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Hal Varian Intermediate Microeconomics Chapter Thirteen Risky Assets

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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.;

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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.

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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.

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Mean and Variance Probability Random Variable Values Two distributions with the same variance and different means.

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Mean and Variance Probability Random Variable Values Two distributions with the same mean and different variances.

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Preferences over Risky Assets u Higher mean return is preferred. u Less variation in return is preferred (less risk).

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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.

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Preferences over Risky Assets Preferred Higher mean return is a good. Higher risk is a bad. Mean Return, St. Dev. of Return,

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Preferences over Risky Assets Preferred Higher mean return is a good. Higher risk is a bad. Mean Return, St. Dev. of Return,

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Preferences over Risky Assets u How is the MRS computed?

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Preferences over Risky Assets u How is the MRS computed?

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Preferences over Risky Assets Mean Return, St. Dev. of Return, Preferred Higher mean return is a good. Higher risk is a bad.

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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

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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

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Budget Constraints for Risky Assets x = 0 and x = 1

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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

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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).

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Budget Constraints for Risky Assets u Portfolios rate-of-return variance is

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Budget Constraints for Risky Assets u Portfolios rate-of-return variance is

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Budget Constraints for Risky Assets u Portfolios rate-of-return variance is

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Budget Constraints for Risky Assets u Portfolios rate-of-return variance is

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Budget Constraints for Risky Assets u Portfolios rate-of-return variance is

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Budget Constraints for Risky Assets u Portfolios rate-of-return variance is

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Budget Constraints for Risky Assets Variance so st. deviation

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Budget Constraints for Risky Assets x = 0 and x = 1 Variance so st. deviation

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Budget Constraints for Risky Assets x = 0 and x = 1 Variance so st. deviation So risk rises with x (more stock in the portfolio).

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Budget Constraints for Risky Assets Mean Return, St. Dev. of Return,

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Budget Constraints for Risky Assets Mean Return, St. Dev. of Return,

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Budget Constraints for Risky Assets Mean Return, St. Dev. of Return,

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Budget Constraints for Risky Assets Budget line Mean Return, St. Dev. of Return,

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Budget Constraints for Risky Assets Budget line, slope = Mean Return, St. Dev. of Return,

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Choosing a Portfolio Budget line, slope = Mean Return, St. Dev. of Return, is the price of risk relative to mean return.

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Choosing a Portfolio Budget line, slope = Where is the most preferred return/risk combination? Mean Return, St. Dev. of Return,

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Choosing a Portfolio Budget line, slope = Where is the most preferred return/risk combination? Mean Return, St. Dev. of Return,

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Choosing a Portfolio Budget line, slope = Where is the most preferred return/risk combination? Mean Return, St. Dev. of Return,

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Choosing a Portfolio Budget line, slope = Where is the most preferred return/risk combination? Mean Return, St. Dev. of Return,

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Choosing a Portfolio Budget line, slope = Where is the most preferred return/risk combination? Mean Return, St. Dev. of Return,

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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?

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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

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Choosing a Portfolio Budget line, slope = Mean Return, St. Dev. of Return,

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Choosing a Portfolio Budget line, slope = Mean Return, St. Dev. of Return,

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Choosing a Portfolio Budget line, slope = Mean Return, St. Dev. of Return,

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Choosing a Portfolio Budget line, slope = Higher mean rate-of-return and higher risk chosen in this case. Mean Return, St. Dev. of Return,

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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.

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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 mix of assets A and B.

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Measuring Risk u Asset As risk relative to risk in the whole stock market is measured by

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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.

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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.

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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?

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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.

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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.

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Equilibrium in Risky Asset Markets u Risk adjustment for asset A is u Risk adjusted rate-of-return for asset A is

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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.

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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.

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