Capital Asset Pricing Model – CAPM (W. Sharp) in Excel
A stock index is a composite indicator of price changes for a certain group of securities - the “index basket”. As a rule, the absolute values of the indexes are not important. Changes in the index over time are more important because they provide an indication of the overall direction of the market, even when stock prices within the index basket move in different directions. Depending on the sample of indicators, a stock index may reflect the behavior of a certain group of securities (or other assets) or the market (market sector) as a whole. . According to Dow Jones & Co. Inc. , at the end of 2003 there were already 2,315 stock indices in the world. At the end of the name of stock indexes there may be a number indicating the number of joint stock companies on the basis of which the index is calculated: CAC 40, Nikkei 225, S&P 500.
The RTS Index reflects the current total market capitalization (expressed in US dollars) of shares of a certain list of issuers in relative units. The total capitalization of these issuers as of September 1, 1995 was taken as 100. Thus, for example, an index value of 2400 (mid-2008) means that over almost 13 years the market capitalization (converted into US dollars) of companies on the RTS list has grown 24 times. Every business day, the RTS Index is calculated during the trading session with each change in the price of an instrument included in the list for its calculation. The first index value is the opening value, the last index value is the closing value. The list of stocks for calculating indices is reviewed every three months. There are also the RTS-2 index (second-tier shares), RTS Standard (15 blue chips denominated in rubles), RTSVX (Volatility Index) and 7 industry indices.
The MICEX Index is calculated as the ratio of the total market capitalization of shares included in the index calculation base to the total market capitalization of these shares on the starting date, multiplied by the index value on the starting date. When calculating market capitalization, the price and quantity of the corresponding shares freely traded on the organized securities market are taken into account, which correspond to the share of the issuer’s share capital, expressed by the value of the free-float coefficient. The index is calculated in real time in rubles, thus, the index value is recalculated when each transaction is made on the MICEX Stock Exchange with shares included in the index calculation base. In 2009, more than 450 thousand transactions worth over 60 billion rubles were used daily to calculate the index. , and the total capitalization of shares included in the calculation base of the MICEX Index is more than 10 trillion rubles. , which corresponds to 80% of the total capitalization of issuers whose shares are traded on the stock exchange. The calculation base for the MICEX Index is revised 2 times a year (April 25 and October 25) based on a number of criteria, the main of which are stock capitalization, stock liquidity, the value of the free-float coefficient and the industry of the stock issuer.
Dynamics of the S&P index
In the securities markets, special indicators – stock indices – are used to determine the general trend in changes in stock prices. An exchange (stock) index is a general indicator of changes in prices of a certain group of assets (securities, goods or derivative financial instruments). Depending on the sample of indicators, a stock index may reflect the behavior of a certain group of assets (securities) or the market (market sector) as a whole. To study the nature of the relationship in changes in stock indices and the profitability of securities, market models are built, with the help of which it is possible to evaluate the investment portfolios of enterprises.
C weighted average capital income on securities The increase in a stock index for a certain period is the weighted average capital income on securities whose prices. used to calculate the index Let m r be the weighted average capital income for the group of securities included in, I index 0 - , index value at the beginning of the period I 1 - . index value at the end of the period 0 01 I II K
Problems of using an index. The main problem associated with the use of indices is how accurately - the index characterizes the market portfolio, that is, absolutely all financial assets that are present on the market, while only a certain sample is used to calculate the index from the entire (set of securities, although according to: some indices and quite large, SP 500 so when calculating, prices of 500 are used). shares of the largest US companies
A few more problems. — , First yield of government securities as, . - and any others are subject to fluctuations. The second rate in the capital asset valuation model, 0, is also the rate on risk-free loans, which further complicates the problem of choosing its value for. practical calculations, Thus, here it is already necessary to resort to certain simplifications. Practically, as a risk-free rate, one usually chooses the rate () of return on short-term from three months to a year, (government obligations, the discount rate or), the refinancing rate of the central bank or calculated by a certain Thus, the weighted average rate on loans on (: in the interbank market, the most famous example of LIBOR is London Interbank offered Rate). rate O
Single-factor Sharpe model Let us study the relationship between the profitability of a certain securities– mi and market return () market index -mr. in the same period, a change in the market index can cause a corresponding change in the price of the i-th security, and such changes are random and interrelated and to reflect them a market model is used in the form (regression equation of the characteristic line of a security): m i = i + i m r +i
m i = i + i m r + i where m i and m r are the return on security i and on the market index for the time period t; i is the regression line shift coefficient, characterizing the expected return of the i-th security under the condition of zero return of the market index; i is the slope coefficient and is a risk characteristic; i is random error.
