1. Efficient Diversification
CHAPTER 6

2. Diversification and Portfolio Risk
Suppose your portfolio has only 1 stock, how many sources of risk can affect your portfolio?
Uncertainty at the market level
Uncertainty at the firm level
Market risk
Systematic or Nondiversifiable
Firm-specific risk
Diversifiable or nonsystematic
If your portfolio is not diversified, the total risk of portfolio will have both market risk and specific risk
If it is diversified, the total risk has only market risk

3. Diversification and Portfolio Risk

4. Diversification and Portfolio Risk

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

6. Covariance and Correlation
Why the std (total risk) decreases when more stocks are added to the portfolio?
The std of a portfolio depends on both standard deviation of each stock in the portfolio and the correlation between them
Example: return distribution of stock and bond, and a portfolio consists of 60% stock and 40% bond
state Prob. stock (%) Bond (%) Portfolio
Recession
Normal
Boom

7. Covariance and Correlation
What is the E(rs) and σs?
What is the E(rb) and σb?
What is the E(rp) and σp?
E(r) σ
Bond
Stock
Portfolio

8. Covariance and Correlation
When combining the stocks into the portfolio, you get the average return but the std is less than the average of the std of the 2 stocks in the portfolio
Why?
The risk of a portfolio also depends on the correlation between 2 stocks
How to measure the correlation between the 2 stocks
Covariance and correlation

9. Covariance and Correlation
Prob rs E(rs) rb E(rb) P(rs- E(rs))(rb- E(rb))
Cov (rs, rb) = -114
The covariance tells the direction of the relationship between the 2 assets, but it does not tell the whether the relationship is weak or strong
Corr(rs, rb) = Cov (rs, rb)/ σs σb = -114/(14.92*7.75) = -0.99

10. 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

11. 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
If ρ = 0, no correlation

12. Two Asset Portfolio St Dev – Stock and Bond

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

14. Numerical Example: Bond and Stock
Case 1: Weights Bond = .5 Stock = .5
What is the E(rp) and σp
E(rp) = 8%
σp = 13.87%
Average std = (25+12)/2 = 18.5
By combining stocks, get average return, but the risk is lower than average

15. Numerical Example: Bond and Stock
Case 1: Weights Bond = .75 Stock = .25
What is the E(rp) and σp
E(rp) = 7%
σp = 10.96%
By combining you get higher return than bond but lower risk than bond
This is power of diversification

16. Two Asset Portfolio St Dev – Stock and Bond
Std of the portfolio is always smaller than the weighted average of the 2 std in the portfolio.
Std of the portfolio is maximized when the correlation = 1, and is minimized when correlation = -1
No diversification benefit when the correlation = 1

17. Figure 6.3 Investment Opportunity Set for Stock and Bonds

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

19. 6.3 THE OPTIMAL RISKY PORTFOLIO WITH A RISK-FREE ASSET

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

21. Figure 6.5 Opportunity Set Using Stocks and Bonds and Two Capital Allocation Lines

22. 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

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

24. Figure 6.7 The Complete Portfolio

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

26. 6.4 EFFICIENT DIVERSIFICATION WITH MANY RISKY ASSETS

27. 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

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

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

30. Portfolio Selection Asset allocation Security selection
These two are separable!

31. Asset Allocation
“Asset allocation accounts for 94% of the differences in pension fund performance”
Identify investment opportunities (risk-return combinations)
Choose the optimal combination according to investor’s risk attitude

32. Optimal Portfolio Construction
Step 1: Using available risky securities (stocks) to construct efficient frontier.
Step 2: Find the optimal risky portfolio using risk-free asset
Step 3: Now We have a risk-return tradeoff, choose your most favorable asset allocation

33. Feasible and Efficient Portfolios
Expected
Portfolio
Return, rp
Efficient Set
Feasible Set
Risk, p
Feasible and Efficient Portfolios

34. An efficient portfolio is one that offers:
The feasible set of portfolios represents all portfolios that can be constructed from a given set of stocks.
An efficient portfolio is one that offers:
the most return for a given amount of risk, or
the least risk for a give amount of return.
The collection of efficient portfolios is called the efficient set or efficient frontier.

35. Optimal Portfolios Expected Return, rp Risk p IB2 IB1 IA2 IA1
Investor B
IA2
IA1
Optimal Portfolio
Investor A
Risk p
Optimal Portfolios

36. What impact does rRF have on the efficient frontier?
When a risk-free asset is added to the feasible set, investors can create portfolios that combine this asset with a portfolio of risky assets.
The straight line connecting rRF with M, the tangency point between the line and the old efficient set, becomes the new efficient frontier.

37. Efficient Set with a Risk-Free Asset
Expected
Return, rp Z . B M ^ rM .
The Capital Market
Line (CML):
New Efficient Set . A
rRF M
Risk, p

38. What is the Capital Market Line?
The Capital Market Line (CML) is all linear combinations of the risk-free asset and Portfolio M.
Portfolios below the CML are inferior.
The CML defines the new efficient set.
All investors will choose a portfolio on the CML.

39. . . Expected Return, rp rRF c M CML I1 M ^ rM C ^ rR C = Optimal
Portfolio
rRF
Risk, p c M

40. Disadvantages of the efficient frontier approach
The efficient frontier was introduced by Markowitz (1952) and later earned him a Nobel prize in 1990.
However, the approach involved too many inputs, calculations
If a portfolio includes only 2 stocks, to calculate the variance of the portfolio, how many variance and covariance you need?
If a portfolio includes only 3 stocks, to calculate the variance of the portfolio, how many variance and covariance you need?
If a portfolio includes only n stocks, to calculate the variance of the portfolio, how many variance and covariance you need?
n variances
n(n-1)/2 covariances

41. Single index model

42. Single index model
When we diversify, all the specific risk will go away, the only risk left is systematic risk component
Now, all we need is to estimate beta1, beta2, ...., beta n, and the variance of the market. No need to calculate n variance, n(n-1)/2 covariances as before

43. Estimate beta
Run a linear regression according to the index model, the slope is the beta
For simplicity, we assume beta is the measure for market risk
Beta = 0
Beta = 1
Beta > 1
Beta < 1

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