Greetings, dear friends. Portfolio theory is what we will talk about today.
I consider this topic to be of fundamental importance for a private investor, especially an index investor, since it is how the investment portfolio is compiled that ultimately determines its profitability. That is, it is the distribution of assets, and not the timing of purchases, that determines the final return on the investment portfolio. More than one study has been conducted on this issue. For example, in 1986, American scientists Gary Brinson, L. RandolphHood and Gilbert Beebower conducted a fundamental study “Determinants of Portfolio Performance”. As a result, they found that asset allocation determines from 75.5% to 98.6% of the final return of an investment portfolio. This is such a short introduction
So, let's begin a close examination of portfolio theory, because it is important to understand whether it can be applied in practice to form the most effective investment portfolio or not.
Article outline:
The essence of portfolio theory
The most important thing is to understand the essence of the theory, which we will subsequently apply in practice.
Portfolio theory is a set of rules for compiling an investment portfolio with certain risk/return characteristics.
This is a very simplified definition, but the simpler the better for understanding complex questions filled with scientific terms and boring formulas.
Let's look at what portfolio theory offers the investor.
The main task of portfolio theory is to find a combination of assets that will be as close as possible to the efficiency frontier. What is the efficiency frontier?
I think many people are now asking this question. So, the efficiency frontier is the best combination of investment assets for a given level of return and risk.
Here is a visual illustration of the efficiency frontier:
How to achieve such a combination of assets? In theory, everything is simple.
A list of assets is taken, after which data such as standard deviation, average return and asset correlation are combined in complex mathematical calculations, the output of which is the most effective investment portfolio.
Let's go over the terms.
Standard deviation is the amount of deviation from the average value. That is, we take the average return of an asset, then measure deviations in both positive and negative directions. And the magnitude of this deviation is the standard deviation.
The higher the standard deviation, the greater the spread of returns, price changes, etc.
Accordingly, an asset with a higher standard deviation is more risky. True, it will not be possible to accurately assess the degree of risk, and a little later you will understand why.
Average yield– this is the average value of the returns of a particular asset. Moreover, there are geometric mean returns and arithmetic mean returns. The geometric mean return is always slightly less than the arithmetic mean due to the difference in calculations.
Calculated using a simple interest rate.
The calculation formula is very simple:
D n – profitability for a certain period
n – period for calculating profitability
Geometric average profitability - calculated using a compound interest rate and equal to the nth root of accumulated income, where n is the number of periods for calculating accumulated income.
D n – income for one period (day, month, year);
n – number of settlement periods
Let me explain this effect to you with a simple example:
As you can see, the geometric mean yield lagged behind the arithmetic mean. They will be equal only if the income is the same for all periods studied. But as you understand, constant profitability is possible only on bank deposits and similar assets with fixed income and no volatility. That is, it is volatility that generates this effect of the geometric mean return lagging behind the arithmetic mean.
Correlation.
You have most likely already come across the term correlation, including on my website, since I have repeatedly mentioned what correlation is in my publications
But still I will remind you. Correlation is the dependence of two or more statistically significant quantities.
The maximum correlation value is 1, the minimum is -1.
If the correlation is 1, then the quantities move absolutely synchronously. This degree of correlation is called ideally positive. If – 1, then the values are directly opposite and such a correlation is called ideally negative.
In the context of investment activity, correlation, of course, refers to the coincidence or discrepancy in changes in the value of various assets.
The closer the correlation is to zero or even below zero, the better diversification works. If you've read the article on diversification, you already know that the greater effect of diversification is the increase in returns while reducing risk.
If you find 2 or more assets with a completely negative correlation, then you can get a guaranteed return. That is, both positive and negative correlation of investment assets are useful for the investor.
It is worth noting that in real life, finding 2 assets with zero or negative correlation is very difficult, if not impossible.
Now that we've sorted out the definitions, let's get back to compiling a portfolio. It seems that everything is fine and at the moment such statistical data as correlation, standard deviation and average return are available to everyone, accordingly it will not be difficult to create the most effective investment portfolio, it is enough to insert the data for calculation into specialized software or entrust these calculations to highly qualified specialists.
And some experts (though most likely the majority) will claim that they can determine the efficiency frontier and create an investment portfolio with the best risk/return combination.
I want to disappoint you, such statements are nothing more than beautiful words. In reality, no one is able to determine the efficiency boundary, because it is constantly changing. This means that the most profitable asset classes yesterday become unprofitable today. And this is inevitable.
