(Market) risk premium. Financial asset return model (CAPM) SARM version for bonds

17.03.2024

Model ( SA PM ) describes the relationship between market risk and required return. Model ( CAPM ) is based on a system of strict premises. According to the logic of this model, an investment decision is made under the influence of two factors - expected return and risk, the measure of which is the dispersion or standard deviation of return. Having accepted a number of assumptions (investors behave rationally, measure time in the same units, think in a similar way, borrow and lend funds at a risk-free rate, etc.), the authors of the model showed that if these assumptions are met, an investment portfolio that follows the proportions of the market should be the optimal investment solution for all investors.

The formal representation of the final equation of this model is as follows:

where is the expected income on a specific security subject to market equilibrium;

m f- the rate of return on a risk-free security, which is the most important element of the stock market. Examples of guaranteed fixed income securities include government bonds.

b i - coefficient of stock i (b i) is a measure of a stock's market risk. It measures the volatility of a stock's return relative to the return of the market average portfolio. b-coefficient is related to tilt characteristic lineb-coefficient is related to tilt characteristic line shares, which is a graphical representation of the regression equation constructed using statistical data on the profitability of the i-th share and the average market profitability.

() -market risk premium.

The relationship between a security's return and its beta is linear and is called the Security Market Line (SML). The SML equation can be written in the form:

On the SML chart, the β coefficients are plotted along the horizontal axis, and the efficiency of securities or portfolios is plotted along the vertical axis. But this direct SML reflects the ideal relationship between β and the performance of securities and portfolios. All points lying on the SML line correspond to “fairly” valued securities (portfolios), and those that lie above/below this line correspond to undervalued/overvalued. Graphic representation of the securities market line for example 4.3. shown in Figure 4.7.

Securities market line ( SML) securities reflects the risk-return relationship for individual shares. The required return of any stock is equal to the risk-free rate added to the product of the market risk premium and b - the stock coefficient:

The absence of risk on risk-free securities entails a minimum level of profit. Because of this, risk-free securities are the main regulator of profits and risks.

Let us assume that the yield on guaranteed securities is mf. In this case, any investment portfolio containing securities with varying degrees of risk gives a higher profit than investments of similar volume in guaranteed securities. Therefore, we can conclude that replacing any securities with more profitable ones increases the risk of the portfolio.

It is convenient to calculate the effectiveness of securities from the effectiveness of a risk-free deposit m f.

m i = a i + b i ´m r = m f + b i (m r – m f)+ a i,

Where a i , = a i + (b i -1) m f.

The excess of security efficiency over risk-free efficiency m f called the risk premium. Thus, this risk premium is basically a linear function of the risk premium for the market as a whole, and the coefficient is the beta of the security. This is, however, true if a=0. Such securities are said to be “fairly” valued. The same securities for which a > 0 are undervalued by the market, and if a< 0, то рынком переоценены.

According to E. Dimson, in the economically leading countries of the world, the market premium () is equal to 8% per annum (data obtained through a retrospective analysis of stock markets over 50 years). That is, if, for example, the risk-free investment rate (in dollars) is 5% per annum, and the coefficient b for a company is 0.65, then the long-term return that an investor should require from the shares of this company in a stable economy is:

5% + 0.65 x 8% = 10.2% per annum, dollars.

However, in developing markets, which include the Russian stock market, such use of the model is impossible.

The question is ambiguous: what is the risk-free rate in Russia?

In a stable economic system, for example in the USA or England, the rate m 0 is assumed to be equal to the yield of government obligations, most often treasury bills (treasure bills), under the terms of issue close to Russian GKOs.

However, Russian government obligations are not at all risk-free. This was obvious long before the 1998 crisis: the yield on GKOs was always variable and either rose (during the period of their circulation) to 200% per annum or higher, or dropped (during the relative stabilization of the economic situation) to 15%. If dispersion is a measure of risk, then we can say unequivocally that GKOs were not just risky, but purely speculative securities.

Another question that is not obvious for emerging markets is: what should be the market premium to profitability, i.e. magnitude()in the CAPM model?

There are two problems here. Firstly, if this premium is determined on the basis of any existing Russian stock index, then we risk relying on unreliable data. The Russian stock market is dominated by over-the-counter activity, and, as some studies show, it has a low degree of information efficiency. This may cause an index based on average bids and offers from over-the-counter traders to distort actual trends in the market.

