VIEWS: 89 PAGES: 86 CATEGORY: Business POSTED ON: 4/23/2011
Formulae for Calculation of Expected Return and Risk document sample
Lessons from Capital Market History Shad Valley K. Hartviksen 1 Dollar Returns measured in absolute dollars. less meaningful than percentage returns because they depend on the amount of the original investment. Bonds give rise to two kinds of returns: • Capital gains or capital losses, and • Interest Stock investments give rise to dollar returns as: • Capital gains or capital losses, and • Dividends Holding Period Return (For investments that yield dividend cash flow returns) Ending Price - Beginning Price Dividend HPR Beginning Price P P0 D1 HPR 1 P0 Holding Period Return (For investments that yield dividend cash flow returns) The single period return calculation can be reformulated to show illustrate the two forms of investment income : capital gains/losses and dividends : HPR capital gain yield dividend yield P P D HPR 1 0 1 P0 P0 Holding Period Return …Illustrated (For investments that yield dividend cash flow returns) Price of stock at the beginning of the year $21.25 Price of the stock at the end of the year $22.10 Dividend received each quarter $.10 P P D1 ($22.10 $21.25) (4)($0.10) HPR 1 0 P0 $21.25 $0.85 $0.40 HPR $21.25 $1.25 HPR 0.0588 5.9% $21.25 The Geometric Average Measuring Investment Returns The Bias Inherent in the Arithmetic Average Arithmetic averages can yield incorrect results because of the problems of bias inherent in its calculation. Example – Consider an investment that was purchased for $10, rose to $20 and then fell back to $10. – Let us calculate the HPR in both periods: $20 $10 $10 HPR1 100% $10 $10 $10 $20 $10 HPR 2 50% $20 $20 The Bias Inherent in the Arithmetic Average Example Continued ... – Consider an investment that was purchased for $10, rose to $20 and then fell back to $10. – Let us calculate the HPR in both periods: $20 $10 $10 HPR1 100% $10 $10 $10 $20 $10 HPR 2 50% $20 $20 – The arithmetic average return earned on this investment was: 100% 50% 50% Average 25% 2 2 The Bias Inherent in the Arithmetic Average Example Continued ... – The answer is clearly incorrect since the investor started with $10 and ended with $10. – The correct answer may be obtained through the use of the geometric average: n GeometricAverage n (1 r ) 1 i 1 i 1/ n n (1 ri ) i 1 1 [(1 100%)(1 (50%))]1/ 2 1 [(2)(.5)]1/ 2 1 (1)1/ 2 1 1 1 0 Geometric Versus Arithmetic Average Returns Consider two investments with the following realized returns over the past Holding Period Returns IBM Government few years: Year Stock Bonds 2000 12.0% 6.0% 2001 12.0% 6.0% 2002 12.0% 6.0% 2003 12.0% 6.0% 2004 12.0% 6.0% 2005 12.0% 6.0% If the returns are equal over time, the arithmetic average return will equal the geometric average return. Geometric Versus Arithmetic Average Returns Holding Period Returns IBM Government Arithmetic Average Return : Year Stock Bonds _ HPR1 HPR2 HPR3 HPR4 HPR5 HPR6 R N 2000 2001 12.0% 12.0% 6.0% 6.0% 12% 12% 12% 12% 12% 12% 2002 12.0% 6.0% 2003 12.0% 6.0% 6 2004 12.0% 6.0% 72 12% 2005 12.0% 6.0% 6 SAME ANSWER ! Geometric Average Return : _ 1 G (1 HPR1 )(1 HPR2 )(1 HPR3 )(1 HPR4 )(1 HPR5 )(1 HPR6 )]6 1 1 (1.12)(1.12)(1.12)(1.12)(1.12)(1.12) 6 1 1.973822685.16667 1 12% Geometric Versus Arithmetic Average Returns Holding Period Returns IBM Government Year Stock Bonds Now consider volatile returns: 2000 40.0% 11.0% Arithmetic Average Return : 2001 -30.0% 4.0% 2002 33.0% 8.0% _ HPR1 HPR2 HPR3 HPR4 HPR5 HPR6 R N 2003 5.0% 3.0% 2004 32.0% 6.0% 40% 30% 33% 5% 32% 8% 2005 -8.0% 4.0% 6 72 12% 6 Arithmetic Average = 12.0% 6.0% Standard Deviation = 27.71% 3.03% NOT THE SAME Geometric Average