Formation of adequate Investment Portfolio
for Stochasticity of Profit Possibilities
Aleksandras Vytautas Rutkauskas
Prof. Habil. Dr., Head of Department of Economics,
Vilnius Gediminas Technical University
Sauletekio al. 11, LT - 2040, Vilnius
Tel/fax: (370) 2 76 79 19, e-mail: email@example.com
Abstract: The article focuses on the development problems of modern investment
theories considering stochastic nature of investment profitableness. This is to be realised
by offering investment portfolio adequate to the description of stochasticity of profit
possibility, and motivating the scheme of its formation and use. Formation of adequate
model and its practical use is based on computerised imitative technologies. Here are
presented concrete examples of adequate model and the use of imitative technologies
have been presented there. The work consists of 3 chapters: 1. “Formation of modern
investment theory and development premises” focuses on the most reasonable results of
this theory and actual problems that need to be solved. 2. “Opportunities and limitation
of portfolio as the mean of investment management” indicates the importance of
stochasticity of profit possibilities stressing on some investment issues incompletely
solved by modern (classical) portfolio 3. “Formation and use of portfolio to evaluate
adequate stochasticity of profit possibilities” presents the adequate concept and
possibilities of practical adjustment. Portfolio description favourable for the stochasticity
of profit possibilities is named after adequate model and geometrical picture of the
portfolio geometrical model.
1. Formation of modern investment theory and assumption of development premises.
thThe middle of 20 century is considered to be the beginning of modern investment theory. Before this date it was possible to change the whole complex of former investment criteria, regulations and rules by one proposal – to buy cheaper and sell more expensive. According to modern investment theory each investment is measured by its profitability and risk, and the possibilities of each investment being intercorrelated are analysed as their interaction – portfolio. These circumstances caused the necessity to realise the interrelation of all assets involved in portfolio determining the possibility of investment management as a whole.
H. Markowitz's work about the formation and optimisation of investment portfolio is considered to be a start point of modern investment theory formation . This work of investment switched one – sided standpoint
of investment profitability to two-aspect - investment profitability and its instability (risk) analysing their interdependence and separate investment in their interaction. It speeded up the formation stages of modern investment and portfolio theories.
Right after H.Markowitz’s “Portfolio selection” there were a great avalanche of publications on investing theory and practise. Of course, in its own the object of investment theory is of great importance while analysing equivalency of generation interaction. The right investment strategy influences by well being of separate countries and mankind future. But the portfolio consideration has an indisputable impact on the processes mentioned.
Portfolio theory as one of ways searching for investment decision overgrew the concept of investment portfolio and became constructive concept of systemic analysis in most spheres of research. The works of F. Modigliani, M.H. Miller  and Rubinstein  attempt to revaluate the decisions of corporation finances, selecting them according to portfolio ideology or modern investment theory.
On the way to modern investment theory there were some fundamental results that speeded up the formation of modern theory and determined the discretion for its further development, such as the results of Fisher’s interest rate analysis  and J.B. William’s theoretical fundamentals of investment value. It is essential to mention T.H.Knight’s work, which stressed on non-deterministic processes developing the grounds of utility
Further abundance of research and multi-aspectability limits the possibility to allot the results in consecutive order. Anyway, J. Tobin’s works  spread the ideas of H. Markowitz and adapted them to macroeconomics while N.F.Sharp , , Lintner , J. Mossin  predetermine Markowitz’s idea to be the main investment theory. At the same abundance of research and importance of results caused some doubts concerning the theory.
Need to possess more common analysis of investment decisions as well as laws of social sciences imply more exceptions while this causes more doubts concerning some results of portfolio theory, focusing on adequacy of capital asset pricing model – CAPM. R.Roll  expressed his doubts concerning some results of
portfolio theory and possibilities of CAPM verification. He suggested arbitrage-pricing model – APM  that
refers to prerequisite that relation of profit and risk shouldn’t let achieve the constant utility only from the arbitrage transactions. S. Ross and R. Roll, the supporters of arbitrage theory assume  that it is possible to check APM empirically.
The main tools of arbitrage model is based on – the existence of efficient market which possibilities thwere actively analysed during the 7 decade , , . Acknowledgement of efficient market hypothesis
School model , which refers to the possibility of non-risk could be interpreted under the idea Black –
transaction precondition when an option transaction is employed beside the basic assets. F.Black, M.Shloes and Merton’s  started analysing the possibilities of portfolio profitability as the function of distributions of asset profit possibilities.
There are lots of other doubts concerning the newest assumptions of modern investment theory as well as the whole finance theory, f.e., “chaos theory” which supporters asserts that the laws of modern finance (economics) theory are just the exceptions not the rules , or that there are inconsistencies between. It’s the logic of the theory and practical issues. The reasons of objections should be found among the premises of point estimation, even where processes are described by probability theory.
