analysis 2 Richard E. Caves, Trade and Economic Structure, Harvard Business Studies, Vol. CXV (Cambridge: Harvard University Press, I960), p. 135. ^N. C. Miller, "A General Equilibrium Theory of International Capital Flows," Economic Journal, LXXVIII (June, 1968), 312. 17 is that capital flows from the country with the lower to the country with the higher interest rate. In many other models developed the concept of the optimal tariff on the current account is applied to the capital account. For instance, a model developed by Ronald Jones considers the impact of taxes on both the current and capital accounts 4 to develop an optimum policy. The terms of trade play a key role in many of the capital flow models. The terms of trade for a country are generally defined as the price of exports divided by the price of imports. To begin the analysis let increased foreign investment flow into the borrowing country to expand export production. The increased export production leads to a decline in export prices. Import prices are considered constant as they are generally beyond the con trol of an individual country. Thus, the terms of trade decline causing a decrease in the rate of return on invest ments in the export sector. The previous reflects the 5 general impact of the terms of trade in Kemp’s analysis. A decline in the terms of trade can also reduce the rate ^Ronald W. Jones, "International Capital Movements and the Theory of Tariffs," Quarterly Journal of Economics, LXXXI (February, 1967), 2. ^Murray C. Kemp, The Pure Theory of International Trade (New Jersey: Prentice-Hall, Inc., 1964), p. 193. of return on foreign investments that serve the internal foreign economy. This occurs indirectly as a decline in the terms of trade leads to reduced national income which causes the rate of return of an investment to decline. In this manner the terms of trade has been given a signi ficant place in many models of capital flow theory. Another work that gives emphasis on the terms of trade and investment for the British experience during the 19th century is that of A. K. Cairncross. The prominence of the terms of trade for international capital flow theory suggests the importance of an empirical test of the influence of the terms of trade on the rate of return. The specific hypothesis evaluated in Chapter IV is that the terms of trade are more significant for rates of re turn of less developed regions, and that growth rates have a more significant influence on the rate of return of investments in developed regions. A model by George H. Borts reflects the complexity of the analysis of capital movements as emphasis is placed on the economy under the assumptions of full employment 7 and fixed exchange rates. For this analysis capital ^A. K. Cairncross, Home and Foreign Investment, 1870-1913 (Cambridge: Cambridge University Press, 1953). "^George H. Borts, "A Theory of Long-run Internation al Capital Movements,” Journal of Political Economy, XXII (August, 1964), 341-359. 19 represents a transfer between countries of real goods and services with ownership transferred in the opposite direc tion. The twenty-three equation model is relatively com plex due to its utilization of a production function for exports and another for domestically consumed goods, behavioral relations, and equilibrium conditions. Impor tant characteristics of the model are that the growth rate of labor determines the growth rate of the economy and re sources are allocated between export and domestic goods by world prices. Thus, the model employs both relations in ternal and external to the economy as disturbances that displace the equilibrium are analyzed. While representing an extension of capital flow analysis, the previous theoretical arguments and models possess common weaknesses. They generally view capital movements between countries as the transfer of ownership of homogeneous assets. This formulation greatly restricts applications to analyze real world problems because it does not make the distinction between portfolio and direct inv vestment. Portfolio investment generally transfers assets between countries through loans, bonds, and securities that do not influence control over firms. Portfolio investment approaches a pure capital transfer and is the type of capi tal movement considered by the previously reviewed litera ture. Unfortunately, direct investment that can control 20 plant and equipment expenditures Is more Important for a firm. Moreover, another weakness is the absence of risk and uncertainty in the previous theoretical models. It Is true, that In a great deal of the previous analysis risk and uncertainty are purposefully neglected to simpli fy the analysis. Nevertheless, this simplification greatly restricts the ability of the previous structure to deal with problems faced in reality. As a tool of worldwide investment analysis a model that explicitly considers risk and uncertainty is more helpful than a model that neglects these elements. O A recent article by Hans Brems escapes a lot of the previous criticism as it explicitly analyzes direct investment. The model presented has direct investment flowing between a two country four sector model of firms and households. A commendable aspect of the model is that direct investment represents the movement of money capital as well as technology and management skill. Moreover, the low level of aggregation allows the entrepreneurs of a country to maximize the present value of future profits by allocating investment between the parent company and foreign subsidiary. The model shows that a country can have a net positive balance of direct investment through ®Hans Brems, "A Growth Model of International Direct Investment," American Economic Review, LX (June, 1970), 320-331. a higher propensity to save, better technology, and greater management skill. The framework of the previous model explicitly considers the investment problems of a firm through focusing on direct investment. In the con text of the model resources are allocated rationally. This model and the previous models show that international capital flows are analyzed extensively in the literature. However, a basic problem is the neglect of risk in the analysis. If risk is introduced into the framework of the previous model it could give results with greater application to real problems. In contrast to the general neglect of risk in the previous models from international capital flow theory the succeeding section describes a model that explicitly considers risk. Portfolio Analysis Portfolio analysis is a relatively new area that finds its first application in security selection. Rather than a partial analysis where each security is considered in isolation, the significant new aspect of portfolio analysis is that the interactions of individual securities are considered in relation to the portfolio as a whole. This analysis generally considers the amount of available resources exogenous, or explained outside the system, 22 and focuses on the proportionate allocation between investment alternatives. A portfolio is simply the pro portion of each security or investment from available resources. The great advantage of the portfolio model over other models is the manner in which the rate of re turn and risk are explicitly considered in developing the optimum portfolio proportions. Markowitz set the foundation for portfolio analysis by formulating the basic model as follows:9 n y = E = E y.x. (1) i=l 1 1 9 n n a - V = E E a.,x,x, (2) i=l j=l 1 J The model evaluates the rate of return in equation (1) as the product of the rate of return y^ times the invest ment proportion xi of the ith investment. A summation occurs over the n investments considered. The element risk, neglected by so many other models, is explicitly accounted for by Markowitz's equation (2). Risk is cal culated objectively from past rate of return data as variance when i=j and covariance when i^j. This g Harry Markowitz, "Portfolio Selection," Journal of Finance, VII (March, 1952), 81. 23 dissertation attempts to show that the measure of risk should be expanded when dealing with domestic and foreign investment to consider more properties than just the vari ance of the distribution of rates of return. An objective measure of risk that accounts for the variation and shape of the distribution of rates of return should give a bet ter explanation of worldwide investment. So the model can n give useful results, a proportion constraint E xi=l and a 1=1 1 constraint that allows only positive proportions 0_<x^£l are added. Contemporary applications have modified the basic p model of Markowitz into a form y-Acr where A is an aversion parameter. This dissertation calls the model with the ob jective function p-Ao2 and proportion constraint the simple Markowitz portfolio model. Although the simple Markowitz portfolio model is built on, modified, and extended in this dissertation, this should not detract from the commen dable characteristic that it explicitly accounts for rate of return, and risk measured by variance. An Important aspect of portfolio analysis is how the variance-covariance between rates of return, which is generally neglected by other forms of investment analysis, receives explicit attention. To focus on the important elements of this analysis subdivide covariance ctjj into the correlation of rates of return multiplied by the rate of return standard deviation of the ith investment times the rate of return standard deviation of 24 the jth investment, or briefly c r = Pij0i0j* In this analysis special attention is focused on the correlation of rates of return. The variance of a two asset port- 2 2 2 2 folio model + a2 X2 + ^“2p120l02xlX2^ clearly shows how a negative correlation between rates of return reduces variance. For independent rates of return, when 2 p p 2 the variance increases to tf-[xl + 02x2* In the unf>ortunate case of positive correlation the variance increases to 2 2 2 2 + a2X2 + 2p120l02XlX2' Although positive correlation increases variance, substantial gains through diversi fication are still possible. Gains from diversification occur up to the case of perfect correlation p12=l, particularly if p^2 < ) (where <-a ). This latter case of positive correlation tends to be the prevalent case found in reality. For example, the average corre lation of rates of return for the nine securities con sidered by Markowitz is a positive 0.48.10 Furthermore, a positive correlation occurred between each of the nine securities analyzed. As the previous analysis generally shows positive correlation, a hypothesis to be tested in Chapter V is the existence of positive correlation by tween rates of return of worldwide investments considered "^Harry Markowitz, Portfolio Selection (New York: John Wiley and Sons, Inc., 1959), p. 113. 25 by a firm. Another important concept closely associated with portfolio analysis is the efficient set. Investments or portfolios composed of investments are considered effi cient when they give the greatest return at each level of risk. Risk can be interpreted in a general sense or in the context of the simple Markowitz portfolio model as variance. Conversely, investments are considered effi cient if for each level of rate of return they give mini mum risk. Figure 1 shows efficient set ab that could be generated from the optimization of the simple Markowitz portfolio model as the coefficient of aversion A is incremented. Portfolios P^, P^ and P^ are efficient as they lie on the efficient set and give the greatest rate of return at each level of risk. Portfolio P^ is ineffi cient as it is not a member of the efficient set. The rate of return can be increased with the same level of risk as P^ moves horizontally toward P^. This analysis shows that it is important to construct a portfolio that lies on or near the efficient set. The solid line portion from a to b is referred to as the efficient set in this dissertation and not the dashed segment that is ineffi cient. An implicit assumption of the analysis is that actual portfolios are efficient. If businessmen are to maximize profits, an assumption basic to microeconomic 26 2 y Figure 1. The Efficient Set. theory, they develop investments on the efficient set so they receive the greatest rate of return at each level of risk. This assumption of efficient portfolios is important for the analysis so that given the input parameters of the portfolio model formulated, determinate predictions are possible as outputs from the model. By definition, portfolios beyond or to the right of curve ab are impossible. Shifts in the efficient set can occur due to changes in the input parameters rate of return, variance, and correlations between rates of return. The influence of positive correlation between 27 rates of return to shift the efficient set of domestic and foreign investment is shown in Chapter V. The award winning dissertation, The Investment Decision Under Uncertainty, by Donald Farrar, represents the direct application of the important aspects of port- 11 folio analysis to the study of mutual funds. After a general review of problems related to investment allo cation, a conclusion reached is the basic applicability of portfolio analysis. Mutual funds provide a good area for analysis as data is readily available, they have the resources to make rational decisions, and transaction impedences that restrict liquidity are limited. The first hypothesis tested concerns whether actual mutual fund portfolios resemble optimal portfolios. The analysis of evidence shows support for the previous two hypotheses. An efficient set analysis shows mutual funds positioned near the efficient set, and that mutual funds can be classified as conservative balanced funds, medium risk stock funds, and speculative growth stock funds due to the observation that funds from the previous categories are generally grouped together according to their desire for risk. Thus, the concepts of portfolio analysis were ■^Donald Farrar, The Investment Decision under Uncertainty (New Jersey: Prentice Hall, Inc., 1962). 28 found applicable to the study of mutual funds. Index models represent a form of portfolio analysis that again shows the application directed toward stock market problems. The main reason for an index model is that a security analyst considers hundreds of potential securities for a portfolio and the generation of variance- covariance inputs for Markowitz portfolio analysis is very difficult, if not impossible, to estimate. For example, when N = 1000 securities are considered, this requires (i s(N2-N)) = *199,000 different correlation coef- 12 ficients. To reduce the complexity of analysis the Sharpe index model for the ith investment relates the rate of return to index I plus a random component c 13 by the relation + B^I + ci« This simplifica tion essentially results in a diagonal variance-covariance matrix that greatly reduces the computational task of determining efficient portfolios. However, the difficulty of finding a good index related to worldwide investments restricts applications toward problems of a firm. Further more, the essence of an index model is simplification of portfolio analysis, and in contrast the direction taken by this dissertation is greater sophistication as the 12William Sharpe, Portfolio Theory and Capital Markets (New York: McGraw-Hill Book Company, 1970), p. 118. ■^william Sharpe, "A Simplified Model of Portfolio 29 simple Markowitz portfolio model is built on modified and extended. Thus, to penetrate deeply into properties of the distribution of rates of return demands, a funda mental approach, rather than reliance on simplifying models from stock market applications. A recent article by William Jean points out the l4 theoretical significance of extending portfolio analysis. The general path followed, however, is not new as an article by Gerhard Tintner considers the importance of variance, skewness, and higher moments when choices IS under risk and uncertainty are considered. Furthermore, the work of Farrar discusses the theoretical importance of skewness as the simple Markowitz portfolio model is - i ^ applied to the study of mutual funds. Following stan dard theoretical procedure Jean expands a utility func tion about its mean in terms of a Taylor series to show the importance of moments. In a theoretical context the third moment is analyzed in terms of expectations. Then, a theoretical portfolio analysis is developed for a risk Analysis," Management Science, IX (January, 1963), 28l. ■^William Jean, "The Extension of Portfolio Analy sis to Three or More Parameters," Journal of Finance and Quantitative Analysis, VI (January, 1971), 505-515. ■^Gerhard Tintner, "The Theory of Choice under Subjective Risk and Uncertainty," Econometrica, IX (July- October, 1941), 298-304. 1 6 Farrar, op. cit. , p. 33. 30 free security and a risky security. A consequence of this security market formulation is that only the risky security has moments. A favorable aspect of Jean's ar ticle and the other previously stated works is that the basic path followed in this dissertation represents an area of considerable interest and significance. The Application of Portfolio Analysis to International Investment Rather than analyze the allocation of plant and equipment expenditures of a firm, the preceding discus sion shows that the portfolio model is focused toward security investments. Similarly, applications of port folio analysis to international investment emphasizes the portfolio component of international investment at the expense of studying rates of return and investment allo cation of international firms. The application of portfolio analysis follows two distinct paths. First, an indirect path applies multiple regression to models inspired by portfolio analysis. Rather, than portfolio analysis per se, this technique utilizes proxy variables and a simplified model for empirical specification. A criticism of this tech nique is that the functional form does not represent an explicit solution or application of relations intrinsic 31 to portfolio analysis as least squares is applied to es timate the relations. Second, a direct approach expli citly employs the portfolio model to empirically analyze the diversification of international investment port folios. The context of directly applying the relations of portfolio analysis, efficient set concept, and proper ties of the distribution of rates of return is the approach followed and emphasized in this dissertation. Portfolio analysis is applied indirectly in a study of the portfolio component United States - Canadian capital flows by C. H. Lee.-*-? After describing a two sector portfolio model, the interest rate differential between Canadian and United States rates of return is employed as the basis for a regression analysis. The basic technique applies a simple least squares regres sion where the dependent variable is the stock of foreign securities to total wealth and the interest rate differ ential is the independent variable. A criticism of this analysis is that the interest rate differential emphasized represents only part of the optimal solution to the port folio model. Furthermore, another weakness is that risk is not introduced into the regression analysis. A general 17C. H. Lee, "A Stock-Adjustment Analysis of Capi tal Movements: The United States-Canadian Case," Journal of Political Economy, LXXVII (July/August, 1969), 512-523. 