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**Unformatted text preview: **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 w 1 ,…, w S with probabilities π 1 ,..., π S ( π 1 + · · · + π S = 1). ◆ The mean (expected value) of the distribution is the average value of the r.v.; E [ ] . w w w s s s S = = = ∑ μ π 1 Variance of a Distribution ◆ The distribution’s variance is the r.v.’s av. squared deviation from the mean; ◆ Variance measures the r.v.’s variation. var [ ] ( ) . w w w s w s s S = =- = ∑ σ μ π 2 2 1 Standard Deviation of a Distribution ◆ The distribution’s standard deviation is the square root of its variance; ◆ St. deviation also measures the r.v.’s variability. st. dev [ ] ( ) . w w w w s w s s S = = =- = ∑ σ σ μ π 2 2 1 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 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( μ , σ ). ◆ U ↑ as mean return μ ↑ . ◆ U ↓ as risk σ ↑ . Preferences over Risky Assets Preferred Higher mean return is a good. Higher risk is a bad. Mean Return, μ St. Dev. of Return, σ Preferences over Risky Assets Preferred Higher mean return is a good. Higher risk is a bad. Mean Return, μ St. Dev. of Return, σ Preferences over Risky Assets ◆ How is the MRS computed? Preferences over Risky Assets ◆ How is the MRS computed? dU U d U d U d U d d d U U = + = ⇒ = - ⇒ = - ∂ ∂μ μ ∂ ∂σ σ ∂ ∂μ μ ∂ ∂σ σ μ σ ∂ ∂σ ∂ ∂μ / / . Preferences over Risky Assets Mean Return, μ St. Dev. of Return, σ Preferred Higher mean return is a good. Higher risk is a bad. d d U U μ σ ∂ ∂σ ∂ ∂μ = - / / Budget Constraints for Risky Assets ◆ Two assets....

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