Tempting to realise, the origin of objection for the main assumptions of modern investment theory, or inadequacy of its conclusions and real facts, it’s essential to determine the conditionality of premises about efficiency line. On the one hand, the existence of the efficiency line from the standpoint of intercourse between
logical expression of any investment traditional portfolio risk and average profitability is mathematical –
collection. But real existence of average portfolio profitability is equal to zero, in case any of portfolio asset profit possibility distribution isn’t discrete. Thus, constructive analysis of the situation could be performed just leaning on the expression of efficiency line possibility distribution - efficiency zone that determines all possibilities of portfolio profitability maximum in their distributions achieved for every level of portfolio standard deviation (see fig. 2).
2. The main opportunities and limitation of portfolio as the means of investment management.
Portfolio is a set of various kinds of assets belonging to any of institutions or individual, which principles of formation are redirected to the employment of various kinds of assets and collection proportions seeking for the utility by the owner of portfolio. Financial portfolio – the collection of financial assets. The
content of portfolio is considered to be of great variety. Besides assets portfolio, liabilities portfolio is possible or it can include both.
It is usual to say that, if the portfolio consists of A, A……A assets, we assume that the structure of 12n
portfolio is w, w….w (w>0, w+w+…w=1) and its value v=wa+wa+wa, when ais the value of asset i. 12ni12n1122nni
Theory of securities portfolio is system of knowledge where an investor could gain the highest expected profit from risk and non-risk collections of securities. The main issues by the theory of portfolio are determination of the whole complex of available portfolios, establishment of efficient portfolio line, selection of optimal portfolio for each investor.
To understand the solution of arisen problems more easily it is essential to concentrate on the geometry surface of those problems reflecting their criteria of decisions in order to find better solution. It is usual to fix the average meaning of portfolio profitability on the ordinate axis, while the instability (risk) measure of the same profitability i.e., average standard deviation – on the abscissa. Thus, the average and deviation of the
same probability distribution are set on separate co-ordinates. Having chosen the set of assets while knowing the meanings of their profitability and standard deviation, and assuming that each of assets could take the part altering from 0 to 1, we'll achieve the set of possible portfolios (see fig. 1). This is the way to identify the whole complex of available portfolios or the set of investor's choices.
Mathematical random variables and characteristics of their weighted sums influence such form of available portfolios. Line YB is called efficiency line and is the part of convex curve AB. Those lines are considered to be of reasonable meaning while analysing separate characteristics of portfolio. Efficiency line - the highest profit average line of available portfolios possessing the proper level of risks.
E (R) B
a. Selection of the best alternative from available variants
’ E(R)A ’’ A
’ * AA
* B ’ B
σ b. Envelope curve with two investors’ indifference curves
E(R)* 1B** BB
σ 1B σ
c. Capital market line (CML)
Fig. 1. Variants of the search for portfolio decision
Mission of the portfolio as the instrument of investment theory - the realisation of investment set structure - w, w,…,wallowing to maximise the portfolio profitability at 12n
present risk level or minimise the risk at chosen profit level. For further analysis it is essential to refer to the allocation of general risks: systemic and non-systemic ones. If the portfolio possesses reasonable amount of investments, it experiences systemic risk - that part which is experienced by the whole (country, region, and the world) economic system. The further analysis stresses emphasise the systemic part of general risks.
The last issue is the selection of optimal portfolio for the separate investor from the set of possible portfolios- the selection of portfolio on the efficiency line. It is important to note that in the portfolio theory the investments (their set) are analysed regarding the investor's utility. That is the distinguishing characteristic of portfolio maximisation criteria.
The modern portfolio theory uses the simplified change of utility function - indifference curves. This concept came from the consumption theory where it determines the combination of two goods equally useful for the consumer. In the portfolio theory it expresses the equal reasonable combination of profit and risk for the investor. Figure 1 shows the way the investor should select the most useful portfolio regarding the set of available portfolios and his function of utility (indifference).
Figure 1b. Illustrates the way the investors possessing different indifference curves A'A'' and B'B'' choose separate optimal (maximising their utility) portfolios at the same maximising general utility of investment formation.
It is obvious that such logic of portfolio optimisation justifies in case the portfolio involves only risky investments. In this case, it's decision belongs to the efficiency line and simultaneously in the network of indifference map. That is the point X in fig. 1 and the points A* and B* in fig. 2a.