Beta coefficient - Beta coefficient evaluates changes in the returns of individual stocks compared to the dynamics of market returns: if >0, then the returns of the corresponding securities change in the same direction as the market returns, with 1, 0 are considered aggressive and riskier than the market as a whole; for less risky securities<1, 0. индекс систематического риска вследствие общих условий рынка. i
According to Sharpe, it is convenient to calculate the efficiency of securities from the efficiency of the risk-free deposit m f m i = m f + β i (m r – m f) + α i, m i – m f is called the risk premium. α = 0 – securities are fairly valued; α > 0 – securities are undervalued by the market; α< 0 – бумаги рынком переоценены. Аналогичные утверждения имеют место и для портфелей.
The difference between the linear market model and CAPM: 1) the linear market model is a one-factor model, where the market index acts as a factor. Unlike CAPM, it is not an equilibrium model that describes the process of formation of security prices. 2) the market model uses a market index (for example, the S&P 500), while the CAPM uses a market portfolio. The market portfolio combines all securities traded on the market, and the market index contains only a limited number of them (for example, 500 for the S&P 500 index). Comparison of the market model of the market and the CAPM model
Example. 5. 1. According to the investment company "FINAM" on the actual return on shares and the return on the RTS index (RTSI) for the period from January 2008 to May 2009. see table 1, determine the expected return, risk and parameters of market models (alpha and beta coefficients) for the shares of Gazprom (GAZP), Sberbank (SBER) and Rosneft (ROSN). Based on the calculation results, construct graphs of the dependence of stock returns on the returns on the RTS index.
For GAZP shares For SBER shares For ROSN shares CONCLUSION OF RESULTS Regression statistics Multiple R 0.894 Multiple R 0.898 Multiple R 0.903 R-squared 0.799 R-squared 0.806 R-squared 0.816 Normalized R-squared 0.784 Normalized R-squared 0.792 Normalized R-squared 0.802 Standard error 6.540 Standard error 11.068 Standard error 6.677 Observations 16 Coefficients for GAZP Coefficients for SBER Coefficients for ROSN Y-intercept, - 0. 56 Y-intercept, 0, 72 Y-intercept, 3, 38 Variable X 1, 0, 72 Variable X 1, 23 Variable X 1, 0,
for Gazprom shares m 1 = - 0.56 + 0.72 mr, for Sberbank shares m 2 = 0.72 + 1.23 mr, for Rosneft shares m 3 = 3.38 + 0.76 Mr.
Some conclusions. . Sberbank shares are aggressive securities t to β = 1.23; For Gazprom shares β = 0.72, it practically coincides with the beta coefficient for Rosneft shares β = 0.76, their characteristic lines. almost parallel to each other (With an increase in stock market returns or) the RTS market index, the expected return on all shares increases, and the return on shares of Sberbank grows more intensively than on. for shares of Gazprom and Rosneft (With zero return on the stock market mr = 0) 0.72% profit is expected for shares of Sberbank and 3.38% for shares of Rosneft and shares of Gazprom. will bring a loss
Determining the share of market and non-market risk of assets The total risk of a security i, measured by its dispersion i 2, is usually presented in the form of: two components market () systematic or non-diversifiable (market risk) + own () non-systematic or diversifiable (unique risk). i 2 = i 2 (m r) 2 + 2, where 2 i m r 2 denotes the market risk of security i, 2 is the own risk of security i, the measure of which is the standard deviation of the random error i in the equation
Total risk = Market risk + Own risk(systematic) + (non-systematic) Thus, the variation in the yield of each security consists of two terms: the “own” variation, independent of the market, and the “market” part of the variation, determined by the random behavior of the market as a whole. In this case, the ratio i 2 2 m r / 2 characterizes the share of securities risk contributed by the market; it is denoted by R i 2 and is called the coefficient of determination. Securities with larger R i 2 values may be preferable because their behavior is more predictable.
