1. Efficient Diversification
Ch 6
Efficient Diversification

2. Diversification and Portfolio Risk
Total risk:
Market risk
Systematic or Nondiversifiable
Firm-specific risk
Diversifiable or nonsystematic or unique

3. Figure 6.1 Portfolio Risk as a Function of the Number of Stocks

4. Figure 6.2 Portfolio Risk as a Function of Number of Securities

5. Exercise 42
1. Risk that can be eliminated through diversification is called ______ risk.
A) unique
B) firm-specific
C) diversifiable
D) all of the above
2. The risk that can be diversified away is ___________.
A) beta
B) firm specific risk
C) market risk
D) systematic risk

6. Two Asset Portfolio Return – Stock and Bond

7. Covariance Cov(r1r2) = r1,2s1s2 r1,2 = Correlation coefficient of
returns
s1 = Standard deviation of
returns for Security 1
s2 = Standard deviation of
returns for Security 2

8. 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

9. Two Asset Portfolio St Dev – Stock and Bond

10. In General, For an n-Security Portfolio:
rp = Weighted average of the
n securities
sp2 = (Consider all pair-wise
covariance measures)

11. 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

12. Return and Risk for Example
.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

13. Figure 6.3 Investment Opportunity Set for Stock and Bonds

14. Minimum variance portfolio
Ws = σB2 - Cov(rS, rB) / (σs2 + σB2 -2Cov(rS, rB))

15. Figure 6.4 Investment Opportunity Set for Stock and Bonds with Various Correlations

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

17. Figure 6.5 Opportunity Set Using Stock and Bonds and Two Capital Allocation Lines

18. Dominant CAL with a Risk-Free Investment (F)
CAL(O) dominates other lines -- it has the best risk/return or the largest slope
Slope = (E(R) - Rf) / s
[ E(RP) - Rf) / s P ] > [E(RA) - Rf) / sA]
Regardless of risk preferences combinations of O & F dominate

19. Figure 6.6 Optimal Capital Allocation Line for Bonds, Stocks and T-Bills

20. Figure 6.7 The Complete Portfolio

21. Figure 6.8 The Complete Portfolio – Solution to the Asset Allocation Problem

22. Extending Concepts to All Securities
The optimal combinations result in lowest level of risk for a given return
The optimal trade-off is described as the efficient frontier
These portfolios are dominant

23. Figure 6.9 Portfolios Constructed from Three Stocks A, B and C

24. Figure 6.10 The Efficient Frontier of Risky Assets and Individual Assets

25. Exercise 22
1. Adding additional risky assets will generally move the efficient frontier _____ and to the _______.
A) up, right
B) up, left
C) down, right
D) down, left
2. Rational risk-averse investors will always prefer portfolios ______________.
A) located on the efficient frontier to those located on the capital market line
B) located on the capital market line to those located on the efficient frontier
C) at or near the minimum variance point on the efficient frontier
D) Rational risk-averse investors prefer the risk-free asset to all other asset choices.

26. Exercise33
1. The standard deviation of return on investment A is .10 while the standard deviation of return on investment B is If the covariance of returns on A and B is .0030, the correlation coefficient between the returns on A and B is __________.
A) .12
B) .36
C) .60
D) .77
2. Consider two perfectly negatively correlated risky securities, A and B. Security A has an expected rate of return of 16% and a standard deviation of return of 20%. B has an expected rate of return 10% and a standard deviation of return of 30%. The weight of security B in the global minimum variance is __________.
A) 10%
B) 20%
C) 40%
D) 60%

27. Exercise32
1. Which of the following correlations coefficients will produce the least diversification benefit?
A) -0.6
B) -1.5
C) 0.0
D) 0.8
2. The expected return of portfolio is 8.9% and the risk free rate is 3.5%. If the portfolio standard deviation is 12.0%, what is the reward to variability ratio of the portfolio?
A) 0.0
B) 0.45
C) 0.74
D) 1.35

28. Single Factor Model ri = E(Ri) + ßiF + e
ßi = index of a securities’ particular return to the factor
F= some macro factor; in this case F is unanticipated movement; F is commonly related to security returns
Assumption: a broad market index like the S&P500 is the common factor

29. ( ) ( ) b a r r r r e Single Index Model - = + - + Risk Premf i m f i i
Risk Prem
Market Risk Prem
or Index Risk Prem a
= the stock’s expected return if the
market’s excess return is zero i
(rm - rf) = 0
ßi(rm - rf) = the component of return due to
movements in the market index
ei = firm specific component, not due to market
movements

30. Risk Premium Format Let: Ri = (ri - rf) Risk premium format
Rm = (rm - rf)
Risk premium
format
Ri = ai + ßi(Rm) + ei

31. Figure 6.11 Scatter Diagram for Dell

32. Figure 6.12 Various Scatter Diagrams

33. Components of Risk
Market or systematic risk: risk related to the macro economic factor or market index
Unsystematic or firm specific risk: risk not related to the macro factor or market index
Total risk = Systematic + Unsystematic

34. Measuring Components of Risk
si2 = bi2 sm2 + s2(ei)
where;
si2 = total variance
bi2 sm2 = systematic variance
s2(ei) = unsystematic variance

35. Examining Percentage of Variance
Total Risk = Systematic Risk + Unsystematic Risk
Systematic Risk/Total Risk = r2
ßi2 s m2 / s2 = r2
bi2 sm2 / (bi2 sm2 + s2(ei)) = r2