Chapter Thirteen

1
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 distributions variance is the r.v.s av.
  squared deviation from the mean
• Variance measures the r.v.s variation.

4
Standard Deviation of a Distribution

• The distributions 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
7
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 assets rate-or-return is rf .
• Risky stocks rate-or-return is ms if state s
  occurs, with prob. ?s .
• Risky stocks 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 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

• Portfolios rate-of-return variance is

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Budget Constraints for Risky Assets

• Portfolios rate-of-return variance is

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Budget Constraints for Risky Assets

• Portfolios rate-of-return variance is

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Budget Constraints for Risky Assets

• Portfolios rate-of-return variance is

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Budget Constraints for Risky Assets

• Portfolios rate-of-return variance is

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Budget Constraints for Risky Assets

• 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
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 gt rm and a st. dev. ?y gt ?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 gt rm and a st. dev. ?y gt ?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
Budget line, slope
St. Dev. of Return, ?
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Choosing a Portfolio
Mean Return, ?
Budget line, slope
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 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.

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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
  60 and is 60 when asset As value is 20
  (perfect negative correlation of values).
• Pay up to 40 gt 30 for a 50-50 mix of assets A
  and B.

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Measuring Risk

• Asset As risk relative to risk in the whole
  stock market is measured by

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Measuring Risk

• 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

• 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

• 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 ?A?m.

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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 ?A?m.
• Price of risk is
• So cost of asset As risk is p?A?m.

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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 assets ? 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.