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Risk and Rates of Return New words • • • • • • • • • stand-alone risk:独立风险 portfolio risk:组合风险 Expected Rate of Return:期望报酬率 Probability Distributions:概率分布 Discrete Probability Distribution:离散概率分布 Continuous Probability Distribution连续概率分布 Variance:方差 Standard deviation: 标准偏差 Coefficient of Variation:变异系数 Investment returns The rate of return on an investment can be calculated as follows: Return = (Amount received – Amount invested) ________________________ Amount invested For example, if $1,000 is invested and $1,100 is returned after one year, the rate of return for this investment is: ($1,100 - $1,000) / $1,000 = 10%. Defining and Measuring Risk Risk is the chance that an outcome other than expected will occur. stand-alone risk: the risk associated with an investment when it is held by itself or in isolation, not in combination with other assets. portfolio risk: the risk associated with an investment when it is held in combination with other assets ,not by itself. Expected Rate of Return The rate of return expected to be realized from an investment The mean value of the probability distribution of possible returns The weighted average of the outcomes, where the weights are the probabilities Expected Rate of Return Probability of This State State of the Economy Occurring (Pr i) (1) Boom Normal Recession (2) 0.2 0.5 0.3 1.0 Martin Products Return if This State Product: Occurs (ki) (2) x (3) (3) 110% 22% -60% ^ = km = (4) 22% 11% -18% 15% U. S. Electric Return if This Product: State Occurs (ki) (2) x (5) (5) 20% 16% 10% ^ = km = (6) 4% 8% 3% 15% Expected Rate of Return k̂ Pr1k1 Pr2 k 2 ... Prn k n n Pri k i i 1 Probability Distributions Probability distribution is a listing of all possible outcomes,or events, with a probability(chance of occurrence) assigned to each outcome. [must sum to 1.0 (100%)] Probability Distributions It either will rain, or it will not – only two possible outcomes Outcome (1) Probability (2) Rain 0.40 = 40% No Rain 0.60 = 60% 1.00 100% Probability Distributions Martin Products and U. S. Electric State of the Economy Boom Normal Recession Probability of This State Occurring 0.2 0.5 0.3 1.0 Rate of Return on Stock if This State Occurs Martin Products U.S. Electric 110% 22% -60% 20% 16% 10% Continuous versus Discrete Probability Distributions Discrete Probability Distribution: the number of possible outcomes is limited, or finite Continuous Probability Distribution: the number of possible outcomes is unlimited, or infinite Discrete Probability Distributions a. Martin Products Probability of Occurrence b. U. S. Electric Probability of Occurrence 0.5 - 0.5 - 0.4 - 0.4 - 0.3 - 0.3 - 0.2 - 0.2 - 0.1 - 0.1 - -60 -45 -30 -15 0 15 22 30 45 60 75 90 110 Rate of Expected Rate Return (%) of Return (15%) -10 -5 0 5 10 16 20 Expected Rate of Return (15%) 25 Rate of Return (%) Continuous Probability Distributions Probability Density U. S. Electric Martin Products -60 0 15 110 Rate of Return (%) Expected Rate of Return Measuring Risk: The Standard Deviation Calculating Martin Products’ Standard Deviation Expected Payoff Return ki k^ (2) (1) 15% 110% 15% 22% 15% -60% ki - k^ (1) - (2) = (3) 95 7 -75 ^2 (ki - k) (4) 9,025 49 5,625 Probability (5) ^ 2Pr (ki - k) i (4) x (5) = (6) 1,805.0 0.2 24.5 0.5 1,687.5 0.3 Variance 2 3,517.0 Standard Deviation m m2 3,517 59.3% Measuring Risk: The Standard Deviation n Expected rate of return k̂ Pri k i i 1 n Variance k i - k̂ Pri 2 i 1 2 Standard deviation 2 k - k̂ Pr n i1 2 i i Measuring Risk: The Standard Deviation • Standard deviation (σ) measures total, or stand-alone, risk. • The larger σ is, the lower the probability that actual returns will be closer to expected returns. • Larger σ is associated with a wider probability distribution of returns. • Difficult