Sharpe index model. Theoretical aspects of the formation of optimal investment portfolios using risk-free loans and borrowed funds Market Sharpe model

The rules for constructing the frontier of efficient portfolios derived by Markowitz make it possible to find the optimal (from the investor’s point of view) portfolio for any number of securities in the portfolio. The main difficulty in applying the Markowitz method is the large amount of calculations required to determine the weights Wi of each security. Indeed, if the portfolio combines P securities, then to construct the frontier of efficient portfolios it is necessary to first calculate P values ​​of expected (arithmetic average) returns E(r i) each security, P values ​​of variances of all returns and P(P– 1)/2 expressions of covariances σi,j of shares in the portfolio. As the number of securities in a portfolio increases, the number of required covariance values ​​becomes prohibitively large. For example, if an investor wants to create a portfolio of 30 stocks, then he needs to calculate 435 covariances, 30 expected returns and 30 variances, i.e. only about 500 values! If the number of securities is doubled (to 60), then the investor will need 1770 covariance values ​​plus 120 values E(r i) and σ j. And with 100 securities in the portfolio, the required number of initial data will exceed 5000.

In 1963, the American economist W. Sharp ( William Sharpe) proposed a new method for constructing the frontier of efficient portfolios, which allows one to significantly reduce the amount of necessary calculations. This method was subsequently modified and is currently known as the single-index Sharpe model. The following are the main stages of constructing this model.

General description of the model

The Sharpe model is based on linear regression analysis method , allowing you to relate two random dependent variables X And Y linear expression like

In the Sharpe model, as a dependent variable Y profitability is taken r i,t some i-th portfolio shares measured over selected calculation steps. Independent variable X The value of some market indicator that affects the return on the portfolio shares is considered. Such an indicator could be, for example, the growth rate of gross domestic product, the inflation rate, the consumer goods price index, etc. Sharpe himself considered the return on the market portfolio as an independent variable. r t,t calculated in the same calculation steps based on the index Standard and Poor "s (S&P500). Expression (3.12) is called the linear regression equation, and the constant coefficients A and β are considered linear regression parameters.

In Russian conditions, profitability r t,t of the market portfolio can be assessed using domestic RCB indices (for example, the MICEX index or the RTS index). If the duration of the holding period is specified and the index values ​​are known I at first I beginning and end I con holding period, then the return on the market portfolio for this period is found by the formula

Building a regression model

For a clear presentation of the content of the Sharpe model, let us assume that the portfolio is formed from the previously discussed shares of companies A, B And WITH. Let the duration of the future holding period be given (for subsequent comparison of the Sharpe model with the Markowitz model, we will assume that this duration coincides with the selected duration in the Markowitz model). Let's also set N= 10 calculation steps in the past (which coincides with the initial conditions introduced in the last chapter for the example according to G. Markowitz). Based on data on changes in the market index (obtained from open sources), we calculate the profitability r t,t market portfolio for selected N calculation steps. We will enter the obtained data into the table. 3.5, which also shows the yields r s,t shares WITH, previously calculated.

Table 3.5

Conditional returns of the market portfolio and stocks WITH

In this case, for the action WITH linear regression equation (3.14) should take the form

Strictly speaking, you can choose any values ​​of the parameters αC and βC, understanding that the theoretical values ​​obtained from this expression r C,t will differ from the actually observed values ​​(see Table 3.5).

For example, if you choose αC = 0.1 and βC = 0.5, then the theoretical value r C,1theor will be

what is different from the observed value r C,1obs = 0.110. To equalize theoretical and observed values, it is necessary to correct the theoretical value r C,1theor. This is achieved by adding to the value r C,1theor of error εС,1, which is εС,1 = -0.0505, since (0.1605 – 0.0505 = 0.110).

You can verify that for the second step of calculation

also does not coincide with the observed value εС,2 = 0.320, therefore it is necessary to correct r With,2theoretic error εС,2 = + 0.074.

Since the quantities r m,t and r C,t are random, then, most likely, the rest of the theoretical values ​​are too r C,t obtained using the linear regression equation will differ from the actually observed values r C,t, given in table. 3.5. In this regard, the values r C,t theory must be corrected by error ε C,t at each calculation step. Since the quantities r m,t , and r C,t are random, then so are the error values ε C,t must also be random variables. As a result, the linear regression equation for the stock WITH should look like this:

Where ε C,t – random error.

In general, if the portfolio includes P shares, then for any ith stock in the portfolio the linear regression equation looks like this:

Where r i,t – profitability i-portfolio shares per step t;αi is a linear regression parameter called coefficient "alpha", showing what part of the profitability i- portfolio shares are not associated with changes in the return of the market portfolio r m,t; βi is a linear regression parameter called beta coefficient, characterizing the sensitivity of the return of the ith stock of the portfolio to changes in market returns r m,t ; r m,t – profitability of the market portfolio at the moment t;εm,t – random error, indicating that real, observed values r i,t deviate from theoretical values r i,ttheor obtained using linear dependence (3.13).