Return : 1 ANSWER ! _ G (1 HPR1 )(1 HPR2 )(1 HPR3 )(1 HPR4 )(1 HPR5 )(1 HPR6 )]6 1 1 (1.40)(0.70)(1.33)(1.05)(1.32)(.92) 6 1 1.661991408.16667 1 8.84% Volatility of returns over time eats away at your realized returns!!! The greater the volatility the greater the difference between the arithmetic and geometric average. Arithmetic average OVERSTATES the return!!! Measuring Returns When you are trying to find average returns, especially when those returns rise and fall, always remember to use the geometric average. The greater the volatility of returns over time, the greater the difference you will observe between the geometric and arithmetic averages. Of course, there are limitations inherent in the use of geometric averages. Historical Returns Average Standard Risk Return Deviation Premium Canadian Equities 13.20% 16.62% 7.16% U.S. Equities 15.59% 16.86% 9.55% Long-Term bonds 7.64% 10.57% 1.60 Treasury bills 6.04% 4.04% 0.00 Small cap stocks 14.79% 23.68% 8.75 Inflation 4.25% 3.51% -1.79 The historical pattern of returns exhibit the classic risk-return tradeoff Historical Returns Historical Averages - Risk and Return 16 Small cap 14 Canadian Equities Percent Return 12 10 8 Long-Term Bonds 6 4 Treasury Bills 2 0 0 5 10 15 20 25 Standard Deviation Capital Asset Pricing Model Return Required return = Rf + bs [kM - Rf] % km Market Security Market Premium Line for risk Rf Real Return Premium for expected inflation BM=1.0 Beta Coefficient 6 Measurement of Risk in an Isolated Asset Case The dispersion of returns from the mean return is a measure of the riskiness of an investment. This dispersion can be calculated using: Variance (an ‘absolute’ measure of dispersion expressed in units squared) Standard Deviation (an ‘absolute’ measure of dispersion expressed in the same units as the mean) Coefficient of Variation (this is a ‘relative’ measure of dispersion…it is a ratio of the standard deviation divided by the mean) Ex Post and Ex Ante Calculations Returns and risk can be calculated after-the- fact (ie. You use actual realized return data) This is known as an ex post calculation. Or you can use forecast data…this is an ex ante calculation. Standard Deviation The formula for the standard deviation when analyzing sample data (realized returns) is: n (ki ki ) 2 i 1 n 1 Where k is a realized return on the stock and n is the number of returns used in the calculation of the mean. Standard Deviation The formula for the standard deviation when analyzing forecast data (ex ante returns) is: n (k i 1 i k i ) Pi 2 it is the square root of the sum of the squared deviations away from the expected value. Forecasting Risk and Return for the Individual Asset S toc k A D e via t io n s S q u a re d W e ig h t e d P ro b a b ilit y E x . R e t . W t d . R e t . fro m m e a n D e via t io n s S q . D e v. R e c e s s io n 0.150 0.02 0.0030 -8 . 2 0 0 % 0.006724 0.001009 N o rm a l 0.600 0.09 0.0540 -1 . 2 0 0 % 0.000144 8 . 6 4 E -0 5 B oom 0.250 0.18 0.0450 7.800% 0.006084 0.001521 E x p e c t e d R e t u rn = 10.20% V a ria n c e = 0 . 0 0 2 6 1 6 S t d . D e v. = 5.11% K. Hartviksen A Normal Probability Distribution The area under the curve bounded by Probability -1 and +1 σ is equal to 68% - 1 standard + 1 standard deviation away from deviation away from the mean the mean 13.2% Return on Large Cap Stocks Finding the Probability of an Event using Z-value tables You can find the number of standard deviations away from the mean that a point of (point of interest) - (mean) interest lies using the z following ‘z’ value formula: standard deviation x z Once you know ‘z’ then you can find the areas under the normal curve using the z value table found