Accordingly, the investor’s task is not to make the portfolio as efficient as possible, but to avoid making too many mistakes when forming it.
Yes, I already said that there is software that allows you to determine the most effective investment portfolio with a given risk/return combination in a few clicks.
Mathematical calculations are carried out according to portfolio theory and in theory everything should work. But as I already said, theory is one thing, but practice is completely different.
In fact, investment portfolio optimizers are error maximizers, showing disgusting results with real investment portfolios if you blindly rely on their calculations and do not set limits on the maximum share of a particular asset in the portfolio.
The very need to set restrictions in the calculations of the program is in fact a confirmation that it is not possible to mathematically calculate the most effective investment portfolio. Determining the composition of an investment portfolio by eye, so to speak, manually and based on calculations of a special program, but with certain restrictions, is essentially no different. This means that, guided by common sense, private investors are able to create an investment portfolio with good risk/return indicators.
Perhaps in the future there will be a mathematician who will find a solution to this problem, similar to how the main character derived the formula for calculating stock quotes in the film called “Pi”. Well, for now we move on.
Assumptions of Portfolio Theory
When developing the theory, the following assumptions were made:
- There are no costs. Moreover, all types of costs: brokerage commissions for transactions, the spread between the purchase and sale prices of securities and, of course, taxes.
- Market liquidity is maximum. No one is able to influence the price of a security, and accordingly, you can open a position of any size.
- Investors are completely rational, they are aware of all the risks that accompany investment activities, and form their investment portfolio according to the level of risk acceptable to them.
- Investors make asset purchase and sale transactions in an identical manner and only if assets with the best risk/return combination appear in their particular situation. Investors do not consider dividends received or capital gains when making investment decisions.
- Investors control the risk of an investment portfolio only through asset diversification
- Investors make a choice between the highest return for a given level of risk and the highest return for the lowest risk.
- Political events do not affect the market in any way. Psychology has no role in asset pricing.
- Investment portfolio risk refers to the instability of income.
History of creation and refinement of portfolio theory
The main provisions of portfolio theory were formulated by Harry Markowitz while preparing his doctoral dissertation in 1950-1951.
The birth of Markowitz’s portfolio theory is considered to be the article “Portfolio Selection” published in the Financial Journal in 1952. In it, he first proposed a mathematical model for the formation of an optimal portfolio and presented methods for constructing portfolios under certain conditions. Markowitz's main contribution was to propose a probabilistic formalization of the concepts of “return” and “risk,” which made it possible to translate the problem of choosing an optimal portfolio into a formal mathematical language. It should be noted that during the years of creation of the theory, Markowitz worked at RAND Corp., together with one of the founders of linear and nonlinear optimization, George Danzig, and he himself participated in solving these problems. Therefore, my own theory, after the necessary formalization, fit well into the indicated direction.
Markowitz is constantly improving his theory and in 1959 he published the first monograph dedicated to it, “Portfolio Selection: Effective Investment Diversification.”
In 1990, when Markowitz was awarded the Nobel Prize, the book “Mean-variance analysis in portfolio selection and the capital market” was published.
Information taken fromWikipedia
Problems of portfolio theory
When I explained the essence of the portfolio theory “at a glance,” I casually mentioned the difficulties that stand in the way of a portfolio investor when trying to implement theoretical calculations.
And the problems are really serious. Now I will try, without unnecessary pathos and disdain for the works of Harry Markowitz, to open your eyes to portfolio theory.
Not long ago I was a proponent of portfolio theory, but after much thought and reading the book The Black Swan by Nassim Nicholas Taleb, my views have changed.
We truly live in the extreme country where black swans are found.
A black swan is a metaphorical definition of a completely unexpected event that carries with it negative consequences.
Extreme countries and middle countries are the definition of two opposite worlds. Extreme countries are a world where there are black swans. But the average country is a world without black swans, which can be squeezed into the framework of some theory, portfolio theory, for example.
But unfortunately, sometimes very harsh reality does not fit into the framework of portfolio theory. How is it shown? The point is that statistics such as correlation and standard deviation do not work in extreme countries. How do you feel about receiving losses 2, 3 times higher than the calculated standard deviation for any asset, or a change in the correlation of two assets from about zero to 0.7-0.8 or even 0.9 at the time of a stock market collapse, which makes diversification practically useless ?
Unpleasant right? I think many people will not be happy with such events, but in fact this is exactly what happens in life.
As a clear example of such events, I will take perhaps the most severe crisis in US history – the Great Depression. And then we’ll look at how standard deviation works as a measure of risk in the domestic market.
