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![Mean of a Distribution
A random variable (r.v.) w takes
values w1,…,wS with probabilities
π 1,...,π S (π 1 + · · · + π S = 1).
The mean (expected value) of the
distribution is the av. value of the
r.v.;
S
E[ w] = µ w = ∑ wsπ s .
s =1](https://image.slidesharecdn.com/ch13-140131134437-phpapp01/75/Ch13-2-2048.jpg)
![Variance of a Distribution
The distribution’s variance is the
r.v.’s av. squared deviation from the
mean;
S
2
2
var[ w] = σ w = ∑ ( ws − µ w ) π s .
s =1
Variance measures the r.v.’s
variation.](https://image.slidesharecdn.com/ch13-140131134437-phpapp01/75/Ch13-3-2048.jpg)
![Standard Deviation of a
Distribution
The distribution’s standard deviation
is the square root of its variance;
2
st. dev[ w] = σ w = σ w =
S
2
∑ ( ws − µ w ) π s .
s =1
St. deviation also measures the r.v.’s
variability.](https://image.slidesharecdn.com/ch13-140131134437-phpapp01/75/Ch13-4-2048.jpg)
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Ch13
- 1.
- 2. Mean of aDistribution A random variable (r.v.) w takes values w1,…,wS with probabilities π 1,...,π S (π 1 + · · · + π S = 1). The mean (expected value) of the distribution is the av. value of the r.v.; S E[ w] = µ w = ∑ wsπ s . s =1
- 3. Variance of aDistribution The distribution’s variance is the r.v.’s av. squared deviation from the mean; S 2 2 var[ w] = σ w = ∑ ( ws − µ w ) π s . s =1 Variance measures the r.v.’s variation.
- 4. Standard Deviation ofa Distribution The distribution’s standard deviation is the square root of its variance; 2 st. dev[ w] = σ w = σ w = S 2 ∑ ( ws − µ w ) π s . s =1 St. deviation also measures the r.v.’s variability.
- 5. Mean and Variance Probability Twodistributions with the same variance and different means. Random Variable Values
- 6. Mean and Variance Probability Twodistributions with the same mean and different variances. Random Variable Values
- 7. Preferences over RiskyAssets Higher mean return is preferred. Less variation in return is preferred (less risk).
- 8. Preferences over RiskyAssets 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 ↑.
- 9. Preferences over RiskyAssets Mean Return, µ Preferred Higher mean return is a good. Higher risk is a bad. St. Dev. of Return, σ
- 10. Preferences over RiskyAssets Mean Return, µ Preferred Higher mean return is a good. Higher risk is a bad. St. Dev. of Return, σ
- 11. Preferences over RiskyAssets How is the MRS computed?
- 12. Preferences over RiskyAssets How is the MRS computed? ∂U ∂U dU = dµ + dσ = 0 ∂µ ∂σ ∂U ∂U ⇒ dµ = − dσ ∂µ ∂σ dµ ∂U / ∂σ ⇒ =− . dσ ∂U / ∂µ
- 13. Preferences over RiskyAssets Mean Return, µ Preferred Higher mean return is a good. Higher risk is a bad. dµ ∂U / ∂σ =− dσ ∂U / ∂µ St. Dev. of Return, σ
- 14. Budget Constraints forRisky 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. π s . Risky stock’s mean rate-of-return is S rm = ∑ msπ s . s =1
- 15. Budget Constraints forRisky 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-ofreturn is r = xr + (1 − x )r . x m f
- 16. Budget Constraints forRisky Assets rx = xrm + (1 − x )r f . x=0⇒ rx = r f and x = 1 ⇒ rx = rm .
- 17. Budget Constraints forRisky Assets rx = xrm + (1 − x )r f . x=0⇒ rx = r f and x = 1 ⇒ rx = rm . Since stock is risky and risk is a bad, for stock to be purchased must have rm > r f .
- 18. Budget Constraints forRisky Assets rx = xrm + (1 − x )r f . x=0⇒ rx = r f and x = 1 ⇒ rx = rm . Since stock is risky and risk is a bad, for stock to be purchased must have rm > r f . So portfolio’s expected rate-of-return rises with x (more stock in the portfolio).