32 criticism of the indirect approach is that a unique relation does not exist between the portfolio model and the selected independent variables that could be cogently established from alternative economic and financial theoretical structures. Throughout this type of analysis explicit references to the significant portfolio concepts of rate of return, variance and correlation, as well as the efficient set are generally neglected. The indirect approach lacks explicit application and utilization of concepts intrinsic to portfolio analysis. Herbert Grubel directly applies portfolio analysis 18 to study international stock market portfolio investment. The author feels that the basic elements of portfolio analysis are a part of orthodox economic theory. In this study rates of return are calculated from stock mar ket indexes for eleven industrial countries composed of the Atlantic Community plus South Africa, Australia, and Japan. An important conclusion derived from efficient set analysis is the greater the number of investment alternatives the greater the rate of return at each level of risk implying an efficient set that shifts to the right. For example, the efficient set for the eleven ^Herbert Grubel, "Internationally Diversified Portfolios: Welfare Gains and Capital Flows," American Economic Review, LVIII (December, 1968), 1299-1314. 33 industrialized countries is to the right of the effi cient set for the Atlantic Community which in turn is to the right of the point representing complete investment in the United States. The methodology of the present article is the direction in which portfolio analysis should strive. Important elements of the portfolio model enter directly into the analysis. The direct approach is applied in the article "International Diversification of Investment Port- 19 folios." Readily available data on common stock indexes for 28 countries for the period 1951-1967 are employed to determine the mean rates of return, variances, and covariances of the portfolio model. The observed negative correlations for rates of return between many of the countries studied Inplies that large potential benefits are possible from internationally diversified stock market portfolios. Positive correlations, so char acteristic of rates of return between national indus tries, are less evident due to the reduced interdependence of a world economy. This contrasting result of the presence of negative correlations between rates of return of international stock market investments emphasizes the 19 Haim Levy and Marshall Sarnat, "International Diversification of Investment Portfolios," American Economic Review, LX (September, 1970), 668-675• importance of testing the hypothesis of the existence of positive correlation between rates of return of world wide investments of a firm. The good return and limited risk allowed the United States’ proportion to dominate the international security portfolio. For example, at mean portfolio rates of return of 0.095 and 0.125 the largest share of a country is the United States with investment proportions of 0.3657 and 0.5106 respectively. The efficient set, however, of the international diver sified portfolio clearly gives advantage over solely investing in the United States. The problems for invest ment in developing countries is reflected by an efficient set that gives less return at each level of risk than effi cient sets established by investments in various other countries. A general conclusion derived from the analysis is that portfolios of investments in both developed and developing countries gives the efficient set with greatest return at each level of risk. This result occurs due to the benefits of international diversification. The direct approach applied in this article makes full use of intrinsic concepts behind portfolio analysis as the mean rates of return, variances, correlations between rates of return, composition of portfolios, and efficient sets for the countries considered are presented exten- 35 The former indirect approach receives limited applications in this dissertation. Only a section of Chapter IV is concerned with this technique. In contrast, the latter direct approach is vigorously emphasized and extended to worldwide investment of a firm. The succeed ing Chapter III will build on, modify, and extend the simple Markowitz portfolio model to deal with domestic and foreign investment problems encountered by a firm. Chapter V will give support to the basic hypothesis that the three dimensional objective function constructed in Chapter III, which considers rate of return, variance, and skewness, is an improvement over the objective function of the simple Markowitz portfolio model presented in this chapter. The direct approach of this dissertation makes full use of important relations intrinsic to portfolio analysis, utilizes the efficient set, and presents input data on the mean rate of return, variance, correlation, and skewness for various investment classifications to fully analyze the distribution of rates of return. CHAPTER III THE MODEL Objective Function The objective function is a key element for for mulating a sound portfolio model. This function intro duces the rate of return, risk, uncertainty, and other variables that are significant for a firm considering investments in various parts of the world. Variables that a firm would desire to increase, such as the investment's rate of return, and variables to be avoided such as an investment's risk, are incorporated into the objective function through coefficients of aversion. Coefficients of aversion allow the trade-off between variables of the objective function. The sophistication of the objective function is reflected by its more complex nonlinear form. Various elements of the nonlinear objective function will be specified in greater detail as the modelling develops. At this point the structure of risk, uncertainty, and intertemporal choice will be introduced into the objective function. Both risk and uncertainty will be - 36 - 37 defined in the context of the work by Prank Knight. Risk occurs when, ...the distribution of the outcome in a group of instances is known (either through past calculations a priori or from statistics of past experience), while in the case of uncertainty this is not true . .. 1 The general character of Knight’s work also emphasizes the importance of judgement in inferring only an approxi mate classification of the outcomes in a set of instances. The case of risk is also associated with judgement of expected outcomes as shown by the following: ...the point we wish to emphasize is that these "risks" do not relate to objective external probabilities, but to the value of the judgement and executive powers of the person taking the chance. 2 Thus, risk exists when the distribution of anticipations is known or can be inferred, and uncertainty prevails when the distribution of anticipations is unknown. Re flecting the usefulness of this structure, a recent article "Stochastic Linear Programming with Application to Planning in India," by Gerhard Tintner and Jati Sengupta, has utilized this basic distinction between risk and ■^Frank H. Knight, Risk, Uncertainty, and Profit (New York: Sentry Press, 1964), p. 233. 2Ibid., p. 365. 38 3 uncertainty. Time is introduced into the objective function through the impact of uncertainty. A necessary defini tion on which to develop uncertainty is the accounting period. An accounting period is the time interval when the revenues and expenses are calculated to derive the profit of an investment. Traditionally, the accounting period is one year as this is the interval to analyze income and expenses. The profit that corresponds to an accounting period will be used to calculate rate of re turn. Use of profit as a cash flow assumes depreciation or non-cash expenses are reinvested in the business. This assumption is accepted for the analysis. Symbolically, tt^j denotes the profit of investment i in accounting period j. In this context profit is an ex ante concept of expected or planned gain. To reflect the initial small or negative profit when the project is first started and the large increase in profit when the investment is fully operating, the profit will generally vary from one accounting period to another. Uncertainty is introduced into the model through ^Gerhard Tintner and Jati Sengupta, "Stochastic Linear Programming with Application to Planning in India," Yearbook of East-European Economics, Vol. I (Munich: Gtinter Olzog Verlag, 1970), p. 190. 39 the concept of the investment period. The investment period is a length of time when the capital expenditures and revenues for the investment are measured in terms of present value. An investment period is composed of several accounting periods. The length or number of accounting periods that compose an investment period is an inverse function of uncertainty. The greater the uncertainty the smaller the length of the investment period. Uncertainty represents an anxiety to the indi viduals analyzing future capital investments as they find it extremely difficult or impossible to evaluate future conditions. This causes a reduction in the length of time for evaluating the investment as the accounting periods further in the future become impossible to evaluate. The rate of return for the ith investment with an investment period of ni accounting periods is defined as 0 = Uil + *12 + ... + X i <3) (l+y ±) (i+hL±) ^ (i+p^)111 The internal rate of return can be employed as an ex ante input for the portfolio model. This input is concerned with the linear element Ey^xi of the Markowitz portfolio model, where denotes the rate of return and x^ the proportion of the ith investment. Thus, time through uncertainty influences the investment period which ko is composed of several accounting periods to have an ultimate impact on the internal rate of return. The length of the investment period can have important consequences for the type of investment under taken. Figure 2 shows an investment period composed of ten accounting periods. Suppose that during the first four accounting periods, profit is negative. There after, the levels of are positive with a slight de cline after the eighth accounting period. Some invest ments with characteristics similar to those illustrated by the previous are heavy manufacturing that produce durable goods and the construction of power plants. Because of the uncertainty of the ith area where the investment occurs, profit of equation (3) is only measured for the investment period composed of ten ac counting periods. Let the rate of return for this invest ment be considered acceptable with an investment period of ten accounting periods. However, if uncertainty increases in the ith area the investment period will fall to, say, five accounting periods shown by the ver tical line at ni = 5 in figure 2. In this instance the rate of return derived will make the investment unaccep table. The large negative values of profit, caused by initial capital expenditures, are only partially offset by a positive profit in the last accounting period. Thus, 41 Profit ni=5 0 Accounting Periods ni=10 ni=5 Figure 2. The Profit of an Investment Suitable for Limited Uncer tainty. H2 investments with the characteristics of figure 2 will generally prove unacceptable when great uncertainty re duces the length of the investment period. Characteristics of an investment acceptable for a limited investment period caused by great uncertainty are small initial expenditures with a rapid positive in crease in profit. This type of investment with an invest ment period ni=5 is shown in figure 3* Generally, mer chandising or market oriented investments that rely on the manufacture of simple consumer goods or imports have characteristics similar to those in figure 3. Uncertainty has been shown by the previous analy sis to have a significant effect on the type of invest ment undertaken. Limited uncertainty is conducive to manufacturing and power plant investments that are con sidered favorable for economic development. In contrast, great uncertainty encourages consumer oriented investments that are not considered conducive to economic development. Thus, uncertainty has a significant impact on the type of investment undertaken in a given area. The previous analysis of the rate of return can be summarized into a functional form P^=G^(y’ i, y_^, t) which will be of great convenience as the development of the portfolio model continues. The rate of return y^ employed at various stages of the analysis is a function Profit ni=5 43 0 Accounting Periods ni=5 Figure 3. The Profit of an Investment Suitable for Great Uncertainty. of the stated independent variables. The Independent variable is the estimated mean rate of return from empirical data that serves as a point of reference. Rate of return y^ is the internal rate of return that reflects the impact of uncertainty through the investment period, as well as estimates of future profit. The independent variable, time, t, will be employed to define the HH logarithmic normal diffusion process. A direct relation ship occurs between the dependent variable y^ and inde pendent variables y^ and For example, an increase in y^ increases the reference point so that changes in ^ can be employed to correspondingly increase y . In areas of analysis concerned with the testing of the port folio model an objective measure of return is desired. The mean rate of return from past data y is an objective measure defined when u_^=0 and t=0 so that y^ = G^Cy^jOjO) = y\ . The elements of the functional relation empiri cally estimated rate of return, internal rate of return, and time will receive varying degrees of emphasis in the succeeding chapters to suit the needs of that area of analysis. The previous analysis has been concerned with the linear element Zyix^ of the portfolio model. Uncer tainty influenced the rate of return y^ through the length of the investment period. Risk will be introduced into the portfolio model through the nonlinear element SEoijxixj where for the case when i=j, a.. is the variance of the ith investment and when 1?j, denotes the covariance between the ith and jth investments. In the fortunate case, when data on the rate of return of previous similar investments are available, risk can be roughly inferred by calculating the variance from past data. However, the measure of risk through calculating the variance of the rate of return has many shortcomings. First, the problem of a small sample size reduces the significance of the calculated statistic. For example, the limited time period covered may not allow the data to measure the impact of important variables influencing investment rate of return. For instance, a period of relatively stable foreign exchange rates, improvements in the terms of trade for the output of an investment, and a rapidly growing economy that an investment serves will show relatively stable data with a possible improve ment in the rate of return. Yet, a longer time interval where the previous variables can become adverse will in duce great changes in the observed rates of return. Second, the major problem of relying solely on calculations from past data is the implicit assumption that conditions which held in the past will also hold in the future. This assumption is especially confining when considering domes tic and foreign investment as the world environment is undergoing constant change through both political and economic forces. Thus, agreeing with Markowitz, the statistical calculation of variance s£ is a first step to be modified through the consideration of neglected 46 Z | conditions. This first step serves as a foundation which reflects past conditions in the estimation of risk. Risk employed as an input into the portfolio model will 2 2 2 be given a general functional definition a^ = P^(s^,a^t), where s| is the variance calculated from past data, p a£ is a measure of the expected future risk, and t is time. Time t will be used primarily to define the logarithmic normal diffusion process. For the previous functional relation changes in the independent variables induce changes in the dependent variable. For example, 2 if future risk is expected to increase relative to 2 2 calculated s., then would be correspondingly increased. P ? 2 A direct relation occurs between cr£ and where s£ serves as a pivot point. In areas of the dissertation where an objective measure of risk is desired, then a^=0 and t=0 so that = F^(s^, 0, 0) = s^. This ob jective measure will be important for empirical testing of the portfolio model. The functional specification of risk allows a convenient summary of elements that will be further utilized as the analysis proceeds. The previous analysis has added sophistication to the Markowitz portfolio model so it can better deal ^Harry Markowitz, "Portfolio Selection," The Journal of Finance, VII (March, 1952), 77-91. with problems of worldwide investment. This model has to its credit an objective function with a linear element for the rate of return and a nonlinear element that mea sures risk in the form of the variance-covariance of rates of return. This objective function is a significant improvement over the commonly assumed criteria of maximi zing the mathematical expectation which Marschak has 5 shown is not always the proper criteria. The Markowitz portfolio model, however, does not go far enough, as it neglects important properties of the distribution of re turns. What is required, as shown in an article by Gerhard Tintner, is a preference function that accounts for "the total shape of the probability distribution."^ In this article the mathematical expectation, variance, skewness, and even higher moments are accounted in the preference functional. A significant step in the direc tion of this previously stated article is to explicitly in troduce skewness, or phrased alternatively, deviations from symmetry of the distribution of returns into the objective function of the portfolio model. Although 5 J. Marschak, "Money and the Theory of Assets," Econometrica, VI (October, 1938), 320. ^Gerhard Tintner, "The Theory of Choice under Subjective Risk and Uncertainty," Econometrica, IX (July-October, 19*11), 301. 48 interest has been expressed in the importance of skew ness on the portfolio model, theoretical and empirical applications have been limited. For instance, Farrar only mentions the importance and applicability of intro ducing skewness or the third moment into portfolio ana lysis, but does not apply skewness to his. analysis of 7 mutual funds. More recently, Hirshleifer in his des cription of portfolio analysis briefly discusses the significance of skewness without explicitly showing its 8 importance. A theoretical formulation by Stone has, however, utilized the third and higher moments in his Q formulation of a market equilibrium portfolio model. Unfortunately, in this latter work, empirical applications are absent. Thus, the explicit introduction of skewness into portfolio analysis is an area that demands greater theoretical attention and explicit empirical application. A significant aspect of the objective function formulated in this dissertation for the portfolio model 7 Donald E. Farrar, The Investment Decision Under Uncertainty (Englewood Cliffs: Prentice Hall, Inc., 19b2), p. 33. 