Therefore, the premises of risky investments do not show the investor's possibilities in the real world of investments involving non-risk securities, such as government bonds. In this case, the investor is trying to achieve the higher utility, the one in the points A* and B*. Those are the points A** and B* indicated in the investors' indifference maps (see ex. , , , ). It is illustrated in fig. 1c.
Rate of non-risk short-term government securities is indicated at the point R. f
Under certain circumstances, any investor could allocate his means as the risk investments at the point M on the efficiency line, and non- - risk securities - at point R. f
There is a rectilinear expression:
where M - parameters of risk investments
B - parameters of non-risk investments
W - part of risk investments. f
All points achieved as rectilinear combination of risk and non-risk securities lay on the straight line connecting the points R and M. If the investor prefers the lower risk, f
this position shifts closer to R; if his priority is given to the higher risk, the position f
approaches above the point M.
Let's go back to the investor B in order to realise his choice. If there is no possibility to invest in non-risk assets, the investor chooses portfolio B* with risk σ and 1B
higher profit E (R). In case there is a possibility to acquire non-risk asset, the investor B 1B
can combine his choice between M and B so that he could gain higher profit E (R') 1B
while experiencing the same risk. Actually, the investor B while leaning upon his indifference map, should prefer the point B*, where the profit decreases but the risk is reasonably lower. Assuming the latter assumption, the investor prefers this position (a little bit lower profit but reasonably lower risk) because indifference curve crossing line R M at the point B** is indifference curve of higher level than the one crossing the point f
B*. The same could be seen by graphical view of investor's indifference curves. Let's go back to investor A. He takes the higher growth of risks. His indifference curve shifts his choice to the right of point M. he doesn't allocate his money between M and B. Instead, he borrows for non-risk price R, and invests the disposed means in the way he could 1
reach the point A**.
As far as we know, the straight line is named as capital market line (CML). This separation of risk and non-risk capital causes the result known as J. Tobin's separation theorem .
Let's analyse the portfolio M. the investor A as well as investor B uses the portfolio of risky asset. Realising the advantages of this portfolio, any of investors will include this portfolio into the combination of his investments. It's the only equilibrium portfolio experienced by any of risk averse investor. That means that in case of two investments of the same profit but different risks, the priority will be given to the lower risk. Thus, the portfolio M becomes the market portfolio and, in case of equilibrium, it should involve all available risk investments in the way of proportional ones.
3. Formation and Use of Portfolio Adequate for the Evaluation of Stochasticity of Profit Possibilities
3.1. Development Opportunities and limitations of Traditional Portfolio Analysis Outline
Classical scheme of portfolio analysis, management or other use is convenient for practical utilisation. Functional expression of efficiency line and envelope curve one should be used in practical portfolio application, but it is not obvious in general case. Formation and management of portfolio requires effective evaluation of various portfolio conditions existing on the efficiency curve, description of their interaction, or analysis of other portfolio characteristics. Portfolio decisions should be achieved when it is impossible to describe the profit possibilities of portfolio as point estimated but as their distribution of probability.
It is essential to turn to geometric model of portfolio (geometric picture). In case of traditional portfolio, expectation (averages) of investment profit as random quantity are set on ordinate axes while their standard deviations are set on the abscissa ones. That is an obvious geometric illustration of the main portfolio analysis results. Geometric evidence wouldn't disappear if we used rectilinear functions instead of expectations or standard deviations, f. e. we shifted them along the abscissa and/or ordinate axes.
Considering the stochasticity of portfolio investment profitability, average value of portfolio profit is not the most appropriate indicator. The expected or average profitability - generalised condition of profit possibilities for the real period. However, this is only one of the set of possibilities, usually without paying such a big attention as, let’s say, quintile of a certain (95%) level etc. In every concrete case, the concrete profit
will be one of appropriate profit possibilities described by the distribution of their probability. The necessity of interpreting portfolio profitability as the random quantity is confirmed by the circumstance that the price of separate investments (bonds, securities, projects, etc.) and portfolio is the random quantity in the market. Thus, the whole picture of portfolio profit possibilities is possible requesting logic of random quantity as the most adequate financial-mathematical model of the profit.
Interpreting investment portfolio in classical way and leaning on the concept of average profit, the problem of profit adequacy management arises. Considering the profitability average in the future, it is possible to foresee and describe it under the categories of random quantity. Such description of portfolio profitability explains the interaction of risk as the instability of profit possibilities and investor's function of utility that is essential striving for systemic risk evaluation and formation of its adequate management model.