Equation (3.13) is basic in linear regression analysis and is taken as a basis in the Sharpe model. In linear regression analysis, it is assumed that the arithmetic mean (expected) value of observation errors Ε (ε i,t) = 0, i.e. actual values r i,t are on average evenly distributed above and below the values ​​obtained from linear regression.

The Sharpe model examines the relationship between the return of each security and the return of the market as a whole.

Basic assumptions of the Sharpe model:

As profitability security is accepted mathematical expectation of profitability;

There is a certain risk-free rate of return, i.e., the yield of a certain security, the risk of which Always minimal compared to other securities;

Relationship deviations return of a security from the risk-free rate of return(Further: security yield deviation) With deviations profitability of the market as a whole from the risk-free rate of return(Further: market return deviation) is described linear regression function ;

Security risk means degree of dependence changes in the yield of a security from changes in the yield of the market as a whole;

It is believed that the data past periods used in calculating profitability and risk fully reflect future profitability values.

According to the Sharpe model, deviations in security returns are associated with deviations in market returns using a linear regression function of the form:

where is the deviation of the security's yield from the risk-free one;

Deviation of market returns from risk-free ones;

Regression coefficients.

The main drawback of the 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 that describe b O most of the relatively stable stock market.

41.Market risk premium and beta coefficient.

Market risk premium- the difference between the expected return of the market portfolio and the risk-free rate.

Beta coefficient(beta factor) - indicator calculated for securities or a portfolio of securities. Is a measure market risk, reflecting variability profitability security (portfolio) in relation to the portfolio return ( market) on average (average market portfolio). For companies that do not have publicly traded shares, a beta can be calculated based on a comparison with the performance of peer companies. Analogues are taken from the same industry, whose business is as similar as possible to the business of a non-public company. When calculating, it is necessary to make a number of adjustments, in particular, for the difference in the capital structure of the companies being compared (debt to equity ratio).

Beta coefficient for an asset in a securities portfolio, or an asset (portfolio) relative to the market is a relation covariances of the quantities under consideration to variances reference portfolio or market, respectively :

where is the estimated value for which the Beta coefficient is calculated: the return on the asset or portfolio being evaluated, - the reference value with which the comparison is made: the return on the securities portfolio or market, - covariance estimated and reference value, - dispersion reference value.

Beta coefficient is a unit of measurement that gives a quantitative relationship between the movement of the price of a given stock and the movement of the stock market as a whole. Not to be confused with variability.

Beta coefficient is an indicator of the degree of risk in relation to an investment portfolio or specific securities; reflects the degree of stability of the price of these shares in comparison with the rest of the stock market; establishes a quantitative relationship between fluctuations in the price of a given stock and the dynamics of market prices as a whole. If this ratio is greater than 1, then the stock is unstable; with a beta coefficient less than 1 – more stable; This is why conservative investors are primarily interested in this ratio and prefer stocks with a low level.

The Sharpe model examines the relationship between the return of each security and the return of the market as a whole.

Basic assumptions of the Sharpe model:

As profitability security is accepted mathematical expectation of profitability;

There is a certain risk-free rate of return, i.e., the yield of a certain security, the risk of which Always minimal compared to other securities;

Relationship deviations return of a security from the risk-free rate of return(Further: security yield deviation) With deviations profitability of the market as a whole from the risk-free rate of return(Further: market return deviation) is described linear regression function ;

Security risk means degree of dependence changes in the yield of a security from changes in the yield of the market as a whole;

It is believed that the data past periods used in calculating profitability and risk fully reflect future profitability values.

According to the Sharpe model, deviations in security returns are associated with deviations in market returns using a linear regression function of the form:

where is the deviation of the security's yield from the risk-free one;

Deviation of market returns from risk-free ones;

Regression coefficients.

Based on this formula, it is possible, based on the predicted profitability of the securities market as a whole, to calculate the profitability of any security that constitutes it:

where , are regression coefficients characterizing this security.

Theoretically, if the securities market is in equilibrium, then the coefficient will be zero. But since in practice the market is always unbalanced, it shows excess return of a given security (positive or negative), i.e. the extent to which a given security is overvalued or undervalued by investors.

The coefficient is called -risk, because it characterizes the degree of dependence of deviations in the profitability of a security from deviations in the profitability of the market as a whole. The main advantage of the Sharpe model is that the interdependence of profitability and risk is mathematically substantiated: the greater the risk, the higher the profitability of the security.