- 19. Budget Constraints forRisky Assets Portfolio’s rate-of-return variance is S σ 2 = ∑ ( xms + (1 − x )r f − rx ) 2 π s . x s =1
- 20. Budget Constraints forRisky Assets Portfolio’s rate-of-return variance is S σ 2 = ∑ ( xms + (1 − x )r f − rx ) 2 π s . x s =1 rx = xrm + (1 − x )r f .
- 21. Budget Constraints forRisky Assets Portfolio’s rate-of-return variance is S σ 2 = ∑ ( xms + (1 − x )r f − rx ) 2 π s . x s =1 rx = xrm + (1 − x )r f . S 2 2 σ x = ∑ ( xms + (1 − x )r f − xrm − (1 − x )r f ) π s s =1
- 22. Budget Constraints forRisky Assets Portfolio’s rate-of-return variance is S σ 2 = ∑ ( xms + (1 − x )r f − rx ) 2 π s . x s =1 rx = xrm + (1 − x )r f . S 2 2 σ x = ∑ ( xms + (1 − x )r f − xrm − (1 − x )r f ) π s s =1 S 2 = ∑ ( xms − xrm ) π s s =1
- 23. Budget Constraints forRisky Assets Portfolio’s rate-of-return variance is S σ 2 = ∑ ( xms + (1 − x )r f − rx ) 2 π s . x s =1 rx = xrm + (1 − x )r f . S 2 2 σ x = ∑ ( xms + (1 − x )r f − xrm − (1 − x )r f ) π s s =1 S 2 2 S 2 = ∑ ( xms − xrm ) π s = x ∑ (ms − rm ) π s s =1 s =1
- 24. Budget Constraints forRisky Assets Portfolio’s rate-of-return variance is S σ 2 = ∑ ( xms + (1 − x )r f − rx ) 2 π s . x s =1 rx = xrm + (1 − x )r f . S 2 2 σ x = ∑ ( xms + (1 − x )r f − xrm − (1 − x )r f ) π s s =1 S 2 2 S 2 2 2 = ∑ ( xms − xrm ) π s = x ∑ (ms − rm ) π s = x σ m . s =1 s =1
- 25. Budget Constraints forRisky Assets Variance so st. deviation 2 σ 2 = x 2σ m x σ x = xσ m .
- 26. Budget Constraints forRisky Assets Variance so st. deviation x=0⇒ 2 σ 2 = x 2σ m x σ x = xσ m . σx =0 and x = 1 ⇒ σ x = σ m.
- 27. Budget Constraints forRisky Assets Variance so st. deviation x=0⇒ 2 σ 2 = x 2σ m x σ x = xσ m . σx =0 and x = 1 ⇒ σ x = σ m. So risk rises with x (more stock in the portfolio).
- 28. Budget Constraints forRisky Assets Mean Return, µ St. Dev. of Return, σ
- 29. Budget Constraints forRisky Assets Mean Return, µ rf rx = xrm + (1 − x )r f . σ x = xσ m . x = 0 ⇒ rx = r f ,σ x = 0 0 St. Dev. of Return, σ
- 30. Budget Constraints forRisky Assets rx = xrm + (1 − x )r f . σ x = xσ m . Mean Return, µ x = 1 ⇒ rx = rm ,σ x = σ m rm rf x = 0 ⇒ rx = r f ,σ x = 0 0 σm St. Dev. of Return, σ
- 31. Budget Constraints forRisky Assets rx = xrm + (1 − x )r f . σ x = xσ m . Mean Return, µ x = 1 ⇒ rx = rm ,σ x = σ m rm Budget line rf x = 0 ⇒ rx = r f ,σ x = 0 0 σm St. Dev. of Return, σ
- 32. Budget Constraints forRisky Assets rx = xrm + (1 − x )r f . σ x = xσ m . Mean Return, µ x = 1 ⇒ rx = rm ,σ x = σ m rm Budget line, slope = rf rm − r f σm x = 0 ⇒ rx = r f ,σ x = 0 0 σm St. Dev. of Return, σ
- 33. Choosing a Portfolio MeanReturn, µ rm Budget line, slope = rm − r f σm is the price of risk relative to mean return. rf 0 σm St. Dev. of Return, σ
- 34. Choosing a Portfolio MeanReturn, µ Where is the most preferred return/risk combination? rm Budget line, slope = rm − r f σm rf 0 σm St. Dev. of Return, σ
- 35. Choosing a Portfolio MeanReturn, µ Where is the most preferred return/risk combination? rm Budget line, slope = rm − r f σm rf 0 σm St. Dev. of Return, σ
- 36. Choosing a Portfolio MeanReturn, µ Where is the most preferred return/risk combination? rm Budget line, slope = rx rm − r f σm rf 0 σx σm St. Dev. of Return, σ
- 37. Choosing a Portfolio MeanReturn, µ Where is the most preferred return/risk combination? rm Budget line, slope = rx rm − r f σm rf 0 σx σm St. Dev. of Return, σ = MRS
- 38. Choosing a Portfolio MeanReturn, µ Where is the most preferred return/risk combination? rm Budget line, slope = rx rm − r f σm rf 0 σx σm St. Dev. of Return, σ ∂U / ∂σ =− ∂U / ∂µ
- 39. Choosing a Portfolio Supposea new risky asset appears, with a mean rate-of-return ry > rm and a st. dev. σ y > σ m. Which asset is preferred?