Secondly, even if we take the most trustworthy stock index as a basis and consider it a fairly reliable indicator of the dynamics of the market portfolio, then there is an acute lack of information.

In deriving his average market premiums, E. Dimson was based on a 50-year historical analysis. However, an emerging market tends to be young and unstable. A period of instability is detrimental to investment activity and should not last long. Therefore, the trend of the developing market is: uncertain due to the shallow depth of history and general volatility; heterogeneous, since the government of a developing country will try to attract investors, stabilize the market and increase its predictability. Along the way, it will try different strategies, which will affect the dynamics of the stock market.

For example, taking the time interval 1995-1997 as the basis for the calculation. for the Russian market, we will receive an average annual return of about 80% (in dollars). It is absolutely clear that we cannot demand such profitability from long-term projects of industrial corporations; this would make the majority of good and real projects in the Russian Federation unprofitable, and therefore calculations of this kind would be incorrect.

The capital market line (CML) reflects the risk-return relationship for efficient portfolios, i.e. for portfolios combining risky and risk-free assets.

Note that not only securities have betas, but also portfolios, and the beta of a portfolio is equal to the weighted sum of the betas of the securities included in the portfolio. As with securities, the portfolio is said to be “fairly” valued, undervalued, or overvalued depending on a p.

From the foregoing follows a relationship known as the capital market line (CML), which connects performance indicators and the degree of portfolio risk, i.e. m r And ( m p £ , s p £ s m r)

m p = m f+ ´ , (4.10)

Where m p- profitability (efficiency) of the stock portfolio;

mf- return on risk-free securities;

Standard deviation of return on market securities;

s p- Standard deviation of return on portfolio shares.

Consider two statements about security risk and portfolio risk:

· Market risk takes into account the majority of a well-diversified portfolio.

· The beta of an individual security measures its sensitivity to market fluctuations.

Let's try to explain this. Suppose we have obtained a portfolio containing a large number of securities, say 100, by randomly selecting them from the market. What will we have then? The market itself, or the portfolio, is very close to the market. The beta of the portfolio will be 1, and the correlation with the market will be 1. If the standard deviation of the market is 20%, then the standard deviation of the portfolio will be 20%.

Let us now assume that we have received a portfolio from a large group of securities with an average beta of 1.5. And this portfolio will be tightly linked to the market. However, its standard deviation will be 30%, 1.5 times that of the market. A well-diversified portfolio with a beta of 1.5 will amplify every market move by 50% and will have 150% of the market risk.

Of course, the same thing can be repeated with securities with a beta of 0.5 and get a well-diversified portfolio that is half as risky as the market. The general statement is that the risk of a well-diversified portfolio is proportional to the beta of the portfolio, which is equal to the average beta of the securities included in that portfolio. This shows how portfolio risk is determined by the betas of individual securities.

Values of beta coefficients in the model SARM And in the market model are similar in meaning. However, unlike the CAPM, the market model is not an equilibrium model of the financial market. Moreover, the market model uses a market index, which generally does not capture the market portfolio used in SARM.

There are a number of reasons why the required and expected returns do not match. These include: 1) a change in the risk-free rate due to a revision of the expected inflation rate, 2) a change in b; 3) reassessment of the investor's attitude to risk.

The CAPM is well founded in theory, but it cannot be confirmed empirically, it parameters are difficult to estimate. Therefore, the use of CAPM in practice is limited.

In order for it to “work,” it is necessary to comply with such obviously unrealistic conditions as the presence of an absolutely efficient market, the absence of transaction costs and taxes, equal access of all investors to credit resources, etc. Nevertheless, such an abstract logical construction has received almost universal recognition in the world of real finance. Major market institutions such as investment bank Merril Lynch regularly calculate β - coefficients of all major companies listed on stock exchanges. The lack of a developed financial infrastructure in Russia still prevents the use of the full potential inherent in this model.

Therefore, let’s consider an example of calculating the level of expected return using the capm approach on the US stock market.

