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Good day, readers of the blog about trading. Beta coefficient shows what level of volatility can be expected from a particular stock. This is worth knowing if you want to avoid panic selling stocks immediately after buying them. High volatility makes the paper unstable, like an unbridled horse. To pacify her, you need to practice for a long time. Scientists have come up with a numerical measure to determine the volatility of a stock - the beta coefficient.

What is beta?

Do you remember the basic rule of trading? The greater the risk, the greater the return. In any case, we can say: profits grow with increasing volatility. So, the beta coefficient allows you to estimate this amount of risk/return of the securities you have chosen.

The S&P 500 index is taken as a standard, the beta coefficient of which is equal to 1. If the stock you selected also has this coefficient of one, then you bear the same risks as when trading the index. The profit will be similar. If the S&P 500 is up 10%, you can expect the same return from your company.

If, for example, a stock's beta is 0.7, that means that if the S&P 500 rises by 10%, your return will be 7%. Accordingly, the risks are reduced. And volatility also becomes lower.

If the beta coefficient is greater than 1, say 2, then with the same index growth, your return will be 20%. But be aware of the increased risks. The price can not only rise, but also fall by the same 20%. Moreover, high volatility will force you to set large stop losses.

Well, finally, a stock may have a beta of 0 or even minus. In the first case, the security moves independently of the S&P 500 index. Its risk/return cannot be measured in this way.

In the second case, when the beta is negative, your risks remain the same (as with a positive coefficient), but the profitability changes in the opposite direction. For example, if the beta coefficient of a stock is -1.5, then if the index grows by 10%, this security will bring its holder -15%.

Conclusions: if you want to earn more, then choose stocks with a beta coefficient greater than one, but be prepared for high volatility = risks. All stocks in the American stock market are linked and depend in direct proportion to the S&P 500 index. There are a few "exceptions", but my advice is, do not look for securities with a beta of zero or minus.

Returning to Earth

I can guess what you're thinking because I've been through it myself. Why not earn exponentially more by choosing stocks with the highest beta. We have learned to tame volatility (read position size – taming volatility). All that's left is to practice a little.

I will now list several provisions that I learned from personal practice that make beta simply a coefficient that needs to be remembered sometimes.

1. Beta is originally designed to compare the performance of mutual funds with the S&P 500 index. Clearly, at a minimum, annual results are compared. That is, if you selected a security with a coefficient of 2 and hope to make double the profit of the index in a few days, then you are mistaken. What you will definitely have is a stock that is more volatile than the index.
2. Beta is calculated based on a company's past performance. Accordingly, if performance changes in the future, then beta will not remain the same.
3. The beta coefficient shows how a security reacts to movements in the index. But they do not reflect a company's strength in its sector and the economy as a whole. That is, the stock with the highest beta is not the best in its industry (at least not necessarily).

Beta coefficient rarely used in my practice. I prefer to control volatility using the same ATR. But, it can be successfully used in combination with other tools. This is just my opinion and my position. Experiment for yourself and draw conclusions for yourself.

Let us analyze such an investment indicator as the beta coefficient, and calculate it by real example using Excel and consider various modern modifications.

Beta coefficient. Definition

Beta coefficient (EnglishBeta,β, beta coefficient) – determines the measure of risk of a stock (asset) in relation to the market and shows the sensitivity of changes in the stock’s profitability in relation to changes in market profitability. Beta can be calculated not only for an individual stock, but also for an investment portfolio. The coefficient is used as a measure of systematic risk, and is used in the W. Sharpe model - valuation of capital assets CAPM ( CapitalAssetsPriceModel). First, the beta coefficient was considered by G. Markowitz to assess the systematic risk of stocks, which was called the non-diversifiable risk index. The beta coefficient allows you to compare shares of different companies with each other based on their degree of risk.

Beta Calculation Formula

β – beta coefficient, a measure of systematic risk (market risk);

r i – profitability of the i-th acacia (investment portfolio);

r m – market return;

σ 2 m – dispersion of market returns.

