CML shows the risk-return profile of efficient portfolios, but says nothing about how inefficient portfolios or individual assets should be valued. This question is answered by the asset market line or SML (Security Market Line). 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. SML is shown in Fig. 3.4. It is a straight line passing through two points whose coordinates are equal to and . Thus, knowing the risk-free rate and the expected return of the market portfolio, one can construct the SML. In market equilibrium, the expected return of each asset and portfolio, whether efficient or not, should be located on the SML. It should be emphasized again that if only efficient portfolios are located on the CML, then both widely diversified and inefficient portfolios and individual assets should be located on the SML.
The SML equation is:
With its help, you can determine the expected return of an asset (portfolio).
The risk-free rate is 15%, the expected market return is 25%. Determine the expected return on an asset with a beta of 1.5. Solution. Beta is equal to:
The slope of the SML is determined by investors' risk tolerance in different conditions market conditions. If investors have optimistic forecasts for the future, then the slope of the SML will be less steep, since in good conditions investors agree to more low premium for the risk, since the risks, in their opinion, are less likely (see Fig. 3.5, SML 1). In other words, in terms of expected return, the price of risk is less. On the contrary, in anticipation of an unfavorable environment, the SML will take on a steeper slope, since in this case investors will demand a higher risk premium as compensation (see Fig. 3.5, SML 2), i.e. in units of expected return, the price of risk is higher.
This dynamics of the SML slope can also be explained from the point of view of discounting future income. As you know, the value of a security is determined by discounting the future income it will bring. Let's imagine the reasoning in general view based on the formula for a security that expects only one payment at the end of period t:
Let an investor predict the level of income for the security of a certain enterprise. It has a certain performance potential and is characterized by a certain level of expected income (P(). In a bad market environment, the probability of receiving such income decreases. Therefore, the investor is ready to buy the paper, but at a lower price (P0). Since the value of the expected income (Pt), as the production potential of the enterprise remains unchanged, and the value (P0) decreases, then, according to formula (3.8), the value r, i.e., the expected profitability, should increase in order to equate the values of Pt and P0. As a result, the slope angle SML in Fig. 3.5 . will increase. On the contrary, under favorable conditions, the probability of receiving the expected income increases, and the investor is ready to buy the security at a higher price (P0). Therefore, in formula (3.8), the value of r decreases. Accordingly, the slope angle SML in Fig. 3.5 decreases.
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If investors' expectations about the risk-free rate change, this will lead to shifts in the SML. If rf increases, the SML will move up, and if rf decreases, it will move down, as shown in Fig. 3.6.
Above we presented formula (3.3), which allows you to calculate the beta coefficient of an asset based on historical data. The value of beta can also be determined using the SML equation by writing it for the actual data received.
Market line valuable papers (English Security Market Line, SML) is a graphical interpretation of the relationship between the risk of an individual security, the measure of which is the beta coefficient, and the rate of return that investors will demand for accepting it. At the same time, the higher the level of accepted risk, the greater compensation should be offered to the investor.
The graphical construction of the securities market line is based on an equation based on the capital asset valuation model ( English Capital Assets Price Model, CAPM).
Where k i– required rate of return for the i-th security;
β i – beta coefficient of the i-th security.
kM– required return on the market portfolio.
Interpretation of a stock market line chart
If the risk-free interest rate and the required return on the market portfolio are known, then the security line graph will look like this:
- For zero-risk securities with a beta of 0, the required rate of return will be equal to the risk-free interest rate. Similarly, the required rate of return on a portfolio of securities with β=0 will also be equal to the risk-free interest rate.
- The slope of the stock market line indicates risk aversion ( English Risk Aversion) in the economy and depends on the value of the risk premium for the market portfolio, which is calculated as the difference between the required return on the market portfolio and the risk-free interest rate ( k M -k RF). Accordingly, the higher the required return on the market portfolio, the stronger its slope.
- Both the line of the securities market as a whole, and the position of an individual security on it, can change over time under the influence of various factors, for example, changes in interest rates, investor risk appetite, changes in the beta coefficient of individual securities, etc.
Example
Suppose that the current risk-free interest rate is 5% and the required return on the market portfolio is 12%. In this particular case SML equation will look like:
k i = 5+ β i (12-5), or
Graphically, this dependence will look like this:
Let's consider two securities: shares of Company A with β=0.5 and shares of Company B with β=2. Substituting these values into the equation, we find that for shares of Company A with relatively low level risk, the required rate of return will be 8.5%, and for shares of Company B 19%.
k A = 5 + 7*0.5 = 8.5%
k B = 5 + 7*2 = 19%
Problems with use
The main problem practical application line of the securities market is that it is based on the same premises as the capital valuation model CAPM assets (You can read more about them). Due to the fact that real markets are not characterized by an absolute degree of efficiency, different investors have different opportunities to attract additional financing (both in terms of volume and interest rates), and taxes and transaction costs have a significant impact on the formation of an individual portfolio, there are many available in the securities market they are not a straight line, but a kind of fuzzy aggregate. If you draw the SML line on this chart, then some securities will appear above it, and some below it.
Also, one of the main reasons for this situation is that beta is used as a complete measure of the risk associated with investing in a particular security. On real markets There are other risks that affect the required rate of return and cause an individual security to move away from the securities market line. However, if we accept the assumption that beta is a complete measure of risk, then securities located above the SML line will be undervalued by the market because they offer investors higher returns with lower risk (beta). On the contrary, securities whose yield is below the SML line will be overvalued by the market, since they have a lower required rate of return at higher high level risk.
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 of risk themselves.
Index 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 the securities portfolio in comparison with other securities on 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 based on comparative characteristics with competing firms, in 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 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 derivatives 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. At the end of the 20th century, it was proven that unsystematic variables such as market capitalization or ratio book value to the market, affect the expected return.
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.
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 on any stock is equal to the risk-free rate added to the product of the market risk premium and - 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 m f . 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 + i m r = m f + i (m r –m f )+ i,
Where i , = a i + ( i -1)m f .
The excess of security efficiency over risk-free efficiency m f called the risk premium. Thus, this risk premium is essentially linearly dependent on the risk premium for the market as a whole, and the coefficient is the beta of the security. This is, however, true if =0. Such securities are said to be “fairly” valued. The same securities for which > 0 are undervalued by the market, and if < 0, то рынком переоценены.
According to E. Dimson, the leading economically 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 for a certain 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 of GKOs was always variable and either rose (during their circulation) to 200% per annum or higher, or fell (during relative stabilization economic situation) up 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.
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, investment portfolio, replicating the proportions of the market, should be the optimal investment decision for all investors.
The formal recording 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 a regression equation built using statistical data on profitability i-th stock and 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 essentially linearly dependent on 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 certain 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?
Under conditions of sustainable 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 as a basis the most trustworthy stock index 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 because the government 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 most good and real projects in Russian Federation unprofitable, and therefore a calculation 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. financial market. Moreover, the market model uses a market index, which general case does not cover 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 real finance. The largest market institutions, such as investment bank Merril Lynch, regularly calculated β - coefficients of all large 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, is going to attract additional equity by issuing ordinary shares. Risk-free level 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 profitability investment projects, which the company was going to finance with raised 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 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%.
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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, profitability model financial assets(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, we took into account financial indicators(in particular, balance sheet data) of companies. A new version BARRA, 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 two-level 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-mu 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.