1. Portfolio Theory and Risk
Investment Analysis and Portfolio Management Instructor: Attila Odabasi
Portfolio Theory and Risk
Main Points:
Portfolio expected return and SD
Investment opportunity set
Optimal Risky Portfolio with two risky assets
Power of diversification

2. Risky portfolio: Extending to the General Case, N > 2
Efficient Frontier is the best diversified set of investments with the highest returns
Efficient
frontier
Individual
assets
Minimum
variance
frontier
Individual assets combining them into portfolios, considering different weights.
So looking at many risky assets using the same techniques it is possible to build a minimum variance frontier. We are only concerned with the upper portion of the curve. Any minimum variance point on the bottom of the curve can be dominated by the similar point on the upper portion of the curve.
Global
minimum
variance
portfolio
St. Dev.

3. Mean & Variance: The General Case
Portfolio variance is the sum of weights times entries in the covariance matrix. w1 w2 … wn
σ11
σ12
σ1n
σ21
σ22
σ2n
σn1
σn2
σnn

4. Variance-Covariance Matrix, n=5w1 w2 w3 w4 w5
σ11
σ12
σ13
σ14
σ15
σ21
σ22
σ23
σ24
σ25
σ31
σ32
σ33
σ34
σ35
σ41
σ42
σ43
σ44
σ45
σ51
σ52
σ53
σ54
σ55

5. Variance-Covariance Matrix: The General Case
Covariance matrix contains N2 terms
N terms are variances
(N2 – N) terms are covariances
In a well diversified portfolio (as N gets larger), covariances become more important than variances
A stock’s covariance with other stocks determines its contribution to the portfolio’s overall variance
Investors should care more about the risk that is common to many stocks; risks that are unique to each stock can be diversified away.

6. Power of Diversification: equally weighted portf
11/24/2018
Power of Diversification: equally weighted portf
Consider an equally weighted portfolio: wi = 1/n
For portfolios with many stocks, the variance is determined by the average covariance among the stocks.

7. Equally Weighted Diversification Example
The average stock has a monthly standard deviation of 10% and the average correlation between stocks is If you invest the same amount in each stock, what is the variance of the portfolio?
What if the correlation is 0.0? 1.0?

8. Example (cont):

9. Naïve Diversification
The power of diversification
Just randomly picking stocks gets rid of 60% of the risk of the typical individual security by naive diversification
Most of the diversifiable risk eliminated at 25 or so stocks

10. Eventually, diversification benefits reach a limit:
Variance of (randomly picked) equally weighted portfolio
Note: 100%= risk when holding only one asset
Total risk of portfolio of size N
Risk eliminated by diversification: Diversifiable risk, Idiosyncratic Risk
Remaining risk: Non-diversifiable risk, Market Risk
Ave Cov n
Number of assets in portfolio

11. Classification of Risk
Part of the risk that cannot be diversified away:
Covariance risk, systematic risk, or non-diversifiable risk.
E.g., market risk, macroeconomic risk, industry risk
Part of the risk that can be diversified away (in a well diversified portfolio):
Idiosyncratic risk, non-systematic risk, diversifiable risk, unique risk.
E.g., individual company news

12. The Efficient Frontier
Given portfolio expected returns and variance:
How should we choose the best weights?
All feasible portfolios lie inside a bullet-shaped region, called the minimum-variance boundary or frontier
The efficient frontier is the top half of the minimum-variance boundry.
Rational investors should select portfolios from the efficient frontier.

13. The Efficient Frontier of Risky Assets, N>2

14. The efficient frontier, N=3
Example: You can invest in any combination of Stock 1, Stock 2, and Stock 3. What portfolio would you choose? (Annual data)
Variance / Covariance
Stock
Mean SD
Stock 1
Stock 2
Stock 3 1
13.6
15.4
237.16
95.63
149.7 2
15.0
23.0
529
45.54 3
14.0
18.0
324

15. The efficient frontier, N=3
Example: You can invest in any combination of Stock 1, Stock 2, and Stock 3. What portfolio would you choose?
Variance / Covariance
Stock
Mean SD
Stock 1
Stock 2
Stock 3 1
13.6
15.4
237.16
95.63
149.7 2
15.0
23.0
529
45.54 3
14.0
18.0
324

17. Investment opportunity set and the new efficient frontier

18. Drawing Efficient Frontier, N>2
To map out the efficient frontier for an n-asset portfolio (n>2) we use an optimization technique that minimises portfolio variance (or SD) for any given level of expected portfolio return. Hence we undertake the following constrained optimisation:

19. Excell Example: Portfolio Optimizer
This optimisation can be done in Excel using Solver.
For the details see the following xcl file on the course web-page: ‘007_14_PortfolioSelection’