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Analysis of the risk level by the value of the beta coefficient (β)

Beta measures the market risk of a stock and reflects the sensitivity of a stock's changes to changes in market returns. The table below shows the risk level estimate based on beta. Beta can have either a positive or negative sign, which shows a positive or negative correlation between a stock and the market. Positive sign reflects that the returns of shares and the market change in the same direction, negative – multidirectional movement.

Data for constructing beta coefficient by information companies

The beta coefficient is used by many information and investment companies to assess systematic risk: Bloomberg, Barra, Value Line, etc. To construct the beta coefficient, monthly/weekly data over several years is used. The table shows the main parameters for assessing the indicator by various information companies.

You can see that Bloomberg uses a short-term assessment of the indicator, while Barra and Value Line use monthly data on stock and market returns over the past five years. Long-term assessment can be greatly distorted due to the influence of various crises and negative factors on the company's shares.

Beta coefficient in the capital asset pricing model –CAPM

Formula for calculating stock returns using the CAPM capital asset model ( CapitalAssetsPriceModel, model by W. Sharpe) has the following form:

Where:

r is the future expected return on the company's shares;

r f – return on a risk-free asset;

r m – market profitability;

β – beta coefficient (a measure of market risk), reflects the sensitivity of changes in the value of a company’s shares depending on changes in market profitability (index);

The CAPM model was created by W. Sharp (1964) and J. Linter (1965) and allows you to predict the future value of the return on a stock (asset) based on linear regression. The model reflects the linear relationship between the planned return and the level of market risk, expressed by the beta coefficient.

To calculate market returns use the return of an index or index futures (MICEX index, RTS index for Russia, S&P500 index for the USA).

Example of calculating beta coefficient inExcel

Let's calculate the beta coefficient in Excel for the domestic company OJSC Gazprom. This company has ordinary shares, the quotes of which can be viewed on the website finam.ru in the “Data Export” section. For the calculation, we took monthly quotes for the shares of OJSC Gazprom (GAZP) and the RTS index (RTSI) for the period from 01/31/2014 to 01/31/2015.

To calculate the beta coefficient, it is necessary to calculate the linear regression coefficient between the return on shares of OJSC Gazprom and the RTS index. Let's consider two options for calculating the beta coefficient using Excel.

Option #1. Calculation via formulaExcel

Calculation via Excel formulas as follows:

INDEX(LINEST(D6:D17,E6:E17),1)

Option #2. Calculation via the Data Analysis add-on

The second option for calculating beta uses the Data Analysis Excel add-in. To do this, go to the “Data” section in the main menu of the program, select the “Data Analysis” option (if this add-in is enabled) and select “Regression” in the analysis tools. In the “Input interval Y” field, select the returns of the Gazprom OJSC shares, and in the “Output interval X” field, select the returns of the RTS index.

Next, we will receive a regression report on a separate sheet. Cell B18 shows the value of the linear regression coefficient, which is equal to beta = 0.46. We will also analyze other parameters of the model, for example, the R-squared indicator (determinism coefficient) shows the strength of the relationship between the profitability of the Gazprom share and the RTS index. The coefficient of determinism is 0.4, which is quite low for accurately predicting future profitability using the CAPM model. Multiple R is a correlation coefficient (0.6), which shows the existence of a relationship between the stock and the market.

A value of 0.46 beta coefficient for a stock indicates moderate risk and at the same time the co-directionality of changes in returns.

★ (calculation of Sharpe, Sortino, Treynor, Kalmar, Modiglanca beta, VaR) + forecasting course movements

Disadvantages of Using Beta in the CAPM Model

Let's consider a number of disadvantages inherent in this coefficient:

1. The difficulty of using beta to value low-liquid stocks. This situation typical for developing capital markets, in particular: Russia, India, Brazil, etc.
2. It is not possible to evaluate small companies that do not issue ordinary shares. Most domestic companies have not gone through the IPO procedure.
3. Instability of beta coefficient forecast. Using linear regression to estimate market risk from historical data does not provide accurate risk forecasts. Generally, it is difficult to predict beta for more than 1 year.
4. It is not possible to take into account the company’s unsystematic risks: market capitalization, historical profitability, industry affiliation, P/E criteria, etc., which influence the expected profitability.