on the following slide. Values of the Standard Normal Distribution Function Z 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.0 .0000 .0040 .0080 .0120 .0160 .0199 .0239 .0279 .0319 .0359 0.1 .0398 .0438 .0478 .0517 .0557 .0596 .0636 .0675 .0714 .0753 0.2 .0793 .0832 .0871 .0910 .0948 .0987 .1026 .1064 .1103 .1141 0.3 .1179 .1217 .1255 .1293 .1331 .1368 .1406 .1443 .1480 .1517 0.4 .1554 .1591 .1628 .1664 .1700 .1736 .1772 .1808 .1844 .1879 0.5 .1915 .1950 .1985 .2019 .2054 .2088 .2123 .2157 .2190 .2224 0.6 .2257 .2291 .2324 .2357 .2389 .2422 .2454 .2486 .2517 .2549 0.7 .2580 .2611 .2642 .2673 .2704 .2734 .2764 .2794 .2823 .2852 0.8 .2881 .2910 .2939 .2967 .2995 .3023 .3051 .3078 .3106 .3133 0.9 .3159 .3186 .3212 .3238 .3264 .3289 .3315 .3340 .3365 .3389 1.0 .3413 .3438 .3461 .3485 .3508 .3531 .3554 .3577 .3599 .3621 1.1 .3643 .3665 .3686 .3708 .3729 .3749 .3770 .3790 .3810 .3830 1.2 .3849 .3869 .3888 .3907 .3925 .3944 .3962 .3980 .3997 .4015 1.3 .4032 .4049 .4066 .4082 .4099 .4115 .4131 .4147 .4162 .4177 1.4 .4192 .4207 .4222 .4236 .4251 .4265 .4279 .4292 .4306 .4319 1.5 .4332 .4345 .4357 .4370 .4382 .4394 .4406 .4418 .4429 .4441 1.6 .4452 .4463 .4474 .4484 .4495 .4505 .4515 .4525 .4535 .4545 1.7 .4554 .4564 .4573 .4582 .4591 .4599 .4608 .4616 .4625 .4633 1.8 .4641 .4649 .4656 .4664 .4671 .4678 .4686 .4693 .4699 .4706 1.9 .4713 .4719 .4726 .4732 .4738 .4744 .4750 .4756 .4761 .4767 2.0 .4773 .4778 .4783 .4788 .4793 .4798 .4803 .4808 .4812 .4817 2.1 .4821 .4826 .4830 .4834 .4838 .4842 .4846 .4850 .4854 .4857 2.2 .4861 .4864 .4868 .4871 .4875 .4878 .4881 .4884 .4887 .4890 2.3 .4893 .4896 .4898 .4901 .4904 .4906 .4909 .4911 .4913 .4916 2.4 .4918 .4920 .4922 .4925 .4927 .4929 .4931 .4932 .4934 .4936 2.5 .4938 .4940 .4941 .4943 .4945 .4946 .4948 .4949 .4951 .4952 2.6 .4953 .4955 .4956 .4957 .4959 .4960 .4961 .4962 .4963 .4964 2.7 .4965 .4966 .4967 .4968 .4969 .4970 .4971 .4972 .4973 .4974 2.8 .4974 .4975 .4976 .4977 .4977 .4978 .4979 .4979 .4980 .4981 2.9 .4981 .4982 .4982 .4982 .4984 .4984 .4985 .4985 .4986 .4986 3.0 .4987 .4987 .4987 .4988 .4988 .4989 .4989 .4989 .4990 .4990 z is the number of standard deviations from the mean. Some area tables are set up to indicate the area to the left or right of the point of interest. In this table, we indicate the area between the mean and the point of interest. MPT – Modern Portfolio Theory Shad Valley K. Hartviksen 1 Risk and Return - MPT Prior to the establishment of Modern Portfolio Theory, most people only focused upon investment returns…they ignored risk. With MPT, investors had a tool that they could use to dramatically reduce the risk of the portfolio without a significant reduction in the expected return of the portfolio. 3 Correlation The degree to which the returns of two stocks co-move is measured by the correlation coefficient. The correlation coefficient between the returns on two securities will lie in the range of +1 through - 1. +1 is perfect positive correlation. -1 is perfect negative correlation. 10 Perfect Negatively Correlated Returns over Time Returns % A two-asset portfolio made up of equal parts of Stock A and B would be riskless. There would be no variability of the portfolios returns 10% over time. Returns on Stock A Returns on Stock B Returns on Portfolio 1994 1995 1996 Time 11 Ex Post Portfolio Returns Simply the Weighted Average of Past Returns n R p xi Ri i 1 Where : xi relative weight of asset i Ri return on asset i K. Hartviksen 5 14 Ex Ante Portfolio Returns Simply the Weighted Average of Expected Returns Relative Expected Weighted Weight Return Return Stock X 0.400 8.0% 0.03 Stock Y 0.350 15.0% 0.05 Stock Z 0.250 25.0% 0.06 Expected Portfolio Return = 14.70% K. Hartviksen 5 14 Grouping Individual Assets into Portfolios The riskiness of a portfolio that is made of different risky assets is a function of three different factors: – the riskiness of the individual assets that make up the portfolio – the relative weights of the assets in the portfolio – the degree of comovement of returns of the assets making up the portfolio The standard deviation of a two-asset portfolio may be measured using the Markowitz model: p w w 2 wA wB A, B A B 2 A 2 A 2 B 2 B Risk of a Three-asset Portfolio The data requirements for a three-asset portfolio grows dramatically if we are using Markowitz Portfolio selection formulae. We need 3 (three) correlation coefficients between A and B; A and C; and B and C. A ρa,b ρa,c B C ρb,c p A wA B wB C wC 2wA wB A, B A B 2wB wC B ,C B C 2wA wC A,C A C 2 2 2 2 2 2 Risk of a Four-asset Portfolio The data requirements for a four-asset portfolio grows dramatically if we are using Markowitz Portfolio selection formulae. We need 6 correlation coefficients between A and B; A and C; A and D; B and C; C and D; and B and D. A ρa,b ρa,d ρa,c B D ρb,d ρb,c ρc,d C Diversification Potential The potential of an asset to diversify a portfolio is dependent upon the degree of co-movement of returns of the asset with those other assets that make up the portfolio. In a simple, two-asset case, if the returns of the two assets are perfectly negatively correlated it is possible (depending on the relative weighting) to eliminate all portfolio risk. This is demonstrated through the following chart. Example of Portfolio Combinations and Correlation Perfect Expected Standard Correlation Positive Asset Return Deviation Coefficient Correlation – A 5.0% 15.0% 1 no B 14.0% 40.0% diversification Portfolio Components Portfolio Characteristics Expected Standard Weight of A Weight of B Return Deviation 100.00% 0.00% 5.00% 15.0% 90.00% 10.00% 5.90% 17.5% 80.00% 20.00% 6.80% 20.0% 70.00% 30.00% 7.70% 22.5% 60.00% 40.00% 8.60% 25.0% 50.00% 50.00% 9.50% 27.5% 40.00% 60.00% 10.40% 30.0% 30.00% 70.00% 11.30% 32.5% 20.00% 80.00% 12.20% 35.0% 10.00% 90.00% 13.10% 37.5% 0.00% 100.00% 14.00% 40.0% Example of Portfolio Combinations and Correlation Positive Expected Standard Correlation Correlation – Asset Return Deviation Coefficient weak A 5.0% 15.0% 0.5 diversification B 14.0% 40.0% potential Portfolio Components Portfolio Characteristics Expected Standard Weight of A Weight of B Return Deviation 100.00% 0.00% 5.00% 15.0% 90.00% 10.00% 5.90% 15.9% 80.00% 20.00% 6.80% 17.4% 70.00% 30.00% 7.70% 19.5% 60.00% 40.00% 8.60% 21.9% 50.00% 50.00% 9.50% 24.6% 40.00% 60.00% 10.40% 27.5% 30.00% 70.00% 11.30% 30.5% 20.00% 80.00% 12.20% 33.6% 10.00% 90.00% 13.10% 36.8% 0.00% 100.00% 14.00% 40.0% Example of Portfolio Combinations and Correlation No Expected Standard Correlation Correlation – Asset Return Deviation Coefficient some A 5.0% 15.0% 0 diversification B 14.0% 40.0% potential Portfolio Components Portfolio Characteristics Expected Standard Weight of A Weight of B Return Deviation 100.00% 0.00% 5.00% 15.0% Lower 90.00% 10.00% 5.90% 14.1% risk than 80.00% 20.00% 6.80% 14.4% asset A 70.00% 30.00% 7.70% 15.9% 60.00% 40.00% 8.60% 18.4% 50.00% 50.00% 9.50% 21.4% 40.00% 60.00% 10.40% 24.7% 30.00% 70.00% 11.30% 28.4% 20.00% 80.00% 12.20% 32.1% 10.00% 90.00% 13.10% 36.0% 0.00% 100.00% 14.00% 40.0% Example of Portfolio Combinations and Correlation Negative Expected Standard Correlation Correlation – Asset Return Deviation Coefficient greater A 5.0% 15.0% -0.5 diversification B 14.0% 40.0% potential Portfolio Components Portfolio Characteristics Expected Standard Weight of A Weight of B Return Deviation 100.00% 