So, the great depression of the 30s of the 20th century. The US stock market fell by more than 80% and such a catastrophic development of events was not expected at all. And if one were to rely on the standard deviation calculated before the Great Depression, one would not be able to adequately measure the riskiness of a stock investment.
And now I will clearly demonstrate this to you. Consider the first stock index in US history - the DowJones. First, as I said, let's take the time period 1900-1929, just before the crisis began. And let’s find out what the standard deviation could have prepared us for.
As you can see from the graph, these 29 years have been quite good for investors, with an annual standard deviation of 26.72%. And in fact, investors who invested money in companies included in the dowjones index lost more than 80% of their capital. These losses exceeded the standard deviation by almost 4 times.
Proponents of portfolio theory may object and suggest choosing a larger sample in order to accurately determine the level of risk for your chosen investment asset.
Okay, let's take the same dowjones index, but only from 1900 to 2016.
Now do you understand that standard deviation is useless for measuring the degree of risk for any investment asset? Because the sample is large, huge events like the Great Depression don't look all that epic. That is, such events cannot have the necessary impact for the standard deviation to fulfill its purpose as a real measure of risk. Moreover, as an example, I took the stock index, the composition of which is constantly changing. But what if we look at individual stocks? It turns out that only a small handful of companies that were founded 100 years ago have survived to this day.
And how do you measure the risk of bankruptcy of the company of the stock you purchased using the same standard deviation? The answer is no. It is only possible to make a forecast based on the fundamental indicators of this company. And the probability that the forecast will come true will decrease as the period that we are going to forecast increases.
Let's go back to the schedule.
The Great Depression does not look that great, and in fact, this period does not stand out on the chart at all. Why is such a monumental event in the entire history of the US stock market almost invisible on the chart? And it all depends on the time scale. The longer the period we want to analyze, the less significant the growth in quotes at the beginning of this period will be. This is due to the effect of compound interest, which makes the stock market move exponentially over the long term.
Well, we found out how things are going abroad, but what about the Russian stock market?
Our situation is a little different. Now I will explain why. The thing is that the beginning of the calculation of the MICEX index coincided with the 1998 crisis, which in scale was comparable to the scope of the Great Depression in the United States. Accordingly, due to the small sample, such a significant event as the 1998 crisis in Russia had a significant impact on the value of the standard deviation.
I calculated the standard deviation for the MICEX index from 1997 to 2007, it turned out to be 81%. That is, the 2008 crisis on the Russian stock market fell within the calculated standard deviation.
You've probably noticed that the longer the time period you look at, the lower the standard deviation turns out to be. As the sample for calculating the standard deviation increases, such black swans (crises: the Great Depression, the 1998 default in Russia, etc.) become “exceptions to the rules of average return” and, accordingly, they cannot greatly influence the final result of the calculations. Although, in fairness, I would like to note that there have been too many such exceptions recently?
Since in Russia the history of the stock market is, in principle, not yet great, this “pathology of the standard deviation in an extreme country” is invisible, but I am absolutely sure that in the future it will be visible to the naked eye.
Now regarding the correlation. It gives the portfolio investor too little information, because it is a dynamic quantity, that is, it is subject to constant fluctuations. And the saddest thing is that asset correlation increases precisely during an economic crisis, precisely at the moment when we need low correlation to reduce the risk of an investment portfolio. Accordingly, calculations based on such a variable value as asset correlation become useless.
And as an example, I took the 2 main components of any investment portfolio: stocks and bonds. For shares – MICEX index ( MICEXINDEXCF), and for bonds - the corporate bond total return index ( MICEXCBITR)
First of all, I marked on the chart the global financial crisis of 2008, the 2nd largest (after the 1998 debt crisis in Russia). As you can see, during stock market declines, the correlation between stocks and bonds increases to about 80%, meaning diversification fails us at the most important time for us.
Well, for contrast, I noted the period of growth of the stock market. The naked eye can see that the correlation gradually decreases when the market is prosperous.
It is easy to draw the following conclusion. The beneficial effect of diversifying an investment portfolio is best seen during periods of rising stock markets, while during periods of falling stock market diversification is very weak.
Well, finally we come to the answer to the main question: what to do with all this?
How to create an investment portfolio for a private investor?
Well, if portfolio theory cannot calculate a sustainable investment portfolio that can withstand crisis situations like the global financial crisis or the Great Depression, then what should a private investor do? How to create an investment portfolio?
