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**Hal Varian Intermediate Microeconomics Chapter Thirteen**

Risky Assets

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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.;

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**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.

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**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.

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**Mean and Variance Two distributions with the same**

variance and different means. Probability Random Variable Values

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**Mean and Variance Two distributions with the same**

mean and different variances. Probability Random Variable Values

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**Preferences over Risky Assets**

Higher mean return is preferred. Less variation in return is preferred (less risk).

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**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 .

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**Preferences over Risky Assets**

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

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**Preferences over Risky Assets**

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

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**Preferences over Risky Assets**

How is the MRS computed?

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**Preferences over Risky Assets**

How is the MRS computed?

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**Preferences over Risky Assets**

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

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**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

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**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

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

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**Budget Constraints for Risky Assets**

Portfolio’s rate-of-return variance is

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**Budget Constraints for Risky Assets**

Portfolio’s rate-of-return variance is

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**Budget Constraints for Risky Assets**

Portfolio’s rate-of-return variance is

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**Budget Constraints for Risky Assets**

Portfolio’s rate-of-return variance is

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**Budget Constraints for Risky Assets**

Portfolio’s rate-of-return variance is

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**Budget Constraints for Risky Assets**

Portfolio’s 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**

Variance so st. deviation x = 0 and x = 1

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**Budget Constraints for Risky Assets**

Variance so st. deviation x = 0 and x = 1 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**

Mean Return, Budget line St. Dev. of Return,

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**Budget Constraints for Risky Assets**

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

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**Choosing a Portfolio Mean Return, Budget line, slope =**

is the price of risk relative to mean return. St. Dev. of Return,

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**Choosing a Portfolio Mean Return, Where is the most preferred**

return/risk combination? Budget line, slope = St. Dev. of Return,

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**Choosing a Portfolio Mean Return, Where is the most preferred**

return/risk combination? Budget line, slope = St. Dev. of Return,

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**Choosing a Portfolio Mean Return, Where is the most preferred**

return/risk combination? Budget line, slope = St. Dev. of Return,

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**Choosing a Portfolio Mean Return, Where is the most preferred**

return/risk combination? Budget line, slope = St. Dev. of Return,

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**Choosing a Portfolio Mean Return, Where is the most preferred**

return/risk combination? Budget line, slope = St. Dev. of Return,

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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?

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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

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**Choosing a Portfolio Mean Return, Budget line, slope =**

St. Dev. of Return,

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**Choosing a Portfolio Mean Return, Budget line, slope =**

St. Dev. of Return,

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**Choosing a Portfolio Mean Return, Budget line, slope =**

St. Dev. of Return,

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**Choosing a Portfolio Mean Return, Budget line, slope =**

Higher mean rate-of-return and higher risk chosen in this case. St. Dev. of Return,

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**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.

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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.

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

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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.

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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.

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**Equilibrium in Risky Asset Markets**

At equilibrium, all assets’ risk-adjusted rates-of-return must be equal. How do we adjust for riskiness?

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**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.

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**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. Price of risk is So cost of asset A’s risk is pAm.

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**Equilibrium in Risky Asset Markets**

Risk adjustment for asset A is Risk adjusted rate-of-return for asset A is

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**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.

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**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.

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© 2010 W. W. Norton & Company, Inc. 12 Uncertainty.

© 2010 W. W. Norton & Company, Inc. 12 Uncertainty.

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