Getting as much income as possible while risking as little as possible is the dream of every investor. Not only millions of stock market participants, but also major scientists, including Nobel Prize laureates, are puzzling over the possibility of realizing this "blue dream". For example, the Americans G. Brinson, L. Randolph Hood and G. Bibauer conducted a fundamental study "Determinants of portfolio efficiency" and found that it is not the choice of time to buy, but the distribution of assets in the portfolio determines 75.5% - 98.6% profitability.
Portfolio theories, of which there are dozens today, are devoted to the correct selection of an investment portfolio. The later a theory is created, the more complex mathematics it applies, as a rule, and requires more and more calculations. However, each of the theories has certain assumptions that are applicable to a limited number of cases and tools. But no one has yet come up with an ideal theory for all occasions.
With all the diversity, any portfolio theory is based on the basic ideas and concepts set out in the article "Portfolio Selection", which was published in 1952 by a modest graduate student Harry Markowitz. Subsequently, the provisions of this article were developed by him in a monograph published in 1959. Having mastered the basic ideas of Markowitz, every investor will be able to understand any portfolio theory more easily, better understand the basics of forming investment portfolios and understand that many investment recommendations that are given, so to speak, in a "ready-made form" are actually the result of serious scientific research.
The essence of Markowitz's theory
With the word investment, the concepts of income and risk are constantly used. And the statement "the higher the income, the greater the risk" has become an axiom. We always talk about this at our webinars.
But, what is income? It would seem an elementary question: you subtract the purchase price from the sale price – that's the income. And how to determine income when we haven't sold the stock yet, and the purchase price is unknown, because the entry point is still questionable?
And the risk? Clearly, this is something terrible and terrible, but how to define it? Statements that risk is an undesirable phenomenon for an investor are unlikely to suit us. How to really measure the risk, to establish where it is greater and where it is less – that's what an investor requires from theorists.
Markowitz's historical merit is to give future profitability and risk a mathematical expression. Based on his work, it became possible to calculate both future profitability and risk.
In his article, Markowitz divides the formation of a portfolio into 2 stages:
- Analysis of historical data. Here, on the basis of existing securities quotations, the future profitability and risk for each share are determined.
- Final portfolio formation - securities with better profitability and lower risk are selected for the portfolio.
Then the portfolio is optimized according to the principle:
- the best return at a given, i.e. acceptable for us, risk;
- minimal risk at a given, i.e., a yield that suits us.
Future profitability is a random variable, the exact value of which is not known. There is only a probability that a random variable will take one or another value. The most probable value of a random variable is called its mathematical expectation. Markowitz identified it as a future income.
So, let's summarize:
- The future profitability of a stock is its mathematical expectation, which is approximately equal to the average value of the return of shares for a certain period.
- Risk is the volatility of a stock, which is measured by its variance, approximately equal to the standard (standard deviation) of returns over the same period.
Thus, Markowitz was the first to define an approach to the numerical expression of future profitability and risk based on mathematical calculations. Thus, investments on the stock exchange were taken out of the field of guesses, assumptions, average ceiling forecasts into the field of specific calculations.
Calculations, subsequent conclusions and investment decisions – we are supporters of this approach. This is the most rational way in comparison with actions based on emotions. The participants of our webinars are well aware of this.
Having determined the future profitability and risk for each stock, you can proceed to compiling a portfolio. The profitability of the portfolio as a whole is very easy to calculate. This is the sum of the returns included in the portfolio of shares ("How to calculate the yield").
The risks are more complicated. The fact is that in the economy all subjects (firms, organizations, enterprises) and objects (products, raw materials, etc.) are interconnected. Oil prices are rising – the shares of oil companies are also rising, but the shares of transport companies are falling. During the crisis, the shares of non-cyclical companies feel good and the shares of cyclical ones fall, but the prices of gold and shares of companies producing it are rising, etc. Naturally, we are talking only about general trends. In specific cases, these dependencies may be violated.
Let's say someone prefers investing in oil and invested in oil futures and shares of oil producers. The risks of these instruments will reinforce each other. Of course, with an increase in prices, you can get a kind of "double" income, but with a fall – a "double" loss.
And someone is a supporter of industry diversification, and in addition to oil companies' shares, they will acquire shares of transport companies. The income will still be equal to the amount of income for each instrument, but the overall risk will decrease due to partial mutual compensation of risks for each stock separately.
Taking into account this universal relationship, Markowitz introduced the correlation and risk covariance functions of individual stocks into the portfolio risk calculation. This is his second revolutionary idea. Before Markowitz, it was believed that for the "right" portfolio, it was enough to choose the most profitable stocks, or you could limit yourself to one, but "the best, the best". Markowitz, not on his fingers, but by specific mathematical calculations, proved the infidelity of this approach: yes, in this case, you can get a good income, but, at the same time, the investor assumes an exorbitant risk.