0.00% 5.00% 15.0% 90.00% 10.00% 5.90% 12.0% 80.00% 20.00% 6.80% 10.6% 70.00% 30.00% 7.70% 11.3% 60.00% 40.00% 8.60% 13.9% 50.00% 50.00% 9.50% 17.5% 40.00% 60.00% 10.40% 21.6% 30.00% 70.00% 11.30% 26.0% 20.00% 80.00% 12.20% 30.6% 10.00% 90.00% 13.10% 35.3% 0.00% 100.00% 14.00% 40.0% Example of Portfolio Combinations and Correlation Perfect Negative Expected Standard Correlation Correlation – Asset Return Deviation Coefficient greatest A 5.0% 15.0% -1 diversification B 14.0% 40.0% potential Portfolio Components Portfolio Characteristics Expected Standard Weight of A Weight of B Return Deviation 100.00% 0.00% 5.00% 15.0% 90.00% 10.00% 5.90% 9.5% Risk of the 80.00% 20.00% 6.80% 4.0% portfolio is almost 70.00% 30.00% 7.70% 1.5% eliminated at 60.00% 40.00% 8.60% 7.0% 70% asset A 50.00% 50.00% 9.50% 12.5% 40.00% 60.00% 10.40% 18.0% 30.00% 70.00% 11.30% 23.5% 20.00% 80.00% 12.20% 29.0% 10.00% 90.00% 13.10% 34.5% 0.00% 100.00% 14.00% 40.0% Diversification of a Two Asset Portfolio Demonstrated Graphically The Effect of Correlation on Portfolio Risk: The Two-Asset Case Expected Return B AB = -0.5 12% AB = -1 8% AB = 0 AB= +1 A 4% 0% 0% 10% 20% 30% 40% Standard Deviation An Exercise using T-bills, Stocks and Bonds Base Data: Stocks T-bills Bonds Expected Return 12.73383 6.151702 7.007872 Standard Deviation 0.168 0.042 0.102 Correlation Coefficient Matrix: Stocks 1 -0.216 0.048 T-bills -0.216 1.000 0.380 Bonds 0.048 0.380 1.000 Portfolio Combinations: Weights Portfolio Expected Standard Combination Stocks T-bills Bonds Return Variance Deviation 1 100.0% 0.0% 0.0% 12.7 0.0283 16.8% 2 90.0% 10.0% 0.0% 12.1 0.0226 15.0% 3 80.0% 20.0% 0.0% 11.4 0.0177 13.3% 4 70.0% 30.0% 0.0% 10.8 0.0134 11.6% 5 60.0% 40.0% 0.0% 10.1 0.0097 9.9% 6 50.0% 50.0% 0.0% 9.4 0.0067 8.2% 7 40.0% 60.0% 0.0% 8.8 0.0044 6.6% 8 30.0% 70.0% 0.0% 8.1 0.0028 5.3% 9 20.0% 80.0% 0.0% 7.5 0.0018 4.2% 10 10.0% 90.0% 0.0% 6.8 0.0014 3.8% 11 0.0% 100.0% 0.0% 6.2 0.0017 4.2% Results Using only Three Asset Classes Attainable Portfolio Combinations and Efficient Set of Portfolio Combinations 14.0 Efficient Set Portfolio Expected Return (%) 12.0 Minimum Variance 10.0 Portfolio 8.0 6.0 4.0 2.0 0.0 0.0 5.0 10.0 15.0 20.0 Standard Deviation of the Portfolio (%) Plotting Achievable Portfolio Combinations Expected Return on the Portfolio 12% 8% 4% 0% 0% 10% 20% 30% 40% Standard Deviation of the Portfolio The Efficient Frontier Expected Return on the Portfolio 12% 8% 4% 0% 0% 10% 20% 30% 40% Standard Deviation of the Portfolio The Capital Market Line Capital Market Line Expected Return on the Portfolio 12% 8% 4% Risk-free rate 0% 0% 10% 20% 30% 40% Standard Deviation of the Portfolio The Capital Market Line and Iso Utility Curves Highly A risk- Risk taker Expected Return on Averse the Portfolio Investor 12% Capital 8% Market Line 4% Risk-free rate 0% 0% 10% 20% 30% 40% Standard Deviation of the Portfolio The Capital Market Line and Iso Utility Curves The risk- A risk-taker’s taker’s utility curve Expected Return on the Portfolio optimal portfolio 12% combination Capital 8% Market Line 4% Risk-free rate 0% 0% 10% 20% 30% 40% Standard Deviation of the Portfolio CML versus SML Please notice that the CML is used to illustrate all of the efficient portfolio combinations available to investors. It differs significantly from the SML that is used to predict the required return that investors should demand given the riskiness (beta) of the investment. Data Limitations Because of the need for so much data, MPT was a theoretical idea for many years. Later, a student of Markowitz, named William Sharpe worked out a way around that…creating the Beta Coefficient as a measure of volatility and then