8 J. Hirshleifer, Investment, Interest and Capital (Englewood Cliffs: Prentice Hall, Inc., 1970), pp. 28l- 283. ^Bernell Stone, Risk, Return, and Equilibrium (Cambridge: The M.I.T. Press, 1970), pp. 19-22. is the introduction of a measure of deviations from sym metry for the distribution of returns. Before the theore tical formulation is specified, an examination of the areas where symmetry is significant and the measure to be employed will be explored. For symmetrical distri butions, a model formulated solely in terms of mean return and its variance-covariance is adequate. For example, the mean and variance completely define a normal distri bution. The normal distribution is symmetrical implying no skewness and thus a measure of deviations from symmetry is unimportant. Yet, when a distribution lacks symmetry and possesses skewness the skewness should not be ignored, but measured. The measure of skewness employed is the third moment in dimensionless form, Ni - z (Yik-yi) M3± = k=l_______ 1____ Nlsi where y ^ is the ith investment's kth rate of return observation, y^ is the mean rate of return for the ith investment, Ni represents the number of observations of the ith investment, and the ith investment's standard deviation is s^. The introduction of a measure of skew ness into the portfolio model represents an increase in sophistication that greatly improves the ability of the model to deal with problems of real investment. 50 Skewness can be summarized functionally in a man ner similar to the rate of return and variance.10 Let the dimensionless third moment be defined functionally as M = E.(M3., M3.,) where M3, is the expected future i 1 1 i third moment. A direct relationship occurs between the independent and dependent variables of the previous func tional relation. For example, a given value of M3^ with a decline expected in M3i will result in a lower value of M.». When M3 = 0 then M = M3 = E.(M3., 0) i i i i i and the empirically calculated dimensionless third moment is employed exclusively. Applications of the portfolio model will show the usefulness of this functional speci fication . To represent variance and dkewness for the distri bution of rates of return a Taylor series characterization will be employed. Let the Taylor series E(U(x)) = U(y) + U'(u)E(x-u) + U"(u)E(x-y)2 21 + U"1(u)E(x-u)3 + .... 3! 10Until this point in the analysis when speaking of the simple Markowitz portfolio model risk and variance were treated synonymously following accepted convention. However, when skewness is introduced the whole term that reflects the variance and skewness of a probability dis tribution is called risk. This specification is consis tent with the previous discussion as all terms other than rate of return are specified as risk. 51 characterize an investor's expected utility E(U(x)) of money x about its mean y. As the first moment is zero, E(x-y) =0, and the previous relation reduces to E(U(x)) = U(u) + U"(u)E(x-y)2 + U'" (u)E(x-y)3+... 2 ! 3 . ' This characterization is important as it shows the sig nificance of higher moments. In particular, the moments of greatest interest are the second moment, variance, which has received considerable attention in the litera ture, and the third moment, skewness, whose importance is shown by this dissertation for domestic and foreign investment. The objective function of the portfolio model that accounts for skewness as well as variance can be expressed as Z = y- Act + BSm< Following standard notation for n investments, n y = E y-jx., i=l and where 0 n n a - E E i=l j=l °ijxixj aij pij0iaj ’ For the rates of return of the ith and jth investments a^ and denote standard deviation and represents 52 correlation. The coefficient of variance aversion is denoted by A. The measure of skewness of a portfolio is introduced operationally as n BS = £ B4M4x i=l m iii where B^ is a coefficient of skewness aversion for the ith investment and M the dimensionless third moment. An i assumption of the analysis is that efficient investment proportions x^ are generated as A and B are adjusted for the optimized objective function. This formulation in troduces uncertainty through the effect of the investment period upon the rate of return y and risk through both 2 variance Ac and skewness BS . The objective or pre- m ference function is explicitly formulated when coefficients of variance aversion and skewness aversion are explicitly stated at given levels. This formulation of the objec tive function embraces more variables than previously and concentrates on variables of importance for the analy sis of investments throughout the world. The previous formulation implies that investors prefer positive skewness, as the coefficient of skewness aversion B.^ is positive. An investor's preference for positive skewness, as shown by Hirshleifer, is generally accepted in economics. However, a discussion of posi tive skewness and world investment is appropriate. Figure 4 shows a positively skewed distribution of rates of return with mean, median, and mode. The median is the central point as one half the distribution lies to the right and left of the median. Vertical line ab divides the horizontal rate of return axis into positive and negative rates of return. For the positively skewed distribution there is only a relatively small probability measured by the shaded area from d to line ab, of a nega tive rate of return. However, if the distribution were rotated about its median so that it would become nega tively skewed the probability of a negative rate of return would greatly increase. Thus the context of the figure suggests that a dislike by investors for a nega tive return causes a preference for positive skewness. Whether positive skewness shown by figure 4. exists or not for data generated by reality is an important hypothe sis to be tested. This hypothesis of the existence of positive skewness will be tested in Chapter V which deals with empirical analysis of the portfolio model. The impact of adding skewness to the efficient set of greatest return at each level of risk expressed l-'-Hirshliefer, op. cit. , pp. 282-283. Frequency 54 Rates of Return positive return negative return a Figure 4. A Positively Skewed Distribution of Rates of Return. in terms of variance and skewness for the three dimensional preference function is illustrated effectively by a surface. Figure 5 shows a three dimensional graph of skewness, variance, and rate of return. If the measure of skewness is neglected the efficient set is represented in only two dimensions of variance and rate of return. The two dimensional preference function can greatly con strain the application of the portfolio model when working with rate of return and variance inputs calculated from empirical data of foreign and domestic investment. The efficient set of the three dimensional preference function shown in figure 5 represents an effort to allow the port folio model to measure properties of the total distribution of rates of return that are neglected by the simple Markowitz portfolio model. The importance of this exten sion will be tested by the basic hypothesis that the three dimensional objective function of rate of return, variance, and skewness represents a significant improve ment over the two dimensional objective function of the simple Markowitz portfolio model. The power of the sophis ticated three dimensional objective function lies in its ability to analyze a greater number of variables signi ficant to investments. To represent the three dimensional objective function in two dimensions the efficient set should shift Variance 56 Rate or Return Skewness Figure 5. A Three Dimensional Efficient Set. with different degrees of skewness. The impact of skew ness to shift the efficient set is completely lost by the preference function of the simple portfolio model. For example, figure 6 describes an efficient set for invest ments with symmetrical rate of return distributions by curve ab. In this case both the sophisticated and simple portfolio models will generate the same identical effi cient sets as the dimensionless third moment is zero. However, when skewness is present the simple objective function neglects it while the sophisticated objective function takes full account of skewness. Following 57 V ar i ance— Skewne s s a Rate of Return Figure 6. A Two Dimensional Representation of the Efficient Set when Skewness is Present. standard economic representation the impact of changes in skewness in this two dimensional figure cause the curve to shift. When positive skewness is present, as it represents a benefit, the efficient set shifts to the right. Conversely, when negative skewness occurs, which represents an adversity, the efficient set shifts to the left. Figure 6 characterizes the case of positive skew ness as the efficient set is shifted to the right by curve a’b1. Only the three dimensional objective function can reflect the impact of varying degrees of skewness. Thus, the two dimensional objective function represents a special case of the general three dimensional objective 58 function. Investment Search To further develop the analysis of worldwide investment significant aspects of investment search will be considered. The search for investment opportunities is important as it is the basis for future operations of the firm. Errors in the appraisal of investment oppor tunities will not only adversely affect present operations, but will also reduce the future opportunities of the firm. Search implies the location of both markets and resources that can be profitably developed by utilizing the resources of a company. Resources imply not only plant and equip ment, but management technique. The search that evaluates both present and especially future market conditions is very important because it will influence the estimate of profit of the ith investment during accounting period j. When profit is estimated the length of the investment period must be considered so that the internal rate of return y^ of the investment can be determined. For a given class of future investments the factors that Influence the extent of searching the population rates of return should be considered. Also important for investment search is the value of experience with pre vious investments. An obvious influence on search is the 59 cost of search for a given class of investments. The analysis of the extent of search, value of past infor mation, and search cost will be introduced into the portfolio model by considering the rate of return net of search effects. The large number of possible investment altera tives confronting a firm implies a lack of complete in formation on rates of return. Therefore, a search of investment rates of return becomes necessary. Figure 7 shows the rates of return for distributions I and II. The dispersion of distribution II, a^, exceeds distribu tion I, a-j^, although both have the same mean r. Let the search begin at the mean rate of return for both distributions. By searching, the goal is to acquire infor mation on higher rates of return. As the sequence of search continues, progressing to two standard deviations from the mean, the rate of return from distribution II exceeds that of distribution I as r + 2a2 > r + 20^. Thus, the greater the relative variation, ceteris paribus, the greater the benefits to investment search. This example shows that the extent of investment search is directly dependent on the relative variation of the rate of return of an investment. Relative variation can be measured effectively by the coefficient of variation of rate of return for a given category of 60 Frequency Frequency 2a r Rate of 2 o r Rate of Return Return Distribution I Distribution II Figure 7. A Comparison of Relative Variation and Investment Search investments. The coefficient of variation is the rate of return standard deviation divided by its mean. A hypothesis suggested by the previous analysis, and to be tested in the succeeding chapter, is that foreign indus tries have greater coefficients of variation for rates of return than American industries. The experience of a firm with previous investments, whether domestic or foreign, will have a great impact on investment policy. The approach to be followed is to measure the information content of data on past rates of return. A similar approach has been suggested by Stigler for analyzing the value of information of previous wage 61 12 offers. However, no real effort has been devoted to the appraisal of the value of previous rates of return for worldwide investments. A simple autocorrelation model y.. = a + X.y that regresses a current rate it i i it—1 of return y for the ith area on the rate of return for a previous period y^t-I gives a good measure of the value of past information. In the model the rates of return are calculated net of search costs and taxes. Following Tintner the autocorrelation between ylt and y^t_-^ will be measured by the autocorrelation coefficient L " I Q which for large samples becomes r = A.. The first L 1 order autocorrelation coefficient r^ is defined with the lag L is one and higher order autocorrelation coefficients are specified by greater lags. Higher order autocorrela tion coefficients are calculated under the assumption that the time series follows the first order stochastic difference equation.The absolute size and sign of the coefficient X^, which generally should be positive, shows the magnitude and direction of impact of successive rates ■^George Stigler, "The Economics of Information," Journal of Political Economy, LXIX (June, 1962), 118. ■^Gerhard Tintner, Econometrics (New York: John Wiley and Sons, Inc., 1965), p. 256. 121 Ibid., p. 259. 62 of return. The information content of previous rates of return is directly reflected by the value of X^. A hypothesis developed from the previous analysis is the more stable the area the greater the information content of rates of return from previous investments. For instance, in an area characterized by great change, rates of return based on past conditions can only have limited value in estimating the rate of return derived from new conditions. In this case the unstable area should have a smaller positive value of X^ than a more stable area. The suc ceeding chapter where the hypothesis is tested will fur ther specify the areas of application. One of the most obvious influences on investment search for an area is the cost of search SC^ in the ith area. Search cost is composed of two dimensions, business expenses and time. The more important business expenses for search are the cost of travel for an on the spot investigation, executive salaries for the investigation, and operating expenses to conduct the investigation. The costs of just on the spot investigation has generally been estimated from $15,000 to $100,000 and in some cases 15 greater than $300,000 for the part actually carried out. ■^Yair Aharoni, The Investment Decision Process (Boston: Harvard Business School Division of Research, 1966), p. 109. 63 In general, the business expense for foreign search is greater than domestic search due to the unfamiliar en vironment. The second dimension of search cost is related to the time required to acquire a given amount of infor mation. A concise statement of the cost relationship is the following: ...dissemination and acquisition (i.e., the production) of information conforms to the ordinary laws of costs of production — viz., faster dissemination or acquisition costs more. 16 Characteristics of information transmission in the foreign environment are time delays in making telephone calls, slower transportation, problems of interpreting foreign laws, and difficulties in communication between cultures. These factors imply that the time element of search cost should be greater in the foreign environment than the domestic. Thus, search cost should generally be greater for foreign investment than domestic invest ment . Search will be introduced into the portfolio model through a rate of return that reflects the impact of investment search in the ith area. Investment search is carried to the point where the marginal benefits Armen Alchian, "Information Costs, Pricing and Resource Unemployment," Western Economic Journal, VII (June, 1969), 110. 64 of search related to the relative variation and the value of past information equals marginal search cost. Symbols useful for a functional representation are relative variation the information value of past infor mation X^, and search cost SC^. Expressed functionally, the rate of return that considers investment search effects is yis = Fis^si ^ i * SCiJ yi^» where (sVy. ), X represent direct relationships, SC i l l i has an inverse relationship, and is the rate of return described by the functional relationship of the previous section that serves as a reference point. The purpose of p^s is to show the importance of factors that in fluence the estimates of rates of return for real invest ments. Rates of return that reflect the elements of investment search are useful for developing corporate investment analysis. Taxes Tax policies relevant to both domestic and for eign investment are considered from the vantage point of their impact on the investment allocation of a firm in the context of the portfolio model. This approach can be contrasted with an analysis of tax laws of different 65 countries with the objective of benefitting from tech nicalities of the tax laws. Here, interest focuses on taxes and their impact on the portfolio model. The introduction of taxes entices the firm with both domestic and foreign investments to distribute the income between its system of branches and subsidiaries in a manner that reduces the tax impact on the whole system. Important aspects of United States and foreign tax law influences profit tt which in turn influences ij the rate of return of the portfolio model y or Uis. The income received (dividends) from the subsidiary and not the total profit from the subsidiary influences taxes 17 in the United States. Thus, a multinational corpora tion may find it advantageous to retain earnings in the foreign subsidiary. For a lower foreign tax rate the income tax rate paid to foreign governments is generally deducted from the tax rate liability to the United States. Thus, a foreign "tax holiday" in many cases only reduces the taxes paid to the foreign government and makes more returned income liable to the United States tax law. When foreign tax rates exceed the United States tax rate great difficulties are encountered in trying to ^Corporations PaY taxes on only 15 per cent of the dividends from a subsidiary. 66 regain the added tax. Although differences in treatment exist, the goal of the United States tax law can be considered as equitable treatment of domestic and foreign investment. Tax benefits, however, exist for investment in less developed countries and thereby promotes economic development. For example, a corporation that can quali fy as a Western Hemisphere Trade Corporation with a majority of its business in the less developed areas of North and South America through special tax deductions can experience an effective tax rate of 34 per cent when 18 regular corporations pay 48 per cent. In general, for less developed countries, when the United States tax rate is greater than the foreign tax rate, the tax on income from the foreign subsidiary "is computed by multi plying (1) the difference between those rates by (2) the net income after foreign taxes received by the parent Consider the example of a tax rate of a less developed country one half the United States tax rate tuS that ^Arthur Stonehill, ed., Readings in International Financial Management, "U.S. Corporations Doing Business Abroad." (Pacific Palisades: Goodyear Publishing Company, 1970), p. 83. 