The Capital Market in our country is having a significant rise in recent years, driven by strong private investments and the decrease in country risk due to the growth in the level of the national economy. Such growth not only attracts large investors, but also micro-investors who invest in the Secondary Market, directly through Stock Brokers (SAB) or indirectly through Fund Management Companies (SAF), be they Mutual Funds or investment.
The most important attraction that this Market presents is that the average profitability reported by a capital is much higher than the passive rate offered by the Banking Market; however the risk can be high.
Due to this, good portfolio management must maintain an optimal level of profitability-risk that allows the portfolio to be maintained over time in a sustainable manner.
development-on-portfolio-theoryThe general objective of this work is to demonstrate that the Risk Level, a consequence of portfolio diversification, is much lower than the risk generated by the assets individually.
This work is divided into three parts in order to adequately explain the process that represents the entire development of a model.
The first shows a theoretical framework on the most important equilibrium models in the
Portfolio Theory, describing more broadly Harry Markowitz's Portfolio Selection, and Sharpe's capital market equilibrium at the point of tangency of the Efficient Frontier with the Capital Market Line.
The second part shows the entire mathematical procedure in the development of the theories described in the previous section.
The last part shows the simplest way to build a Market Portfolio, on the development of the first step of the Separation Theorem. The Portfolio presented is POLARIS made up of seven shares that are listed on the Lima Stock Exchange and a bond from the Financial System. The production of POLARIS is basically divided into two stages. The first is data collection and the second is the actual construction of the portfolio.
We assume that POLARIS is based on Harry Markowitz's Portfolio Theory that seeks to maximize the expected return in a given period at a level of risk, or to obtain the minimum risk for a given return.
Making use of Excel for statistical calculations and the generation of graphics, and the analysis of the results based on the aforementioned theory, the present work is concluded with the demonstration of the objective that motivated it.
PART I THEORETICAL FRAMEWORK
THEORETICAL FRAMEWORK
Portfolio valuation models are basically divided into two groups:
- Models that are based on investors' fundamentals against risk. Models of returns with arbitrage arguments.
Of the first stands out; the Portfolio Selection Model with a mean-variance selection criterion, developed by Harry Markowitz (1952, 1959) and deepened by Markowitz and Levy himself in 1978. Another model belonging to this group is the CAPM Securities Valuation Model (Capital Asset Pricing Model) developed by William Sharpe in 1964 and further deepened by Lintner (1965), Mossin (1966). Treynor studies in 1964 also help to understand the model. Until now, this is the most widespread and studied model, and different variations and / or extensions to it have been born from it; thus, for example, the CAPM and M -CAPM with several periods (Intertemporal CAPM) developed since 1973 by Merton (I-CAPM), Champbell (1996), Chen's studies in 2002 (BWX-CAPM) together with Brennan, Xiang and Wang,and most recently in 2004 by Campbell and Vuolteenaho. There are also the consumption-based CAPM models (Consumption-CAPM) developed by Litzenberg, Breeden and
Gibbson in 1978. A year earlier Lanskroner postulated the In -CAPM (Inflation CAPM). Fama and French since 1992 have developed studies on the Multifactorial CAPM. More recent models on the CAPM are the L-APM (Liquidity Asset Pricing Model) based on liquidity developed by Holsmtröm and Tirole in 2001. Santos and Veronesi in 2004 developed the conditional M-CAPM subject to more global market restrictions. A third type of model is more eclectic with respect to the two previously mentioned, since its selection criterion is based on semi-variance (MSB) and stochastic moments of different order, for example Hogan and Warren in 1974 developed the CAPM model with negative deviations (Dowside CAPM) and Smith in 2003 made studies on the CAPM model with conditional moments (Conditional Moment CAPM).
In the second group of models we find the Arbitrage Pricing Model APM, developed by Ross in 1976 and again revised and expanded in 1980 by Roll and Ross himself, Chamberlain and Rohschlild (1984), Ingersoll (1984), Connor (1984); Roll, Ross and Chen (1986), Reisman (1992) and Nawalkha (1997, 2004).
This work will only be framed within Harry Markowitz's Portfolio Selection Theory and the Capital Market Line as part of the CAPM Model.
The Theory of Portfolio Selection that H. Markowitz postulated in 1952 is based on the following hypotheses:
- The profitability of any security or portfolio is a random variable, obtained from the prices of said security or portfolio, whose probability distribution for the historical base period is known by the investor. The model accepts as a measure of return on investment the mathematical expectation of said asset. The risk measure is the dispersion of the return series of a transferable security or portfolio, measured by the variance or standard deviation. The investor will tend to choose those portfolios with higher profitability and lower risk .