to compare standard deviations, because return has not been accounted for. Measuring Risk: Coefficient of Variation Standardized measure of risk per unit of return Calculated as the standard deviation divided by the expected return Useful where investments differ in risk and expected returns Risk Coefficient of variation CV Return k̂ Portfolio Returns k̂ p w 1k̂ 1 w 2 k̂ 2 ... w N k̂ N ^ N w j1 j k̂ j Expected return on a portfolio, kp The weighted average expected return on the stocks held in the portfolio ^ k p 0.10 (3.0%) 0.20 (6.4%) 0.40 (10.0%) 0.20 (12.5%) 0.10 (15.0%) 9.6% Amount in each stock is equal Economy Prob. HT Coll Port. Recession 0.1 -22.0% 28.0% 3.0% Below avg 0.2 -2.0% 14.7% 6.4% Average 0.4 20.0% 0.0% 10.0% Above avg 0.2 35.0% -10.0% 12.5% Boom 0.1 50.0% -20.0% 15.0% 20% 13.4% σ Calculating portfolio standard deviation and CV 0.10 (3.0 - 9.6) 0.20 (6.4 - 9.6)2 2 p 0.40 (10.0 - 9.6) 0.20 (12.5 - 9.6)2 2 9.6) (15.0 0.10 2 3.3% 0.34 CVp 9.6% 1 2 3.3% Comments on portfolio risk measures • σp = 3.3% is much lower than the σi of either stock (σHT = 20.0%; σColl. = 13.4%). • σp = 3.3% is lower than the weighted average of HT and Coll.’s σ (16.7%). • Portfolio provides average return of component stocks, but lower than average risk. • Why? Negative correlation between stocks. Investment 1 million on A and B, fifty-tofity case A B portfolio year return Rate of return return Rate of return return Rate of return 20×6 20 40% -5 -10% 15 15% 20×7 -5 -10% 20 40% 15 15 % 20×8 17.5 35% -2.5 -5% 15 15 % 20×9 -2.5 -5% 17.5 35% 15 15 % mean 7.5 15% 7.5 15% 15 15 % σ case A B portfolio year return Rate of return return Rate of return return Rate of return 20×6 20 40% 20 40% 40 40% 20×7 -5 -10% -5 -10% -10 -10% 20×8 17.5 35% 17.5 35% 35 35% 20×9 -2.5 -5% -2.5 -5% -5 -5% mean 7.5 15 7.5 15 15 15 σ Returns Distribution for Two Perfectly Negatively Correlated Stocks (r = -1.0) and for Portfolio WM: Stock W Stock M Portfolio WM 25 25 25 15 15 15 0 0 0 -10 -10 -10 Returns Distributions for Two Perfectly Positively Correlated Stocks (r = +1.0) and for Portfolio MM: Stock MM’ Stock MM’ Stock M 25 25 25 15 15 15 0 0 0 -10 -10 -10 Portfolio Risk Correlation Coefficient, r A measure of the degree of relationship between two variables Positively correlated stocks rates of return move together in the same direction Negatively correlated stocks rates of return move in opposite directions Portfolio Risk Risk Reduction Combining stocks that are not perfectly positive correlated will reduce the portfolio risk by diversification The riskiness of a portfolio is reduced as the number of stocks in the portfolio increases The smaller the positive correlation, the lower the risk Portfolio Risk • Most stocks are positively correlated with the market (rk,m 0.65). • σ 35% for an average stock. • Combining stocks in a portfolio generally lowers risk. because they would not be perfectly correlated with the existing portfolio. • Eventually the diversification benefits of adding more stocks (after about 10 stocks), and for large stock portfolios, σp tends to converge to 20%. Firm-Specific Risk versus Market Risk Firm-Specific Risk That part of a security’s risk associated with random outcomes generated by events, or behaviors, specific to the firm It can be eliminated by proper diversification Firm-Specific Risk versus Market Risk Market Risk That part of a security’s risk that cannot be eliminated by diversification because it is associated with economic, or market factors that systematically affect most firms Firm-Specific Risk versus Market Risk Relevant Risk The risk of a security that cannot be diversified away, or its market risk This reflects a security’s contribution to the risk of a portfolio Risk Aversion and Required Returns • Risk aversion – assumes investors dislike risk