19 Raymond Vernon, Manager in the International Economy. (Englewood Cliffs’ ! Prentice Hall, Inc., 1968), pp. 212-213. reduces the aggregate United States and foreign tax by (1/4)(t£s)(I), where I Is before taxes subsidiary profits for basing remissions to parent.20 The tax advantages for less developed countries have an important impact on the efficient set of the portfolio model. Investments in less developed countries have the typical character of greater risks and greater rates of return. The higher return to an American cor poration is due in part to the tax advantages. The efficient set abc of figure 8 represents the case where tax advantages do not exist for less developed countries. With the existence of tax advantages for less developed countries, the efficient set shifts and rotates to ab'c* due to the greater proportion of investments in develop ing areas with their higher risks and rates of return. The investments of greater risk and return enter the solution of the portfolio model at greater values of variance and return in figure 8. The impact of tax policy is introduced into the portfolio model by con sidering and net of taxes. The empirical data reflects the differential tax policy for developed and less developed countries. Thus, the existence of tax advantages toward less developed countries shifts and 20Ibid., p. 213. 68 2 o ^is Figure 8. The Impact of Differential Tax Policies for Developed and Less Developed Countries. rotates the efficient set to the right and increases the proportion of investments in less developed countries. Constraints The purpose of this section, like the previous sections, is to modify the simple portfolio model so that it can cope with the problems faced by a firm with both foreign and domestic investment. Reality places bounds of a managerial and political nature on the previously developed sophisticated objective function. This sec tion will formulate constraints for the model that reflects bounds on the policy freedom of the firm. The introduction of constraints is a good point to show how the model can account for various forms of corporate structure. If the model is to be a useful tool of analysis it should have the flexibility to repre sent the major forms of business organization. When analyzing the investment decisions of a worldwide corpora tion the forms of organization of the international division, worldwide product structure, and geographic structure are prevalent.21 Figure 9 utilizes flow charts to describe the three previous forms of corporate organi zation in tie context of traditional line and staff management. The subscript i that usually defines the two sectors domestic and foreign investment of x^ and other coefficients can be formulated to specify the forms of organizational structure for the portfolio model. For example, when the international division that controls all foreign operations is one of many operating divisions the subscript i is defined with respect to each operating division. The subscript i would be defined from i = 1, 2, 3, ..., p when the international division structure of figure 9 is modelled. To model the worldwide product 21Gilbert Clee and Wilbur Sachtjen, "Organizing a Worldwide Business," Harvard Business Review, XLII (November-December, 1964), 57-64. International Division Structure 70 Line Management Domestic Division 1=2 Domestic Division i=l International Division i=p Corporate Staff Worldwide Product Structure Worldwide Line Management Product B 1=2 Product A i=l Product Q Corporate Staff Geographic Area Structure Geographic Area Line Management, Africa 1=2 Europe 1=1 Asia i=u Corporate Staff Figure 9* The Major Forms of Corporate Structure for an Internationally Oriented Firm. 71 structure of the figure, i denotes each product line that is marketed throughout the world. The flow chart shows i = 1, 2, 3, q product lines. A geographic struc ture is modelled by defining i = 1, 2, 3, • > u where i denotes each of the managerial geographic regions. The two sector formulation of domestic and foreign investment can be considered a geographic managerial structure. When greater complexity is added to the portfolio model the elements of subscript i can be defined with respect to the various forms of corporate structure. Thus, by specifying the subscript i, the various types of organi zational structure can be modelled into portfolio analysis. Control of a vast network of foreign investments serves as a powerful constraint when the firm varies its investment proportions. The profitability and risk of foreign investment may cause a firm to desire increasing this proportion of investment. However, a constraint on the control and coordination of foreign investment can restrain a firm from achieving its desired investment proportions. The addition of the constraint, akJxj - bk allows the model to reflect managerial problems of control and coordination. The subscript k denotes the type of corporate organization and j describes the investment. The coefficient akj signifies the cost of controlling changes in the proportion of the jth investment. 72 Coordination for a network of investment proportions is reflected by coefficient b^. Different forms of corporate organizational structure k and managerial policies re quires different values of a^ and b^. For example, a firm with a small proportion of foreign investment due to its reliance on licensing agreements and export-import activities would have one set of values for control and coordination. A firm with more extensive foreign invest ment with a corporate structure based on the international division k=l will have different coefficients of control and coordination. By changing from the previous form to a form of corporate structure lying on the range between organization with respect to geographic area k=2 or according to the type of product line k=3 a firm can con trol a larger proportion of foreign investment. The constraint of control allows the model to reflect how "a firm removes a constraint to continued growth abroad p p when it abandons its international division.' The abandonment of the international division toward a more integrated form of corporate structure, such as product line, can be reflected in the model by increasing the coordination coefficient bk and thereby allowing greater ^^John M. Stopford, "Growth and Organizational Change in the Multinational Firm." (unpublished D.B.A. Thesis, Harvard University, 1968), p. 84. 73 foreign investment. If superior management policies are found for controlling investments in the jth sector as well as a change in organizational structure, the model can show this by reducing the control coefficient a^.. Thus, the control constraint allows the model to reflect problems encountered by a firm in the coordination and control of worldwide investments. Investment proportions can be constrained by the physical process of production. As an example, let pro duction of one unit in the jth sector require g^ units of output from sector h. Assume that equal dollars of investment are necessary to produce a unit of output for sectors j and h. Then investment proportions between sectors are related by the constraint x^ = Sixh* This constraint can be generalized when the jth investment proportion is linked to the output of several other investments by a linkage constraint xj = slxh + S2xh+1 + • • • + V w - 1 + aaj • The parameter GG represents a minimal required propor- tion. For some cases an inequality rather than equality for the previous constraint may be more appropriate when production-investment linkages are not rigidly fixed. The linkage constraint allows the model to capture im portant interrelationships between production and invest- 74 ment. Unlike the previous constraints that were con cerned with problems internal to a firm, the constraint to be introduced is external to the firm as it is based on political considerations. The influence of political considerations on investments cannot be denied. For instance, the nationalization of United States oil interests in Mexico during the 1930’s due to the dislike of foreign domination of the petroleum industry shows the significance of political constraints. Today especi ally, with threatened nationalization of oil in the middle east, current nationalization of foreign investments in Chile, and Canada's concern for the extent of United States direct investment show the importance of politics. A constraint to reflect the impact of the political environment on the investment of a firm is d. < x. < e . 3 ~ 3 “ j The lower bound on the jth proportion dj can represent a lower limit that a foreign government may not want the jth investment proportion to fall below as this could retard plans for economic development. For a firm to receive tax benefits and subsidies from a foreign govern ment it cannot go below the lower bound d^. The upper bound e is a proportion of the jth investment that J cannot be exceeded. For example, a foreign government may not want too great an amount of investment in the jth 75 sector and will enforce this upper bound with the threat of nationalization. The previously developed constraint allows the introduction of political realities into the model. The Complete Model This section will synthesize the individual ele ments previously developed into a complete model. The analysis will show how the added sophistication to the Markowitz portfolio model greatly improves the ability of the model to deal with problems of worldwide invest ment . The previously developed objective function and constraints formulated individually are shown together in Table I to define the improved portfolio model. Optimization of the objective function is achieved through incrementing the coefficients of risk and skewness aver sion subject to the constraints. For different values of the coefficients of risk and skewness aversion an optimum solution of x occurs subject to the constraints. The J constraints shown in Table I are expressed generally so that they can be modified or amended to fit the needs of worldwide investment. However, the nature of the solution to problems of real investment requires that the proportion and positive proportion constraints are always 76 TABLE I The Complete Model Objective Function Maximize Z = p -Acr^ + BS s m n n n n Z = £ p-f^x.-A £ £ a..x.x,+ £ B.M.x J-l JS 3 l-lj-l 13 1 3 1=1 1 1 * Constraints n Control £ a, .x._ < b. J=1 kj j - k where k denotes a form of organizational structure . < m Linkage Xj - ^ Political Bounds d . < x < e. j - j “ J n Proportion £ x = 1 j=l J Positive Proportion 0, <_ x^ _< 1 for all j 77 binding so that meaningful solutions are obtained. The added sophistication reflected in the objective function and constraints should greatly improve the capacity of the model over the simple Markowitz portfolio model to more effectively deal with domestic and foreign invest- 2? ments. J A graphic Illustration will more clearly show the properties of the optimal solution to the sophis ticated portfolio model. Figure 10 illustrates the con straints and objective function for the complete model shown in Table I. The line pp denotes the proportion constraint. Lines dd and ee represent political bounds on Xp and x^. In the case shown, bound dd is effective on Xj and ee does not restrain the proportion of x^. The control constraint is shown by line cc and Is binding where it cuts the proportion constraint pp and political bound dd. An effective linkage constraint x^ £ S^x2 is denoted by line HZ. The feasible region for solutions is shown by the area within the bounds of 0, 1, 2, 3, arid 4. The feasible region implies that the positive propor tion constraint is satisfied. To obtain an optimal solu tion within the feasible region the objective function 23 JFor empirical work reference to the sophisticated portfolio model refers to the objective function and proportion constraint for a two sector model. 78 b 19 ft H i H I t! •ftli» j J I ; ; » r »TJ JJ iJ fJ j rT TT I * / t m Figure 10. An Illustration of the Solution to the Complete Portfolio Model. 79 must be considered. The straight lines E^E^ represent the return and the ellipses on the axis aa-bb 2 denotes the risk Aa of the Markowitz portfolio model. As the coefficient of risk aversion increases the size of the ellipse increases. Tangencies between lines E^E^ 2 and Aa define a critical path fQfi of optimal portfolios. Once a constraint is reached the critical path follows the constraints in a direction of increasing return. For the simple Markowitz portfolio model the critical path is f0f^2,3. The previous solution only allows a maximum proportion for the first investment of x°. This solution, however, for many cases is inadequate as the actual proportion exceeds x°. A corresponding proportion on the efficient set cannot be determined when the input parameters of the model are taken as given. However, a property of the distribution of rates of return not ac counted for by the simple Markowitz portfolio model can be utilized to generate an efficient investment propor tion. This is achieved by introducing skewness through BSm into the objective function. The addition of BS^ to the objective function shifts the axis of the ellipses to a'a’-b'b1. This occurs as BS influences the axis m of each ellipse, but does not effect the obliqueness or form. Then for a given measure of skewness the critical line path becomes f£,f^,2,3. This increases the maximum proportion of investment one in an efficient portfolio from x° to x|. The ability to increase the maximum proportion of an investment can become very significant3 and this ability depends on the calculated values of variance and skewness. A hypothesis developed from this analysis is that the introduction of skewness into the simple portfolio model greatly improves the ability of the model to deal with the analysis of worldwide invest ments . CHAPTER IV ECONOMETRIC ANALYSIS Relative Dispersion This section utilizes a simple technique to test the hypothesis suggested in the previous chapter that foreign rates of return show greater relative variation than American rates of return. Although the applications of econometrics in this and other beginning sections are relatively simple, the degree of complexity gradually increases as the analysis of a stochastic process is approached. The collection of data on rates of return is the primary obstacle to testing the hypothesis of relative variation. Fortune serves as the source for the compon ents of measuring rate of return as earnings divided by total assets for cross section data of the years 1963, 1964 and 1965 for both foreign and American firms.1 A 1Fortune, "Fortune's 500 and 200," (Chicago: Time Inc., July-August, 1964; July-August, 1965; and July- August, 1966). - 81- basic problem encountered is the lack of comparability between American and foreign accounting techniques. Fortune states that, "the problem of comparability arises, of course, out of the differing methods of financial reporting in different countries, and even within some 2 countries." The extent of the problem of comparison is shown by the fact that domestic rates of return gener ally exceed foreign rates of return. Thus, errors of measurement are present in this measure of rate of return. This problem, however, can be overcome by employing a measure that reflects relative variation, rather than greater variation due to the larger absolute size of the variable. The coefficient of variation, which is the standard deviation of the variable divided by its mean, overcomes the problem and allows a test of the hypothe sis concerning relative variation. Relative variation is measured for nine industries that follow the standard industrial classification. Thirty observations were taken for each industry, except for cases where only a few observations were available, such as foreign rubber with fourteen observations, domestic rubber with twenty-nine observations, and p Fortune, "The Fortune Directory," (Chicago: Time Inc., August, 1964), p. 153. domestic machinery with twenty-four observations. There are a total of 517 observations for the cross section data of nine foreign and domestic industries. Table II presents the results for testing the hypothesis that foreign industries have greater relative variation than American industries. For the nine industries, Table I shows the foreign and domestic coefficients of variation. The hypothesis is tested by the last column of Table I that shows the foreign coefficient of varia tion divided by the domestic coefficient of variation. When the ratio of foreign to domestic coefficients of variation exceeds one the hypothesis is supported. The results of Table I give general support to the hypothesis as the ratio of foreign to domestic coefficients of varia tion exceeds one in seven of the nine cases. Food products and transportation equipment are the only industries that do not confirm the hypothesis. The results from this section support the hypothesis that foreign industries have greater relative variation of rates of return than American industries. 8.4 TABLE II Relative Variation Industry Coefficients of Variation Relative Foreign (s2/y2) Domestic (si/?i) Variation s2yi 81*2 Pood products .3117 .4796 .6499 Paper and allied products .5752 .2733 2.1046 Chemicals .4882 .2818 1.7321 Rubber products .4378 .2619 1.6716 Primary and fabri cated metals .6432 .5357 1.2007 Machinery, except electrical .7611 .4755 1.6008 Electrical machinery .5126 .4176 1.2274 Transportation equipment . 66l6 1.8016 .3672 Mining and petroleum .8069 .3361 2.4005 Indirect Portfolio Analysis Some indirect portfolio analysis is presented in this section. A regression on the dependent variable, the proportion of foreign investment, is run for inde pendent variables concerned with the mean and variance of rates of return. This technique can be criticized as it only represents an average least squares fit be tween dependent and independent variables, rather than intrinsically employing the properties and relations of the model formulated by Markowitz. Moreover, the de pendent and independent variables chosen for the regres sion are taken from general business and economic theory rather than portfolio analysis per se. However, indirect portfolio analysis is presented so that a complete coverage of techniques applied to worldwide investment are presented. The same basic problems of errors of observation exist for the data in this section as in the previous section due to the utilization of the same basic cross section data for 1963, 1964, and 1965 to measure mean rate of return and variance. To overcome these problems relative measures of rate of return and variance are applied. In contrast, the measure of the foreign in vestment proportion represents a good measure as the 86 Department of Commerce conducted accurate surveys of the nine industries for the years 1963, 1964,^ and 1965.