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Measuring the Beta using Historical Stock Prices 2039 In this slide set The beta coefficient The linear regression approach to beta measurement using historical return data – Normalizing the data – Normalized holding period returns – Running the regression using MS Excel – Relevant regression statistics and their interpretation – Different regression charts The Beta Coefficient Under the theory of the Capital Asset Pricing Model total risk is partitioned into two parts: – Systematic risk – Unsystematic risk Total Risk of the Investment Systematic Risk Unsystematic Risk Systematic risk is the only relevant risk to the diversified investor The beta coefficient measures systematic risk The Term – “Relevant Risk” What does the term “relevant risk” mean in the context of the CAPM? – It is generally assumed that all investors are wealth maximizing risk averse people – It is also assumed that the markets where these people trade are highly efficient – In a highly efficient market, the prices of all the securities adjust instantly to cause the expected return of the investment to equal the required return – When E(r) = R(r) then the market price of the stock equals its inherent worth (intrinsic value) – In this perfect world, the R(r) then will justly and appropriately compensate the investor only for the risk that they perceive as relevant…hence investors are only rewarded for systematic risk…risk that can be diversified away IS…and prices and returns reflect ONLY systematic risk. The Proportion of Total Risk that is Systematic Each investor varies in the percentage of total risk that is systematic Some stocks have virtually no systematic risk. – Such stocks are not influenced by the health of the economy in general…their financial results are predominantly influenced by company-specific factors – An example is cigarette companies…people consume cigarettes because they are addicted…so it doesn‟t matter whether the economy is healthy or not…they just continue to smoke Some stocks have a high proportion of their total risk that is systematic – Returns on these stocks are strongly influenced by the health of the economy – Durable goods manufacturers tend to have a high degree of systematic risk The Regression Approach to Measuring the Beta • You need to gather historical data about the stock and the market • You can use annual data, monthly data, weekly data or daily data. • You need at least thirty (30) observations of historical data. • Hopefully, the period over which you study the historical returns of the stock is representative of the normal condition of the firm and its relationship to the market. • If the firm has changed fundamentally since these data were produced (for example, they have merged with another firm or have divested itself of a major subsidiary) there is good reason to believe that future returns will not reflect the past…and this approach to beta estimation SHOULD NOT be used….rather, use the ex ante approach. Historical Beta Estimation In this example, we have determined the quarterly returns on the stock and the market and using Excel…ran a regression to produce the accompanying chart. Period HPR(Stock) HPR(TSE 300) 1990.1 -0.04 0.012 Characteristic Line (Regression) 0.3 1990.2 -0.16 -0.07 Returns on Stock 1990.3 0.32 0.12 0.2 1990.4 0.16 0.08 1991.1 -0.22 -0.11 0.1 1991.2 0.15 0.16 1991.3 0.28 0.13 0 1991.4 0.19 0.07 -0.5 0 0.5 1992.1 -0.16 -0.04 -0.1 1992.2 0.08 0.16 -0.2 1992.3 -0.03 -0.11 1992.4 0.34 0.25 Returns on TSE 300 Characteristic Line The characteristic line is a regression line that represents the relationship between the returns on the stock and the returns on the market over a period of time. The slope of the Characteristic Line is the Beta Coefficient The degree to which the characteristic line explains the variability in the dependent variable (returns on the stock) is measured by the coefficient of determination. (also known as the R2 (r-squared or coefficient of determination)). If the coefficient of determination equals 1.00, this would mean that all of the points of observation would lie on the line. This