Using this theory, the Efficient Frontier is obtained, consisting of all the portfolios with a maximum expected return for a given risk level, in the absence of a Risk Free Rate. All the portfolios that are located under this border will be considered inefficient, since for the same risk level they have a lower return than that found on the border.
According to the Theory of the Capital Market Line based on the CAPM as an extension of the Markowitz model with the presence of a Risk Free Rate, this constitutes the intercept with the ordered profitability, and whose slope of the line is the Ratio of Sharpe.
The interaction of both borders will constitute a point of equilibrium known as the Market Portfolio. In this way we can speak of portfolios with loan (Lending Portfolios) when a part of the budget is invested by granting a loan at the interest rate of the asset without risk, and of portfolios with indebtedness (Borrowing Portfolios) when funds are borrowed to invest in the Market portfolio, at the same interest rate. So, in the first case, part of the available capital is invested in the “Market Portfolio” and part in a risk-free asset; in the second case, the available capital plus funds received through indebtedness are invested in the Market Portfolio.
In order to follow an adequate theoretical framework, we continue to describe the procedure to follow to obtain a portfolio that satisfies the expected profitability of an investor compared to their level of risk aversion. This portfolio must follow an exercise in normative economics according to the “Separation Theorem”.
The first step is to obtain the Optimal Market Tangency Portfolio.
At this stage, the investor needs to estimate the expected returns and variances of all contemplated securities. Additionally, you need to estimate all the covariances between these values, as well as determine the risk-free rate. Once this is done, the investor can identify the composition of the tangency portfolio as well as its expected return and its standard deviation (level of risk). In doing so, all investors would obtain the same tangency portfolio in equilibrium, under the following assumptions:
Investors evaluate portfolios by judging the expected returns and standard deviations of the portfolios over a one-period horizon.
- Investors are never satisfied, so when a choice is made between two portfolios with identical risk levels, they will choose the one with the highest expected return (choice criterion of the mean variance: maximum return). Investors are adverse to the risk, so that when a choice is given between two portfolios with identical expected returns, they will choose the one with the lowest level of risk (choice criterion of the mean variance: minimum risk). Individual assets are infinitely divisible, which means that an investor can buy a fraction of a share if he wishes There is a risk-free rate at which the investor can lend or borrow money, that is, the rate is one-size-fits-all Taxes and transaction costs they are irrelevant.Since all investors have the same one-period horizon, they face the same risk-free rate, and they get the same information. Havehomogeneous expectations; that is, they share the same perceptions regarding expected returns, risk levels, and covariances of values.
Therefore, the linear efficient set is the same for all investors because it simply involves the combinations of the tangency portfolio and the agreed risk-free borrowing or risk-free loan.
The specific objective of this work is to present the model and the resolution of an illustrative example for this first stage.
The separation theorem states that "the optimal mix of risky and risk-free assets for any given investor can be determined without any knowledge of the investor's risk and return preferences." That is, the optimal mix of risky assets is determined without any knowledge of the shape of an investor's indifference curves.
The second step is the determination of the Optimal Portfolio of each Investor.
Since all investors face the same efficient set, the only reason they choose different portfolios is that they have different preferences for risk and return, resulting in particular indifference curves.
You can then identify the investor's optimal portfolio by observing where one of his indifference curves touches but does not cut through the efficient set. At that point of tangency, the investment is determined with a certain amount of debt or loan at the risk-free rate, because the efficient set is linear.
However, although the chosen portfolios will be different, each investor will choose the same combination of risky securities, as determined in the previous step.
ANALYTICAL MODEL PART
ANALYTICAL MODEL
General Objective:
Obtain the Market Portfolio; that is, the one that constitutes the point of tangency between the efficient frontier and the capital market line.
Specific objectives:
- Elaborate, from a mathematical model, the efficient frontier based on Harry Markowitz's portfolio selection theory Elaborate the capital market line, based on Sharpe's theory and determine the Sharpe ratio Solve the first step of the Separation Theorem; that is, select the optimal portfolio from the linear set of portfolios that tangents the efficient portfolios of the market, from a set of risk-sos transferable securities (equity assets, mainly equities) in an environment of risk aversion by investors, combined with a risk-free asset (mainly bonds), for the case in which short sales can be made.