** The first indirect portfolio analysis runs a regression on the nine industries of Table II. The dependent variable p is the average investment proportion for the years 1963, 1964, and 1965. For the cross sec tion data over the period 1963, 1964, and 1965 the mean and variance are calculated. Relative measures of the independent variables are employed. The first independent variable is the ratio of the mean foreign to the mean domestic rate of return (y /y^). Similarly the second independent variable is the ratio of the foreign variance 2 9 to the domestic variance (s /st). It is assumed that the 2 1 influence of investment search is reflected by the third independent variable, the ratio of the foreign to domes tic coefficients of variation (SgjF /s^yg). The problem of multicolinearity is a possibility as the ratio of the coefficients of variation have elements of the other independent variables. If this problem becomes serious, the program for calculating the regressions will not operate. As a criterion for evaluating the regression ^Department of Commerce, Survey of Current Business (Washington, D.C.: Office of Business Economics, United States Department of Commerce, October 1964), p.11. ^Ibid., September 1966, p. 33. results, the coefficients must agree with what would be expected from economic theory. Economic theory would dictate that a direct relation holds for (y^/y^) as greater relative foreign return should increase the for eign investment proportion. As the variance of foreign rates of return increases relative to domestic rates of return the foreign investment proportion should decrease. This can be reflected in the regression equation by a negative coefficient before the independent variable 2 2 (s /Bi). A direct relation should occur between invest ment search, reflected by relative variation, and the proportion of foreign investment. The Cochrane-Orcutt autoregressive transformation is applied to the data. In the following equation, and subsequent equations that follow, the t values are shown in parentheses under their respective coefficients. Results of the regression are shown as follows: p= -.3166 + .6190(^/3^)-.1330(s2/sJ-) + .2l84(s2y1/s1y2) (2.38) (-2.38) (2.58) R2 = 0.466 Satisfactory results are obtained from the regression as the sign of the coefficients agrees with the dictates of economic theory. One apparent problem for the regres sion is the limited sample size. The results show a low 86 2 value for the coefficient of determination R . The t values in parentheses under their respective coefficients show that the coefficients are significant by this test. In general, the results of the regression should be considered satisfactory as the association between the dependent and independent variables agrees with the re sults expected from economic theory. To increase the number of observations to twenty- seven, a regression equation similar to the previous one is presented based on the assumptions that risk is re flected by the variance of rates of return for the period 1963, 1964, and 1965 treated as an aggregate, and that mean rate of return can best be measured by treating each year individually. In this manner nine industries for three years gives twenty-seven observations. The depen dent variable proportion of foreign investment is calcu lated for each year, and the independent variable relative variation is based on a three year average. In this case the possible problem of multicolinearity is greatly re duced as the independent variables are developed from data with different time bases. A Cochrane-Orcutt auto regressive transformation is applied to the data. The results of the regression equation are shown as follows: 89 p = .lSsa+voiis^/^)-.0019(52/8^)+.oinscs^/s^) (.550) (.101) (1.52) R2 = 0.583 Unfortunately, the t values show that the coefficients 2 are not very significant. However, a fair value of R is obtained. The equation is supported by the criterion that the independent variables are associated with the dependent variable foreign investment proportion p in a manner that agrees with economic theory. A direct relation holds between relative rate of return (y^/y^) and relative variation (Sgy^/s^y ). The adverse variable 2 2 relative variance (s^/s^ has an indirect influence that is supported by theory. The direction of influence is the same for the regression based on twenty-seven observations with special assumptions concerning variance and mean rate of return and the similar regression based on nine observations. As a whole, the degression analysis should be considered satisfactory as the direction of influence agrees with the dictates of economic theory. An Autocorrelation Model A more elaborate form of analysis is developed in this section to evaluate the hypothesis concerning the information content of previous rates of return for various areas of the world. This hypothesis, developed 9o in the previous chapter, states that the greater the stability of an area the greater the information content of previous investments1 rate of return. To evaluate this hypothesis requires the formu lation of a criterion to order the areas considered with respect to stability. The areas under consideration are the United States, Canada, Latin America, Europe, and other areas. A subjective approach of ordering on a line the extreme areas of stability and instability, then plac ing the remaining areas with respect to the extremes is the criterion followed. Of the five areas analyzed, Europe should be considered the most stable. The coopera tion and success of the common market, general economic prosperity, and lack of involvement in major foreign wars since the Korean War are reasons for placing Europe in the position of greatest stability. On the other extreme, the position of greatest instability is reserved for Latin America due to its general character of underdeveloped nations, armed conflicts like the Cuban revolution, a right wing regime in Brazil, a socialist regime in Chile, and general terrorist activity throughout the area. Canada should be positioned below Europe when ordered with respect to stability. Reasons for this position are the relative advance after the early 1950's of Europe with respect to Canada and internal problems between Englidi and French segments of the population. Canada should be considered a relatively stable area due to its lack of involvement in major foreign wars since the Korean War. The United States should be placed between Canada and Latin America when ordered with respect to stability. The United States has changed from a basically stable area toward instability. This is generally the result of extensive foreign involvement like the Vietnam mili tary conflict at the expense of the domestic society. The United States has been involved directly and indirectly in the major military and political crises since World War II. Domestic problems of employment, minority groups, and the environment are not absent. The last area, other areas, is difficult to place due to its ambiguous nature. Other areas is composed of Africa, Asia and influenced greatly by large American international oil companies. Due to the influence of rates of return from established international oil companies other areas should be position ed in the middle slightly above the United States and below Canada. The position of other areas is made with reservation due to its ambiguous nature and special influence of international oil corporations. In summary, the five areas are ordered in terms of decreasing stabili ty as Europe, Canada, other areas, United States, and Latin America. 9 2 An autocorrelation model = a^ + where y^ denotes the rate of return for time t of the ith area was formulated in Chapter III to test the hy pothesis that in more stable areas the information content of previous rates of return is greater. Speci fically, the greater the stability, the greater the first order autocorrelation coefficient of the model. Support is given to the hypothesis in relation to how well the ranking of the areas with respect to ^i conforms to the ordering with respect to stability of the previous paragraph. The results of the regression analysis for the five areas is shown as follows: Europe, i = 1 ylt =.0193 + .3325ylt_1 R2 = .553 (4.56) other areas, i = 2 y2t =.0571 + •7535y2t_1 R2 = .520 (4.28) Canada, 1 = 3 y3t =.0263 + •6694y3t_1 R2 = .406 (3.45) United States, i = 4 y4t =.0538 + .5226yllt_1 R2 = .197 (2.22) Latin America, i = 5 y_. = .0828 + .3542 y R2 = .098 5t bt-1 (1.66) The equations presented are ranked with respect to X , beginning with the largest value and continuing to the smallest value of X^. Except for the ranking of other areas the results support the previous ordering with respect to stability. For example, the regression results show that Europe holds the position of greatest stability and Latin America has the greatest instability. Between the two extremes lies Canada with greater stability than the United States. Thus, the hypothesis that the more stable areas have greater information content of rates of return from previous investments is supported by the empirical results of this section. An Econometric Model In this section the approach followed is to ex plain the rate of return for a given area in terms of an economy's growth rate, terms of trade, and other indepen dent variables. This analysis implies that there is enough order in the world so that an econometric model is capable of explaining reality. The model developed to test the hypothesis concerning the importance of the terms of trade and growth rate of national product repre- 94 sents a more intricate form of analysis than previously presented. The hypothesis evaluated is that the rate of return is more significantly influenced by the growth rate of an economy in developed regions and that the terms of trade are more significant for less developed regions. The developed regions considered are the United States, Canada, and Europe. Latin America represents the less developed region. The basic model, which is varied for particular areas or regions, is represented for the ith area as follows: yit = ci + Vit-l + e±GRi + f’ iTTi where i = 1 United States i = 2 Canada i = 3 Europe i = 4 Latin America For the ith area, rate of return is denoted by y^; is a constant; y^t-1 a ^-aS the rate of return; GR^ represents a region's growth rate, and the variable TT^ denotes the terms of trade. The terms of trade is defined conventionally as the price of exports divided by the price of imports, and growth rate is defined as the change in national product divided by the midpoint of the change in national product. The above regression equation repre sents the basic form of the model and in various cases will be varied for individual areas. The equation is varied to give better results. One real problem encoun tered is the presence of multicolinearity between the independent variables so that the basic regression equa tion would not run. A criterion to evaluate the terms of trade and growth rate is to see if the sign of the coefficient agrees with the dictates of economic theory. If the sign of a coefficient does not agree with the dictates of theory the. variable is considered to have limited impor tance. The equation as a whole will be evaluated in terms of the level of significance of coefficients and by the coefficient of determination. In this manner rate or return is explained and the importance of growth rates and terms of trade is evaluated. The importance of the rate of growth and terms of trade demands explicit consideration of their direc tion of influence on the rate of return. The growth rate of national product represents a demand factor where greater growth implies the need for more investment which in turn implies a higher rate of return. Thus, a direct relation is expected between the rate of growth of national product and the rate of return. A positive coefficient for the variable growth rate is necessary to show the direct relation. Theory also requires a direct relation occur between the terms of trade and rate of return. Therefore, the terms of trade should have a positive coef ficient. An examination of the terms of trade defined as the price of exports divided by the price of imports will show reasons for the direct relationship. First, as the price of exports rises relative to the price of imports, rates of return increase in the export sector. This occurs for either a relative rise in the price of exports or a relative decline in the price of imports that are utilized as inputs to help produce the outputs of the export sector. Then, the increase in the price of exports can eventually lead to greater aggregate demand for the whole economy whereupon the rate of return of other sec tors of the economy should increase. Thus, a direct relation occurs between the terms of trade and rate of return. An important criterion for evaluating regression equations that are variations of the basic form, equation (4), is significant coefficients whose sign agrees with economic theory. Also, to be considered acceptable, the equations should have the power to explain reality. To build the succeeding regression equations that evaluate the hypothesis of this section requires data on the terms of trade and rate of growth of national product. The terms of trade data for the four areas considered is found in International Financial Statistics of the 5 International Monetary Fund. The growth rate of national product is found in a large variety of sources. For the United States growth rate data is calculated from 6 1969 Business Statistics. A source of data to calculate the Canadian growth rate is the Canadian Statistical Review.^ The growth rate of Europe up until the formation of the Common Market is derived from the Statistical Handbook. After the formation of the Common Market, European growth rate is calculated from Basic Statistics Q of the Community. The Latin American growth rate of 5 ^International Monetary Fund, International Financial Statistics: Supplement to the 1966/67 Issues (Washington, D.C.: Statistics Bureau of the International Monetary Fund, 1967), pp. XIV-XV. ^Department of Commerce, 1969 Business Statistics (Washington, D.C.: Office of Business Economics, U.S. Department of Commerce, 1969), p. 1. ^Dominion Bureau of Statistics, Canadian Statis- tical Review (Ottawa: Minister of Trade and Commerce, 1950-1967),XXV-XXXVII. o European Coal and Steel Community, Statistical Handbook (Luxembourg: Publications of the European Community, 1958), p. 72. 9 Statistical Office of the European Communities, Basic Statistics of the Community (Saarbrucker Zeitung, Verlag und Druckerei Gmbh, 1967), p. 169. national product is derived from publications of the Economic Commission for Latin America. Growth rate data up until 1956 is found in the Economic Survey of Latin America, 1957.10 For the period 1957-1962, the Economic Survey of Latin America, 1963 supplied data to calculate the growth rate.11 Thereafter, a source for deriving the growth rate is the Economic Survey of Latin America, 12 1967. The previous sources of data are employed to develop the time series of the period 1950-1966 for ana lyzing the hypothesis concerned with the relative impor tance of the terms of trade and growth rate of national product. The United States is the first area analyzed to evaluate the hypothesis concerning the terms of trade and growth rate. In the first regression equation (5) below the coefficients of lagged rate of return and growth rate are significant and agree with economic theory. A coefficient of determination equal to 0.651 shows good -^United Nations, Economic Commission for Latin America, Economic Survey of Latin America, 1957 (E/CN.12/ 489), 1959," p. 53. ■^United Nations, Economic Commission for Latin America, Economic Survey of Latin America, 1963 (E/CN.12/ 696), 1965, p. 8. 12 United Nations, Economic Commission for Latin America, Economic Survey of Latin America, 1967 (E/CN.12/ 808), 1969, p. 5. explanatory power. However, equation (6) directly below the previous equation (5) adds the variable terms of trade and reduces the ability to explain reality by lowering the coefficient of determination to 0.624. Al though the variable terms of trade has a sign that agrees with theory, the t value shows very limited significance. In both equations the variable growth rate has the proper sign that agrees with economic theory. Thus, the regres sion equations show that growth rate is important, but terms of trade generally is not significant and lacks explanatory power. The results for the United States confirm the hypothesis evaluated in this section. ylt = .0407 + .4443ylt_1 + .4851 GR (5) R2 = .651 (2.84) (4.53) y1+. = .0334 + . 4564y + .4841 GR J-U ±Z — X X (6) (2.41) (4.34) + .0061 TT R2 = .624 (.124) The hypothesis is given further support in the regression equations for Canada. Growth rate is signi ficant and has a positive sign that agrees with economic theory in both equations (7) and (8). However, the addition of terms of trade to equation (8) has the wrong 100 sign and is not significant. In both equations the coefficients of determination are approximately the same. Thus the regression results show that the growth rate is more significant than the terms of trade for Canada and thereby supports the hypothesis. y_. = .0210 + .6l53y„ n + .2104GR. (7) 2t-l ^ R2 = .536 (1.44) (3.55) y2t = .1066 + .6059y2t_1 + .2013GR2 (1.36) (3.52) - .0842 TT„ (8) r2 = .5H4 (-1.11) For Europe the regression equation is modified by adding the variable time t to improve the results by introducing a variable of particular importance for Europe. The introduction of the time captures the declin ing trend in the rate of return for Europe. The growth rate variable of equation (9) has a positive coefficient and agrees with economic theory. In contrast, the variable terms of trade of equation (10) has a negative coefficient and therefore is not consistent with economic theory. The equation with the terms of trade has a lower value for the coefficient of determination. The following equations support the hypothesis as they show that the 101 growth rate is more important than the terms of trade for Europe. y3t = .1101 + .2474y3t_1 + .3443GR3 (2.15) (1.80) (9) - .0401 t R2 = .905 (-7.18) y3t = .2827 + .2450y3t_1 - .l657TT3 (1.98) (-1.06) (10) - .0288 t R2 = .891 (-2.23) A second order lag of rate of return is added to the regression equation for Latin America. Equation (11) shows the importance of the terms of trade due to its positive coefficient and large value for the coefficient of determination. In equation (12) the sign of the growth rate variable is positive, but the coefficient of deter mination has fallen to one half that of the previous equation. This latter equation with the low value of the coefficient of determination lacks the power to explain reality. Therefore, -the equation with the variable terms of trade can explain reality and supports the hy pothesis in the case of Latin America that the terms of trade is more significant than the growth rate for less 102 less developed countries. + . 