Specific risk is associated with such phenomena as changes in legislation, strikes, successful or unsuccessful marketing policies, the conclusion or loss of important contracts and other events that have consequences for the company. The impact of such events on a stock portfolio can be eliminated by diversifying the portfolio. Market risk arises from factors that affect all stocks. Such factors include war, inflation, decline in production, rising interest rates, etc. Since such factors affect most stocks in one direction, market and systematic risk cannot be eliminated through diversification.
Sharpe model n i iim n i iipxx 1 222 2 1 2 minmin p n i iimxm 1 1 1 n i ix
Portfolio optimization according to Sharpe
t 1 2 3 4 5 6 7 8 9 10 market index 10 9 9 10 10 11 11 12 10 8 stock A 10 11 9 12 13 12 14 12 15 13 stock B 23 21 20 22 23 24 25 27 25 20 Example. The returns of two stocks and the return of the market index for 10 months are known: Determine: 1. Characteristics of each security: coefficients of dependence on the index, own (or unsystematic) risk, market risk and the share of risk contributed by the market. 2. Create a portfolio minimal risk of two types of securities, provided that the portfolio returns are not less than for risk-free securities (5%) taking into account the market index.
date OFZ index, % year. RBC index RTKM (Rostelecom) EESR (RAO UES) KMAZ (KAMAZ) SBER (Sberbank) LKOH (LUKOIL) 1 Nov 07 6, 16 195, 93 112, 46 -27, 92 -24, 14 103, 14 551, 36 2 Nov 07 6, 12 -158, 76 -298, 98 501, 65 -230, 55 -397, 67 -268, 26 6 Nov 07 6, 13 228, 40 -435, 60 -97, 05 37, 90 460, 97 1071, 51 7 Nov 07 6, 05 349, 90 -71, 70 -272, 71 -778, 55 17, 11 332, 93 14 Jan 08 6, 01 -32, 50 494, 78 211, 67 689, 43 97, 81 -585, 93 15 Jan 08 5, 98 310, 83 179, 85 301, 95 2254, 86 376, 25 -134, 32 16 Jan 08 5, 94 -1, 68 -261, 76 -980, 08 576, 80 -1331, 03 -1717, 19 17 Jan 08 5, 98 -1471, 25 -1087, 70 -289, 08 1254, 74 -440, 19 -854, 21 average 6, 14 39, 81 205, 36 59, 83 516, 15 33, 50 -104, 21 SKO total. risk 0.09 450. 60 556. 84 382. 06 1101. 37 501. 22 554. 98 correlation 0.27 1.00 0. 51 0. 24 0. 11 0. 44 0. 51 alpha 6.14 0. 00 180, 31 51, 62 505, 73 14, 05 -129, 20 beta 0, 00 1, 00 0, 63 0, 21 0, 26 0, 49 0, 63 own. risk 412, 51,359, 44,1088, 74,404, 51,410, 90 market. risk 144, 34 22, 62 12, 63 96, 71 144, 08 market share. risk 100, 00% 25, 92% 1, 15% 19, 30% 25, 96% Dynamics of returns on stocks and bonds
portfolio RTKM (Rostelecom) KMAZ (KAMAZ) portfolio market share 44.31% 55.69% 100.00% avg. income 205, 36 516, 15 378, 43 39, 81 avg. risk 556, 84 1101, 37 381, 81 450, 60 SML portfolio RTKMKMAZ
Good afternoon, dear community of traders, investors and everyone interested in the securities market!