I published an article: where I answered this question in sufficient detail.
Let's sum it up
The article turned out to be voluminous as always, but it couldn’t have been any other way, because portfolio theory is a very interesting topic that occupies many scientific minds. I hope I didn't bore you too much. I also hope that you got the main idea that I tried to convey throughout the article. And it lies in the fact that it is impossible to create a “normal” investment portfolio using portfolio theory. If you like, putting together an investment portfolio is a kind of art. In addition, capital markets are chaotic financial energy and it is unlikely that anyone will soon be able to curb it with the help of yet another coherent theory. Today you have learned a lot of new things: the essence of portfolio theory and the assumptions that formed its basis, you also got acquainted with the history of the development of this theory, the problems of its application in practice, and finally learned briefly about how to implement a personal investment plan.
That's probably enough for today. If you find inaccuracies or outright errors in the article, or if you don’t understand something, write in the comments and ask your questions. All the best.
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Modern portfolio theory assumes the diversification of investment risks. The theory allows you to create a collection of assets with a lower level of risk than any single asset. Thus, a portfolio with maximum profitability for a given risk or a portfolio with minimal risk for a given profit is formed. Thus, modern portfolio theory is a strategic tool for investment diversification.
Model Harry Markowitz
Modern Portfolio Theory is a mathematical formulation of risk diversification in investing that aims to select a group of investment assets that collectively have lower risk than any one asset on its own. This is possible because the values of different types of assets often move in opposite directions. In fact, investing, being a trade-off between risk and return, involves high expected returns on high-risk assets.
Thus, modern portfolio theory shows how to select a portfolio with the highest possible expected return for a given level of risk. It also describes how to select a portfolio with the lowest possible risk for the expected return. Therefore, modern portfolio theory is considered as a form of diversification that explains how to find the best diversification strategy.
Model Harry Markowitz, also known as Medium Dispersion Model based on the expected returns (average) and standard deviations (variance) of different portfolios. Using this model, it is possible to make the most effective choice by analyzing different portfolios of certain assets. The method clearly shows investors how to reduce risk if they have chosen assets that do not “move” synchronously.
Basic Provisions of Modern Portfolio Theory
Modern portfolio theory is based on the following key concepts:
- There are no transaction costs for buying and selling securities. There is no spread between the purchase and sale prices. No tax is paid, the only thing that plays a role in determining which securities an investor will buy is risk.
- An investor has the opportunity to open any position of any size and for any asset. Market liquidity is infinite and no one can move the market. So there is nothing stopping an investor from opening a position of any size for any asset.
- When making investment decisions, the investor does not take into account taxes, dividends received or capital gains.
- Investors are generally rational and risk averse. They are aware of all investment risks, and take positions whose risk level is known and expect increased returns in the face of increased market volatility.
- Risk and return relationships are considered over the same period of time. Long-term and short-term speculators have the same motives: expected profit and time frame
- Investors have the same views on risk assessment. All investors are given information and their sale or purchase is subject to identical valuations of the investment and all have the same expectations from the investment. The seller is motivated to sell only because another asset has a volatility level that matches his desired profit. The buyer will make the purchase because the asset has a level of risk that matches the desired return.
- Investors seek to control risk only through asset diversification
- In the market, all assets can be bought and sold, including human capital.
- Politics and investor psychology do not influence the market.
- Portfolio risk directly depends on the volatility of the portfolio's income.
- The investor gives preference to increasing the level of recycling.
- An investor either maximizes his return with minimal risk or maximizes the return of his portfolio for a given level of risk.
- The analysis is based on a single periodic investment model.
Choosing the Best Portfolio
To choose the best portfolio from a range of possible ones, you need to make two significant decisions:
- Determine a set of efficient portfolios
- Select the best portfolio from the efficient set.
While an important advance in finance, the theory has also found application in other fields. In the 1970s, it was widely used in the field of regional science to determine the relationship between variability and economic growth. It has also been used in the field of social psychology to form the concept of self. Currently, it is used to model project portfolios of both financial and non-financial instruments.
Practicing managers even before the advent of portfolio theory talked about profitability and risk, but the inability to quantify these quantities made their principles for constructing portfolios extremely unstable. At the same time, their attention was focused primarily on the risks of individual assets, without understanding how a combination of assets could affect the risk of the entire portfolio. Portfolio theory revolutionized the world of financial management by allowing managers to quantify the return and risk of investments. The importance of these theories was confirmed by the award in 1990 of the Nobel Prize in Economics to Professor Harry Markowitz and Professor William Sharp for the development of portfolio theory.