1276TTf j (11) (7.11) y4t = .9211 - .2642 y ^ _ 2 + .9574GR4 (12) R2 = .321 (5.56) (-1.47) (4.28) The previous empirical results support the hypothe sis tested in this section. The growth rate of national product is found to have the most significant impact on the rate of return for investments of the developed regions United States, Canada, and Europe. In contrast, the terms of trade is shown to have a more important influence on the rate of return for the less developed region Latin America. Thus, the hypothesis that the growth rate of an economy has more importance for developed regions and the terms of trade more importance for less developed regions is supported by the empirical evidence of this section. Application of the Logarithmic Normal Diffusion Process The stochastic model utilized in this section represents a more complex form of analysis than previously presented in this chapter. In this approach reality is 103 considered too complex to be modelled by a regression equation that attempts to explain the real world. Perhaps, the best way to characterize an invest ment's rate of return and risk is through a stochastic process. A stochastic process is a set of probability distributions whose characteristics change with time. Reality, especially the reality of world investments, is always changing with time and is influenced greatly by large number of random factors changing simultaneously. The logarithmic normal diffision process is a stochastic process that can be defined by the notation when rate p of return = G ( 0, 0, t) and variance = F^( 0, 0, t). A logarithmic normal distribution is appropriate as an economy's growth rates, labor conditions, cost of capital, political environment, and government policy represent a few of the large number of coincident events influencing the observed rate of return of an investment. A closer examination of the important characteris tics of the logarithmic normal diffusion process will help show its capacity to estimate rates of return. Three important properties are (1) temporal dependence reflected by the Markov property, (2) continuous variables, and (3) the proportionate effect of a probability distribution 104 IS that is logarithmic normal. J These properties allow interdependence between succeeding time periods, the reflection at the aggregate level of the continuous nature of individual acts of firms, and the consideration of a large number of complex effects acting simultan eously. The previous properties are all characteristic of the rates of return of worldwide investments. The mathematical expectation is utilized to apply the stochastic process to rates of return. Derived from the log-normal distribution the mathematical expectation of y(t) at time t is bQt E(y(t)) = yQe ° and variance p 2b ^ a t V(y(t)) = yV°<> (e ° -1) Rather than standard regression procedure, the coefficient bQ is estimated by applying equations of the logarithmic normal diffusion process due to the appropriateness of this process shown by the work of Gerhard Tintner and ■ ^ G e r h a r d Tintner and Malvika Patel, "A Lognormal Diffusion Process Applied to the Growth of Yield of Some Agriculture Crops in India," Journal of Development Studies, VI (October, 1969), p. 49. li{G. Tintner and R. C. Patel, "A Log-normal Dif fusion Process Applied to the Economic Development of India," Indian Economic Journal, XIII, No. 3 (1965), 466. ------------------------- 105 Malvlka Patel.^ The parameters of the previous rela tions are estimated for n+1 observations of data by the following equations: a = “ y o ' A A b0 = M ^ y n 3 = (l/n) E (Log y - Log ylt-1) t— 1 n 2 16 y = (l/n) E (Log y±t- Log yit-1) - $ t—1 In the analysis of rate of return of domestic and foreign investment the estimation of the expected value E(y(t)) is of primary interest. A favorable theoretical property of E(y(t)) from the log-normal distribution is that nega tive values of bQ cause the slope to diminish as time increases. This is seen by the derivative y' = keU(du/dx). Whether this property is going to play a helpful role in explaining rates of return depends on the values of yQ and bQ. If bQ is estimated as a small value, then the graphic presentation will most likely appear linear. Testing the hypothesis that the logarithmic normal ■^Gerhard Tintner and Malvika Patel, op. cit., pp. 49-57. ■I £ Gerhard Tintner and R. C. Patel, op. cit., pp. 467-468. ----- 106 diffusion process is superior to a regression model utilizing a graphic approach. The "naive" models employed for comparison are a first order autocorrelation model and a time trend model. Greatest interest focuses on the time trend model as the expected value of the log-normal diffusion process represents a trend. The reason for decreased emphasis on the autocorrelation model is that it produces short run movements rather than a trend. In the figures that follow, the solid fluctuating line denotes the actual rate of return, the time trend is described by the straight line TT, the dashed line describes the first order autocorrelation model, and the straight line LN shows the expected value of the logarithmic normal diffusion process. After employing the previous formulas for the mathematical expectation of domestic investment the result derived is E(y(t)) = .1540 e*001^. The parameter required to calculate variance V(y(t)) for domestic investment is aQ = .0193. The mathematical expectation for foreign investment is E(y(t)) = .1645 e“*°l80t^ and the parameter for variance V(y(t)) is aQ = .0091. A graphic comparison is relied on in the succeeding para graphs to evaluate the ability of the mathematical expec tation of the logarithmic normal diffusion process to reflect the trend for rate of return data. 107 Results from analyzing domestic rate of return are shown In figure 11. The first order autocorrelation model ylt = *0533 + *6255yit-l R2 = .197 (3.32) Is estimated from domestic rate of return data. The domestic time trend equation where t denotes time is estimated as ylt = .1168 - .0059 t R2 = .038 (-.643) Figure 11 shows that the diffusion process LN lies above the actual data and regression equation results. A large initial value relative to succeeding observations is the main reason for this result. Nonlinear aspects do not appear in the figure due to the relatively small value of coefficient bQ. Focusing on the end of the time period, both the time trend and autoregressive models underesti mate the actual rate of return, while the diffusion process overestimates rate of return. Underestimation of the rate of return at the end of the time period is a serious problem for the trend regression equation as the actual data showed a clear upward movement. Other problems for the regression models are Reflected by their low values of coefficients of determination. However, viewed as a whole the results of figure 11 do not present a strong Rate of Return 108 .15- .10 I960 1966 1950 Figure 11. A Comparison of the Results of Various Models for Domestic Investment. case for accepting the hypothesis. Judgement on whether the log-normal diffusion process is superior to the "naive" regression models should be withheld until the case of foreign investment is investigated. The foreign investment results of figure 12, present a case supporting the hypothesis that the logari thmic normal diffusion process is superior to a regression model. A comparison of the results TT for the time trend regression model y2t = .1798 - .0421 t R2 = .698 (-6.16) with the diffusion process LN shows that the latter pro 109 cess Is more appropriate than the former as the slope of LN is less than TT. This allows the diffusion process to give a better explanation as the downward trend of the rate of return is moderated by the log-normal diffusion process. Unfortunately, the line LN appears linear as the estimated coefficient b is small so that the decline o in the slope does not appear in the graphic presentation. Projections of the diffusion process trend into the future do not decline as greatly as the time trend model and thus allow more useful results. The previous discussion shows for the case of greatest intersst the superiority of the log-normal diffusion process over the time trend model. A comparison of the diffusion process with the autocorrelation model is more difficult. The foreign autocorrelation model is y2t « -0533 + •6275y2t_1 R2 - .385 (3.32) Figure 12 shows a favorable aspect of the autocorrelation model: its fluctuations are similar to the actual data. Unfortunately, the estimated autocorrelation results generally lag by one period changes in the actual data. Due to the fluctuations of the autoregressive, results along with the results of the actual data about the trend LN of the diffusion process it can be concluded that the diffusion process gives the best representation of trend. 110 Rate of Return _ .10 - I960 1966 1950 Figure 12. A Comparison of the Results of Various Models for Foreign Investment. Taken as a whole the foreign investment results support the superiority of the log-normal diffusion process. Although weakness in the case are apparent, this section tends to support the hypothesis, especially for capturing the trend, that the logarithmic normal diffusion process is more appropriate than simple regression models. CHAPTER V EVALUATION OF THE PORTFOLIO MODEL The Markowitz Portfolio Model To facilitate the establishment of a feasible region for the sensitivity analysis of a two sector domes tic and foreign portfolio model tests of hypotheses con cerning the input parameters rate of return, variance, and correlation of rates of return are considered. The sensitivity analysis points out the need for a more sophisticated objective function. The first set of hypotheses tested are concerned with the mean rate of return and variance. Comparisons are made between similar domestic and foreign activities. The first hypothesis evaluated Is that foreign rate of return exceeds domestic rate of return. Column one of Table III shows the mean rate of return for the period 1950-1966 for various domestic and foreign industries and areas. A comparison of similar categories shows the hypothesis is supported in nine of the eleven cases. The only exceptions are foreign manufacturing and Canada. - Ill - 112 TABLE III Rate of Return Means and Variances Classification Mean Variance x 10"^ Aggregate Domestic .11147 .31595 Foreign .14189 .59469 Industries Manufacturing Domestic .12676 .34218 Foreign .11800 .38118 Petroleum Domestic .12700 .41529 Foreign .16241 1.89824 Areas Domestic .11147 .31595 Canada .08127 .25749 Latin America .12691 .36274 Europe .12209 .67171 Other .23368 2.02487 113 Thus, column one of Table III presents evidence that supports the hypothesis of greater foreign than domes tic rate of return. A second hypothesis evaluated is that foreign variance exceeds domestic variance of rates of return. A second hypothesis evaluated is that foreign variance exceeds domestic variance of rates of return. Column two of Table III describes the variance for the period 1950-1966. In every case, except Canada, the hypothesis is supported. Thus, ten of the eleven cases support the hypothesis of greater foreign than domestic variance. The third hypothesis to be tested is the existence of positive correlation between rates of return of world wide investments. A positive correlation of cr^2 = .39332 occurred between domestic and foreign investment. Further support is given to the hypothesis when a greater number of sectors and areas are considered for the period 1950-1966. The hypothesis of positive correlation is supported when i=l, 2, 3, 4, 5 denotes the respective areas United States, Canada, Latin America, Europe, and other areas of Table IV below. In this table every corre lation is positive. The classification of manufacturing with respect to domestic i = 1, foreign i = 2, and petroleum according to domestic i = 3 and foreign 1 = 4 of Table V supports the hypothesis as all correlations 114 TABLE IV Area Correlation Matrix i 1 correlation P.. 2 3 4 5 1 1.0 .76111 .44259 .09554 .21485 2 1.0 .64477 .58480 .69249 3 1.0 .45858 .78595 4 1.0 .80386 5 1.0 TABLE V Industry and Area Correlation Matrix i correlation 1 2 3 4 1 1.0 .67578 .75307 .47687 2 1.0 .79689 .77348 3 1.0 .88651 4 1.0 are positive. Thus, the evidence for various classifi cations supports the hypothesis of positive correlation 115 between rates of return of worldwide investments. The sensitivity analysis that follows investigates changes in the optimum solution of x^ and in the effi cient set as the input parameters . cr , p,0 and A are i i varied for a simple two sector domestic and foreign Markowitz portfolio model. Variation of the input para meters is conducted in the context of greater foreign than domestic rate of return and variance. Greater foreign rate of return and variance is supported by general economic theory and the empirical results of the previous paragraphs. To concentrate on the relevant changes of the parameters of the simple Markowitz portfolio model, a feasible region is defined. The feasible region for the exogeneous or input parameters of the simple portfolio model are the foundation for deriving results from the sensitivity analysis of the two sector model. This feasible region is specified as follows: -1 < p12 < 1 (13) y2 1 Vi (I'D o\ > of (15) By defining foreign greater than or equal to domestic rate of return and variance the feasible region reflects 116 reality arid accepted economic theory. Relations (14) and (15) show that changes for the foreign Investment paramteres are specified with respect to domestic invest ment parameters. Objective measures of rate of return and variance calculated from the period 1950-1966 serve as starting points for the sensitivity analysis.1 Thus, domestic rate of return and variance are generally defined as follows: Mi = y"i = G1^i» °» °) 2 ai = si = Fi(sij 0j 0) ° i = When the starting point is objectively defined with respect to reality the sensitivity analysis can meaning fully show the impact of changes in the input parameters on the outputs of endogenous investment proportions generated from the optimized portfolio model. A feasible region 0. <_ x <_ 1 for the outputs generated from the optimized model is important as only positive proportions Although the actual rate of return calculated accounts for elements of investment search, the rate of return employed for this chapter is specified as so that the objective nature of the calculations becomes explicit. 117 are meaningful. The sensitivity analysis of the simple portfolio model will be conducted with respect to the previous feasible regions for both inputs and outputs. Optimization of the two asset simple Markowitz portfolio model Z = ylXl + y2x2 " A(aixi+a2X2+2pi2ala2XlX2^ (16) x + x = 1 1 2 is a significant step in developing the sensitivity analysis. A classical indirect method of applying differ ential calculus is employed to obtain the optimum solution. By substituting the proportion constraint x = 1 - x^ into equation (16) and then differentiating with respect to x the optimum solution is obtained as follows: 1 xn = U1~P2 + a2 -p12glq2 2A$ $ * = o* + o1-2 p12a1a2 The sensitivity analysis, with respect to the previously defined feasible region, investigates changes in the optimum solution as the input parameters y , a , P ^ 3 and A are varied. Through incrementing A and employing ob jective measures where possible for the other input parameters, the sensitivity analysis is begun. Then, 118 the foreign input parameters are varied with respect to the domestic input parameters. The focus of the sensi tivity analysis on the outputs will vary from x^ to x^, but remains uniform as x^ is indirectly related to by the relationship x^ + x = 1. For example, when x^ increases x2 falls, and conversely when x2 increases x^ declines. To reduce the computation task of the sensi tivity analysis a computer is employed for the calculations. The presentation of the sensitivity analysis results relies on calculus and graphs to show the calculations of the computer. In this way general rates of change derived from calculus are portrayed graphically to ex plicitly show the impact of varying input parameters. For objective measures of y^ y2> cr^, a2, and p^2 calculated from empirical data the changes in the outputs x and x are observed as the input A is incre- 1 * mented. From calculus the rate of change of x^ with respect to A is (3x1/8A) = (- (u1-y2 )/A2$ ) This shows that the rate of change of x decreases greatly as A Increases. The graph of figure 13 confirms the derivative (8x^3 A) as x1 changes by smaller amounts as A increases. A similar problem occurs for x^ as it is the converse of x^. The limit of x.^ as A approaches infinity is 119 50 100 200 300 500 Figure 13. The Impact of Changes in. A on the Investment Proportions. This limit can present a serious problem for the appli cation of the simple portfolio model when objective inputs Pi = y± * G±(y.*0,0) °± = si = °> °)» and p12 are employed as the model may not be able to generate efficient investment proportions for x^ and x^ found in reality. This is in fact the solution shown in figure 13 for objectively measured inputs from the period 1950- 1966 as the Lim x = 0.2554 A-voo d when the foreign investment proportions of plant and equipment expenditures are 0.1166, 0.1210, and 0.1213 for 1964, 1965, and 1966, respectively. Thus, the simple Markowitz portfolio model underestimates the domestic investment proportion, overestimates the foreign invest ment proportion and has problems explaining reality when objective inputs are utilized; The efficient set concept is helpful to gain 121 further Insights into the difficulties and problems en countered when the simple Markowitz portfolio model is applied to explain reality when objectively determined inputs are employed. The solid line gh of figure 14 describes the efficient set generated from objective in puts over the period 1950-1966. The point P represents o the position of the portfolio composed of the 1964, 1965 3 and 1966 average foreign investment proportion of 0.1196. The simple Markowitz portfolio model shows that portfolio Pq could be inefficient. Since the succeeding section shows the presence of skewness, rather than accept the result from the simple Markowitz portfolio model of a possible inefficient portfolio, the more sophisticated portfolio model is applied to give a better explanation of reality. A better explanation occurs as the sophis ticated portfolio model makes the portfolio efficient so that realistic investment proportions are generated by adjusting coefficients of aversion. For the case of figure 14 the curves shift up to establish a new effi cient set. In figure 14, point e^ represents a point on the new efficiency frontier as a lower bound less than P and a new upper bound on the efficiency frontier o is represented by e^. By this implicit assumption of an efficient portfolio the model determines operational outputs when the input parameters are objectively given. 122 cr2 x 10“^ 6.0 5.0 4.0 3.0 2.0 1.0 • o 12 Figure 14. The Efficient Set Generated from Objective Inputs. 123 The problems and difficulties encountered by the applica tion of the simple Markowitz portfolio model are due at least in part to the failure to include skewness as reality shows skewness. One reason why the model may not explain reality is that the estimate of the foreign variation may be too small. If foreign variation equalled a2=^a1//pi2^ = 0.0*151, then the Lim x^ = 1 as so that the portfolio model can explain reality. The limited sample period and assumption of past conditions prevailing in the future can be employed to present a case explaining the under estimation of foreign variation. Possibly, a measure of variation different than variance could solve the problem. Markowitz suggests the application of semi variance, which p can reflect skewness, as a possible alternative measure. Semi variance calculates variation for only a particular range of the distribution. This range is generally the variation below a critical minimum level. Other measures of variation are possible that can greatly increase the foreign dispersion relative to domestic dispersion. For instance, a different measure of variation that sums the squared deviations of rate of return from an average 2 Harry Markowitz, Portfolio Selection (New York: John Wiley and Sons, Inc., 1959), pp. 188-201. 