later developing the CAPM. CAPM The Capital Asset Pricing Model was the work of William Sharpe, a student of Harry Markowitz at the University of Chicago. CAPM is an hypothesis … Capital Asset Pricing Model Return Required return = Rf + bs [kM - Rf] % km Market Security Market Premium Line for risk Rf Real Return Premium for expected inflation BM=1.0 Beta Coefficient 6 CAPM This model is an equilibrium based model. It is called a single-factor model because the slope of the SML is caused by a single measure of risk … the beta. Although this model is a simplification of reality…it is robust (it explains much of what we see happening out there) and it enjoys widespread use in a great variety of applications. Although it is called a ‘pricing model’ there are not prices on that graph….only risk and return. It is called a pricing model because it can be used to help us determine appropriate prices for securities in the market. Risk Risk is the chance of harm or loss; danger. We know that various asset classes have yielded very different returns in the past: Historical Returns and Standard Deviations 1948 - 941 Average Return Standard Deviation Canadian common stock 12.73% 16.81% U.S. common stock (Cdn $) 14.09 16.60 Long term bonds 7.01 10.20 Small cap stocks 15.67 24.40 Inflation 4.52 3.54 Treasury bills 6.15 4.17 ___________________ 1The Alexander Group Risk and Return The foregoing data point out that those asset classes that have offered the highest rates of return, have also offered the highest risk levels as measured by the standard deviation of returns. The CAPM suggests that investors demand compensation for risks that they are exposed to…and these returns are built into the decision- making process to invest or not. Capital Asset Pricing Model Return Required return = Rf + bs [kM - Rf] % km Market Security Market Premium Line for risk Rf Real Return Premium for expected inflation BM=1.0 Beta Coefficient 6 CAPM The foregoing graph shows that investors: – demand compensation for expected inflation – demand a real rate of return over and above expected inflation – demand compensation over and above the risk-free rate of return for any additional risk undertaken. We will make the case that investors don’t need compensation for all of the risk of an investment because some of that risk can be diversified away. Investors require compensation for risk they can’t diversify away! Beta Coefficient The beta is a measure of systematic risk of an investment. Systematic risk is the only relevant risk to a diversified investor according to the CAPM since all other risk may be diversified away. Total risk of an investment is measured by the securities’ standard deviation of returns. According to the CAPM total risk may be broken into two parts…systematic (non-diversifiable) and unsystematic (diversifiable) TOTAL RISK = SYSTEMATIC RISK + UNSYSTEMATIC RISK The beta can be determined by regressing the holding period returns (HPRs) of the security over 30 periods against the returns on the overall market. 7 Measuring Risk of the Individual Security Risk is the possibility that the actual return that will be realized, will turn out to be different than what we expect (or have forecast). This can be measured using standard statistical measures of dispersion for probability distributions. They include: – variance – standard deviation – coefficient of variation Standard Deviation The formula for the standard deviation when analyzing population data (realized returns) is: n (ki ki ) 2 i 1 n 1 Standard Deviation The formula for the standard deviation when analyzing forecast data (ex ante returns) is: n (k i 1 i k i ) Pi 2 it is the square root of