Harry Markowitz is considered the father of modern “portfolio theory,” which deals with methods for balancing risk and economic benefit when choosing risky investments. In his seminal paper "Portfolio Selection," published in 1952, he developed a mathematical model demonstrating how investors can maximize risk reduction for a given rate of return. The Markowitz model is part of the fundamentals of finance and is widely used in practice by specialists in investment portfolio management.
This article was the first to propose a mathematical model for the formation of an optimal securities portfolio, and presented methods for constructing such portfolios under certain conditions. Markowitz's main contribution was the probabilistic theoretical formalization of the concept of profitability and risk proposed in this short article. This immediately made it possible to translate the problem of choosing the optimal investment strategy into strict mathematical language. It was he who first brought attention to the common practice of portfolio diversification and precisely showed how investors can reduce the standard deviation of portfolio returns by choosing stocks that move differently in price. He also continued to develop the basic principles of portfolio formation. These principles have provided the basis for many studies describing the relationship between risk and return.
Markowitz argues that an investor should base his portfolio selection decision solely on expected return and standard deviation. This means that an investor must estimate the expected return and standard deviation of each portfolio, and then choose the “best” one based on the relationship between these two parameters. Intuition plays a decisive role in this. Expected return can be thought of as a measure of the potential reward associated with a particular portfolio, and standard deviation can be thought of as a measure of the risk associated with that portfolio. Thus, after each portfolio has been examined in terms of potential reward and risk, the investor must select the portfolio that is most suitable for him.
Markowitz quantifies the total risk of a stock portfolio, which consists of systemic (market) and non-systemic (specific) risk. Unsystematic risk in a stock portfolio can be eliminated through diversification. When managing a stock portfolio, it is necessary to quantify not only the risk of each asset included in the portfolio, but also the risk of the entire stock portfolio.
However, the portfolio theory proposed by G. Markowitz has some disadvantages. The main drawback of Markowitz's portfolio theory was that the theory was developed only for stocks, which are known to be quite risky assets. Another disadvantage of the Markowitz model is that the expected return on securities is assumed to be equal to the average return based on historical data. Therefore, it is rational to use the Markowitz model in a stable state of the stock market, when it is desirable to form a portfolio of securities of a different nature that have a more or less long life on the stock market.
In the first half of the 60s, Markowitz’s student W. Sharp proposed the so-called one-factor model of the capital market, in which the later famous “alpha” and “beta” characteristics of shares first appeared. Based on the one-factor model, Sharpe proposed a simplified method for selecting the optimal portfolio, which reduced the quadratic optimization problem to a linear one. In the simplest cases, for small dimensions, this problem could be solved practically “manually”. This simplification made portfolio optimization methods applicable in practice. By the 70s. The development of programming, as well as the improvement of statistical techniques for assessing the “alpha” and “beta” indicators of individual securities and the market return index as a whole, led to the emergence of the first software packages for solving problems of securities portfolio management.
William Sharp used the results of G. Markowitz's research as a starting point for further research, during which he determined the impact of the Markowitz model on the prices of financial assets. By making the assumption that at any given time the prices of financial assets would change to balance the supply and demand of each risky asset, he demonstrated that the expected returns of risky assets should have a very specific structure. The asset structure derived from Sharp's theoretical constructs is now very widely used as a basis for regulating the degree of risk in many areas of the theory and practice of finance.
Developing the approach of G. Markowitz, W. Sharp divided the “entire” risk of an asset into two types: the first is systematic (or market) risk for equity assets, the second is unsystematic.
For a common stock, systematic risk is always associated with changes in the value of securities outstanding in the market. In other words, the return on one share constantly fluctuates around the average return of the entire asset of securities. There is no way to avoid this, since the blind market mechanism operates. The important point about systematic risk is that increasing the number of stocks or bonds cannot eliminate it. However, increasing purchases of securities may entail the elimination of unsystematic risk, that is, those risks that are associated with the influence of all other factors specific to the corporation issuing securities. This means that the investor cannot avoid the risk associated with fluctuations in the stock market. The task in forming a market portfolio is to reduce risk by purchasing various securities. And this is done in such a way that factors specific to individual corporations balance each other. Thanks to this, the portfolio's return approaches the average for the entire market.
The main disadvantage of the Sharpe portfolio model is the need to predict stock market returns and the risk-free rate of return. The model does not take into account fluctuations in risk-free returns. In addition, if the relationship between the risk-free return and the stock market return changes significantly, the model becomes distorted. Thus, the Sharpe model is applicable when considering a large number of securities representing a large part of a relatively stable stock market.