124 taken at the end of the period, rather than from the mean of the period in calculating variance, will greatly Increase foreign variation. This measure emphasizes the impact of a downward trend in rates of return for increas ing the measure of variation. As foreign rate of return declined over the period, and domestic rate of return stabilized at values near the average for the end of the period, foreign variation would be estimated at a much greater value than domestic variation. Thus, both quali tative and quantitative cases can be established for a relative increase in foreign variation. However, this approach is not pursued. Rather, variation is measured conventionally as variance, and the addition of skewness is introduced to reflect important properties of the dis tribution of rates of return not reflected by the measure of variation. The specific formulation is to build a 2 sophisticated objective function Z = \i-Ao + BSm where y reflects rate of return and risk is considered by the 2 terms Aa + BS where each term reflects variance and m skewness respectively. In this manner the simple Markowitz portfolio model is built on, modified, and extended as dispersion and skewness of a distribution are directly introduced into portfolio analysis. The formulation of the sophisticated portfolio model in the succeeding sec tion employs objective inputs to generate efficient 125 portfolios where the simple Markowitz portfolio model lacked the capacity to do this as it neglects skewness. Thus, problems occur for the application of the simple two sector domestic and foreign portfolio model when inputs are defined objectively over the period 1950- 1966. In the sensitivity analysis that follows domestic inputs will always be defined objectively, but foreign inputs will deviate from objective measures for cases where the exposition is improved. This will allow the interest ing properties of the portfolio model to emerge clearly. An objective of the sensitivity analysis is to show in a general sense the impact of variations in the input para meters . A more elaborate presentation of results from the sensitivity analysis is portrayed in figure 15. Here the impact of varying a a s shifts in A occur are shown to have an impact on X2« Although the proportion x^ is the dependent variable in figure 15, changes in x can be inferred by the relation x^ = 1 - Xg. By emphasizing either x^ or X2, figure 15 is simpler to interpret, but still contains implicit information on the proportion not emphasized. Figure 15 shows that as A increases by increments and varies, the curve shifts down, implying a smaller proportion of x^. Larger values of the coef ficient of variance aversion A implies a desire to avoid 126 . 60 .50 .40 10 .30 .20 .10 .08 .06 .05 .07 Figure 15. The Influence of Shifts in A as Foreign Variance Varies on the Foreign Investment Proportion. 127 risky investments and move into more conservative investments. This result is clearly shown for objec tive measures of inputs as o2 varies and A increases by increments. For a given value of A, figure 15 shows that x decreases as a increases. A negative derivative 2 2 (dx2/da2) confirms that as a2 increases, x declines. Correspondingly, (dx^/bc2) is positive so x^ increases as a2 increases. Allowing risk to approach infinity and observing the impact on investment proportions gives results that agree with intuition as follows: Lim x = 0 Lim x„ = 1 a1->“ Lim x1 = 1 Lim x = 0 c ^ - * - 0 0 a^ - * 0 0 The limits show that as risk for a given sector increases, its investment proportion goes to zero and the alternative investment proportion becomes one. The previous results of the sensitivity analysis show that the portfolio model satisfactorily captures the relationship between variance which reflects risk and the investment proportions. Figure 16 presents a direct relationship as varies for increasing values of A on the dependent vari able x^. The impact of this direct relation, however, declines as the values of A increase. By taking the first derivative, 128 A=1 1.00 . 80 60 A=10 20 A=25 A=89 .11 .12 .13 .14 .15 .16 .17 .18 .19 .20 Figure 16. The Impact on the Foreign Investment Proportion of Shifts in A as p2 Varies. 129 G x2/&u2) = (1/2 A$ ) , the decrease in the direct relation is evident as A increases. Both figure 16 and the above derivative show that the slope of the relationship becomes zero as A becomes large. Thus, for large values of variance aver sion A the influence of p^ to increase the proportion of x2 grealy diminishes. The utilization of a three dimensional surface representation is another helpful method for illustrating the relationship between variables. When x^ is the depen dent variable with p2 and a2 as independent variables, the relationship shown in figure 17 is derived. The three dimensional representation for A = 101 shows x2 declining rapidly as a2 increases and x^ increasing slowly as p^ increases. For lower values of A the surface shifts upward, and for larger values of A the surface shifts down. The three dimensional characterization allows the sensitivity analysis to show the impact on one graph of a large number of variables. Three dimensional figures can also be employed to show the impact on the dependent variable x^ as independent variables p^2 and A vary. In this analysis the importance of changes in p12 should be emphasized as the significance of p-^2 is developed further in succeeding analysis of 130 21 Figure 17. The Three Dimensional Surface for a Single A Parameter. 131 this section. Figure 18 describes the situation for negative values of p- j^ j and figure 19 shows the case when o Is positive. In both cases as the value of p 12 v 12 increases the proportion of x declines. This result 2 is reasonable as the high return and variance investment proportion x^ has a greater correlation to the investment proportion x_^, the proportion x^ becomes less desirable as benefits from diversification are reduced. Similarly, as the desire to avoid large variance investments through increasing A occurs the proportion of the large variance investment x„ declines. The surface shifts downward 2 as foreign variance increases relative to domestic variance. Conversely, when foreign variance decreases relative to domestic variance the surface shifts upward. Increases in y relative to y, cause the surface to shift 2 1 upward, but as the value of A increases the impact of the shift greatly diminishes. When Ug decreases relative to y^ the surface shifts downward, but the extend of the downward shift is greatly reduced as A increases. The three dimensional representation has halped to show the importance of the coefficient of variance aversion and correlation between rates of return with shifts in rela tive rate of return and variance on the investment pro portion. Variation in the correlation between rates of Og • x2 133 1.00 t-.20 100 150 200 4 - 1.00 P12 Figure 19. The Influence of Positive Variations in p}p and Variations in.A on the Dependent Variable Xg. 134 return can have a significant influence on the position of the efficient set. The efficient set describes for given inputs the greatest rate of return for each level of variance. Figure 20 shows efficient sets where all inputs p., y , c , and c , are objectively determined 1 d 1 2 from foreign and domestic rate of returns over the period 1950-1966. The efficient set specified by R is generated by employing the correlation of rates of return determined from actual data. This efficient set can be contrasted with the remaining efficient sets denoted by their rate of return correlations. Figure 20 shows that the efficient sets shift to the right as the correlation of rates of return decreases from a positive to a negative range. The relative position of efficient sets with posi tive correlations behind those with negative correlations of rates of return clearly shows how the existence of positive correlation reduces the gains derived from diver sification. For any given efficient set the proportion of foreign investment increases as the movement toward increasing variance and rate of return continues. Figure 20 shows that the spread between the curves is larger for large values of variance aversion and the spread diminishes as variance aversion decreases. A comparison of efficient set R generated by an objective correlation input and the efficient sets with 0.0, -0.5, and -1.0 correlations 135 .0006 .0005 .0004 .0003 .0002 .0001 0.00 - 1.0 12.0 13.0 Figure 20. Efficient Sets with Various Values of Correlation. 136 of rates of return shows how the existence of positive correlation shifts the efficient set R to the left. The previous analysis shows how changes In correlation of rates of return can significantly shift the efficient set. The sensitivity analysis helps to describe the important properties of the simple Markowitz portfolio model. An examination of real world data shows that on a comparative basis foreign generally exceeded domestic rates of return and variance. The prevalence of positive correlation between rates of return is the general case that exists in reality. In agreement with intuition, the sensitivity analysis shows that when variance aversion is very significant, variance plays a much more important role in determining the composition of a portfolio than rate of return. The reduction in benefits derived through diversification is shown graphically by presenting effi cient sets with different degrees of positive and negative correlation. A serious problem emerged that the simple Markowitz portfolio model has problems explaining the investment proportions that occur in reality as efficient investments when objective inputs from domestic and for eign rates of return are utilized. The existence of this problem shows the need for a more sophisticated portfolio model based on an extension and modification of the simple Markowitz portfolio model. o 137 The Sophisticated Portfolio Model The previous sensitivity analysis shows that the simple Markowitz portfolio model encounters problems and difficulties when it attempts to generate realistic efficient portfolios with objectively determined inputs. Thus, the need for greater sophistication emerges. Through Introducing the measure of skewness into the objective function for a distribution of rates of return the sophisticated portfolio model will attempt to over come the previous problems and develop useful efficient portfolios. Before applying the objective function of the sophisticated portfolio model, testing the hypothesis of the prevalence of positive skewness should be con ducted. The objective function was formulated in a con text of preference for positive skewness. The existence of positive skewness represents a beneficial characteris tic of the distribution of rates of return not accounted for in the simple Markowitz portfolio model. Table VI shows that positive skewness is the prevalent case found in reality. Positive skewness represents the only case that occurs for aggregate and industry classified domestic and foreign investment. A single case of area classifi cation shows negative skewness. In only one of the eleven cases is the dlmensionless third moment negative. Thus, 138 TABLE VI The Dimensionless Third Moment Classification Dimensionless Third Moment Aggregate Domestic 0.61430 Foreign 0.49283 Industries Manufacturing Domestic 0.43463 Foreign 1.06712 Petroleum Domestic 0.13799 Foreign 0.44741 Areas United States 0.61827 Canada 0.94304 Latin America 0.64568 Europe -0.38657 Other 0.70108 positive skewness is generally present in rate of return distributions of United States domestic and foreign investment. 139 The prevalence of positive skewness is established. Now, the sophisticated portfolio model that accounts for variance and skewness is applied to the case of primary interest dealing with input parameters from aggregate domestic and foreign investment over the period 1950- 1966. For this case the previous section shows that the simple Markowitz portfolio model encounters problems when employing objectively determined input parameters to generate efficient portfolios. Contrastingly, the sophisticated portfolio model is capable of explaining reality when the inputs of the objective function rate 2 of return and risk, +BSm are objectively determined. Optimization of the sophisticated portfolio model that consists of the sophisticated three dimensional objective function plus the constraint x.. + x = 1 is shown in the 1 2 Appendix, pages 163-164, for generating the domestic and foreign investment proportions, x^ and x^ respectively. These proportions are measured from available data as the average proportion of plant and equipment investment for the years 1964, 1965, and 1966. The average invest ment proportion of 0.1196 from the previous three years serves as a general point of reference for a framework to evaluate the sophisticated portfolio model. To es tablish the capability of the sophisticated objective function the coefficients of variance and skewness 140 aversion are set at levels that underestimate and over estimate the investment proportion. As the sophisticated three dimensional objective function plus a proportion constraint can both under and overestimate the reference investment proportion it can explain any proportion as an efficient set lying between the previous range by further adjiasting the coefficients of aversion: a result that the simple Markowitz portfolio model could not achieve. In the empirical results that follow for aggre gate foreign and domestic investment, explicit reference is given to the foreign investment proportion x^. This does not, however, neglect the domestic investment propor tion x^ due to the relation x^ = 1 - x^. By the previous relation the ability to explain x^ automatically Implies an ability to explain x^. The foreign reference propor tion is x2 = 0.1196 for the three year average. The lower bound of the range is a foreign investment propor tion x^ = 0.0538 with coefficients of aversion parameters A = 140, = 0.06 and B2 = 0.06. Coefficient of aversion parameters A = 140, B^ = 0.15, and = 0.10 give an upper bound for the range of the foreign investment pro portion as x^ = 0.1774. The sophisticated objective function can explain any investment proportion as effi cient between the range 0.054 to 0.177 by adjusting the coefficients of aversion. Thus, the sophisticated port 141 folio model with objectively determined inputs is capable of explaining a wide range of investment proportions as efficient where the simple Markowitz portfolio model encounters problems. This application shows the super iority of the sophisticated portfolio model as the simple Markowitz portfolio model is modified and extended to deal with worldwide investment. In particular, the greater sophistication of the three dimensional objective function of rate of return, variance, and skewness is shown, as hypothesized, to be an improvement over the objective function of the simple Markowitz portfolio model. Further cases where the simple Markowitz portfolio model encounters difficulty explaining the foreign refer ence proportion as an efficient portfolio by adjusting the coefficient of aversion are explained by the sophis ticated portfolio model. Throughout this analysis the Markowitz portfolio model serves as a foundation as it is sophisticated by modifications and extensions of its objective function. The general procedure followed is similar to the previous of explaining a range of invest ment proportions that enclose a reference investment proportion. Table VII shows the results for the previous case of aggregate domestic and foreign investment plus four cases that the sophisticated portfolio model could TABLE VII Estimated Range of Investment Proportions X1 Classification x_ Foreign Parameters and Range of Reference Investment Proportion Proportion the Estimated 2 Lower Bound Upper Bound X„ A B1 B^ X„ A B1 B2 X Domestic-Foreign .1196 140 .15 .06 .0538 140 .15 .10 .1774 Domestic Manufacturing-Petroleum .1747 200 .10 .20 .1013 200 .10 .26 .2105 Domestic-Latin America .0192 200 .15 .018 .0037 200 .15 .029 .0506 Domestic-Europe .0468 200 .15 . 06 .0090 200 .15 .001 .0724 DomeStic-Other .0111 200 .3 .009 .0006 200 .3 .057 .0428 explain as efficient portfolios, where the simple Marko witz portfolio model could not explain as efficient portfolios. For a stated industrial classification the reference investment proportion is the average for 1964, 1965, and 1966. The inputs of mean rate of return, vari ance, skewness, and correlation are objectively deter mined from actual data over the period 1950-1966 for the stated type of investment classification. The invest ment proportion outputs are determined by incrementing the coefficients of aversion for the optimized sophis ticated objective function, with the proportion constraint x-^ + X2 = Coefficients of aversion A and are shown in Table VII that corresponds to their respective invest ment proportions. The table is described in terms of foreign investment x , but domestic investment is impli- 2 cit as x = 1 - Xg. In all cases the sophisticated portfolio model can explain the range that includes the reference investment proportion by adjusting the coef ficients of aversion. Thus, Table VII shows four more cases utilizing objective inputs where the simple Marko witz portfolio model encounters problems explaining reality as an efficient portfolio, but the sophisticated portfolio model with the three dimensional objective function is capable of explaining reality as an efficient portfolio. The analysis of this section gives clear 144 support to the basic hypothesis that the sophisticated three dimensional objective function is an improvement over the two dimensional objective function of the simple Markowitz portfolio model. Predictions of the Sophisticated Portfolio Model A previous section has shown the ability of the sophisticated portfolio model to explain reality as effi cient portfolios for different types of investment pro portions that occurred for the period 1964, 1965, and 1966. As the model has the capacity to explain reality, an important extension is to predict the future. The application of the sophisticated portfolio model to pre dict aggregate domestic and foreign investment propor tions not only serves as a means for evaluating the model, but also synthesizes the theoretical structure. The prediction is carried on in the context of a multinational firm with a geographic area structure similar to that of Figure 9, page 70. When i = 1 the domestic area is defined and i = 2 defines the foreign area. For the 1969 prediction, when predicted and actual investment propor tions can be compared, the input parameters are generally objectively determined from past data. In contrast, the 1973 prediction does not allow a comparison with actual investment proportions and employs input parameters based 145 on the previously developed theoretical and empirical structure. The input parameters for the 1969 prediction are based on empirical data. Both domestic and foreign rates of return are the mean values that actually occurred over the period 1950-1969. The implies that u. = p. = Y. = 1S X X G^CY^, 0j 0) where Y^is the actual mean value for the per iod 1950-1969 for areas i = 1 domestic and i = 2 foreign. In a similar manner actual observations over the period 1950-1969 are employed to generate the inputs for variance and skewness. By employing the functional notation for the ith area the variance and skewness are defined as follows: ck = s2 = F (s^ 0, 0) 1 i 1 1 M± = M3± = Ei(M31, 0) The input parameter of correlation between rates of return is also an empirically determined value for the period 1950-1969. The objectively determined input parameters' mean rate of return, variance, skewness, and correlation for domestic and foreign investment over the period 1950-1969 are shown in Table VIII below. 