W. Sharp's model or as it is often called, the market model was first proposed by an American economist, Nobel Prize winner William Forsythe Sharp in the mid-60s of the last century.
William F. Sharp is currently professor emeritus High school business from Stanford University.
In 1990 he received Nobel Prize in Economics, which he received for developing the theory of financial asset valuation.
The Sharpe model represents the relationship between the expected return of an asset and the expected return of the market. It is assumed that the return of an ordinary share for a certain period is related to the return for a similar period with the return of a market index. In this case, as the market index rises, the share price is likely to rise and vice versa.
Thus, this model is assumed to be linear. And the equation of the proposed model has the following form:
The main difference between W. Sharpe’s model and G. Markowitz’s model is as follows:
The Sharpe model examines the relationship between the returns of each security and the returns of the market as a whole, while the Markowitz model considers the relationship between the returns of securities.
It was in order to avoid the high complexity of the Markowitz model that William Sharp proposed a market (index) model. At the same time, the Sharpe model is not new method compilation of a securities portfolio is a simplified Markowitz model, where the solution to the problem of choosing an optimal portfolio is carried out with less effort. The Sharpe model is usually used when considering large quantity securities that represent a significant part of the market.
It is very interesting to compare the results obtained using the Markowitz model and the Sharpe model.
For this purpose, I developed an application in Microsoft Office Excel* called "".
In my recent post, I demonstrated the result of calculating the determination of the optimal settlement on the Russian stock market using the Markowitz model with the following inputs:
- shares included in the calculation of the main index of the Moscow Exchange - the MICEX Index - 50 of the most liquid and capitalized securities on the Russian stock market were taken;
- the historical period for analysis for the instruments under consideration was selected from January 9, 2007 to October 24, 2013;
- level of expected profitability - maximum;
- the level of acceptable risk is minimal;
- diversification (maximum share of investments in a financial instrument) - 15% of existing assets;
- The minimum level of daily liquidity for shares is 6 million rubles.
You can see the results obtained for these models below:
Markowitz model:
Sharpe model:
As you can see, the difference in the composition of the proposed optimal securities portfolios is small. In the Sharpe model, the share of Severstal securities was 11% versus 2.8% in the Markowitz model; Bashneft shares in the Sharpe model are less than 1%, in the Markowitz model - 5.8%; in the Sharpe model, NLMK shares are -13.3%, in the Markowitz model - 15%; in the Sharpe model there are no Tatneft shares at all, in the Markowitz model - 1.5%. The remaining shares of papers are the same for the described models.
The final parameters are as follows:
Markowitz model:
Sharpe model:
Here we observe that, with the same level of risk, the return on Sharpe’s portfolio turns out to be slightly higher than the return on the Markowitz model - 26.75% versus 24.32% per annum, respectively. At the same time, we see that the beta of the portfolio according to the Sharpe model is also higher than the beta obtained according to the Markowitz model (0.64 versus 0.59), and this, in turn, suggests that the Sharpe portfolio is slightly less defensive (protective) than Markowitz's briefcase.
Market model W. Sharpe's optimal portfolio ultimately looks like this:
All other calculated indicators are in the presented application " Portfolio investments in the Russian stock market according to the W. Sharpe model (market model)" are the same as in the Markowitz model.
Application " Portfolio investments in the Russian stock market according to the W. Sharpe model (market model)" also contains the same specifications, as the application “Portfolio investments in the Russian stock market according to the Markowitz model”
The transaction journal has a convenient, quick transition from one page to another using internal hyperlinks. Hyperlinks to graphs will allow you to quickly go to the desired summary table on the basis of which they are built. In stock detailed instructions to work with the application.
In total there are more than 65 different charts, more 75 pivot tables and everything is clearly structured.