Since the coefficient proposed by U. Sharpov did not have proper stability and could not be used to predict future profitability in the CAPM model, various scientists proposed modifications and adjustments to this indicator ( Englishadjusted betamodifiedbeta).Let's look at the adjusted betas:

Modification of the beta coefficient from M. Blum (1971)

Marshall Bloom showed that over time, the beta coefficients of companies tend to 1. The formula for calculating the adjusted indicator is as follows:

Using these weights allows for a more accurate prediction of future systematic risk. So this modification is used by many news agencies, such as: Bloomberg, Value Line and Merrill Lynch.

Beta modification from Bava-Lindsberg (1977)

In his adjustment, Lindsberg proposed calculating a one-sided beta coefficient. The main postulate was that most investors do not consider changes in profitability above a certain level as a risk, and only what is below the level is considered a risk. The minimum level of risk in this model was the return on a risk-free asset.

Where:

r i – stock return; r m – market profitability; r f – return on a risk-free asset.

Beta modification from Scholes-Willims

β -1, β, β 1 – beta coefficients for the previous (-1) current and next (1) period;

ρ m – autocorrelation coefficient of market returns.

Beta modification from Harlow-Rao (1989)

The formula reflects one-sided beta, with the assumption that investors view risk only as a downward deviation from average market returns. In contrast to the Bava-Lindsberg model, the level of average market profitability was taken as the minimum level of risk.

where: μ i – average share return; μ m – average market profitability;

Summary

The beta coefficient is one of the classic measures of market risk for assessing the performance of stocks, investment portfolios and mutual funds. Despite the complexity of using this tool to evaluate domestic low-liquid stocks and the instability of its changes over time, the beta coefficient is a key indicator for assessing investment risks. The considered modifications of the coefficient allow us to adjust and give a more accurate assessment of the systematic risk. Ivan Zhdanov was with you, thank you for your attention.

One of the most important indicators for a stock is the beta coefficient - it shows the change in the share price relative to the market situation. When the coefficient β increases, we can talk about an increase in the price of the asset, and a decrease in β indicates a fall in price. With a low beta coefficient, there is almost zero dependence of the price of a given asset on the general market trend.

Beta can be calculated for a single stock or for a selected set of stocks. Using β, you can evaluate the risks and returns of both an individual asset and a selected portfolio of investments relative to a similar portfolio. In other words, a stock's beta indicates the degree of risk associated with a selected portfolio or individual security.

Description

The first person to propose using the beta coefficient of a portfolio to calculate systemic risk was the American economist Harry Markowitz, back in the early 50s of the last century. He first characterized such ratios as “indices of non-diversifiable risk.” The basis is the direct dependence of the profitability of a particular exchange instrument on the average profitability of the market where the asset is traded. For example, IBM shares - when calculating their beta coefficient, we will need the profitability of the stock itself and the profitability of the exchange platform itself where they are traded. Similarly, to calculate the profitability of a corporation or even an entire industry: we take the profitability indicator of a specific company or industry and the average profitability ratio of the entire industry.

If we get β = 1, then the conclusion will be simple: the risk of a particular instrument that is not subject to diversification coincides with the general market risk. If β = 0, which means we have come across an absolutely risk-free asset - relative to the risk that is not subject to diversification. The higher the beta value, the higher the risks for the selected investment. An important advantage of the β-coefficient is the ability to calculate the portion of risk to be diversified for a specific investment object in both macro- and microeconomics.

But the investor, as a rule, tries to find general meaning risk, so relying only on the β coefficient when forming an investment portfolio will be a dubious decision. This picture can be observed when investing in production, when there are not enough funds for full-fledged capital investments or there is no option to distribute investments. Often there is a need to calculate the risks for specific investment objects consisting of different niches, at the same time, the β-coefficient evaluates the risks of an asset relative to a specific market. That is, you will not be able to compare the risk of purchasing shares with the risk of investing in the construction of a manufacturing factory.

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Beta coefficient calculation

For an asset that is part of a selected set or relative to other securities, or an asset in the form of a stock market index of a relative reference portfolio, the coefficient βa is calculated in linear regression for the time period Ra,t relative to the portfolio return for the time period Rp,t:

Ra,t = a + βаrp,е+ Еt

To calculate the beta of a security:

βa = Cov (ra,rp) : Var(rp)

Now let's look at the components of the formulas:

• ra is the profitability of the asset in question or the size of the assessment for which the asset is calculated;
• rp – the return of a security or market is compared with this value;
• Cov – covariance of the estimated value and the standard;
• Var – the magnitude of the possible deviation of the indicator.