Company having β - coefficient 2.5, intends to attract additional equity capital by issuing ordinary shares. The risk-free interest rate is 6.25%, the average market return calculated using the S&P 500 index is 14%. In order to make its securities attractive to investors, the company must offer an annual income of at least 25.625% (6.25 + 2.5 * (14 – 6.25)). The risk premium will be 19.375%. Such significant restrictions imposed by the market on the possibility of reducing the price of capital set a limit on the profitability of investment projects that the company was going to finance with attracted capital: the internal rate of return of these projects should not be lower than 25.625%. Otherwise, the NPV of projects will be negative, that is, they will not provide an increase in the value of the enterprise. If β -the company's ratio was equal to 1.5, then the risk premium would be 11.625% (1.5 * (14 – 6.25)), that is, the price of new capital would be only 17.875%.

m f = 6.25%

2.5

Drawing. Level Relationship β - coefficient and required profitability

In order to overcome the noted shortcomings of the CAPM, attempts were made to develop alternative risk-return models; theory of arbitrage pricing(ART) – the most promising from new models.

Example 4.3.

The table provides information on the profitability of the GLSYTr stock (m i) and the market index (m r) for ten quarters:


m i
m r

It is known that the efficiency of risk-free investments is 4%.

(market model, financial asset return model (CAMP), securities market line (SML) papers) .

Required:

1) build market model, Where m i – dependent variable, m r - explanatory variable;

2) determine the characteristics of the security: market (or systematic) risk, own ( or unsystematic) risk, R2,a.

3) provide a graph of the constructed model;

4) construct a security market line (SML).

Solution

1) We will find the model parameters using the tool Regression Analysis Package EXCEL.

1. Data entry (Fig. 4.4. – 4.5.).

Rice. 4.4. Regression - choice of analysis tool.

Rice. 4.5. Input data intervals are specified.

2. Calculation results (Tables 4.3 – 4.5).

Table 4.3.

Table 4.5.

WITHDRAWAL OF THE REST
Observation	Predicted m i	Leftovers
	23.000	0.000
	21.167	-0.167
	21.167	-1.167
	23.000	-1.000
	23.000	0.000
	24.833	-0.833
	24.833	0.167
	26.667	0.333
	23.000	2.000
	19.333	0.667

Using the data in Table 4.3, the resulting market model can be written as m i = 4.667 + 1.833 ´m r . Hence, b- GLSYTr stock ratio is 1.833.

b i = =2.2/1.2=1.833,

where 230/10=23, =100/10=10,

· To calculate your own risk let's use the formula = .

7.667/10 = 0.77 (7.667 of table 4 .)

Table 4.

Explanations for Table 4.

	Df – number of degrees of freedom	SS – sum of squares	MS
Regression	k =1		/k
Remainder	n-k-1 = 8		/(n-k-1)
Total	n-1 = 9

To calculate systematic risk (or market) must first be calculated b i 2 = 1.833*1.833=3.36, and now you can determine the amount of market risk: b i 2 s mr 2 = 3.36*1.2= 4.03.

General risk s i 2 = b i 2 s mr 2 +s e 2 = 4.03+0.77=4.8

· R-squared equals 0.840 (from table 5)

Explanations for calculations without a PC.

R i 2 =b i 2 s mr 2 / = 4.03 /4.8=0.84

This ratio characterizes the share of risk of these securities contributed by the market. The behavior of GLSYTr shares is 84% predictable using the market index.

Table 5.

· a i, = a i + (b i - 1)m f = 4.667 +(1.833 –1) ´4=8

GLSYTr shares can be classified as “aggressive” securities, since the beta coefficient is 1.833.

· The graph of the regression model of the dependence of the return on GLSYTr shares on the market index is shown on rice. 8.

3) The graph of the regression model of the dependence of the return on GLSYTr shares on the market index is shown in Figure 4.6.

4) Rice. 4.7. Securities Market Line (SML).

4.4 Multifactor models. The theory of arbitrage pricing.

In factorial(or index) models (factor models) the return of a security is assumed to respond to changes in various factors (or indices).

The CAPM is a one-factor model. This means that risk is a function of one factor - b - a coefficient that expresses the relationship between the return of a security and the return of the market. In reality, the relationship between risk and return is more complex. In this case, it can be assumed that the stock's required return will be a function of more than one factor. Moreover, it is possible that the relationship between risk and return is multifactorial. Stephen Ross proposed a method called theory of arbitrage pricing(Arbitrage Pricing Theory, ART). The ART concept allows for the inclusion of any number of risk factors, so that the required return may be a function of three, four, or even more factors.

To accurately estimate the expected returns, variances, and covariances of a security, multifactor models are more useful than a market model. This is because actual security returns are sensitive to more than just changes in the market index, and there is more than one factor in the economy that influences security returns.