THE EFFICIENT BORDER
Starting from the fundamental features of the rational behavior of an investor, the Portfolio Selection Theory that H. Markowitz postulated in 1952 is based on the following hypotheses:
- The profitability of any security or portfolio is a random variable, obtained from the prices of said security or portfolio, whose probability distribution for the base historical period is known by the investor. The model accepts as a measure of return on investment the mathematical expectation of said asset. The model accepts as a measure of risk the dispersion, measured by the variance or standard deviation of the transferable securities from the profitability. The investor will tend to choose those portfolios with a higher return and lower risk .
Based on the above, an objective function is formulated that must be optimized, subject to certain restrictions, for which it is necessary to choose the variables on which the function must be optimized. Thus we have:
- Minimize portfolio risk.
- Obtain a certain expected return on the portfolio. All the available capital must be used to be invested in the different transferable securities that are part of the portfolio, there being no prohibitions to carry out short sales.
- The proportions of capital to be invested in each of the portfolio assets.
For the elaboration of the Efficient Frontier, the case will be assumed that the transferable securities analyzed are risky in their entirety.
Formalization of the optimization problem with risky transferable securities:
(See PDF)
THE CAPITAL MARKET LINE
The introduction of a non-risky movable asset to the group of risky assets generates a risk-free rate that is equal to the rate paid by the non-risky security. It is therefore necessary to form new investment portfolios, starting from the combination of efficient portfolios with risk-free assets, which maximize the investor's profitability. This new
-Minimize the risk of the portfolio that combines risk-free assets with an efficient portfolio consisting only of random assets.
-Obtaining a certain expected return on the combined portfolio.
-The total available capital must be used to acquire the efficient portfolio and / or the risk-free asset.
-The proportion of the total capital destined to the efficient portfolio must be fully invested among the assets that make up said portfolio (there are no prohibitions to carry out short sales).
-The proportions of capital to be invested in the risk-free asset and the efficient portfolio made up of random assets and the internal composition of said portfolio.
Formalization of the problem of optimization of risky values combined with a risk-free asset:
(See PDF)
OPTIMAL MARKET TANGENCY PORTFOLIO
Commonly known as the Market Wallet. It is that which results from the equalization of the expected profitability of a portfolio that is on the efficient frontier curve and the expected return of a portfolio that is located on the Capital Market Line. In this way we can speak of portfolios with loan (Lending Portfolios) when a part of the budget is invested by granting a loan at the interest rate of the asset without risk, and of portfolios with indebtedness (Borrowing Portfolios) when funds are borrowed to invest in the Market portfolio, at the same interest rate. So, in the first case, part of the available capital is invested in the “Market Portfolio” and part in a risk-free asset; in the second case, the available capital is invested plus funds received through indebtedness,in the Market Portfolio. Mathematically it is solved as follows:
(See PDF)
PART I
DATA COLLECTION METHODOLOGY
A very important stage and on which the research will be based is data collection. It is not an easy task to group all the information that the market can provide, that is why by prioritizing the information needs it is decided to consider the relevant data.
The main data that are needed for the elaboration of an Investment Portfolio are the following:
- The closing prices. The returns, which are obtained from the closing prices. o The amounts traded for each share per day The number of shares traded per day.
In this work, exclusive use is made of the daily closing prices.
Not many years ago Alvin Tofler wrote in his "Third Wave" that we are living in the information age. Under this precept and making proper use of it, we proceed to detail from which sources and forms the data used in the preparation of an Investment Portfolio can be extracted.
The first is the written source. The newspaper by the way. They stand out among this group: Management, in its Business and Finance section, where the daily prices of the shares listed on the Stock Market appear. Another important newspaper is El Comercio in its Business section, Finance subsection, which shows the same data and many others from the Market.
A second source is through specialized software called Economática, which has exquisite data on both national and most Latin American stock prices. This software is available to the general public at the Information Department of the National Supervisory Commission for Companies and Securities (CONASEV) free of charge. Economática also has a command through which portfolios can also be optimized based on their data, subject to various restrictions.
A third way to obtain the series of closing prices is through a request addressed to the Commercial Management of the BVL, after depositing their cost in a current account of said entity.
A Four mechanism for such series is through the website of the Lima Stock Exchange, www.bvl.com.pe. The advantage of this web page is that you can obtain the quotes in real time and its main disadvantage is that it does not keep a historical record of them; However, on the CONASEV website, www.conasev.gob.pe, such data can be found, share by share, company by company.