would mean that the characteristic line would explain 100% of the variability of the dependent variable. The alpha is the vertical intercept of the regression (characteristic line). Many stock analysts search out stocks with high alphas. Characteristic Line for Imperial Tobacco Characteristic Returns on Line for Imperial Imperial Tobacco Tobacco % • High alpha • R-square is very low • Beta is irrelevant Returns on the Market % (TSE 300) High R2 An R2 that approaches 1.00 (or 100%) indicates that the characteristic (regression) line explains virtually all of the variability in the dependent variable. This means that virtually of the risk of the security is „systematic‟. This also means that the regression model has a strong predictive ability. … if you can predict what the market will do…then you can predict the returns on the stock itself with a great deal of accuracy. Characteristic Line General Motors Characteristic Returns on Line for GM Imperial Tobacco % (high R2) • Positive alpha • R-square is very high • Beta is positive and close to 1.0 Returns on the Market % (TSE 300) An unusual Characteristic Line Returns on a Characteristic Line for a stock Stock % that will provide excellent portfolio diversification • Positive alpha (high R2) • R-square is very high • Beta is negative and < 1.0 Returns on the Market % (TSE 300) Diversifiable Risk (non-systematic risk) Examples of this type of risk include: – a single company strike – a spectacular innovation discovered through the company‟s R&D program – equipment failure for that one company – management competence or management incompetence for that particular firm – a jet carrying the senior management team of the firm crashes – the patented formula for a new drug discovered by the firm. Obviously, diversifiable risk is that unique factor that influences only the one firm. OK – lets go back and look at raw data gathering and data normalization A common source for stock of information is Yahoo.com You will also need to go to the library a use the TSE Review (a monthly periodical) You want data for at least 30 months. For each month you will need: – Ending stock price – Number of shares outstanding for the stock – Dividend per share paid during the month for the stock – Ending value of the market indicator series you plan to use (ie. TSE 300 composite index) Demonstration Through Example The following slides will be based on Alcan Aluminum (AL.TO) Five Year Stock Price Chart for AL.TO Spreadsheet Data From Yahoo Process: – Go to http://ca.finance.yahoo.com – Use the symbol lookup function to search for the company you are interested in studying – Use the historical quotes button…and get 30 months of historical data – Use the download in spreadsheet format feature to save the data to your harddrive Spreadsheet Data From Yahoo The raw downloaded data should look like this: Date Open High Low Close Volume 01-May-02 57.46 62.39 56.61 59.22 753874 01-Apr-02 62.9 63.61 56.25 57.9 879210 01-Mar-02 64.9 66.81 61.68 63.03 974368 01-Feb-02 61.65 65.67 58.75 64.86 836373 02-Jan-02 57.15 62.37 54.93 61.85 989030 03-Dec-01 56.6 60.49 55.2 57.15 833280 01-Nov-01 49 58.02 47.08 56.69 779509 Spreadsheet Data From Yahoo The raw downloaded data should look like this: Date Open High Low Close Volume 01-May-02 57.46 62.39 56.61 59.22 753874 01-Apr-02 62.9 63.61 56.25 57.9 879210 Volume of Opening price per share, the trading done The day, highest price per share during the in the stock month and month, the lowest price per share on the TSE in year achieved during the month and the the month in closing price per share at the end numbers of of the month board lots Spreadsheet Data From Yahoo From Yahoo, the only information you can use is the closing price per share and the date. Just delete the other columns. Date Close 01-May-02 59.22 01-Apr-02 57.9 01-Mar-02 63.03 01-Feb-02 64.86 02-Jan-02 61.85 Acquiring the Additional Information You Need In addition to the closing price of the stock on a per share basis, you will need to find out how many shares were outstanding at the end of the month and whether any dividends were paid during the month. You will also