1. Efficient Diversification
Chapter 7
Efficient Diversification

2. Two-Security Portfolio: Return
rp = W1r1 + W2r2
W1 = Proportion of funds in Security 1
W2 = Proportion of funds in Security 2
r1 = Expected return on Security 1
r2 = Expected return on Security 2 W i S =1 n = 1

3. Two-Security Portfolio: Risk
sp2 = w12s12 + w22s22 + 2W1W2 Cov(r1r2)
s12 = Variance of Security 1
s22 = Variance of Security 2
Cov(r1r2) = Covariance of returns for
Security 1 and Security 2

4. 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

5. 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

6. Three-Security Portfolio
rp = W1r1 + W2r2 + W3r3
s2p = W12s12
+ W22s22
+ W32s32
+ 2W1W2
Cov(r1r2)
+ 2W1W3
Cov(r1r3)
+ 2W2W3
Cov(r2r3)

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

8. Two-Security Portfolio
E(rp) = W1r1 + W2r2
sp2 = w12s12 + w22s22 + 2W1W2 Cov(r1r2)
sp = [w12s12 + w22s22 + 2W1W2 Cov(r1r2)]1/2

9. TWO-SECURITY PORTFOLIOS WITH DIFFERENT CORRELATIONS
13% 8%
12%
20%
St. Dev
TWO-SECURITY PORTFOLIOS WITH DIFFERENT CORRELATIONS
r = 0

10. Portfolio Risk/Return Two Securities: Correlation Effects
Relationship depends on correlation coefficient
-1.0 < r < +1.0
The smaller the correlation, the greater the risk reduction potential
If r = +1.0, no risk reduction is possible

11. Minimum Variance Combination2
E(r2) = .14
= .20
Sec 2 12
= .2
E(r1) = .10
= .15
Sec 1 s r 1
s 2
- Cov(r1r2) 2 = W1
s 2
s 2
- 2Cov(r1r2) + 2 1 W2
= (1 - W1)

12. Minimum Variance Combination: r = .2
(.2)2 - (.2)(.15)(.2) = W1
(.15)2 + (.2)2 - 2(.2)(.15)(.2) W1
= .6733 W2
= ( ) = .3267

13. Minimum Variance: Return and Risk with r = .2
rp = .6733(.10) (.14) = .1131 s
= [(.6733)2(.15)2 + (.3267)2(.2)2 + p
1/2
2(.6733)(.3267)(.2)(.15)(.2)]
1/2 s
= [.0171]
= .1308 p

14. Minimum Variance Combination: r = -.3
(.2)2 - (.2)(.15)(.2) = W1
(.15)2 + (.2)2 - 2(.2)(.15)(-.3) W1
= .6087 W2
= ( ) = .3913

15. Minimum Variance: Return and Risk with r = -.3
rp = .6087(.10) (.14) = .1157 s
= [(.6087)2(.15)2 + (.3913)2(.2)2 + p
1/2
2(.6087)(.3913)(.2)(.15)(-.3)]
1/2 s
= [.0102]
= .1009 p

16. 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

17. The minimum-variance frontier of risky assets
Efficient
frontier
Individual
assets
Global
minimum
variance
portfolio
Minimum
variance
frontier
St. Dev.

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

19. ALTERNATIVE CALS s CAL (P) CAL (A) E(r) M M P P CAL (Global
minimum variance) A A G F s P
P&F M
A&F

20. Dominant CAL with a Risk-Free Investment (F)
CAL(P) 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 P & F dominate

21. 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

22. Single Index Model 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

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

24. Estimating the Index Model
Excess Returns (i) . . . . . . . .
Security
Characteristic
Line . . . . . . . . . . . . . . . . . . .
Excess returns
on market index . . . . . . . . . . . . . . . . . . . . . . .
Ri = a i + ßiRm + ei

25. 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

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

27. 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

28. Advantages of the Single Index Model
Reduces the number of inputs for diversification
Easier for security analysts to specialize