The application is configured so that you can easily print all the sheets (there is no need to specially format them) in order to make special folders for yourself where you can file your calculations, etc. and so on. All pages are numbered.
You can also, if desired, convert it into a convenient, readable PDF format (if you have a special program for creating PDF files).
For clarity, I posted the final data file, converted to PDF format, on a shared disk. You can follow the link and watch or download:
All formulas in the application are open so that you can look into the depths of the calculations themselves in terms of the various indicators used in the application.
If desired, the original database of the application about the price parameters of the financial instruments already included in it can be changed, expanded (both according to the list of securities under consideration and according to the horizon of their research) and, of course, periodically update the application for the current date.
In conditions of developed and stably functioning stock markets, the above-mentioned classical Markowitz and Sharpe models work quite effectively. At the same time, in modern conditions Using just one model alone is not correct. The models of W. Sharpe and G. Markowitz can be a good addition to other factors when compiling an optimal portfolio of securities.
"Portfolio investments in the Russian stock market according to the W. Sharpe model (market model)" is an excellent tool for a professional approach to investing in the securities market.
If you are interested in the application, you can purchase it either on the website.
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're talking about about securities, then, therefore, it is necessary 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 value 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 price. Thus, even a portfolio consisting of a very large number of stocks will have high degree risk, although the risk will be significantly 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. 12.9, which shows how portfolio risk is reduced if
the number of shares in the portfolio increases. The standard deviation for an "average portfolio" made up of one NYSE stock (AD) is approximately 28%. An average portfolio made up of two randomly selected stocks will have a smaller standard deviation of about 25%. If the number of stocks in the portfolio to 10, then the risk of such a portfolio decreases 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, which is usually called market portfolio would have a standard deviation of about 15.1%. Thus, almost half the risk inherent in the average individual stock could be eliminated if the stocks were held in a portfolio of 40 or more stocks. However, some risk always remains, no matter how widely diversified the 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).
Firm-specific risk is associated with such phenomena as changes in legislation, strikes, successful or unsuccessful marketing programs, the conclusion or loss of important contracts and other events that have consequences for a particular company. 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.
It is known that investors demand a risk premium, and the higher the degree of risk, the higher the required rate of return. However, since investors hold a portfolio of stocks and face portfolio risk rather than individual stock risk in the portfolio, the question arises; How to assess the risk of each individual stock?
The answer to this question is provided by the financial asset valuation model. The relevant risk of an individual stock is its contribution to the risk of a broadly diversified portfolio. For example, the risk of a Delta stock for an individual investor with a portfolio of 40 stocks, or for an investment fund with a portfolio of 300 stocks, will be measured by the contribution that the Delta stock makes to the portfolio risk. A stock can have a very high degree of risk if held on its own. However, if a significant portion of its risk can be eliminated through diversification, then its relevant risk, that is, its contribution to the risk of the portfolio, may be very small.
The question arises: aren't all stocks equal in risk in the sense that adding them to a broadly diversified portfolio has the same impact on the portfolio's risk? The answer is clear - no. Different stocks will impact portfolio risk differently. How can this risk be measured? The risk that remains after portfolio diversification is the risk inherent in the market as a whole, or market risk. Therefore, the relevant risk of an individual stock can be measured by the extent to which the stock tends to move up and down with the market.
The concept of "beta"
The tendency of a stock to “move” with the entire market is measured using the beta coefficient (^-coefficient), which characterizes the degree of its volatility in relation to the “average stock,” which is considered a stock that tends to “move” in sync with the entire stock market. Such a promotion, by definition, will have a (3-ratio equal to 1.
