# Hal Varian Intermediate Microeconomics Chapter Thirteen

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Hal Varian Intermediate Microeconomics Chapter Thirteen
Risky Assets

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 the av. value of the r.v.;

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.

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.

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

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

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(,). U  as mean return  . U  as risk  .

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

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

Preferences over Risky Assets
How is the MRS computed?

Preferences over Risky Assets
How is the MRS computed?

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

Budget Constraints for Risky Assets
Two assets. Risk-free asset’s rate-or-return is rf . Risky stock’s rate-or-return is ms if state s occurs, with prob. s . Risky stock’s 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 portfolio’s av. rate-of-return is

Budget Constraints 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 have So portfolio’s expected rate-of-return rises with x (more stock in the portfolio).

Budget Constraints for Risky Assets
Portfolio’s rate-of-return variance is

Budget Constraints for Risky Assets
Portfolio’s rate-of-return variance is

Budget Constraints for Risky Assets
Portfolio’s rate-of-return variance is

Budget Constraints for Risky Assets
Portfolio’s rate-of-return variance is

Budget Constraints for Risky Assets
Portfolio’s rate-of-return variance is

Budget Constraints for Risky Assets
Portfolio’s rate-of-return variance is

Budget Constraints for Risky Assets
Variance so st. deviation

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

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

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

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

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

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

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

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

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

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

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

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

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

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 Mean Return,  Budget line, slope =
St. Dev. of Return, 

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

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

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

Measuring Risk Quantitatively, how risky is an asset?
Depends upon how the asset’s value depends upon other assets’ values. E.g. Asset A’s value is \$60 with chance 1/4 and \$20 with chance 3/4. Pay at most \$30 for asset A.

Measuring Risk Asset A’s value is \$60 with chance 1/4 and \$20 with chance 3/4. Asset B’s value is \$20 when asset A’s value is \$60 and is \$60 when asset A’s value is \$20 (perfect negative correlation of values). Pay up to \$40 > \$30 for a mix of assets A and B.

Measuring Risk Asset A’s risk relative to risk in the whole stock market is measured by

Measuring Risk Asset A’s risk relative to risk in the whole stock market is measured by where is the market’s rate-of-return and is asset A’s rate-of-return.

Measuring Risk asset A’s return is not perfectly correlated with the whole market’s return and so it can be used to build a lower risk portfolio.

Equilibrium in Risky Asset Markets
At equilibrium, all assets’ 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 Am.

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 Am. Price of risk is So cost of asset A’s risk is pAm.

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, all risk adjusted rates-of-return for all assets are equal. The risk-free asset’s  = 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.