the sum of the squared deviations away from the expected value. Using Forecasts to Estimate Beta The formula for the beta coefficient for a stock ‘s’ is: Cov (k s k M ) Bs Variance (k M ) Obviously, the calculate a beta for a stock, you must first calculate the variance of the returns on the market portfolio as well as the covariance of the returns on the stock with the returns on the market. Systematic Risk The returns on most assets in our economy are influenced by the health of the ‘system’ Some companies are more sensitive to systematic changes in the economy. For example durable goods manufacturers. Some companies do better when the economy is doing poorly (bill collection agencies). The beta coefficient measures the systematic risk that the security possesses. Since non-systematic risk can be diversified away, it is irrelevant to the diversified investor. Systematic Risk We know that the economy goes through economic cycles of expansion and contraction as indicated in the following: Canada’s Business cycles from 1873-1992 Trough to ExpansionPeak to Contraction (months from trough to peak)(months from peak to trough) Nov 1873 66 May 1879 38 July 1882 32 Mar 1885 23 Feb 1887 12 Feb 1888 29 July 1890 9 Mar 1891 23 Apr 1893 13 Mar 1894 17 Aug 1895 12 Aug 1896 44 Apr 1900 10 Feb 1901 22 Dec 1902 18 June 1904 30 Dec 1906 19 July 1908 20 Mar 1910 16 July 1911 16 Nov 1912 25 Jan 1915 36(WWI) Jan 1918 15 Apr 1919 14 June 1920 15 Sep 1921 21 June 1923 14 Aug 1924 56 Apr 1929 47 (Depression) Mar 1933 52 July 1937 15 (Depression) Oct 1938 80(WWII) June 1945 8 Feb 1946 33 Oct 1948 11 Sep 1949 44(Korean War) May 1953 14 July 1954 31 Feb 1957 12 Feb 1958 26 Apr 1960 10 Feb 1961 160 June 1974 10 Apr 1975 58 Feb 1980 6 July 1980 12 July 1981 6 Nov 1982 89 Apr 1990 22 Feb 1992 Companies and Industries Some industries (and by implication the companies that make up the industry) move in concert with the expansion and contraction of the economy. Some lead the overall economy. (stock market) Some lag the overall economy. (ie. automotive industry) Amount of Systematic Risk Some industries may find that their fortunes are positively correlated with the ebb and flow of the overall economy…but that this relationship is very insignificant. An example might be Imperial Tobacco. This firm does have a positive beta coefficient, but very little of the returns of this company can be explained by the beta. Instead, most of the variability of returns on this stock is from diversifiable sources. A Characteristic line for Imperial Tobacco would show a very wide dispersion of points around the line. The R2 would be very low (.05 = 5% or lower). Characteristic Line for Imperial Tobacco Characteristic Returns on Line for Imperial Imperial Tobacco Tobacco % 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 General Motors % (high R2) 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. Partitioning Risk under the CAPM Remember that the CAPM assumes that total risk (variability of a security’s returns) can be separated into two distinct components: Total risk = systematic risk + unsystematic risk 100% = 40% + 60% (GM) or 100% = 5% + 95% (Imperial Tobacco) Obviously, if you were to add Imperial Tobacco to your portfolio, you could diversify away much of the risk of your portfolio. (Not to mention the fact that Imperial has realized some very high rates of return in addition to possessing little systematic risk!) Using the CAPM to Price Stock The CAPM is a ‘fundamental’ analyst’s tool to estimate the ‘intrinsic’ value of a stock. The analyst needs to measure the beta risk of the firm by using either historical or forecast risk and returns. The analyst