This means that if the profitability of the market as a whole increases by 10%, then the profitability of the “average stock” increases to the same extent, and vice versa - if it falls, it falls. A portfolio of stocks with a 3-ratio of one will have the same degree of risk as the entire market. If a stock has p = 0.5, this means that its return will rise or fall at half the rate of the entire market. A portfolio of stocks with such a coefficient will have half the risk compared to a portfolio with P ~ 1. At the same time, if a stock has P = 2, then its mobility is twice as high as that of the average stock. A portfolio consisting of such stocks, will be twice as risky as a portfolio of “average stocks.” The value of a portfolio of stocks with p = 2 rises or falls much faster than the value of the entire stock market.
Suppose there are three stocks A, B and C, the returns of which for three years are presented! in table 12.5.
Table 12.5
Dynamics of profitability of shares A, B, C and the market portfolio
The returns of all three stocks move in the same direction, but at different rates. In 2000, all three stocks had the same return of 15%, which was in line with the return of the market portfolio. In 2001, the return on the market portfolio went down and became negative (-10%), the return on shares B fell to zero, and shares A experienced the greatest decline - the return reached -20%. In 2002, the return on share C increased in full accordance with the market portfolio, while on share B it increased to a lesser extent, and on share A to a greater extent.
In Fig. Figure 12.10 shows graphs of the relative mobility of three stocks. The slope of the line relative to the horizontal axis shows how each stock moves relative to the overall market. The slope of this line is nothing more than (V coefficient.
In the US, well-known companies such as Merrill Lynch and Value Line calculate 3-ratios for many hundreds of companies. For most stocks, the 3-ratio varies from 0.5 to 1.5, and its average value for all stocks is, by definition, 1.
Theoretically, the 3-coefficient can be negative; this occurs if the return on the market portfolio rises, but on an individual stock it falls, and vice versa. In this case, the regression line in Fig. 12.10 will have a downward slope. In fact
This happens extremely rarely. Thus, out of 1,700 stocks for which 3_ coefficients are calculated by Value Line, there is not a single stock with a negative 3 coefficient.
If the 3-ratio of a stock is higher than its average market value (3 1), and this stock is added to the portfolio with 3 = 1, then the 3-ratio of the portfolio increases, and the risk of the portfolio increases accordingly. On the contrary, if to the portfolio with (3=1 add a share with рlt; = r f + β i E (r m ) − β i r f
E (r i ) = r f (1 − β i ) + β i E (r m )
Since the term βi E(rm) is common to the SML and the Sharpe model, then:
y i = r i (1 − β i ) |
Equation (198) implies that for an asset with a beta of one, y will be approximately zero. For an asset with β
The CAPM model is an equilibrium model, i.e. it talks about how prices for financial assets are set in an efficient market. The Sharpe model is an index model, meaning it shows how the return of an asset is related to the value of a market index. Theoretically, the CAPM assumes a market portfolio, and therefore the value of β in the CAPM assumes the covariance of the asset's return with the entire market. In the index model, only a market index is taken into account, and beta indicates the covariance of the asset's return with the return of the market index. Therefore, theoretically, β in the CAPM is not equal to β in the Sharpe model. However, in practice it is impossible to create a truly market portfolio, and such a portfolio in the CAPM is also some kind of broad-based market index. If the same market index is used in the CAPM and the Sharpe model, then β will be the same value for them.