There are several factors that influence all areas of the economy:

1. Growth rate of gross domestic product.

2. Level of interest rates.

3. Inflation rate.

4. Oil price level.

When constructing multifactorial X models try to take into account the main economic factors that systematically affect the market value of all securities. in practice, all investors explicitly or implicitly use factor models. This is due to the fact that it is impossible to consider the relationship of each security with each other separately, since the amount of calculations when calculating the covariances of securities increases with the number of analyzed securities.

If we assume that security returns are influenced by one or more factors, then the initial goal of security analysis is to determine these factors and the sensitivity of security returns to their changes. Unlike single-factor models, a multi-factor model of security returns that accounts for these various influences may be more accurate.

· The best known is the BARRA multifactor model, which was developed in the 1970s by Barr Rosenberg and has been constantly improved since then. At the same time, in addition to market indicators, when developing BARRA, financial indicators (in particular, balance sheet data) of companies were taken into account. The new version of BARRA, the so-called E2, uses 68 different fundamental and industrial factors. Although BARRA was originally intended to evaluate American companies, practice has shown that it can be successfully applied in other countries.

· Another type of multifactor models is ART arbitrage pricing model Stefan Ross (1976). ART is a two-tier model. First, sensitivities to pre-selected factors are determined, and then a multifactor model is constructed in which the role of factors is played by returns on portfolios that have unit sensitivity to one of the factors and zero sensitivity to all others.

The model of an analogue of the SML line in arbitration theory is as follows:

where is the required portfolio return with unit sensitivity to j-th economic factor and zero sensitivity to other factors.

The disadvantage of this model is that in practice it is difficult to know which specific risk factors should be included in the model. Currently, the following indicators are used as such factors: the development of industrial production, changes in the level of bank interest, inflation, the risk of insolvency of a particular enterprise, etc.

Having considered the main issues related to the calculation of interest rate risk, we can draw some conclusions. The securities market is divided into many different groups with different levels of income and risk, and usually the relationship between these values is direct (note that in the case of an inverse relationship, the dominance of the most profitable and safe paper will be observed, as was the case with GKOs). The increased return is a kind of risk premium. Thus, the investor has to choose between risk and return.

A line of graphs that is systematic, or market risk versus the return of the overall market at a certain time, and shows all risky securities.

Also referred to as a "feature line".

SML basically plots the results from the Capital Asset Pricing Model (CAPM) formula. The X-axis represents risk (beta) and the Y-axis represents expected return. Market risk premium is determined on the slope of the SML.

The equity market line is a useful tool in determining which assets are being considered for a portfolio that offers a reasonable expected return on risk. Individual securities plot on SML chart. If the security risk compared to the expected return is higher, the SML is underestimated because the investor can expect a greater return for the inherent risk. The security chart below the SML is inflated because the investor will accept less return on the amount risked themselves.

Indicator beta coefficient- is one of the units of measurement that provides a quantitative comparison between the exchange rate movement of the value of shares and the movement of the stock market in general terms.

Application of beta coefficient

In economics, there is also the concept of beta coefficient - this is a certain indicator of the level of risk that is used for an investment portfolio or applied to securities.

As an indicator, this coefficient indicates the following factors:

Determines the degree of stability of a securities portfolio in comparison with other securities on the stock market.

Indicates the quantitative relationship between the rise and fall of prices for a specific share, and price fluctuations in the market in general.

The value of the beta coefficient ranges from 1; if the beta coefficient of a stock is less than one, the stock is stable; if the value is more than 1, the stock is unstable. Therefore, investors prioritize buying shares with low ratios.

Beta Calculation

For an asset Beta coefficient as part of a portfolio of certain securities, or an asset in the form of a stock index relative to a reference portfolio, the coefficient is applied β and in linear regression (asset return) for the period Ra,t in relation to the return for the period Rp,t of the market portfolio

Ra,t = a + βаrp,е+ Еt

The formula for a security's beta is:

βа=Cov(ra,rp) : Var(rp)

Where are the indicators:

ra- this is the value of the assessment for which the coefficient or profitability of the analyzed asset is calculated.

rp- the value with which the profitability of securities or the market is compared.

Cov– means the covariance of the reference and estimated values.

Var- dispersion (measure of deviation of the indicator) of the reference value.