Today, the Markowitz model is used mainly at the first stage of forming a portfolio of assets when distributing invested capital among various types of assets: stocks, bonds, real estate, etc.
The one-factor Sharpe model is used in the second stage, when the capital invested in a certain segment of the asset market is distributed among the individual specific assets that make up the selected segment (i.e., specific stocks, bonds, etc.).
The development of Sharpe's one-factor model was a model for assessing the profitability of financial assets - the Capital Asset Pricing Model (CAPM), proposed in 1964 by W. Sharpe, J. Lintner and J. Mossin. The main result of the CAPM was the establishment of the relationship between the profitability and risk of an asset for an equilibrium market. At the same time, it is important that when choosing an optimal portfolio, an investor should take into account not the “entire” risk associated with an asset (Markowitz risk), but only part of it, called systematic or non-diversifiable risk. This portion of an asset's risk is closely related to the overall risk of the market as a whole and is quantified by the beta coefficient introduced by Sharpe in his one-factor model. The rest (the so-called unsystematic or diversifiable risk) is eliminated by choosing the appropriate (optimal) portfolio.
The influence of Markowitz's "portfolio theory" increased significantly after its appearance in the late 50s and early 60s. works by J. Tobin on similar topics. J. Tobin developed the theory of choice and “portfolio investment”. According to this theory, investors seek to make both higher-risk and less-risky investments in order to secure their investment portfolios. That is, they combine a high degree of risk with guaranteed investment security and only in rare cases strive to obtain the highest profit. The “portfolio investment” model, developed by J. Tobin, combines many securities and represents a much richer arsenal of tools for carrying out economic policy than all the models that preceded it.
There are some differences to note here between Markowitz's and Tobin's approaches. Markowitz's approach is in line with microeconomic analysis, since it focuses on the behavior of an individual investor who forms an optimal, from his point of view, portfolio based on his own assessment of the profitability and risk of the selected assets. In addition, initially the Markowitz model concerned mainly a portfolio of shares, that is, risky assets. Tobin also suggested including risk-free assets, such as government bonds, in the analysis. His approach is essentially macroeconomic, since the main object of his study is the distribution of total capital in the economy in its two forms: cash (money) and non-cash (in the form of securities). The emphasis in Markowitz's works was not on the economic analysis of the initial postulates of the theory, but on the mathematical analysis of their consequences and the development of algorithms for solving optimization problems. In Tobin's approach, the main topic is the analysis of the factors that force investors to form portfolios of assets, and not to hold capital in any one form, such as cash.
In 1976, Yale University professor S. Ross proposed arbitration theory, which is an alternative to the CAPM general equilibrium model in the financial market. The main assumption of the theory is that each investor seeks to take advantage of the opportunity to increase the profitability of his portfolio without increasing risk. Ross's arbitrage theory states that stock returns depend partly on macroeconomic factors and partly on factors affecting specific (diversifiable) risk.
The model is called arbitrage because it imposes arbitrage restrictions on asset returns. This means that in the event of an imbalance in the market, that is, the emergence of non-linear relationships between risk and return on assets, arbitrage profits can be earned. In turn, the actions of the arbitrageurs will restore the balance. Arbitrage profits are obtained as a result of the formation of an arbitrage portfolio. portfolio return Markowitz
Investors in the market seek to increase portfolio returns without increasing risk. This opportunity can be realized through an arbitrage portfolio, that is, the formation of a portfolio by simultaneously selling shares at a relatively high price and buying the same shares elsewhere at a relatively low price. This operation will allow the investor to receive risk-free income without investing. Arbitrage opportunities arise when stocks or portfolios with the same sensitivity to factors have different expected returns. Investors are rushing for risk-free returns, and arbitrage opportunities are running out. Thus, in equilibrium, stocks and portfolios with the same sensitivity to factors have the same expected return (adjusted for specific risk). The advantage of the arbitrage model is that there are fewer assumptions about investor behavior in the market compared to the CAPM model. In addition, building an arbitrage portfolio means no additional investment (money for the purchase of securities is generated through the sale of other securities) and no risk.