In addition, the Sharpe model has a peculiarity: there is a danger that the estimated deviation of the security's return will not belong to the constructed regression line. This risk is called residual risk. Residual risk characterizes the degree of dispersion of the deviation values ​​of a security's return relative to the regression line. Residual risk is defined as the standard deviation of the empirical points of a security's return from the regression line. The residual risk of the i-th security is denoted by .

In other words, the risk indicator of investing in a given security is determined by risk and residual risk.

In accordance with the Sharpe model, the return on a securities portfolio is the weighted average of the return indicators of the securities and its components, taking into account risk. The portfolio return is determined by the formula:

where is the risk-free return;

Expected profitability of the market as a whole;

The risk of a securities portfolio can be found by estimating the standard deviation of the function and is determined by the formula:

,

where is the standard deviation of the profitability of the market as a whole, i.e., an indicator of the risk of the market as a whole;

Risk and residual risk of the i -th security;

Using the Sharpe model to calculate portfolio characteristics, the direct problem takes the form:

The inverse problem looks similar:

In the practical application of the Sharpe model to optimize the stock portfolio, the following assumptions and formulas are used.

1). Usually, the yield on government securities, for example, domestic government loan bonds, is taken as the risk-free rate of return.

2). Expert estimates of market returns from analytical companies, the media, etc. are used as the profitability of the securities market as a whole in period t. In conditions of a developed stock market, it is customary to use any stock indices for these purposes. For a stock market that is not very large in terms of the number of securities, the average return on the securities making up the market for the same period t is taken:

where is the return on the securities market in period t;

The initial data for the calculation (the yield of securities) remains unchanged (see Table 4.9.1). In addition, the Sharpe model uses the return of the market as a whole and the risk-free return. The profitability of the market as a whole was taken on the basis of expert estimates, due to the lack of data from external sources. The weekly yield of three-month government short-term bonds was taken as the risk-free yield. Data on the profitability of the market as a whole and on risk-free profitability are presented in table. 4.9.5.

As noted above, the Markowitz model does not make it possible to choose the optimal portfolio, but rather determines a set of efficient portfolios. Each of these portfolios provides the highest expected return for determining the level of risk. However, the main disadvantage of the Markowitz model is that it requires a very large amount of information. A much smaller amount of information is used in W. Sharpe's model. The latter can be considered a simplified version of the Markowitz model. While the Markowitz model can be called a multi-index model, the Sharpe model is called a diagonal model or a single index model.

According to Sharp, the earnings per individual stock are highly correlated with the overall market index, making it much easier to find an efficient portfolio. The use of the Sharpe model requires significantly less calculations, so it turned out to be more suitable for practical use.

Analyzing the behavior of stocks on the market, Sharp came to the conclusion that it is not at all necessary to determine the covariance of each stock with each other. It is quite enough to establish how each stock interacts with the entire market. And since we are talking about securities, it follows that we need to take into account the entire volume of the securities market. However, one must keep in mind that the number of securities and, above all, shares in any country is quite large. A huge number of transactions are carried out with them every day both on the exchange and over-the-counter markets. Stock prices are constantly changing, so it is almost impossible to determine any indicators for the entire market volume. At the same time, it has been established that if we select a certain number of certain securities, they will be able to fairly accurately characterize the movement of the entire securities market. Stock indices can be used as such a market indicator.

Considering above the relationship between the behavior of stocks with each other, we have established that it is quite difficult or almost impossible to find such stocks whose returns have a negative correlation. Most stocks tend to rise in value when the economy is growing and fall in value when the economy is down.

Of course, you can find a few stocks that rose in price due to a special set of circumstances when other stocks fell in price. It is more difficult to find such stocks and give a logical explanation for the fact that these stocks will increase in value in the future, while other stocks will decrease in value. Thus, even a portfolio consisting of a very large number of stocks will have a high degree of risk, although the risk will be much less than if all the funds were invested in the shares of one company.

In order to understand more precisely what effect the portfolio structure has on the risk of the portfolio, let us turn to the graph in Fig. 7, which shows how the risk of a portfolio decreases if the number of stocks in the portfolio increases. The standard deviation for the "average portfolio" made up of one stock listed on the New York Stock Exchange is approximately 28%. An average portfolio made up of two randomly selected stocks will have a smaller standard deviation—about 25%. If the number of shares in the portfolio is increased to 10, then the risk of such a portfolio is reduced to approximately 18%. The graph shows that the risk of the portfolio tends to decrease and approaches a certain limit as the size of the portfolio increases. A portfolio consisting of all stocks, commonly called the market portfolio, would have a standard deviation of about 15.1%. Thus, nearly half the risk inherent in the average individual stock can be eliminated if the stock is held in a portfolio of 40 or more stocks. However, some risk always remains, no matter how widely diversified a portfolio is.

That part of the stock risk that can be eliminated by diversifying the stocks in the portfolio is called diversifiable risk (synonyms: unsystematic, specific, individual); that part of the risk that cannot be eliminated is called non-diversifiable risk (synonyms: systematic market risk).