For the data collection of this work, the aforementioned web pages were used. To the BVL to compile the Directory of ISIN Codes, from where it is possible to know which is the mnemonic of each action to be analyzed; in addition to the conformation of the different Portfolios of the BVL Market Indices. From the CONASEV website, the different closing prices of each share to be analyzed were compiled.
All the series obtained are recorded in software specialized in Statistics. Among them, the SPSS and E-Views stand out, but for a more didactic use Excel and all its functions will be used, in addition to the elaboration of graphs, both linear and dispersion.
The next step is the actual development of the portfolio.
CHOICE AND INDIVIDUAL DESCRIPTION OF THE SECURITIES
For the elaboration of the POLARIS INVESTMENT PORTFOLIO, the following transferable securities listed on the Lima Stock Exchange have been chosen, whose sample period was from November 3, 2006 to February 28, 2007 :
- ALICORC1 which is a capital share of the company ALICORP SAA EDELNOC1 which is a capital share of the company EDELNOR SAA MOROCOI1 which is an investment share of COMPAÑÍA MINERA SAN IGNACIO DE MOROCOHA SA ATACOI1 which is an investment share of the COMPANY MINERA ATACOCHA SAA MINSURI1 which is an investment share of Compañía Minera MINSUR SA
| VOLCABC1 | He leads the portfolio of the General Index of the Lima Stock Exchange with the | 10.8046% |
| CVERDEC1 | It ranks third in the same portfolio with an effective share of the | 7.0956% |
| ATACOI1 | It is ranked fourth in the aforementioned portfolio with a | 5.2206% |
| MOROCOI1 | It ranks fifth in the same portfolio with an effective share of | 4.4632% |
| MINSURI1 | It is located in sixth place in the aforementioned portfolio with a proportion of | 4.4168% |
| ALICORC1 | It ranks fifth in the portfolio of the Industrial Sector Market Index with the | 7.8507% |
| EDELNOC1 | It ranks fifth in the Services Sector Market Index portfolio with the | 13.8033% |
- CVERDEC1 which is a capital share of SOCIEDAD MINERA CERRO VERDE SAA
- VOLCABC1 which is a capital share of the company VOLCÁN COMPAÑÍA MINERA SAA
- PEP01000CT06 than the ISIN code of a bond issued in Nuevos Soles by the Ministry of Economy and Finance, whose mnemonic is SB10AGO11. It has a nominal value of 1000 Nuevos Soles and pays a nominal annual rate of 16.08%.
The stock mnemonics can be found in the ISIN CODE DIRECTORY located on the Lima Stock Exchange portal.
The common denominator of the chosen shares is that they have participation in the different portfolios of the Market Indicators of the Lima Stock Exchange in force as of January 2, 2007. Thus:
It should be noted that of the seven stocks considered, five of them belong to the mining sector, which in recent times, up to the date of analysis, has contributed a significant volume to GDP growth and their shares have reached historical peaks and profitable returns that have increased. the level of earnings of its shareholders.
For a simple descriptive analysis, graphs of the Evolution of the Closing Price of each selected share are presented, followed by their minimum and maximum prices and profitability.
(See PDF)
SOURCES
- VAN HORNE, James C.; "Financial administration"; Dec. Ed. Edited by Prentice HallHispanoamericano, S.ANICHOLSON, Walter; "Macroeconomic Theory"; Thomson Editores Spain; 8th Edition; 2004. MELLI MUNDI, José; "All about the Stock Market" Notes of the Course of Microeconomics II, dictated for the 5th cycle (year 2007) of the Specialty of Economy of the Universidad Católica Sedes Sapientiae.http: // bvl.com.pe http: // www.conasev.gob.pe
Investment Fundamentals, Theory and Practice; Alexander & Sharpe & Bailey; Pearson; Mexico; 2003. Cap. 7, 8, 9 and 10
They are Random Variables with their expectations, variances, covariances, linear correlation between them.
They are not Random Variables, they are a certain constant. Its variance, covariance, and correlation with risky transferable securities is zero.
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To complement what is stated in this document on portfolio theory, we suggest the following couple of video-lessons, which are part of the master's degree in stock market and capital markets at the ENyd Business and Management School, in which risk issues are addressed and profitability, diversification and portfolio management procedure is taught. (1 hour and 55 minutes)
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