want to find the end-of-the-month value of the S&P/TSX Total Return Composite Index (look in the green pages) You will find all of this in The TSE Review periodicals (HG 5160.T6T6) found on the second floor of the library. Raw Company Data Closing Price Cash Issued for Alcan Dividends Date Capital AL.TO per Share 01-May-02 321,400,589 $59.22 $0.00 01-Apr-02 321,400,589 $57.90 $0.15 01-Mar-02 321,400,589 $63.03 $0.00 01-Feb-02 321,400,589 $64.86 $0.00 02-Jan-02 160,700,295 $123.70 $0.30 01-Dec-01 160,700,295 $119.30 $0.00 Number of shares doubled and share price fell in half – this is indicative of a 2 for 1 stock split. Normalizing the Raw Company Data Closing Cash Issued Price for Dividends Adjustment Normalized Normalized Date Capital Alcan per Share Factor Stock Price Dividend 01-May-02 321,400,589 $59.22 $0.00 1.00 $59.22 $0.00 01-Apr-02 321,400,589 $57.90 $0.15 1.00 $57.90 $0.15 01-Mar-02 321,400,589 $63.03 $0.00 1.00 $63.03 $0.00 01-Feb-02 321,400,589 $64.86 $0.00 1.00 $64.86 $0.00 02-Jan-02 160,700,295 $123.70 $0.30 0.50 $61.85 $0.15 01-Dec-01 145,000,500 $111.40 $0.00 0.45 $50.26 $0.00 The adjustment factor is just the value in the issued capital cell dividend by 321,400,589. Calculating the HPR on the stock from the normalized data Normalized Normalized Date Stock Price Dividend HPR ( P P0 ) D1 HPR 01-May-02 $59.22 $0.00 2.28% 1 01-Apr-02 $57.90 $0.15 -7.90% 01-Mar-02 $63.03 $0.00 -2.82% P0 01-Feb-02 $64.86 $0.00 4.87% 02-Jan-02 $61.85 $0.15 23.36% 01-Dec-01 $50.26 $0.00 Use $59.22 as the ending price, $57.90 as the beginning price and during the month of May, no dividend was declared. Now Put the data from the S&P/TSX Total Return Composite Index in Ending Normalized Normalized TSX Date Stock Price Dividend HPR Value 01-May-02 $59.22 $0.00 2.28% 16911.33 01-Apr-02 $57.90 $0.15 -7.90% 16903.36 01-Mar-02 $63.03 $0.00 -2.82% 17308.41 01-Feb-02 $64.86 $0.00 4.87% 16801.82 02-Jan-02 $61.85 $0.15 23.36% 16908.11 01-Dec-01 $50.26 $0.00 16881.75 You will find the Total Return S&P/TSX Composite Index values in TSE Review found in the library. Now Calculate the HPR on the Market Index Ending Normalized Normalized TSX HPR on Date Stock Price Dividend HPR Value the TSX 01-May-02 $59.22 $0.00 2.28% 16911.33 0.05% 01-Apr-02 $57.90 $0.15 -7.90% 16903.36 -2.34% 01-Mar-02 $63.03 $0.00 -2.82% 17308.41 3.02% 01-Feb-02 $64.86 $0.00 4.87% 16801.82 -0.63% 02-Jan-02 $61.85 $0.15 23.36% 16908.11 0.16% 01-Dec-01 $50.26 $0.00 16881.75 Again, you simply use the HPR formula using the ending values for the total return composite index. Regression In Excel If you haven‟t already…go to the tools menu…down to add-ins and check off the VBA Analysis Pac When you go back to the tools menu, you should now find the Data Analysis bar, under that find regression, define your dependent and independent variable ranges, your output range and run the regression. Now Use the Regression Function in Excel to regress the returns of the stock against the returns of the market SUMMARY OUTPUT Regression Statistics Multiple R 0.05300947 R Square 0.00281 R-square = coefficient of Adjusted R Square -0.2464875 Standard Error 5.79609628 determination Observations 6 ANOVA df SS MS F Significance F Regression 1 0.3786694 0.37866937 0.011271689 0.920560274 Residual 4 134.37893 33.5947321 Total 5 134.7576 CoefficientsStandard Error t Stat P-value Lower 95% Upper 95% Lower 95.0%Upper 95.0% Intercept 59.3420816 2.8980481 20.4765686 3.3593E-05 51.29579335 67.38836984 51.2957934 67.38837 X Variable 1 3.55278937 33.463777 0.10616821 0.920560274 -89.35774428 96.46332302 -89.3577443 96.46332 Alpha Beta Finalize Your Chart You can use the charting feature in Excel to create a scatter plot of the points and to put a line of best fit (the characteristic line) through the points. Finally, you will want to interpret the Beta (X- coefficient) the alpha (vertical intercept) and the coefficient of determination. The Beta Obviously the beta (X-coefficient) can simply be read from the regression output. You will want to interpret it in the context of the firms, its products and the likely relationship that they hold with the health of the overall market.