Correlation shows the degree of interdependence of securities quotations.If it is equal to 1 between two stocks, then the quotes are completely interdependent and move parallel to each other on the chart: the failure of one paper corresponds to exactly the same failure of the other. Accordingly, deviations in their profitability, i.e. risks, will add up. If the correlation is -1, then the quotes are completely opposite: each fall of one paper is accompanied by the growth of another, and vice versa. The overall risk of a portfolio of two securities is reduced at the same time. If the correlation is 0, then the quotes of securities, and, consequently, the risks are absolutely independent.
The correlation values of 1, -1 and 0 are extreme and very unlikely. In reality, the correlation takes values in the ranges from 1 to 0 and from 0 to -1.
For the convenience of calculations, not correlation is usually used, but covariance, which is a measure of correlation.
With the introduction of correlation and covariance risk into the calculation, Markowitz proved the need for broad portfolio diversification. Moreover, such diversification, which is not a simple set of profitable stocks, but securities of companies in various industries with preferably negative correlation. Only in this case there is an opportunity to reduce the risk of the portfolio.
After the portfolio is selected, the next step is to optimize it. And here the shares of shares in the portfolio appear. According to Markowitz's calculations, by changing these shares, you can get a lot of portfolios with the same profitability, but only one of them will provide minimal risk. And, conversely, there are the same mass of portfolios with equal risk, but only one of them will provide the greatest return.
The point of the optimization procedure is to sort through all possible combinations of shares to find a portfolio:
- with minimal risk, having the level of profitability we need;
- or with a risk acceptable to us and the highest return.
Optimization of only one portfolio requires a very large amount of computational work, which must be periodically resumed to account for changes in the market situation. It is no coincidence that few people have paid attention to Markowitz's work for more than 30 years. Only the advent of computer technology decisively changed the situation, and Markowitz was awarded the Nobel Prize in 1990.
Read more: Warren Buffett: 10 golden rules of domination
Limitations of the Markowitz portfolio theory, its development by Sharpe, Teynor, Sortino
Publications about Markowitz's theory often talk about its many shortcomings. But the term "disadvantage" is not quite fair. Yes, indeed, Markowitz did not take into account bonds, futures, options, dividends, trading with leverage, considered only growing stocks. But Markowitz did not set a goal to take all this into account. He did exactly what he did - he reviewed a portfolio of long-growing stocks.
The fact is that any theory has a limited scope of application. Therefore, it is better not to talk about its shortcomings, but about its limitations.
To date, portfolio theories have been developed that consider combinations of stocks and bonds, take into account dividends, leverage trading, etc. But they use such a complex mathematical apparatus and require such a huge amount of calculations that are beyond the capacity of an ordinary investor armed with an Excel. Therefore, in practice, only the main conclusions and some particular points are applied from these theories, for example, in the form of ratios, with the help of which it is possible to assess the risk and potential profitability of the portfolio.
In the theory of Markowitz , significant additions were made by U. Sharpe. Firstly, in the general risk of the stock, he singled out unsystematic, i.e. inherent only in this paper, and systematic risk – the risk of the market as a whole. For more information about risks and how to deal with them, see the article Investment risks and ways to minimize them. Secondly, instead of calculating the interdependence between stocks, as in Markowitz, Sharp proposed to determine the correlation of a particular stock and the market as a whole. In practice, market indices are used to assess market movements. The relationship is established by calculating the coefficient β, which is defined as the ratio of the covariance of stock and market returns to the square of the standard deviation of the market. This approach has greatly simplified the calculation of the profitability and risk of the portfolio.
Read more: The value of the Central Bank's Key Rate for the financial market
And finally, Sharp determined how to compare portfolios that have different risks and returns. This is done by calculating the Sharpe Ratio. The Sharpe Ratio shows how much income an investor will receive per unit of risk in excess of the risk-free one. The higher this ratio, the more profitable the portfolio. If the coefficient is negative, it means that the return on the portfolio is lower than the risk-free one and it is easier to invest in a risk-free asset. As the yield of a risk-free asset, the key rate of the Central Bank, deposit rates in a very reliable bank, and the yield of government bonds with a maturity approximately equal to the investment period can be selected.
D. Trainor proposed to replace the portfolio risk with the coefficient β in Sharpe's formula. The point is that an investor can eliminate unsystematic risk by diversifying the portfolio. Therefore, it is advisable to evaluate the portfolio only in relation to the systematic risk that β reflects. The disadvantage of the Sharpe ratio is that the standard deviation of portfolio returns distorts the investor's real risk. It grows with any increase in the volatility of stock prices, both in a positive and negative direction. But only negative deviations carry the risk for the investor.
Therefore, F. Sortino proposed in Sharpe's formula to take into account only the fall of quotations, replacing their general standard deviation with the standard deviation of losses. He also recommended that instead of the profitability of a risk–free asset, MAR (minimum acceptable return) should be included in the formula - the minimum acceptable level of profitability for an investor. The Sortino Ratio - in its essence shows how excess profitability was obtained for the portfolio. Due to high-quality investment ideas or by taking excessive risk on the portfolio.
