1、Chapter Thirteen,Risky Assets,Main Issue,Mean-Variance Utility Budget Constraints for Risky Assets Measuring Risk Capital Asset Pricing Model,Mean of a Distribution,A random variable (r.v.) w takes values w1,wS with probabilities 1,.,S (1 + + S = 1). The mean (expected value) of the distribution is
2、the average value of the r.v.;,Variance of a Distribution,The distributions variance is the r.v.s av. squared deviation from the mean;Variance measures the r.v.s variation.,Standard Deviation of a Distribution,The distributions standard deviation is the square root of its variance;St. deviation also
3、 measures the r.v.s variability.,Mean and Variance,Probability,Random Variable Values,Two distributions with the same variance and different means.,Mean and Variance,Probability,Random Variable Values,Two distributions with the same mean and different variances.,Preferences over Risky Assets,Higher
4、mean return is preferred. Less variation in return is preferred (less risk).,Preferences over Risky Assets,Higher mean return is preferred. Less variation in return is preferred (less risk). Preferences are represented by a utility function U(,). U as mean return . U as risk .,Preferences over Risky
5、 Assets,Preferred,Higher mean return is a good. Higher risk is a bad.,Mean Return, ,St. Dev. of Return, ,Preferences over Risky Assets,Preferred,Higher mean return is a good. Higher risk is a bad.,Mean Return, ,St. Dev. of Return, ,Preferences over Risky Assets,How is the MRS computed?,Preferences o
6、ver Risky Assets,How is the MRS computed?,Preferences over Risky Assets,Mean Return, ,St. Dev. of Return, ,Preferred,Higher mean return is a good. Higher risk is a bad.,Budget Constraints for Risky Assets,Two assets. Risk-free assets rate-or-return is rf . Risky stocks rate-or-return is ms if state
7、s occurs, with prob. s . Risky stocks mean rate-of-return is,Budget Constraints for Risky Assets,A bundle containing some of the risky stock and some of the risk-free asset is a portfolio. x is the fraction of wealth used to buy the risky stock. Given x, the portfolios av. rate-of-return is,Budget C
8、onstraints for Risky Assets,x = 0 ,and x = 1 ,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,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
9、 have,So portfolios expected rate-of-return rises with x (more stock in the portfolio).,Budget Constraints for Risky Assets,Portfolios rate-of-return variance is,Budget Constraints for Risky Assets,Portfolios rate-of-return variance is,Budget Constraints for Risky Assets,Portfolios rate-of-return va
10、riance is,Budget Constraints for Risky Assets,Portfolios rate-of-return variance is,Budget Constraints for Risky Assets,Portfolios rate-of-return variance is,Budget Constraints for Risky Assets,Portfolios rate-of-return variance is,Budget Constraints for Risky Assets,Variance,so st. deviation,Budget
11、 Constraints for Risky Assets,x = 0 ,and x = 1 ,Variance,so st. deviation,Budget Constraints for Risky Assets,x = 0 ,and x = 1 ,Variance,so st. deviation,So risk rises with x (more stock in the portfolio).,Budget Constraints for Risky Assets,Mean Return, ,St. Dev. of Return, ,Budget Constraints for
12、Risky Assets,Mean Return, ,St. Dev. of Return, ,Budget Constraints for Risky Assets,Mean Return, ,St. Dev. of Return, ,Budget Constraints for Risky Assets,Budget line,Mean Return, ,St. Dev. of Return, ,Budget Constraints for Risky Assets,Budget line, slope =,Mean Return, ,St. Dev. of Return, ,Choosi
13、ng a Portfolio,Budget line, slope =,Mean Return, ,St. Dev. of Return, ,is the price of risk relative to mean return.,Choosing a Portfolio,Budget line, slope =,Where is the most preferred return/risk combination?,Mean Return, ,St. Dev. of Return, ,Choosing a Portfolio,Budget line, slope =,Where is th
14、e most preferred return/risk combination?,Mean Return, ,St. Dev. of Return, ,Choosing a Portfolio,Budget line, slope =,Where is the most preferred return/risk combination?,Mean Return, ,St. Dev. of Return, ,Choosing a Portfolio,Budget line, slope =,Where is the most preferred return/risk combination
15、?,Mean Return, ,St. Dev. of Return, ,Choosing a Portfolio,Budget line, slope =,Where is the most preferred return/risk combination?,Mean Return, ,St. Dev. of Return, ,Choosing a Portfolio,Suppose a new risky asset appears, with a mean rate-of-return ry rm and a st. dev. y m.Which asset is preferred?
16、,Choosing a Portfolio,Suppose a new risky asset appears, with a mean rate-of-return ry rm and a st. dev. y m.Which asset is preferred? Suppose,Choosing a Portfolio,Budget line, slope =,Mean Return, ,St. Dev. of Return, ,Choosing a Portfolio,Budget line, slope =,Mean Return, ,St. Dev. of Return, ,Cho
17、osing a Portfolio,Budget line, slope =,Budget line, slope =,Mean Return, ,St. Dev. of Return, ,Choosing a Portfolio,Budget line, slope =,Budget line, slope =,Higher mean rate-of-return and higher risk chosen in this case.,Mean Return, ,St. Dev. of Return, ,Measuring Risk,Quantitatively, how risky is
18、 an asset? Depends upon how the assets value depends upon other assets values. E.g. Asset As value is $60 with chance 1/4 and $20 with chance 3/4. Pay at most $30 for asset A.,Measuring Risk,Asset As value is $60 with chance 1/4 and $20 with chance 3/4. Asset Bs value is $20 when asset As value is $
19、60 and is $60 when asset As value is $20 (perfect negative correlation of values). Pay up to $40 $30 for a 50-50 mix of assets A and B.,Budget Constraints for Risky Assets,Asset As risk relative to risk in the whole stock market is measured by,Measuring Risk,Asset As risk relative to risk in the who
20、le stock market is measured by,where is the markets rate-of-return and is asset As rate-of-return.,Measuring Risk,asset As return is not perfectly correlated with the whole markets return and so it can be used to build a lower risk portfolio.,Equilibrium in Risky Asset Markets,At equilibrium, all as
21、sets risk-adjusted rates-of-return must be equal. How do we adjust for riskiness?,Equilibrium in Risky Asset Markets,Riskiness of asset A relative to total market risk is A. Total market risk is m. So total riskiness of asset A is Am.,Equilibrium in Risky Asset Markets,Riskiness of asset A relative
22、to total market risk is A. Total market risk is m. So total riskiness of asset A is Am. Price of risk isSo cost of asset As risk is pAm.,Equilibrium in Risky Asset Markets,Risk adjustment for asset A is Risk adjusted rate-of-return for asset A is,Equilibrium in Risky Asset Markets,At equilibrium, al
23、l risk adjusted rates-of-return for all assets are equal. The risk-free assets = 0 so its adjusted rate-of-return is just Hence, for every risky asset A.,Equilibrium in Risky Asset Markets,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.,