Essentially, beta is isolated case relationships between several variables. And the variables here are the securities of the selected company relative to other securities of the stock market.

What will the beta coefficient show?

Upon obtaining the value β = 1, we can conclude that the risk of underdiversification for a given stock is equal to the general market risk indicator. If beta is zero, then you are working with a risk-free asset. In general, the higher your beta value, the more risky the asset is. In this way, it is possible to analyze the distribution of investment risks at both the micro and macroeconomic levels.

To calculate the coefficient β, two quantities are needed:

• The company's profitability level. It represents the difference between the opening and closing of a company's shares on the stock market for a selected period of time.
• Average market level of profitability. This average level the profitability of all securities included in a specific investment portfolio. The portfolio may be composed of shares of the company in question.

Beta coefficient

Beta is a measure of the sensitivity of a stock's rate of return to market movements. In other words, it measures the systemic risk inherent in a stock.
To determine the beta coefficient, it is necessary to plot the relationship between the monthly returns of stocks and the market (S&P 500 index or other market instrument). The graph will illustrate the average change in the stock price relative to the change in the value of the market index. Sloping line and there is a beta coefficient that determines how a stock will react to market movements.

Table 4–8
How to Calculate Standard and Average Deviation Using Excel

Use Excel to calculate these statistics by entering monthly returns in the column.
1. Click on the fx (Functions) icon in the toolbar at the top of the screen.
2. Select the function category Statistical.
3. In the right cell, select Average for the mean and STDEVPA for the standard deviation of the population of data.
4. Enter the data field. For example, if the monthly yield values ​​are indicated in columns C1 to C12, then enter C1: C12.

Table 4–9
How to Use the Internet to Find Stock Betas

1. Log in to the Yahoo website (www.yahoo.com).
2. Click on the Finance link.
3. Enter the stock tickers (separated by commas) of the companies you're interested in, then click on each symbol to get detailed information.
4. Click on the Profile link, which is provided under the block with detailed information, and then select Key Statistics, where you will find the beta value for a particular stock.

Portfolio beta

Even if there is no perfect mechanism for measuring market risk, beta can be used to determine whether a portfolio of stocks will be above or below the market risk. The beta of a stock portfolio is a weighted average of the betas of the individual stocks. For example, a portfolio of 32 stocks, of which 8 stocks have a beta of 1.2, 16 stocks have a beta of 1.1, and another 8 stocks have a beta of 0.8, would have a beta of 1.05. Below is the calculation:

A beta coefficient of 1.05 means that if the market falls or rises by 1%, the value of the portfolio will fall or rise by 1.05%. Such a portfolio has a risk slightly higher than the market one. Although individual beta values ​​do not always accurately predict market price movements, they can still help assess a portfolio's market risk.

Sharpe ratio

The Sharpe ratio is a measure of a portfolio's risk-adjusted return. The risk-free rate is measured using the 90-day Treasury bill rate, which reduces the portfolio's return, which is then divided by the portfolio's standard deviation. When comparing Sharpe ratios for different portfolios, the higher the Sharpe ratio, the greater the return per unit of comparable risk. In other words, by holding the risk of different portfolios constant, you can use the Sharpe ratio to construct a portfolio that will provide higher returns.
You can analyze the risks of your stocks and portfolios using the website: www.riskgrades.com.

How to deal with risks

Risk analysis and awareness are important for two main reasons. First, you can determine your acceptable level of risk through asset allocation (determining the percentage of investments in stocks, bonds, and money market securities). Second, you should have a diversified stock portfolio, as this reduces the individual risk of individual stocks, as well as a long investment horizon (holding period) for your stock portfolio.

List of sources

Faerber E. (Faerber E.) All About Investing. NewYork: McGraw-Hill, 2006.
Malkiel B. G. (Malkiel B. G.) « Random walk"A Random Walk Down Wall Street." New York: W. W. Norton, 1990.
Siegel JJ (Siegel J.J.) Stocks for the long term (Stocks for the Long Run). 3rd ed. New York: McGraw-Hill, 2002.