Sharpe, Teynor and Sortino Ratios are most often used when comparing and evaluating portfolios.
Read more: What is the Sortino Ratio and how to use it
Criticism of portfolio theories
All portfolio theories, as well as all theories of technical analysis, are based on historical data on stock prices. That is, profitability and risk for the future are predicted based on their values in the past. But who said that the yield will necessarily be equal to the mathematical expectation? On the contrary, the theories themselves claim that this is only the most likely result. There is, albeit a smaller probability that the yield will be significantly higher or lower than the mathematical expectation. And what about future profitability in the run-up to or in the conditions of an economic crisis or inflating market "bubbles"? After all, portfolio theories, focusing all investors only on growing stocks, directly provoke such "bubbles".
This means that all our efforts to optimize the portfolio may be useless.
Markowitz himself was aware of the limitations of the historical approach and mentioned in the article that it makes sense to adjust the results of his calculations of profitability and risk based on expert opinions. But Markowitz did not specify how to select trustworthy experts and how to integrate their estimates into the developed formulas. And how can any experts help with the onset of unpredictable events, the so-called "black swans", which can shake the stock market to the ground? A striking example of such a "black swan" is the current pandemic.
Another limitation is that the recommendations for selecting stocks in the portfolio are rather vague. In fact, there are two of them: growing stocks and their diversification by industry. But if we take not the Russian, but, let's say, the American market, where there are a dozen or even dozens of growing stocks in each industry, then which one to choose specifically?
The construction of portfolio theories only on the basis of historical analysis has become the basis for criticism from many scientists and practitioners. A consistent opponent is, for example, W. Buffett, whose views were formed under the influence of the book "A Reasonable Investor" by Benjamin Graham. Neither Graham nor Buffett deny the importance of historical stock price analysis. But Buffett clarifies that if this analysis is used to forecast for a short time interval, then the possibility of predicting the stock price is equal to the probability of an "eagle" or "tails" falling out. But if the investment horizon is expanded to several years, we will get quite acceptable results.
The discrepancy arises in the definition of risk. Following Markowitz, all portfolio theories assess the degree of risk by the volatility of securities. Buffett also points out that the real risk of an investor is an incorrect assessment of the real value of the company and its future profitability, as well as the influence of external and little predictable factors for the stock market: wars, catastrophes, natural disasters, government decisions, taxes, inflation, etc.
Graham emphasized that the principle of "buy while it's cheap, when everyone is selling, sell when it's expensive, when everyone is buying" does not always work. There are periods of prolonged growth or decline in the market. Therefore, you can wait a very long time for the moment to buy, losing money due to inflation, or sell off all the shares, and then buy them at higher prices. It is not so important for a reasonable investor to monitor the market in itself and do not rely on forecasting its dynamics. The main thing, according to Graham, is to buy shares of companies with a high "margin of reliability" and sell them not at any price increase, but only when the margin of reliability begins to run out. In this case, the principle of "buy cheap, sell expensive" begins to work perfectly.
The margin of reliability is determined by the difference between the price of shares and their true value, which is determined by fundamental analysis of financial statements. The margin of reliability consists of the following:
- The ratio of the company's debt to its assets. The lower it is, the better.
- The company's ability to generate profit, high profitability of its business. Moreover, it is important not only to have high profits and profitability in previous periods, but also confidence in their growth for the future.
- Undervaluation of stocks by the market, which is expressed in low values of the P/E (Price/Earnings) and P/B (Price/Book) ratios.
Stocks with a high "margin of safety" we usually call "fundamentally good" stocks.
Graham, like Markowitz, insists on portfolio diversification, but recommends that bonds be included in it. This is an additional way to reduce risk, especially from factors external to the stock market. The fact is that bonds bring stable income, which compensates for stock drawdowns.
Thus, critics do not so much reject portfolio theories as supplement them with criteria for selecting tools for creating the least risky portfolios. In particular, the recommendation on the selection of fundamentally good stocks removes the limitation of Markowitz's theory on the inclusion of exclusively growing securities in the portfolio.
Read more: What is the essence of the Sharpe Ratio and what is it for?
Conclusion
Markowitz's theory had a revolutionary significance for the stock market, creating the basis for the formation and optimization of investment portfolios based on rigorous mathematical calculations.
The introduction of the concept and function of correlation of stock prices showed that the portfolio is not a simple set of securities, but an integral, internally interconnected object, which, with the right selection of tools, reduces the risk of investing.
Markowitz's theory, like any other theory, has its limitations, which have been consistently overcome in the works of followers and opponents. In particular, coefficients were developed to compare portfolios with different risks and returns, as well as criteria for selecting stocks in the portfolio based on fundamental analysis.
The common disadvantage of all portfolio theories is that they require the collection of a huge amount of factual material and the execution of a colossal amount of mathematical calculations. This turns investing in science into an impossible task for an ordinary investor.