For companies that do not trade on the stock market, the beta coefficient is calculated on the basis of comparative characteristics with competing firms; for such calculations, a number of changes are made to the formula/

A coefficient is a special case of assessing the relationship between several variables. The variables are the volatility of own and stock securities.

Criticism of the CAPM.

One of the most famous criticisms is the work of Richard Roll (Roll, 1977). The author focuses on the problem of forming a market portfolio. In reality, it turned out to be impossible to assemble a portfolio that would include absolutely all assets, some of which turned out to be impossible to value, for example, such as intellectual capital, or difficult to link with the prices of shares and other assets, for example, real estate. Therefore, in practice, a well-diversified portfolio is used for calculations, for example, a market index. This approach to building a market portfolio can ultimately distort the results of the study: beta values.

The assumption of the existence of a risk-free asset also raises criticism. In practice, they use the yield of government bonds, the risk of non-payment on which is minimal, but still exists. The problem is that the real return on them is often negative due to inflation.

The CAPM has a number of assumptions associated with ideal investors: everyone has the same investment horizon, everyone values all assets on the market in exactly the same way, and to make such a valuation, every investor has an equal amount of information at any given time (information is disseminated instantly). These assumptions do not hold true in real life, even in the most efficient markets.

The beta coefficient is also a subject of criticism. In their works, Levy (1971) and Blume (1975) pay attention to the problem of the stability of beta over time. The authors came to the conclusion that for any stock the beta coefficient changes over time, however, if portfolios are randomly formed from the same stocks, for example, 10 shares in each, then the beta coefficients of these portfolios become quite stable, which means they can be considered as measures of portfolio risk over a long period of time. Bluma also concluded that in the long term the beta coefficient approaches one, and the company's internal risk tends to the industry average. Using the results of this study, Bluma proposed making adjustments to the so-called “raw beta”, which is obtained from the regression equation. Two types of amendments are most often used:

proposed by Bloom:

βOSL is the beta obtained by estimating the regression equation using the Ordinary Least Squares method.

proposed by Scholes and Williams

where β is the estimated value of the beta coefficient from the regression equation for the present linking the stock returns with the present returns of the market portfolio, β -1 is the estimated beta value relating the stock return to the previous values of the market portfolio return, β +1 is the estimated beta value relating the stock returns with future values of market portfolio return, ρ m is the autocorrelation coefficient of market return.

Also, the problem of beta instability can be solved using the Market Derived Capital Pricing Model (MCPM), in which the model parameters are estimated in the futures asset market and based on expectations for prices of financial assets.

The classical CAPM's premise that only systematic risk factors are important has also been questioned. In the late 20th century, unsystematic variables such as market capitalization or book-to-market ratio were shown to influence expected returns.

The risk measure used in the CAPM: two-way variance has also been criticized. The fact is that in order to use two-way dispersion, a number of conditions must be met: the expected return must have a symmetric distribution and at the same time it must be normal. In practice, these prerequisites are not met. The use of two-way dispersion is also difficult from the point of view of investor psychology. It has been empirically proven that investors tend to invest in assets with positive volatility rather than in assets with negative volatility. And two-way dispersion is a deviation from the average, both negatively and positively, which means that if the stock price rises, then we will consider this asset as risky as if the stock price decreases, which is incorrect taking into account the psychology of investors . Therefore, to solve these problems, it is better to use one-way dispersion. Its use is possible with both symmetric and asymmetric yield distributions. Estrada suggested using this method for calculating beta specifically in emerging markets. (Estrada, 2002).

Hogan and Warren (1974) showed that replacing two-way variance with one-way variance does not change the fundamental structure of the CAPM.

Thus, the classic version of CAPM has many disadvantages. Therefore, various modifications of the CAPM were developed in which the criticism was taken into account.

Security Market Line (SML)

CML shows the risk-return profile of efficient portfolios, but says nothing about how underperforming portfolios or individual assets will be valued. To describe such a relationship characterizing an individual security, it is necessary to carry out some transformations.

The standard deviation of the portfolio is calculated using the formula:

Applying it to the market portfolio, we get:

That is, the standard deviation of the market portfolio is the root of the weighted average covariance of the market portfolio with each security included in it. The amount of acceptable risk of each security is determined by the covariance of this security with the market portfolio, i.e., the greater the covariance of the security with the market portfolio, the more risk it introduces into it. It turns out that the standard deviation of the security itself does not play a significant role in determining the risk of a market portfolio; it can be either high or insignificant. Accordingly, investors will choose those securities that have higher covariances with the market portfolio, since such securities bring higher returns. Equation:

called the security market line (SML) and reflects the relationship between the covariance of a security with the market portfolio and the expected return of the security. This dependence is presented in Fig. 2.