Chapter 5
Stock return

Main themes

Investment section
On the day Caterpillar announced record earnings for the second quarter of 2006, its stock price fell 1.2%.
Alcoa announced a 62 percent jump in net income for the quarter, and its stock price dropped several dollars on the day the news was announced.
Citigroup reported a 4% rise in quarterly profit and its stock price was little changed on the day.

What kind of return can the shares bring?

What is the rate of return that is expected from its shares? The answer to this question depends on many factors, one of which is the time frame for which the question is asked. If we are talking about a “bull” market, the rate of return is often expressed in double digits (about 10%), and if we are talking about a “bear”, then it will be 6-7%. This discrepancy is not that important; It's important to understand exactly how returns are generated and that stocks generally outperform bonds and money market securities over long investment horizons. Professor Jeremy Siegel argues that every 10 years since 1802, stocks have outperformed bonds and Treasury bills; also, during the same time periods, even the lowest stock returns outperformed bonds and Treasury bills (2002, p. 26). Over short investment horizons, stocks are riskier than bonds and treasury bills, but over long investment horizons, the returns of a portfolio of stocks exceed the returns of a portfolio of bonds and treasury bills.
Stock returns come from two sources: dividend income (if the issuer pays dividends) and capital gains. If companies manage to increase profits, they can increase the amount of dividends they pay to shareholders. Companies do not pay out all profits as dividends. Retained earnings (not paid out to shareholders) are invested in expanding the business for future profit growth. Thus, shareholders of companies that do not pay dividends may benefit if the companies' earnings increase. The disappointment that occurs when a company fails to increase earnings to expected levels results in a decline in the share price. Volatility in stock prices over the short term causes stock returns to fluctuate, but over the long term investors can achieve higher returns than bonds or Treasury bills.

Return rate calculation

Return rate - is a measure of the increase (or decrease) in the amount of investment over certain period time in order to generate income in the form of interest and dividends and/or capital gains (if the sale price of the instruments is higher than their purchase price). Some investment vehicles, such as savings accounts and certificates of deposit, provide only income without capital gains; others, such as common stock, offer potential capital gains that may or may not pay dividends. If the stock price falls below the selling price, you will lose some capital if you need to sell the stock. The simple equation for calculating your portfolio's total return includes regular income and capital gains.
Calculating yield is important because it measures the increase or decrease in the value of your investment. It is a measure of the performance of your portfolio based on your goals. Total rate profitability is calculated as follows:

Return rate for the period of holding securities =
= [(Final cost – Initial cost) + Dividends] /
The cost of purchasing securities excluding commissions.

Rate of return = [(1500–1000) + 50] / l000 = 55%.

Share return indicators are reduced to weighted average values ​​and then summed to obtain the weighted average rate of return of the portfolio.
To be able to compare your portfolio's returns to the market's returns, you need to accurately measure your portfolio's returns.

The weighted average return of the portfolio is 2.92%.

The calculation will be more complicated if you buy and sell securities during the holding period. You may remember several years ago that the Beardstown Women's Investment Club was faced with the problem of accurately calculating returns. Club members announced that the long-term average annual return was just under 20%, which exceeded the market's annual return - and discovered that they had calculated incorrectly. In fact, an audit by a professional accounting firm showed that average annual returns were in the single digits during that period.
If during the period of holding securities you do not replenish your portfolio or withdraw funds, then the return for the holding period, calculated at a simple rate, will accurately reflect the results of your investment:

Return for the period of holding securities =
= (Final cost - Initial cost) / Initial cost.

Table 5–1
Calculation of portfolio profitability in case of depositing and withdrawing assets

If you started the year with a portfolio worth $100,000 and ended the year with a portfolio value of $109,000, your return was 9% [(109,000-100,000)/100,000].
If assets were added or written off during the year, their return on the holding period is calculated as follows:

For example, a portfolio whose value at the beginning of the year was $110,500 was replenished with dividends equal to $8,600. Capital gains were $12,000, unrealized losses were $6,000. At the beginning of April, funds in the amount of $10,000 were added to the portfolio ., and at the end of October $4,000 was written off. Thus, the annual return was 12.44%.
The investment portfolio's pre-tax return will be 12.44%, which is comparable to a comparable benchmark index over a similar time period.