The main conclusions that classical portfolio theory has reached today can be formulated as follows:
- - the effective set contains those portfolios that simultaneously provide the maximum expected return at a fixed level of risk, and the minimum risk at a given level of expected return;
- - it is assumed that the investor chooses the optimal portfolio from the portfolios that make up the efficient set;
- - diversification usually leads to a reduction in risk, since the standard deviation of the portfolio will generally be less than the weighted average standard deviations of the securities included in the portfolio;
- - in accordance with the market model, the total risk of a security consists of market risk and own risk;
- - diversification leads to averaging of market risk;
- - diversification can significantly reduce your own risk.
Thus, we can formulate the following basic postulates on which classical portfolio theory is built:
- - the market consists of a finite number of assets, the returns of which for a given period are considered random variables;
- - the investor is able, for example, based on statistical data, to obtain an estimate of the expected (average) values of returns and their pairwise covariances;
- - the investor can create any acceptable (for this model) portfolios;
- - portfolio returns are also random variables;
- - comparison of selected portfolios is based on only two criteria - average return and risk;
- - the investor is risk averse in the sense that, of two portfolios with the same return, he will definitely prefer the portfolio with less risk.
It is obvious that any portfolio theory is an abstract model that ignores many objectively existing patterns and obvious facts in the financial market, that is, it presupposes a number of limitations and simplifications in the description of real economic processes and relationships. However, portfolio theory as an economic and mathematical model should have a minimum number of restrictions, which will allow achieving greater situational flexibility of the model and greater realism of the results obtained.
The limitations, taking into account which it is permissible to formulate a portfolio theory for the modern financial market, may be as follows:
- - the presence at the portfolio manager’s disposal of a certain limited amount of financial resources, which may include both resources transferred to him by the investor and additionally borrowed resources;
- - the rate on risk-free investments is lower than the rate on risk-free borrowings;
- - the assessment of investment assets is carried out according to an unlimited number of parameters using characteristics that have numerical values;
- - transaction costs and the level of liquidity of financial assets are recognized as different and taken into account when assessing their investment attractiveness;
- - investors strive for maximum profit while minimizing all risk factors, however, individual investors may be insensitive to certain risk factors;
- - the portfolio is fixed, that is, changes in its structure are not expected throughout the entire period of its existence;
- - none of the investors can influence the market as a whole, and all investors make transactions only at market prices.
It is clear that in practice strict adherence to these provisions is very problematic. However, the assessment of portfolio theory should be based not only on the degree of adequacy of the initial assumptions, but also on the success of solving investment management problems with its help. In recent decades, the use of portfolio theory has expanded significantly. An increasing number of investment managers and investment fund managers are using its methods in practice, and although it has many opponents, its influence is constantly growing not only in academic circles, but also in practice, including in Russia. The awarding of Nobel Prizes in economics to its creators and developers is evidence of this.
Portfolio theory of G. Markowitz and its development
Note 1
Harry Markowitz is considered the founder of the theory of determining the efficiency of the portfolio set. In 1952, his work was published, in which he, for the first time in the history of economics, formulated the main provisions of portfolio theory and introduced key concepts in this area. In 1990, his achievements were recognized with the Nobel Prize in Economics.
The main provisions of portfolio theory are the following statements:
- The structure of the portfolio influences the degree of risk of the securities included in it;
- The profitability of securities is directly related to the degree of their risk.
The basis of G. Markowitz's portfolio theory is the idea of the possibility of reducing the investor's total risk by combining securities into a portfolio. This theory suggests that the risk of a financial instrument within a portfolio should be assessed by taking into account the impact of other financial instruments in the same portfolio. Markowitz's portfolio theory included mathematical methods applicable to the formation of an optimal investment portfolio, but largely theoretical and difficult to use in practice.
G. Markowitz's theory was developed in the works of his student, W. Sharp, who proposed more practical solutions in the field of effective management of investment portfolios. The portfolio theory was concretized with the advent of the Capital Asset Pricing Model (CAPM), which was developed by several specialists at approximately the same time and independently of each other.
The CAPM model is based on the idea of ideal markets. In accordance with this model, the required return on an investment portfolio is determined by three variables - the risk-free rate of return, the average return on the securities market and the index of the volatility of the return of a particular asset to the average market return (Beta). Thus, CAPM is a one-factor model that reflects the relationship between security returns and average market returns.
Effective portfolio of G. Markowitz
Markowitz's portfolio theory has a number of assumptions, without which it would be impossible to carry out analysis and build an effective portfolio. Within the framework of this theory, in particular, speculative possibilities are not considered, which greatly distances the theory from the conditions of the real market. According to Markowitz, the total return of a portfolio cannot exceed the maximum return of the financial instruments included in it. A portfolio is considered effective if it is balanced in terms of return and risk and tends to grow even in cases where its components lose value. There can be several such portfolios, and their collection is called a set of effective portfolios. The figure below shows the so-called “Markowitz Umbrella”, reflecting the many possible and effective portfolios.
