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Hal Varian Intermediate Microeconomics Chapter Thirteen Risky Assets Slide 2 Mean of a Distribution u A random variable (r.v.) w takes values w 1,,w S with probabilities 1,..., S ( 1 + + S = 1). u The mean (expected value) of the distribution is the av. value of the r.v.; Slide 3 Variance of a Distribution u The distributions variance is the r.v.s av. squared deviation from the mean; u Variance measures the r.v.s variation. Slide 4 Standard Deviation of a Distribution u The distributions standard deviation is the square root of its variance; u St. deviation also measures the r.v.s variability. Slide 5 Mean and Variance Probability Random Variable Values Two distributions with the same variance and different means. Slide 6 Mean and Variance Probability Random Variable Values Two distributions with the same mean and different variances. Slide 7 Preferences over Risky Assets u Higher mean return is preferred. u Less variation in return is preferred (less risk). Slide 8 Preferences over Risky Assets u Higher mean return is preferred. u Less variation in return is preferred (less risk). u Preferences are represented by a utility function U(, ). u U as mean return. u U as risk. Slide 9 Preferences over Risky Assets Preferred Higher mean return is a good. Higher risk is a bad. Mean Return, St. Dev. of Return, Slide 10 Preferences over Risky Assets Preferred Higher mean return is a good. Higher risk is a bad. Mean Return, St. Dev. of Return, Slide 11 Preferences over Risky Assets u How is the MRS computed? Slide 12 Preferences over Risky Assets u How is the MRS computed? Slide 13 Preferences over Risky Assets Mean Return, St. Dev. of Return, Preferred Higher mean return is a good. Higher risk is a bad. Slide 14 Budget Constraints for Risky Assets u Two assets. u Risk-free assets rate-or-return is r f. u Risky stocks rate-or-return is m s if state s occurs, with prob. s. u Risky stocks mean rate-of-return is Slide 15 Budget Constraints for Risky Assets u A bundle containing some of the risky stock and some of the risk-free asset is a portfolio. u x is the fraction of wealth used to buy the risky stock. u Given x, the portfolios av. rate-of- return is Slide 16 Budget Constraints for Risky Assets x = 0 and x = 1 Slide 17 Budget Constraints for Risky Assets x = 0 and x = 1 Since stock is risky and risk is a bad, for stock to be purchased must have Slide 18 Budget Constraints for Risky Assets x = 0 and x = 1 Since stock is risky and risk is a bad, for stock to be purchased must have So portfolios expected rate-of-return rises with x (more stock in the portfolio). Slide 19 Budget Constraints for Risky Assets u Portfolios rate-of-return variance is Slide 20 Budget Constraints for Risky Assets u Portfolios rate-of-return variance is Slide 21 Budget Constraints for Risky Assets u Portfolios rate-of-return variance is Slide 22 Budget Constraints for Risky Assets u Portfolios rate-of-return variance is Slide 23 Budget Constraints for Risky Assets u Portfolios rate-of-return variance is Slide 24 Budget Constraints for Risky Assets u Portfolios rate-of-return variance is Slide 25 Budget Constraints for Risky Assets Variance so st. deviation Slide 26 Budget Constraints for Risky Assets x = 0 and x = 1 Variance so st. deviation Slide 27 Budget Constraints for Risky Assets x = 0 and x = 1 Variance so st. deviation So risk rises with x (more stock in the portfolio). Slide 28 Budget Constraints for Risky Assets Mean Return, St. Dev. of Return, Slide 29 Budget Constraints for Risky Assets Mean Return, St. Dev. of Return, Slide 30 Budget Constraints for Risky Assets Mean Return, St. Dev. of Return, Slide 31 Budget Constraints for Risky Assets Budget line Mean Return, St. Dev. of Return, Slide 32 Budget Constraints for Risky Assets Budget line, slope = Mean Return, St. Dev. of Return, Slide 33 Choosing a Portfolio Budget line, slope = Mean Return, St. Dev. of Return, is the price of risk relative to mean return. Slide 34 Choosing a Portfolio Budget line, slope = Where is the most preferred return/risk combination? Mean Return, St. Dev. of Return, Slide 35 Choosing a Portfolio Budget line, slope = Where is the most preferred return/risk combination? Mean Return, St. Dev. of Return, Slide 36 Choosing a Portfolio Budget line, slope = Where is the most preferred return/risk combination? Mean Return, St. Dev. of Return, Slide 37 Choosing a Portfolio Budget line, slope = Where is the most preferred return/risk combination? Mean Return, St. Dev. of Return, Slide 38 Choosing a Portfolio Budget line, slope = Where is the most preferred return/risk combination? Mean Return, St. Dev. of Return, Slide 39 Choosing a Portfolio u Suppose a new risky asset appears, with a mean rate-of-return r y > r m and a st. dev. y > m. u Which asset is preferred? Slide 40 Choosing a Portfolio u Suppose a new risky asset appears, with a mean rate-of-return r y > r m and a st. dev. y > m. u Which asset is preferred? u Suppose Slide 41 Choosing a Portfolio Budget line, slope = Mean Return, St. Dev. of Return, Slide 42 Choosing a Portfolio Budget line, slope = Mean Return, St. Dev. of Return, Slide 43 Choosing a Portfolio Budget line, slope = Mean Return, St. Dev. of Return, Slide 44 Choosing a Portfolio Budget line, slope = Higher mean rate-of-return and higher risk chosen in this case. Mean Return, St. Dev. of Return, Slide 45 Measuring Risk u Quantitatively, how risky is an asset? u Depends upon how the assets value depends upon other assets values. u E.g. Asset As value is $60 with chance 1/4 and $20 with chance 3/4. u Pay at most $30 for asset A. Slide 46 Measuring Risk u Asset As value is $60 with chance 1/4 and $20 with chance 3/4. u Asset Bs value is $20 when asset As value is $60 and is $60 when asset As value is $20 (perfect negative correlation of values). u Pay up to $40 > $30 for a 50-50 mix of assets A and B. Slide 47 Measuring Risk u Asset As risk relative to risk in the whole stock market is measured by Slide 48 Measuring Risk u Asset As risk relative to risk in the whole stock market is measured by where is the markets rate-of-return and is asset As rate-of-return. Slide 49 Measuring Risk u asset As return is not perfectly correlated with the whole markets return and so it can be used to build a lower risk portfolio. Slide 50 Equilibrium in Risky Asset Markets u At equilibrium, all assets risk- adjusted rates-of-return must be equal. u How do we adjust for riskiness? Slide 51 Equilibrium in Risky Asset Markets u Riskiness of asset A relative to total market risk is A. u Total market risk is m. u So total riskiness of asset A is A m. Slide 52 Equilibrium in Risky Asset Markets u Riskiness of asset A relative to total market risk is A. u Total market risk is m. u So total riskiness of asset A is A m. u Price of risk is u So cost of asset As risk is p A m. Slide 53 Equilibrium in Risky Asset Markets u Risk adjustment for asset A is u Risk adjusted rate-of-return for asset A is Slide 54 Equilibrium in Risky Asset Markets u At equilibrium, all risk adjusted rates- of-return for all assets are equal. u The risk-free assets = 0 so its adjusted rate-of-return is just u Hence, for every risky asset A. Slide 55 Equilibrium in Risky Asset Markets u That at equilibrium in asset markets is the main result of the Capital Asset Pricing Model (CAPM), a model used extensively to study financial markets.