15. 3. 4. Determining a set of efficient portfolios
Considering the question of the efficient frontier, we presented the Markovets method for determining a set of efficient portfolios. Its inconvenience is that to calculate the risk of a widely diversified portfolio it is necessary to make a large number of calculations. The Sharpe model allows you to reduce the number of units of required information. So, instead of units of information according to the Markovets method,
When using the Sharpe model, only 3n + 2 units of information are needed. This simplification is achieved thanks to the following
transformations. The covariance of the i-th and j-th assets based on the Sharpe equation is equal to:
Cov i, j = β i β jσ m 2 + σ i, j (199)
If i =j, then σi, j = σi 2
If i≠j, then σi, j = 0
To determine the portfolio risk, let’s substitute formula (199) into the formula proposed by Markovets:
σ 2 p = ∑∑ θi θ j Cov i , j = ∑∑ θi θ j (βi β j σ 2 m + σ i , j ) = |
||
i =1 j =1 |
i =1 j =1 |
|
= ∑∑ θi θ j βi β j σ 2 m + ∑ θ 2 i σ 2 i ) = |
||
15. 4. MULTIFACTOR MODELS
There are financial instruments that react differently to changes in various macroeconomic indicators. For example, the performance of shares of automobile companies is more sensitive to the general state of the economy, and the performance of shares of savings and loan institutions is more sensitive to the level of interest rates. Therefore, in some cases, a forecast of the profitability of an asset based on a multifactor model, which includes several variables on which the profitability of a given asset depends, may be more accurate. Above we presented W. Sharpe’s model, which is one-factor. It can be turned into a multifactorial one if the term βi E(rm) is represented as several components, each of which is one of the macroeconomic variables that determine the profitability of the asset. For example, if an investor believes that the profitability of a stock depends on two components - total output and interest rates, then the model of its expected profitability will take the form:
E (r) = y + β 1 I 1 + β 2 I 2 +ε
β1, β2 - coefficients that indicate the influence of indexes I1 and I2, respectively, on the stock’s profitability;
ε - random error; it shows that the yield of a security can vary within certain limits due to random circumstances, i.e., regardless of the adopted indices.
Analysts can include any number of factors they deem necessary in the model.
BRIEF SUMMARY
The CAPM model establishes the relationship between the risk of an asset (portfolio) and its expected return. The capital market line (CML) shows the relationship between the risk of a broadly diversified portfolio, as measured by variance, and its expected return. The asset market line (SML) indicates the relationship between the risk of an asset (portfolio), measured by beta, and its expected return.
The entire risk of an asset (portfolio) can be divided into market and non-market. Market risk is measured by beta. It shows the relationship between the return of an asset (portfolio) and the return of the market.
Alpha is an indicator that indicates the amount of misjudgment of an asset's return by the market compared to the equilibrium level of its return. A positive alpha value indicates its underestimation, a negative value indicates its overestimation.
The Sharpe model represents the relationship between the expected return of an asset and the expected return of the market.
The coefficient of determination allows you to determine the share of risk determined by market factors.
Multifactor models establish a relationship between the expected return of an asset and several variables that influence it.
QUESTIONS AND CHALLENGES
1. What is the difference between market and non-market risk. Why should only market risk be considered when assessing the value of a security?
2. What does an asset's beta mean?
3. If an asset's beta is zero, does that mean it is risk-free?
4. What does the coefficient of determination of a security indicate?
5. The risk-free rate is 10%, the expected return of the market is 20%, the beta of the stock portfolio is 0.8. Determine the expected return of the portfolio.
(Answer: 18%)
6. The portfolio consists of five assets. The share and beta of the first asset are equal to 20% and 0.5, respectively, the second - 20% and 0.8, the third - 40% and 1, the fourth - 10% and 1.2, the fifth - 10% and 1.4. Determine portfolio beta.
(Answer: 0.92)
7. The portfolio consists of two shares - A and B. Share share
A in the portfolio is equal to 30%, beta - 0.8, non-market risk - 15%. The share of share B is 70%, beta 1.3, non-market risk - 8%. Market risk is 10%. What is the total portfolio risk represented by the standard deviation?
(Answer: 13.5%)
8. What is the difference between CAPM and market model?
9. What is the difference between CML and SML?
10. Determine the alpha of an asset if its equilibrium expected return is 20% and its actual expected return is 18%.
(Answer: -2)
11. Draw some SML. In relation to it, use new SMLs to show cases where investors' expectations regarding future market returns have become more: a) pessimistic; c) optimistic.
12. The portfolio consists of two assets. The share of the first asset is 25%, the second - 75%, the portfolio alpha - 5, the first asset - 3. Determine the alpha of the second asset.