will then need a forecast for the risk-free rate as well as the expected return on the market. These three estimates will allow the analyst to calculate the required return that ‘rational’ investors should expect on such an investment given the other benchmark returns available in the economy. Required Return The return that a rational investor should demand is therefore based on market rates and the beta risk of the investment. To find this, you solve for the required return in the CAPM: R(k ) R f b s [k M R f ] This is a formula for the straight line that is the SML. Security Market Line This line can easily be plotted. Draw Cartesian coordinates. Plot the yield on 91-day Government of Canada Treasury Bills as the risk-free rate of return on the vertical axis. On the horizontal axis set a scale that includes Beta=1 (this is the beta of the market) Plot the point in risk-return space that represents your expected return on the market portfolio at beta =1 Draw a straight line to connect the two points. Plot the required and expected returns for the stock at it’s beta. Plot the Risk-Free Rate Return % Rf 1.0 Beta Coefficient Plot Expected Return on the Market Portfolio Return % km =12% Rf = 4% 1.0 Beta Coefficient Draw the Security Market Line Return SML % km =12% Rf = 4% 1.0 Beta Coefficient Plot Required Return (Determined by the formula = Rf + bs[kM - Rf] Return SML % R(k) = 13.6% km =12% R(k) = 4% + 1.2[8%] = 13.6% Rf = 4% 1.0 1.2 Beta Coefficient Plot Expected Return E(k) = weighted average of possible returns Return SML % R(k) = 13.6% R(k) = 4% + 1.2[8%] = 13.6% km =12% E(k) Rf = 4% 1.0 1.2 Beta Coefficient If Expected = Required Return The stock is properly (fairly) priced in the market. It is in EQUILIBRIUM. Return SML % R(k) = 13.6% R(k) = 4% + 1.2[8%] = 13.6% km =12% E(k) Rf = 4% 1.0 1.2 Beta Coefficient If E(k) < R(k) The stock is over-priced. The analyst would issue a sell recommendation in anticipation of the market becoming ‘efficient’ to this fact. Investors may ‘short’ the stock to take advantage of the anticipated price decline. Return SML % R(k) = 13.6% R(k) = 4% + 1.2[8%] = 13.6% km =12% E(k) = 9% E(k) Rf = 4% 1.0 1.2 Beta Coefficient Let’s Look at the Pricing Implications In this example: – E(k) = 9% – R(k) = 13.6% If the market expects the company to pay a dividend of $1.00 next year, and the stock is currently offering an expected return of 9%, then it should be priced at: d1 P0 E (k s ) $1.00 P0 $11.11 .09 But, given the other rates in the economy and our judgement about the riskiness of this investment we think that this stock should be worth: $1.00 P0 $7.35 .136 Practical Use of the CAPM Regulated utilities justify rate increases using the model to demonstrate that their shareholders require an appropriate return on their investment. Used to price initial public offerings (IPOs) Used to identify over and under value securities Used to measure the riskiness of securities/companies Used to measure the company’s cost of capital. (The cost of capital is then used to evaluate capital expansion proposals). The model helps us understand the variables that can affect stock prices…and this guides managerial decisions. Rf rises SML2 Return % SML1 ks2 ks1 Rising interest rates will cause all required rates of Rf2 return to increase and this Rf1 will force down stock and bond prices. Bs=1.2 Beta Coefficient The Slope of The SML rises (indicates growing pessimism about the future of the economy) SML2 Return % SML1 ks2 Growing pessimism will cause investors to ks1 demand greater compensation for taking on risk…this will mean prices on Rf1 high beta stocks will fall more than low beta stocks. Bs=1.2 Beta Coefficient