0. Efficient Diversification

1. Risk Premiums and Risk Aversion
Degree to which investors are unwilling to accept uncertainty
Risk aversion
If T-Bill denotes the risk-free rate, rf, and variance, σp2 , denotes volatility of the portfolio returns then:
The risk premium of a portfolio is:

2. Risk Premiums and Risk Aversion
To quantify the degree of risk aversion with parameter A:
Or:

3. The Sharpe (Reward-to-Volatility) Measure

4. ASSET ALLOCATION ACROSS RISKY AND RISK-FREE PORTFOLIOS

5. Allocating Capital
Possible to split investment funds between safe and risky assets
Risk free asset: T-bills
Risky asset: stock (or a portfolio)

6. Allocating Capital Issues Examine risk vs return tradeoff
Demonstrate how different degrees of risk aversion will affect allocations between risky and risk free assets

7. The Risky Asset: Example
Total portfolio value = $300,000
Risk-free value = $90,000
Risky (Vanguard and Fidelity) = $210,000
Vanguard (V) = 54%
Fidelity (F) = 46%

8. The Risky Asset: Example
Vanguard 113,400/300,000 =
Fidelity ,600/300,000 =
Portfolio P 210,000/300,000 =
Risk-Free Assets F 90,000/300,000 =
Portfolio C 300,000/300,000 =

9. Calculating the Expected Return: Example
rf = 7%
srf = 0%
E(rp) = 15%
sp = 22%
y = % in p
(1-y) = % in rf

10. Expected Returns for Combinations
E(rc) = yE(rp) + (1 - y)rf
rc = complete or combined portfolio
For example, y = .75
E(rc) = .75(.15) + .25(.07)
= .13 or 13%

11. Variance on the Possible Combined Portfolios
= 0, then s p c =
Since rf y s s

12. Combinations Without Leverage
= .75(.22) = .165 or 16.5%
If y = .75, then
= 1(.22) = .22 or 22%
If y = 1
= 0(.22) = .00 or 0%
If y = 0 s s s

13. Using Leverage with Capital Allocation Line
Borrow at the Risk-Free Rate and invest in stock
Using 50% Leverage
rc = (-.5) (.07) + (1.5) (.15) = .19
sc = (1.5) (.22) = .33

14. Investment Opportunity Set with a Risk-Free Investment

15. Risk Aversion and Allocation
Greater levels of risk aversion lead to larger proportions of the risk free rate
Lower levels of risk aversion lead to larger proportions of the portfolio of risky assets
Willingness to accept high levels of risk for high levels of returns would result in leveraged combinations

16. ASSET ALLOCATION WITH TWO RISKY ASSETS

17. Covariance and Correlation
Portfolio risk depends on the correlation between the returns of the assets in the portfolio
Covariance and the correlation coefficient provide a measure of the returns on two assets to vary either in tandem or in opposition

18. Two-Asset Portfolio Return: Stock and Bond

19. Covariance and Correlation Coefficient

20. Correlation Coefficients: Possible Values
Range of values for r 1,2
-1.0 < r < 1.0
If r = 1.0, the securities would be perfectly positively correlated
If r = - 1.0, the securities would be perfectly negatively correlated

21. Two-Asset Portfolio Standard Deviation: Stock and Bond

22. Two-Risky-Asset Portfolio
Rate of return on the portfolio:
Expected rate of return on the portfolio:

23. Two-Risky-Asset Portfolio
Variance of the rate of return on the portfolio:

24. Numerical Example: Bond and Stock Returns
Bond = 6% Stock = 10%
Standard Deviation
Bond = 12% Stock = 25%
Weights
Bond = .5 Stock = .5
Correlation Coefficient
(Bonds and Stock) = 0

25. Numerical Example: Bond and Stock Returns
Return = 8% .5(6) + .5 (10) Standard Deviation = 13.87% [(.5)2 (12)2 + (.5)2 (25)2 + … 2 (.5) (.5) (12) (25) (0)] ½ [192.25] ½ = 13.87

26. Investment Opportunity Set for Stocks and Bonds

27. Investment Opportunity Set for Stocks and Bonds with Various Correlations

28. THE OPTIMAL RISKY PORTFOLIO WITH A RISK-FREE ASSET

29. Extending to Include Riskless Asset
The optimal combination becomes linear
A single combination of risky and riskless assets will dominate

30. Opportunity Set Using Stocks and Bonds and Two Capital Allocation Lines

31. Dominant CAL with a Risk-Free Investment (F)
CAL(O) dominates other lines -- it has the best risk/return ratio or the largest slope
Slope =

32. Optimal Capital Allocation Line for Bonds, Stocks and T-Bills

33. The Complete Portfolio

34. The Complete Portfolio – Solution to the Asset Allocation Problem

35. EFFICIENT DIVERSIFICATION WITH MANY RISKY ASSETS

36. Extending Concepts to All Securities
The optimal combinations result in lowest level of risk for a given return
Markowitz Portfolio Theory
a single asset or portfolio of assets is considered to be efficient if no other asset or portfolio of assets offers higher expected return with the same (or lower) risk, or lower risk with the same (or higher) expected return.

37. Portfolios Constructed from Three Stocks A, B and C

38. The Efficient Frontier of Risky Assets and Individual Assets