Strengths and weaknesses of portfolio theory
Each theory has its advantages and disadvantages. The advantages of G. Markowitz’s portfolio theory include the mathematical apparatus he formulated, which allows us to largely automate and simplify the process of forming an investment portfolio, as well as the ability to graphically represent information about the portfolio set.
Note 2
The main weakness of the investment portfolio theory is its complexity from the point of view of practical application. This theory does not define the criteria for inclusion and exclusion of financial instruments from the portfolio. In addition, the methodology of this theory is based on retrospective analysis without forecasting. It can be noted that the Markowitz portfolio theory is not applicable in situations of general deterioration in the market situation.
Despite quite significant shortcomings, Markowitz’s portfolio theory is still used today as a component of the investor’s toolkit.
Portfolio theory(portfolio theory) is a theory of investment management based on statistical methods for optimizing the relationship between the level of profitability and risk formed according to a selected criterion. This theory consists of the following main sections:
- assessment of investment qualities of individual investment objects— description of all types of assets in terms of expected income and risk;
- formation of individual investment decisions- determining how assets should be allocated between different classes of investments, such as or;
- portfolio optimization- balancing risk and return when choosing what will be included in a portfolio of investments, for example determining which stock portfolio offers the highest return for a given level of expected risk;
- cumulative assessment of the investment portfolio based on the ratio of the level of profitability and risk- highlighting the different types of results shown by each of the shares (risk), and classifying them into those related to the market (systematic) and related to a given industrial sector or to a given/type of securities (residual).
Portfolio theory is a comprehensive approach to investment decision making that allows an investor to classify, evaluate and control both the type and amount of expected risk and return; also called portfolio management theory(portfolio management theory) or modern portfolio theory(modern portfolio theory).
Important provisions of portfolio theory are the numerical expression of the relationship between risk and return and the assumption that investors should receive compensation for what they accept. Portfolio investment theory diverges from traditional security analysis in that it shifts the focus from analyzing the characteristics of individual investments to determining the statistical relationships of specific securities that make up the portfolio as a whole.
Portfolio theory is a concept developed by G. Markowitz in 1952, which explores ways to maximize the expected return on the components, while properly allocating risk. Markowitz believed that rational investors would only risk their savings if the expected return sufficiently compensated them for the risk. A significant portion of investors prefer to have an effective investment portfolio, i.e. maximum investment security relative to the expected income or the highest possible income relative to a given degree of risk. The practical implications of portfolio theory are that investors must properly distribute risks by composing their portfolio from different stock market instruments.
The Harry Markowitz model, also known as the mean-variance model, is based on the expected returns (mean) and standard deviations (variance) of different portfolios. Using this model, you can make the most effective choice by analyzing different portfolios of certain assets. The method clearly shows investors how to reduce risk if they have chosen assets that do not “move” synchronously.
Basic provisions of modern portfolio theory
Modern portfolio theory is based on the following key concepts:
- There are no securities for buying and selling. There is no difference between the purchase and sale price. No tax is paid, the only thing that plays a role in determining which securities an investor will buy is risk.
- An investor has the opportunity to open any position of any size and for any asset. Market liquidity is infinite and no one can move the market. So there is nothing stopping an investor from opening a position of any size for any asset.
- When making investment decisions, the investor does not take into account taxes received or capital gains.
- Investors are generally rational and risk averse. They are aware of all investment risks, and take positions whose degree of risk is known and expect increased returns in the face of increased market volatility.
- Risk and return relationships are considered over the same period of time. Long-term and short-term speculators have the same motives: expected profit and time frame.
- Investors have the same views on risk assessment. All investors are given information and their sale or purchase is subject to identical valuations of the investment and all have the same expectations from the investment. The seller is motivated to sell only because the other asset has a volatility level that matches his desired profit. The buyer will make the purchase because the asset has a level of risk that matches the desired return.
- Investors seek to control risk only through assets.
- In the market, all assets can be bought and sold, including human capital.
- Politics and investor psychology do not influence the market.
- Portfolio risk directly depends on the volatility of the portfolio's income.
- The investor gives preference to increasing the level of recycling.
- An investor either maximizes his return with minimal risk or maximizes the return of his portfolio for a given level of risk.
- The analysis is based on a single periodic investment model.