and require higher rates of return to encourage them to hold riskier securities. Risk Aversion and Required Returns Risk Premium (RP) The portion of the expected return that can be attributed to the additional risk of an investment The difference between the expected rate of return on a given risky asset and that on a less risky asset, which serves as compensation for investors to hold riskier securities. Market Risk Premium RPM is the additional return over the risk-free rate needed to compensate investors for assuming an average amount of risk Risk Premium for a Stock Risk Premium for Stock j = RPj = RPM x bj The Concept of Beta Beta Coefficient, b • • Measures a stock’s market risk, and shows a stock’s volatility relative to the market. Indicates how risky a stock is if the stock is held in a well-diversified portfolio. Firm-Specific Risk versus Market Risk CAPM A model based on the proposition that any stock’s required rate of return is equal to the risk-free rate of return plus a risk premium, that reflects the riskiness of the stock after diversification. Portfolio Beta Coefficients The beta of any set of securities is the weighted average of the individual securities’ betas b p w1b1 w 2 b 2 ... w n b n N w jb j j1 The Relationship Between Risk and Rates of Return k̂ j expected rate of return on the jth stock The Relationship Between Risk and Rates of Return k̂ j expected rate of return on the j stock th k j required rate of return on the jth stock The Relationship Between Risk and Rates of Return k̂ j expected rate of return on the j stock th k j required rate of return on the jth stock k RF risk free rate of return The Relationship Between Risk and Rates of Return k̂ j expected rate of return on the j stock th k j required rate of return on the jth stock k RF risk free rate of return RPM k M - k RF market risk premium The Relationship Between Risk and Rates of Return k̂ j expected rate of return on the j stock th k j required rate of return on the j stock th k RF risk free rate of return RPM k M - k RF market risk premium RP j k M - k RF b j risk premium th on the j stock The Required Rate of Return for a Stock k j required rate of return for stock j k j k RF RPM b j k RF k M k RF b j Security Market Line (SML) The line that shows the relationship between risk as measured by beta and the required rate of return for individual securities Security Market Line SML : k j k RF k M k RF b j Required Rate of Return (%) khigh = 22 kM = kA = 14 kLOW = 10 Safe Stock Risk Premium: 4% kRF = 6 Market (Average Stock) Risk Premium: 8% Relatively Risky Stock’s Risk Premium: 16% Risk-Free Rate: 6% 0 0.5 1.0 1.5 2.0 Risk, bj The Impact of Inflation kRF is the price of money to a riskless borrower The nominal rate consists of a real (inflation-free) rate of return an inflation premium (IP) An increase in expected inflation would increase the risk-free rate Changes in Risk Aversion The slope of the SML reflects the extent to which investors are averse to risk An increase in risk aversion increases the risk premium and increases the slope Changes in a Stock’s Beta Coefficient The Beta risk of a stock is affected by composition of its assets use of debt financing increased competition expiration of patents Any change in the required return (from change in beta or in expected inflation) affects the stock price Stock Market Equilibrium The condition under which the expected return on a security is just equal to its required return Actual market price equals its intrinsic value as estimated by the marginal investor, leading to price stability Changes in Equilibrium Stock Prices Stock prices are not constant due to changes in: Risk-free rate, kRF Market risk premium, kM - kRF Stock X’s beta coefficient, bx Stock X’s expected growth rate, gX Changes in expected dividends, D0