System of co-ordinates, the one geometric model of portfolio is analysed in, is considered to be the determined one: on the ordinate axis there are averages of random quantities (processes) or other rectilinear functions of available values while the abscissa axis implies average standard deviations or their rectilinear functions. Thus, the facts analogic to classical (traditional) portfolio such as the set of available portfolios, efficiency lines, existence of envelope curve and their characteristics are true for every possibility of portfolio profit. Considering the possibilities of portfolio profit, it is essential to analyse the entire efficiency zone instead of efficiency line (see fig. 2).
Thus, the further analysis of portfolio should be switched from the co-ordinate system of portfolio profit standard deviation and the expected values (averages) of this profit to the more complex and adequate one where the abscissa axis implies average standard deviations of portfolio profit, and the ordinate axis indicates the efficiency zone (fig. 2b. or fig. 3a.), distributions of all available portfolio maximum profitability (fig. 2a. or fig. 3b.), utility (response) functions of available investors (fig. 3c.), and the values of created utility (fig. 3d.).
For more adequate comprehension complex schemes of investment portfolio risk analysis (fig. 3) should be compared to the outlet of modern portfolio analysis.
According to classical theory of modern portfolio, the investor should be interested in the portfolios located on the efficiency line. The efficiency line could be realised as the whole complex of maximum expected profit (averages) acquired for concrete standard deviation of portfolio set. In classical scheme the set of available portfolios could be formed joining present investments to portfolio in all possible proportions, in this way evaluating profit average (expected value) and standard deviation for each of formed portfolios.
Actually, the investments are observed and realised by some possible values defined by investment market and price. That's why investors should face the entire set of portfolio profit possibilities. He is interested in the whole efficiency zone, which is realised as the whole complex of efficiency lines. In this way the analysis of efficiency line that involve portfolios possessing maximum average switches to the analysis of efficiency zone. In it’s turn investors' indifference curves should be changed (enlarged)
by utility functions. This creates the complex picture of portfolio risk analysis (fig. 3) where the set of possible values of investment portfolio risk is connected to distributions of portfolio profit distributions (fig. 2a. or fig. 3b.), functions of portfolio owner
(recipient of portfolio risk) response (fig. 3c.) and possibilities of portfolio utility for the region or country (fig. 3c.).
Summarising this stage, it is essential to underline the main differences determined by use of classical (modern) portfolio theory and adequate portfolio theory. Let's present them by table:
Classical portfolio theory Adequate portfolio theory
： determines the efficiency line where the ： determines the efficiency zone where
existing portfolios possess maximum each level of possible portfolio risk
expected (average) profitability among possesses the distribution of maximum
the given portfolios of riskiness possibility probability
： each investor's indifference curve ： each investor's utility function could
allows the choice of portfolio where the experience such level of risk and
investor is able to gain the maximum of distribution of the highest possibilities
average profitability that maximise the investor's utility.
Imitative picture of functional-numerical portfolio analysis. Realisation of
complex analysis of investment portfolio risk requires multiple analytic methods including multiple logic operations and optimisation algorithms. This refers to the determination of efficiency zone. The complexity is also illustrated by the problems faced describing efficiency line. When the possibilities of separate asset profitability submit to normal distributions and it is possible to interpret them as independent ones, the determination of efficiency line causes the selection of gravity weights www(when 1, 2,…, n
www= 1) maximising rectilinear form wa+ wa+…wawhere aa a - 1, 2,…, n 11 22 nn. 1, 2,…n
averages of corresponding asset profits.
This simplicity shows up for the average of normally distributed quantity and standard deviation is not interdependent quantities.
Actually, integral I 2，;；x？a？12，2I，edx，1, )，！2？，
is equal to one in the presence of any value of a and σ, and average of asset as the random 2quantity Mξ = a and variance Dξ = σ.
Anyway, this simplicity should disappear when the distributions of portfolio asset profitability possibilities become more complicated. F. i., when those distributions are lognormal - distribution (density) function possesses the form:
?1ln？xm；?exp？(~，2()，px when x>0 ?2SxS2！??(0?
There is non-rectilinear dependence between the average and variance. When this dependence is particularly complicated there are some distributions, and the solutions of discussed objects are multiple ones. Turning to the variety of dependence of separate assets, it becomes obvious that the discovery of efficiency line is the complex issue as well. The complexity doesn't decrease while determining efficiency zone.
In most cases the discovery of portfolio analysis and management decisions is inefficient if the possibilities of imitative technologies are not used for the search of solutions (,. Figure 4 presents general system of portfolio set analysis and management using imitative technologies. This system - computerised functional quantitative and imitative numerical models devoted to the analysis of investments, their individual characteristics and their interaction; characteristics of investment portfolios; the main characteristics of possible investment portfolio set - efficiency zone; investment portfolio risk, possessing the characteristics of real investment portfolios and permitting to fulfil all provided logic operations while selecting portfolios with the required characteristics.