- 40. Choosing a Portfolio Supposea new risky asset appears, with a mean rate-of-return ry > rm and a st. dev. σ y > σ m. Which asset is preferred? ry − r f rm − r f Suppose > . σy σm
- 41. Choosing a Portfolio MeanReturn, µ rm Budget line, slope = rx rm − r f rf 0 σx σm St. Dev. of Return, σ σm
- 42. Choosing a Portfolio MeanReturn, µ ry rm Budget line, slope = rx rm − r f rf 0 σx σ mσ y St. Dev. of Return, σ σm
- 43. Choosing a Portfolio MeanReturn, µ ry Budget line, slope = rm Budget line, slope = rx ry − r f σy rm − r f rf 0 σx σ mσ y St. Dev. of Return, σ σm
- 44. Choosing a Portfolio MeanReturn, µ ry Budget line, slope = rm Budget line, slope = rx ry − r f σy rm − r f σm Higher mean rate-of-return and higher risk chosen in this case. rf 0 σx σ mσ y St. Dev. of Return, σ
- 45. Measuring Risk Quantitatively, howrisky 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.
- 46. Measuring Risk Asset A’svalue 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 50-50 mix of assets A and B.
- 47. Measuring Risk Asset A’srisk relative to risk in the whole stock market is measured by risk of asset A βA = . risk of whole market
- 48. Measuring Risk Asset A’srisk relative to risk in the whole stock market is measured by risk of asset A βA = . risk of whole market covariance( rA , rm ) βA = variance( rm ) where rm is the market’s rate-of-return and rA is asset A’s rate-of-return.
- 49. Measuring Risk −1 ≤β A ≤ +1. β A < +1 ⇒ 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.
- 50. Equilibrium in RiskyAsset Markets At equilibrium, all assets’ riskadjusted rates-of-return must be equal. How do we adjust for riskiness?
- 51. Equilibrium in RiskyAsset Markets Riskiness of asset A relative to total market risk is β A. Total market risk is σ m. So total riskiness of asset A is β Aσ m.
- 52. Equilibrium in RiskyAsset Markets Riskiness of asset A relative to total market risk is β A. Total market risk is σ m. So total riskiness of asset A is β Aσ m. Price of risk is p= rm − r f σm . So cost of asset A’s risk is pβ Aσ m.
- 53. Equilibrium in RiskyAsset Markets Risk adjustment for asset A is pβ A σ m = rm − r f σm β Aσ m = β A (rm − r f ). Risk adjusted rate-of-return for asset A is rA − β A (rm − r f ).
- 54. Equilibrium in RiskyAsset Markets At equilibrium, all risk adjusted rates-of-return for all assets are equal. rf The risk-free asset’s β = 0 so its . adjusted rate-of-return is just r f = rA − β A (rm − r f ) Hence, i.e. rA = r f + β A (rm − r f ) for every risky asset A.
- 55. Equilibrium in RiskyAsset Markets That rA = r f + β A (rm − r f ) at equilibrium in asset markets is the main result of the Capital Asset Pricing Model (CAPM), a model used extensively to study financial markets.