Rice. 2.

The equation represents a straight line intersecting the ordinate at point R f with a slope:

The slope of the SML is determined by investors' risk tolerance under various market conditions.

SML is the main outcome of the CAPM. It says that in equilibrium, the expected return of an asset is equal to the risk-free rate plus the reward for market risk, which is measured by beta.

In market equilibrium, the expected return of each asset and portfolio, whether efficient or not, should be located on the SML. SML takes into account only the systemic risk of the portfolio; the unit of risk is the beta value.

And on the CML, in a state of equilibrium, only efficient portfolios are located, and all other portfolios and individual assets are under the CML, it takes into account the entire risk of the portfolio, the unit of risk is the standard deviation.

The model can also be built by simply calculating the required return for different values of the beta factor, leaving the risk-free asset rate of return and the market return constant. For example, given a risk-free rate of return of 6% and a market return of 10%, the required return would be 11% when beta is 1.25. By increasing the beta factor to 2, the required return will be 14% (6% + ). Similarly, you can find the required return for different values of the beta factor and end up with the following combinations of risk and required return:

Risk (beta)	Required return (in%)

By plotting these values on a graph (beta on the horizontal axis and required return on the vertical axis), one could get a straight line, as in Fig. 2. The graph shows that risk (beta) increases with the required return, and vice versa.

Model ( SA PM ) describes the relationship between market risk and required return. Model ( CAPM ) is based on a system of strict premises. According to the logic of this model, the investment decision is made under the influence of two factors - the expected return and risk, the measure of which is the dispersion or standard deviation of return. Having accepted a number of assumptions (investors behave rationally, measure time in the same units, think in a similar way, borrow and lend funds at a risk-free rate, etc.), the authors of the model showed that if these assumptions are met, an investment portfolio that follows the proportions of the market should be the optimal investment solution for all investors.

The formal representation of the final equation of this model is as follows:

where is the expected income on a specific security subject to market equilibrium;

m f- the rate of return on a risk-free security, which is the most important element of the stock market. Examples of guaranteed fixed income securities include government bonds.

() -market risk premium.

The relationship between a security's return and its beta is linear and is called the Security Market Line (SML). The SML equation can be written in the form:

The absence of risk on risk-free securities entails a minimum level of profit. Because of this, risk-free securities are the main regulator of profits and risks.

It is convenient to calculate the effectiveness of securities from the effectiveness of a risk-free deposit m f.

m i = a i + b i ´m r = m f + b i (m r – m f)+ a i ,

Where a i , = a i + (b i -1) m f.

5% + 0.65 x 8% = 10.2% per annum, dollars.

However, in developing markets, which include the Russian stock market, such use of the model is impossible.

The question is ambiguous: what is the risk-free rate in Russia?

Another question that is not obvious for emerging markets is: what should be the market premium to profitability, i.e. magnitude()in the CAPM model?

Secondly, even if we take the most trustworthy stock index as a basis and consider it a fairly reliable indicator of the dynamics of the market portfolio, then there is an acute lack of information.

The capital market line (CML) reflects the risk-return relationship for efficient portfolios, i.e. for portfolios combining risky and risk-free assets.

From the foregoing follows a relationship known as the capital market line (CML), which connects performance indicators and the degree of portfolio risk, i.e. m r And ( m p £, s p £ s m r)

m p = m f + ´, (4.10)

Where m p- profitability (efficiency) of the stock portfolio;

mf- return on risk-free securities;

Standard deviation of return on market securities;

s p- Standard deviation of return on portfolio shares.

Consider two statements about security risk and portfolio risk:

· Market risk takes into account the majority of a well-diversified portfolio.

· The beta of an individual security measures its sensitivity to market fluctuations.

Values of beta coefficients in the model SARM in the market model are similar in meaning. However, unlike the CAPM, the market model is not an equilibrium model of the financial market. Moreover, the market model uses a market index, which generally does not capture the market portfolio used in SARM.

The CAPM is well founded in theory, but it cannot be confirmed empirically, it parameters are difficult to estimate. Therefore, the use of CAPM in practice is limited.