(Answer: 5, 67)
13. What is R. Roll’s criticism of the CAPM model?
14. The average return on an asset for previous periods is 30%, the average return on the market is 25%. The covariance of the asset return with the market return is 0.1. The standard deviation of the market portfolio return is 30%. Determine the market model equation.
(Answer: E(ri) = 2, 5 + l, l E(rm) + εi)
15. Beta of the asset is 1, 2, the standard deviation of its return is 20%, of the market - 15%. Determine the market risk of the portfolio.
Model equation
The expected return of an asset can be determined not only using the SML equation, but also based on so-called index models. Their essence is that changes in the profitability and price of an asset depend on a number of indicators characterizing the state of the market, or indices.
A simple index model was proposed by W. Sharp in the mid-60s. It is often called the market model. The Sharpe model represents the relationship between the expected return of an asset and the expected return of the market. It is assumed to be linear. The model equation is as follows:
where: E(ri) - expected return on the asset;
Yi is the profitability of the asset in the absence of exposure to market factors;
βi - asset beta coefficient;
E(rm) - expected return of the market portfolio;
εi is an independent random variable (error): it shows the specific risk of an asset that cannot be explained by market forces. Its average value is zero. It has a constant variance; covariance with market returns equal to zero; covariance with the non-market component of the returns of other assets is equal to zero.
Equation (192) is a regression equation. If it is applied to a widely diversified portfolio, then the values of the random variables (εi), due to the fact that they change in both positive and negative directions, cancel each other out, and the value of the random variable for the portfolio as a whole tends to zero. Therefore, for a widely diversified portfolio, specific risk can be neglected. Then the Sharpe model takes the following form:
portfolio;
βp - portfolio beta;
ur - portfolio profitability in the absence of influence of market factors on it.
Graphically, the Sharpe model is presented in Fig. 66 and 67. It shows the relationship between market return (rt) and asset return (ri) and is a straight line. It is called the characteristic line. The independent variable is market profitability. The slope of the characteristic line is determined by the beta coefficient, and the intersection with the ordinate axis is determined by the value of the yi indicator.
YI can be determined from formula (193), taking the average values of market and asset returns for previous periods of time. 1
Average market return.
Determine the market model equation.
model has the form:
shown in Fig. 66. The dots show specific return values of the i-th asset and market for various points in time in the past.
In Fig. 66 and fig. 67 shows the case when beta is positive, and therefore the graph of the market model is directed upward to the right, i.e., as the market return increases, the asset’s return will increase, and if it decreases, it will fall. With a negative beta value, the graph is directed downwards to the right, which indicates an opposite movement in the profitability of the market and the asset. A steeper slope of the graph indicates a high beta value and greater risk of the asset, a less steep slope indicates a lower beta value and less risk (see Fig. 68). When β = 1, the asset's return corresponds to the market's return, with the exception of a random variable characterizing a specific risk.
If we plot the model for the market portfolio itself relative to the market portfolio, then the value of y for it is equal to zero, and beta is +1. Graphically, this model is presented in Fig. 67.
15. 3. 2. Determination coefficient
The market model can be used to divide the entire risk of an asset into diversifiable and non-diversifiable. Graphically, specific and market risks are presented in Fig. 68. According to the Sharpe model, the asset dispersion is equal to:
To calculate the portion of an asset's variance that is determined by the market, the coefficient of determination (R2) is used. It represents the ratio of the market-explained variance of an asset to its total variance.
Substituting this value into formula (196), we obtain a result that indicates that the coefficient of determination is the square of the correlation coefficient.
R2 = (Corri,m)2 (197)
R2 = (Corri,m)2 (197)
In the last example, the R-squared is 0.1699. This means that 16.99% of the change in the return of the asset in question can be explained by changes in market returns, and 83.01% by other factors. How closer value R-squared to one, the more the market movement determines the change in the asset's return. A typical R-squared value in a Western economy is around 0.3, meaning 30% of the change in its return is determined by the market. The R-squared for a broadly diversified portfolio can be 0, 9 or more.
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