Balancing risk and return

If you use standard deviation, return range, and beta to measure risk, you can better assess the risk of an investment and its expected return. However, you now know that choosing riskier instruments does not necessarily mean higher returns. The balance of risk and return is directly related to the expected, or required, rate of return on the investment instruments you buy and hold. To invest in risky instruments, you need to include increased returns in your calculations in advance. Required rate (rate) of return (required return)- This minimum bid yield, taking into account which you can purchase securities. This minimum ratio includes the rate of return on a risk-free investment (usually Treasuries) and the risk premium associated with the investment. Risk premium– this is an additional profitability associated with the riskiness of a certain instrument.

Required rate of return =
Risk-free rate + Risk premium.

Required rate of return = Risk-free rate +
+ Beta coefficient × (Market rate – Risk-free rate).

Historical stock returns, as noted in Chap. 1 confirms the balance of risk and return. In other words, over long investment horizons, the higher the risk, the higher the expected return.
However, this rule is not always followed on short horizons and in falling markets. According to Ibbotson and Sinquefield (1994), small and large companies over the 74-year period - from 1926 to 2000 - showed average annual returns of 7.7 and 9.1%, compared with 2.2% for medium-term government bonds. Real rate profitability- This nominal rate, reduced by the inflation rate. The highest risk is inherent in shares of small companies. They are followed by large company stocks, as evidenced by standard deviations of returns.

Asset Allocation and Investment Selection

Diversification helps reduce some of the risks of investing. For example, if the shares of one company in your portfolio fall in price, others may rise and thereby offset your losses. In the case of market risk, as noted above, diversification will not help. If the stock market falls, the value of the stocks in a diversified portfolio falls along with it. If the stock and bond markets are trending the same way, even a diversified portfolio in a falling market will not be protected from market risk. Another factor that can help combat market risk is time. By choosing securities for a long-term investment period, you can wait until stock prices recover from a decline before selling them.
The choice of securities depends on your goals, circumstances ( Family status, age, dependents, education, income, net worth and portfolio size), acceptable level of risk, expected return and economic situation. Asset Allocation is the placement of your funds into various categories of investment instruments such as stocks, bonds, money market securities, options, futures, real estate and gold. The asset allocation model shown in Fig. Figure 5–2 illustrates how some of these asset allocation factors determine the choice of investment vehicles.

Figure 5–2
Asset Allocation and Investment Selection

shares – 75%;
real estate – 10%;
bonds – 5%;

shares – 60%;
bonds – 30%;
money market instruments – 10%.

shares – 15%;
bonds – 65%;
money market instruments – 20%.

Table 5–2
Asset Allocation Principles

1. Analyze your investment goals and financial situation. If the goal is to generate current income and preserve capital, your asset allocation model should include a higher proportion of bonds and money market securities. If current income is not a priority and you are investing for future capital growth, you should invest more in stocks and real assets (real estate, precious metals and collectibles).
2. Determine your risk tolerance. If you are considering a long investment horizon and are willing to bear the risks inherent in the stock and real estate markets, invest more in stocks and real estate. If your risk tolerance is low, invest in bonds and money market securities.
3. Consider the time frame. If you are young and can invest for the long term (over 25 years), invest most of your money in stocks. If you want to invest for a limited period, a larger proportion of your funds should be invested in bonds and a smaller proportion in stocks.
4. Be realistic in your expectations from investing. Yield for two last decades were quite impressive. In the 1980s long-term bonds returned about 13% annually. Stock returns in the late 1990s was abnormally high due to the technology boom and the Internet bubble, but in the early 2000s. indicators have dropped to more realistic levels. Thus, the S&P 500 index showed the following returns: about 37% in 1995, 22% in 1996 and 33% in 1997. 1980-1990s. have been very favorable for the stock and bond markets due to the fall in interest rates - from approximately 17% in 1980 to the current low of 3-5% by the early 2000s. In the future, you should lower your return expectations to more realistic ones.