Therefore, let’s consider an example of calculating the level of expected return using the capm approach on the US stock market.

m f = 6.25%

2.5

Drawing. Level Relationship β - coefficient and required profitability

In order to overcome the noted shortcomings of the CAPM, attempts were made to develop alternative risk-return models; theory of arbitrage pricing(ART) – the most promising from new models.

In the theory of portfolio analysis, there are approaches that allow you to form an optimal investment portfolio. The optimal portfolio of securities is one that provides the optimal combination of risk and return.

Describing the theory capital market lines (CML) the equation allows you to form an optimal portfolio by maximizing the return for the selected risk value (in this case, the selected risk value must lie on the capital market line). The equation looks like:

where is the profitability of the market portfolio (the market index can be used as such an indicator);

Standard deviation of securities market returns;

Standard deviation of the return on the optimal portfolio.

The overall risk of an investment portfolio (measured by standard deviation) consists of systematic and unsystematic. The systematic risk of assets can be measured by the β-coefficient; it reflects the sensitivity of a particular financial asset to changes in market conditions.

In formalized form, the β-coefficient can be represented

Where COVоr is the covariance between the return on stock j and the return on p.

To estimate the β-coefficient of a securities portfolio, use the weighted average formula; the β-portfolio is the weighted average of the β-coefficients included in its shares, i.e.

where is the share of the i-th asset in the portfolio.

where is the required profitability;

Yield on risk-free securities;

Return on the market portfolio.

From the above it follows the well-known relation like a capital line, connecting performance indicators and the degree of risk of the portfolio, i.e.

And ( ≤ ; ≤ ):

, (5.9)

where is the return (efficiency) of the stock portfolio;

Z – guaranteed interest paid on government securities;

Average market return of shares for period K;

Standard deviation of market securities;

Standard deviation of shares of a securities portfolio.

For and = expression (5.9) will take the following form:

To further analyze the portfolio structure, we use the indicator – beta coefficient (b), calculated using the following formula: .

Beta measures changes in individual stock returns relative to changes in market returns. Securities with this ratio above 1 are characterized as aggressive and more relaxed than the market as a whole. Securities with a beta of less than 1 are characterized as defensive and remain less risky than the overall market. In addition, the beta coefficient can be positive or negative: in the first case, the performance of the securities for which the beta coefficient is calculated will be similar to the dynamics of market performance; If the beta is negative, the security's performance will decrease.

Beta is also used to determine the expected rate of return. The stock pricing model assumes that the expected rate of return on a particular security is equal to the risk-free return (Z) plus β (a measure of risk) times the underlying risk premium (r m -Z).

The rt indicator is usually taken to be a value calculated using some well-known market index.

This model is described by the following formula: ,

where is the expected (average) income for a specific security;

The rate of return on a risk-free security;

Beta - coefficient;

Average market rate of return;

Market risk premium.

The linear relationship described by the formula shown in Fig. 5.1. and is called securities market line (SML).

In order for the return on a security to match the risk, the price of common shares must decline; due to this, the rate of return will increase until it becomes sufficient to compensate for the risk taken by the investor. In an equilibrium market, prices for all common shares are set at a level at which the rate of return on each share balances the investor's risk associated with owning this security. In this case, in accordance with the levels of risk and rate of return, all shares will be placed on the direct securities market.

Capital market theory distinguishes two types of risk: systematic and unsystematic. The total risk is determined by systematic and unsystematic factors. Based on this, the risk of an individual stock can be expressed by the following formula:

where is the risk characteristic of the 1st type of shares;

Characterizes the influence of the general market condition on specific securities;

Characterizes the variation of unsystematic risk, i.e. risk not related to market position.

When considering the issue of optimizing the portfolio structure, it is necessary to dwell on one more indicator - ά (alpha).

The stock price is subject to frequent fluctuations, which are not always adequate to real changes in the affairs of the issuer company. Therefore, many stock market operators try to take advantage of such short-term situations in time to make a profit.

Along with this, there are always securities on the market with persistently overvalued or undervalued prices, and these deviations from the “true” price are long-term in nature. The measure of this deviation is the indicator a, which is calculated as follows:

At<0 действовавшая цена считается завышенной, а при >0 – underestimated. Based on ά-analysis, investors refine the composition of the portfolio, choosing, ceteris paribus, those stocks that have positive ά.