146 TABLE VIII Input Parameters for the 1969 Prediction Input Parameter Domestic Foreign i=l i=2 Mean .11240 .13671 Variance .27431 x 10"3 .66115 x 10-3 Skewness .48936 .58673 Correlation .27552 To generate a prediction, coefficients of aversion are required along with the previously objectively deter mined input parameters. The coefficient of risk aversion is placed at the level A = 140. The first and second coefficients of skewness aversion are set at = 0.16 and Bg = 0.06. By employing the objectively determined input parameters and these coefficients of risk aversion into the optimized sophisticated objective function with a proportion constraint X-^ + X^ = 1 generates, as output, the 1969 prediction of the foreign investment proportion X^ = 0.1284. This result compares very favorably with the actual 1969 foreign investment proportion of 0.1253. The real significance of the 1969 prediction, however, 147 rests In the fact that the simple portfolio model utili zing objective Inputs over the period 1950-1969 could only generate as an efficient portfolio a minimum foreign investment proportion of 0.2241, and in contrast the portfolio model with the sophisticated objective function can give an adequate efficient portfolio prediction. Thus, the 1969 prediction gives further support to the basic hypothesis that the sophisticated three dimensional objective function is an improvement over the two dimen sional objective function of the simple Markowitz port folio model. A prediction further into the future implies greater unknown influences and therefore a more explicit reliance on the theoretical structure of the investment and accounting periods. Furthermore, the ability of the logarithmic normal diffusion process to capture the trend is very important for the prediction of future rate of return. The investment and accounting periods are stressed in the analysis of domestic rate of return. Foreign rate of return is analyzed by relying bn the logarithmic normal diffusion process. Thus, input parameters for the 1973 prediction of domestic and foreign investment pro portions is the area where the previous theoretical elements receive explicit application. The investment period represents the length of time into the future that is considered when evaluating an investment in terms of present value. Greater uncer tainty over future conditions implies a shorter invest ment period. The uncertainty for domestic American invest ments is surely greater for future time periods than dur ing the past. For the late 1950’s and early 1960’s an investment period of about twenty years would seem rea sonable. Unfortunately, an investment period of approxi mately twenty years would seem unreasonable for the early 1970's. The shorter investment period is justified by the uncertainty over turning down of the Vietnam War, the impact of pollution control, and government policies to expand the economy along with the control of inflation. In the rbsence of the previous conditions an investment period of twenty years would seem reasonable. To develop the calculations for illustrating the importance of the twenty year investment period the investment outlay em ployed is the 1969 domestic plant and equipment expendi ture of 75.3 billion dollars and the 1969 domestic net profit of 33.248 billion dollars is the annual profit employed to calculate the internal rate of return. For simplicity, the expenditure of 75*3 billion dollars is assumed for each of the first two yearly accounting periods and profits of 33*248 billion dollars are assumed for each of the remaining eighteen yearly accounting 149 periods. When these expenses and profits for an invest ment period nl = 20 are placed Into a relation of the general form 0 = 1Tl1 + - *12 +....+ 7Tlnl (1+M.l) (1+H.i)2 (l+y1)nl an Internal rate of return of = 0.1310 occurs. Unfor tunately, uncertainty has increased and the investment period declines to an estimated fourteen years. For each of the years the expenditures and profits are kept at their previous levels. Expenditures occur in the first two accounting periods and revenues only occur for the twelve remaining accounting periods. This results in an unacceptable rate of return of 9*54 percent. The previous discussion illustrates the situation of a de clining investment period described in figure 2, page To compensate for the decline in the investment period the level of expenditures must also decline. Let the expenditure in the second accounting period decline to 95 per cent of its original level to 71*535 billion dollars. In this case an acceptable internal rate of return of 0.1000 is established. The notation = 0.1000 = G^(0, 0.1000, 0) shows that the 10 per cent domestic rate of return based on the investment period is employed to generate the prediction. The previous analysis 150 analysis illustrates the significance of the investment period on the rate of return. Another important domestic input for the predic tion is variance. The domestic variance calculated for the period 1950-1969 is 0.27431 x 10“3. However, the larger variance of 0.3159 x 10“3 calculated for the period 1950-1966 is a better estimate of future variance due to the increase in general risk brought about by the current price controls and stabilization polisy. The value of variance employed for the prediction is a com promise between the two previous levels. Expressed by the functional definition the variance utilized is a ^ = 0.295 x 10"3 = F^O. 27431 x 10“3, 0.3159 x 10-3, 0) which represents an increase from the variance calculated from the period 1950-1969 due to greater expected variance _ o 0.3159 x 10 The compromise variance employed repre sents a blending of calculated and expected future variance. An increase in risk due to the impact of stabili zation and price controls is also going to have an impact on domestic skewness. The amount of positive skewness should decline as risk increases. As a point of refer ence the skewness calculated from the domestic rate of return data for 1950-1969 is 0.48936. Expected future skewness could decline by approximately 20 per cent to 151 0.40. In the context of the functional definition the dimensionless third moment (skewness) is = 0.43 = E^(0.48936, 0.40) showing that the input employed in the prediction is less than the empirically calculated value due to greater domestic risk. A blending of the empirical measure of skewness with the expected value produces a skewness input for the prediction approximately 15 per cent below the value calculated from past data. Foreign rate of return is another important input parameter to be analyzed for the 1973 projection. The logarithmic normal diffusion process has been shown in figure 2, page ^1, to capture the trend better than the other models applied. For this reason, foreign rate of return is solely estimated by the logarithmic normal diffusion process for the year 1973- Then, the foreign rate of return input expressed by the notational defini tion is = 0*11008 = 0, t)* The remaining input parameters are objectively determined from the rate of return data over the period 1950-1969. Thus, the correlation between foreign and domestic rates of return is set at the level 0.27522. Foreign variance and skewness are respectively defined 2 _-3 by the notation as 0 = 0.66115 x 10 - F2(0.66ll5 x 10“3, 0, 0) and M2 = 0.58673 = E2(0.58673, 0). A review of the factors Influencing the levels of the input parameters helps to exemplify the genera tion of the 1973 prediction by the sophisticated portfolio model. Greater domestic uncertainty reduced the invest ment period which in turn lowered the domestic rate of return. Reliance on the logarithmic normal diffusion process to capture the trend generates the foreign rate of return input. The importance of risk is reflected by general increase in domestic risk that increases domes tic variance and reduces domestic skewness. The remain ing input parameters are objectively determined over the period 1950-1969. When the model is optimized a greater rate of return for a given sector implies a greater in vestment proportion, a smaller investment proportion when variance increases, and a greater investment proportion with greater positive skewness. To generate a prediction the coefficients of aversion must be specified. The levels of the coefficients of aversion employed for the 1973 prediction are the same as those utilized for the 1969 prediction of A = 140, B = 0.16, and B„ = 0.06. 1 2 By placing the previously described inputs into the optimized sophisticated objective function of the port folio model with a constraint + x2 = 1 generates the prediction output of respective domestic and foreign investment proportions x^ = 0.8700 and X2 = 0.1300. An increase of 0.0047 from the actual 1969 foreign 153 Investment proportion seems reasonable for the 1973 prediction. The 1973 prediction gives a good illustration of the theoretical elements developed in this disserta tion. In general the 1969 and 1973 predictions generate useful and reasonable results, and also help to illus trate and synthesize the theoretical structure of this dissertation. Furthermore, empirical work in this sec tion as well as the analysis of the whole chapter give strong support to the basic hypothesis, that the sophis ticated three dimensional objective function, formulated In this dissertation, is an improvement over the two dimen sional objective function of the simple Markowitz port folio model. CHAPTER VI CONCLUSION Summary • In Chapter II the literature was reviewed from three perspectives: international capital flow theory, portfolio analysis, and the application of portfolio analysis to international investment. A fundamental assumption of portfolio analysis is that actual port folios are efficient. This allows the portfolio model to generate determinate predictions or investment pro portion outputs when the input parameters are objectively given. The simple Markowitz portfolio model y - Aa2, whose respective terms consider rate of return and variance for distributions of rates of return, was built on, modi fied, and extended in Chapter III to explicitly consider 2 skewness by the formulation y - Ac + BSm. In this formulation the impact of uncertainty was introduced through rate of return y; and risk was considered through 2 both variance Act and skewness BSm. The econometric analysis of Chapter IV proceeded - 15** - from the simple to the more complex. For example, rela tive variation was analyzed through a comparison, and a regression equation was developed to show indirect port folio analysis. On a higher level of analysis, an auto correlation model evaluated the information content of previous rates of return for various areas, and an econo metric model analyzed the relative importance of the terms of trade and growth rate of the economy for various areas. The stochastic process, logarithmic normal diffu sion process, was applied to capture the trend for rates of return. The useful results from this stochastic pro cess for foreign investmer were later employed as input for a portfolio model preo '.lion. An empirical evalu ,ion of portfolio analysis was developed in Chapter V. The input parameters mean, variance, and skewness were measured from empirical rate of return data. A sensitivity analysis showed that the simple Markowitz portfolio model encounters difficulties generating efficient realistic portfolios when objective inputs were utilized. These difficulties were overcome when the sophisticated portfolio model with the three dimensional objective function of rate of return, variance and skewness was employed to generate realistic efficient portfolios from objective inputs. This result supported the basic hypothesis that the sophisticated portfolio 156 model with the three dimensional objective function is an improvement over the simple two dimensional Markowitz portfolio model. In a later section the sophisticated portfolio model was applied to predictions of domestic and foreign investment proportions. Results from Testing the Hypothesis This section reviews the results from testing hypotheses developed in this dissertation. First, the results from hypotheses on the first level of analysis dealing with worldwide investment are stated. Then, the conclusions derived from hypotheses on the second level of analysis that explores portfolio analysis are presented. A succeeding section further explores the basic hypothe sis and portfolio analysis. Hypotheses related to the first level of analysis were primarily concerned with rates of return and were tested in Chapter IV. Empirical analysis supported the hypotheses that foreign rates of return have greater relative variation than American rates of return. Good support was found for the hypothesis that more stable areas have greater information content from previous rates of return than less stable areas. The econometric analysis gave support to the hypothesis that rates of return are influenced more significantly by the terms of 157 trade for less developed areas and that the growth rate of an economy Is more significant for developed areas. Evaluation of the hypothesis whether a regression model or the logarithmic normal diffusion process is more appro priate for explaining rates of return showed the superi ority of the latter stochastic process. The second level of hypotheses were concerned with portfolio analysi per se. This analysis was concerned with empirical applications of the simple Markowitz port folio model and the sophisticated three dimensional portfolio model constructed in Chapter III. The empirical analysis of input parameters for the portfolio model sup ported the hypotheses of (1) greater foreign than domestic rates of return; (2) greater foreign variance of rates of return; (3) the existence of positive correlation between rates of return of worldwide investment; and (4) the prevalence of positive skewness for rate of return distributions. An outcome of the sensitivity analysis was the appearance of problems and difficulties when the simple Markowitz portfolio model was applied to generate efficient portfolios with objective inputs. Given the input parameters of the model a reasonable investment proportion was difficult to generate as output. This problem was overcome by applying the sophisticated port folio model. Chapter V gave support to the basic hypo- 158 thesis that the sophisticated portfolio model with the three dimensional objective function of rate of return, variance, and skewness is an improvement over the two dimensional objective function of the simple Markowitz portfolio model. Then, given the support of the basic hypothesis the sophisticated portfolio model was applied to predictions. Revaluation of the Basic Hypothesis In this section an intuitive approach is followed for revaluating and focusing on the basic hypothesis. The basic hypothesis states that the sophisticated port folio model with the three dimensional objective function of rate of return, variance, and skewness is an improvement over the two dimensional objective function of the simple Markowitz portfolio model that only considers the rate of return and variance. Important aspects of the hypothesis are the explicit introduction of skewness and the empiri cal application of the three dimensional objective func tion to domestic and foreign investment. A simple example can effectively show the impor tance of explicitly introducing the measure of skewness into portfolio analysis. Consider the distributions of rates of return for two investments G and H. Both investments have identical variance and mean rates of 159 return. The simple Markowitz portfolio model would, therefore, evaluate both Investments as equivalent. Figure 21 clearly shows that although both investments have the same mean and variance, the distributions are clearly not equivalent. What is the property that makes these distributions different? The neglected property is skewness. Distribution G shows negative skewness, and distribution H is positively skewed. Figure 21 shows that distribution H is preferred to distribution G as the positively skewed distribution gives less chance of a low rate of return and a greater probability of a high rate of return. The preference for distribution H should be introduced into portfolio analysis so that the model can function effectively. The sophisticated objective function of rate of return, variance, and skewness intro duces the skewness property of the distribution of rates of return into portfolio analysis. This allows portfolio analysis to further analyze differentiating characteris tics of the distribution of rates of return. Support is given to the basic hypothesis as skewness is an important element for determining investment decisions. A great many directions for the future extension of portfolio analysis are possible. The best directions include the introduction of higher moments and the development of a more dynamic portfolio analysis. When l6o Frequency Frequency Rate of » « r Rate of Return Return G H Figure 21. The Importance of Skewness for Two Distributions of Rates of Return G and H. higher moments are considered businessmen’s preference for the properties of higher moments should be established. A dynamic programming formulation is a possible dynamic extension of portfolio analysis. When the extensions are pursued, however, they should not make the model so com plex that it becomes difficult to apply and operate. This is especially true of empirical work where many un- forseen problems can be encountered. The trade-offs between a complex theoretical structure and the difficul- ; ties of empirical applications must be fully considered. The development of Sharpe’s Index model for reducing the inputs required to generate efficient portfolios represents evidence of simplification for stock market l6l applications where a very large number of securities are considered.^ The form of the portfolio model con» structed should fit the needs of the problem analyzed and give useful results. Portfolio analysis should remain flexible so that an operational approach can be employed toward empirical problems. ^William Sharpe, "A Simplified Model of Portfolio Analysis," Management Science, IX (January, 1963), 277-293. APPENDIX 163 OPTIMIZATION OP THE TWO SECTOR SOPHISTICATED PORTFOLIO MODEL Objective Function: Z = y — Ac^ + BS m 2 2 2 2 Z = y ^ + u2x2 - A(a1x1 + a2x + 2P12°ia2xix2^ ■f B 1 M 1 x 1 + B 2 M 2 x 2 Proportion Constraint x+x= l 0<x<l 1 2 “ j “ Substitute x^ = 1 - x^ into the objective function. 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Asset Metadata

Creator Noreiko, Gary Victor (author)

Core Title A Portfolio Approach To Domestic And Foreign Investment

Degree Doctor of Philosophy

Degree Program Economics

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag Economics, theory,OAI-PMH Harvest

Language English

Contributor Digitized by ProQuest (provenance)

Advisor Tintner, Gerhard (committee chair), Elliott, John E. (committee member), Maass, Randal (committee member)

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c18-476494

Unique identifier UC11362077

Identifier 7217495.pdf (filename),usctheses-c18-476494 (legacy record id)

Legacy Identifier 7217495.pdf

Dmrecord 476494

Document Type Dissertation

Rights Noreiko, Gary Victor

Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Access Conditions The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the au...

Repository Name University of Southern California Digital Library

Repository Location USC Digital Library, University of Southern California, University Park Campus, Los Angeles, California 90089, USA