Firm-specific risk is associated with such phenomena as changes in legislation, strikes, a successful or unsuccessful marketing program, the winning or loss of important contracts, and other events that have consequences for a particular firm. The impact of such events on a stock portfolio can be eliminated by diversifying the portfolio. In this case, unfavorable events in one company will be offset by favorable developments in another company. The essential point is that a significant portion of the risk of any individual stock can be eliminated through diversification.

Market risk arises from factors that affect all firms. Such factors include war, inflation, decline in production, rising interest rates, etc. Since such factors affect most firms in the same direction, market or systematic risk cannot be eliminated through diversification.

W. Sharp's model (beta coefficient concept). Forecasting the profitability of financial instruments is impossible without taking into account risk factors. First of all, systematic risks. Risks are measured using probability theory methods in the form of dispersion (standard deviation). K i – possible income, p i – probability.

The coefficient of variation shows the risk per unit of return. . .

Average market returns can be measured using stock indexes. W. Sharp set the task of finding a connection between market fluctuations and fluctuations in a particular stock. All stocks listed on an exchange represent the market portfolio.

TO

δ

All points inside the curve are valid portfolios. Points on a line – efficient portfolios– these are portfolios that provide either maximum return for an acceptable risk, or minimum risk for a given return.

But for a particular investor, a portfolio that meets all the requirements is necessary. This combination of profitability and risk that suits a particular investor is optimal portfolio.

CML (capital market line) – capital market line, a linear combination between a risk-free asset and a market portfolio. It shows what relationship exists between the return on a risk-free asset and the return on the market portfolio. The point of intersection of the capital market line with the profitability axis gives the value of the risk-free rate (R f).

CML shows that the expected rate of return of any efficient portfolio is equal to the risk-free rate plus the risk premium. It illustrates that the higher the return, the greater the risk.

This yield can be calculated as follows:
. − market standard deviation. − portfolio of the mean square investor.

Sharpe examined the return relationship between market and asset fluctuations.

Financial analysts are engaged in forecasting average market returns (stock indices), and a specific investor needs to know the profitability of a specific stock.

If you build a regression through points, you can get SML (share market line), and it will look like this: . This is the Sharpe model (Capital Asset Pricing Model - CAMP).

This model allows you to estimate the required return on a specific stock (the cost of raising capital). The β coefficient in this case is the same coefficient that shows how the fluctuations of the market and a specific security are related. From a mathematical point of view, .

These odds are published. For example, on Damodaram.com. For example, at the end of 2007 for Russian companies, β was as follows. For blue chips, β=1. This means that the shares of these companies fluctuate just like the stock market. If β>1, then the range of stock return fluctuations is greater than market fluctuations. For example, construction companies, high-tech industries. The systematic risk for these securities is greater than the market average (greater dispersion). If coefficient β<1, это значит, что размах колебаний доходности конкретной акции меньше среднерыночного и, следовательно, систематический риск тоже меньше среднерыночного. Примером таких компаний являются компании пищевой промышленности.

− risk premium. − market risk premium. The global risk premium and country risk premiums are calculated. In particular Damodaran.com has such data: global risk premium = 3.7%, for Europe – 3.9%. By country, Denmark has the lowest risk premiums, 1.8%, Belgium – 2.6%, Switzerland – 2.1%, USA – 4.2%, Japan – 5.1%, Germany – 5.4%, Ireland – 2.6%, Norway – 2%.

Country risks are determined, incl. also a credit rating.

The advantages of the model are the availability of source data, ease of calculation and wide distribution. Disadvantages: this is a single-factor model, it does not take into account the influence of other factors, there is a large error in calculating the β coefficient due to differences in opinions among different analysts.

The disadvantages of the CAPM model are overcome with the help of some adjustments. The modified CAPM – MCAPM model is calculated. It takes into account factors such as country risk, premiums for small companies, and the risk of introducing new products or new investment projects.

There are other multifactorial models, for example, three-factor model: Fama-Freich. . β is the sensitivity of the asset to market returns, с is the sensitivity to the size of the company, d is the sensitivity of the asset to the Tobin coefficient. K HMB is the market risk premium, K SMB is the expected size premium.

For example, R f = 6.8%, β = 0.9%, c = 0.5, K SMB = 6.3, K HMB = 5%, d = 0.3. R m = 8%.

6.8+(8-6.8)*0.9+6.3*0.5+5*0.3= we get the expected return or required return.

The advantage of the model is that it uses three factors. Its disadvantage is that coefficients c and d are not published anywhere and must be calculated by the analysts themselves.

Multifactor model. In addition to the three mentioned factors, it also takes into account others: expected inflation, GDP growth and other economic and political factors.

In practice, the CAPM model is optimal.