Portfolio Selection

1
Portfolio Selection

• By
• Gunjan Agrawal

2
Portfolio SelectionbyHarry MarkowitzJournal
of Finance (1952)
3
Portfolio Selection Efficient Diversification of
Investments byHarry MarkowitzNew York
Wiley, 1959
4
Portfolio OptimizationUsing Linear
ProgrammingbyMark BroadieColumbia Business
School
5
Historical facts byCampbell HarveyDuke
University
6
Portfolio Selection

• What is Portfolio?
• What is Portfolio Selection?
• Efficient Asset Allocation.

7
Stages of Process

• Observations and Experience
• Beliefs about the future performance
• Relevant beliefs about future
• Choice of portfolio

8
First Stage

• Inputs Historical Data
• Outputs Possible Scenarios
• Scenario Returns ( )
• Scenario Probabilities ( )

9
Example

• Discrete Time Stochastic Process

10
Second Stage

• Inputs Outputs of First Stage
• Scenario Returns and Probabilities
• Assumptions Short sales are not Permitted
• Output Weight of securities in portfolio

11
Objective Functions

• Maximum Discounted Expected Returns
• Maximize Expected Returns
• and
• Minimize Variance of Returns

12
Variables

• Expected return from ith security at time t ( )
• Discount factor for ith security at time t ( )
• Discounted return of the ith security
• Relative amount invested in ith security ( )
• Discounted expected return of the portfolio

13
Optimization Problem 1

• Objective
• Maximize R
• Subject to

14
Optimized Results

• Solution
• if Ri is maximum
• Otherwise
• Conclusion
• Diversified portfolio (R) All Non-Diversified
  portfolios (R)

15
Assumption and Variables

• Assumption
• Static Models
• Variables
• Flow of returns from the ith security ( )
• Relative amount invested in ith security ( )
• Total Return from the portfolio

16
Mean and Variance

• The mean of the return is
• where
• The variance of the return is
• where

17
Optimization Problem 2

• Objective
• Maximize and Minimize
• Subject to

18
Graphical Solution

• Attainable Combination
• Set of Feasible Solution
• Efficient Combination
• Minimum V for given E
• Maximum E for given V

19
Three Securities Case

• Reduce in two dimension.
• Define
• X3 in terms of X2 and X1.
• Iso-mean Curve
• Iso-Variance Curve

20
Feasible Region

• X2

21
Iso-Mean Curve

22
Efficient Portfolio

23
Case II
24
Return

• Return in Scenario s
• Total Expected Return

25
Risk

• Downside Risk
• if
• otherwise
• Average Downside Risk

26
Optimization Problem

• Objective
• Maximize and Minimize ADR
• Subject to

27
Input Data

• Scenario Returns and Probabilities

28
Results

• Max Return Vs. Min ADR

29
Results

• Max Return Min ADR vs. Min ADR Max Return

30
Insights

• What is Diversification?
• Is it Holding of Securities?
• Is it Holding Securities in right proportion?

31
Example

• U.S. Small Stock and SP 500
• SP 500 and U.S. LT Corp.

32
Values of Different Assets
33
Returns from Different Assets
34
E-V Hypothesis

• Diversification
• Right Kind of Diversification
• Right Reason
• Theoretical Analysis
• change in beliefs
• Change in preferences of E vs. V
• Change in and
• Practical Analysis
• Actual Selection

35
Conclusion

• First Stage was not considered
• Formation of relevant beliefs
• Considered Second Stage
• Relevant beliefs about the assets involved
• Selection of portfolio
• Third Moment M3 is not included

36

• The process of analyzing one's portfolio,
  maximizing expected returns for given risks, and
  rebalancing when necessary, to ensure optimal
  risk-adjusted returns.

37
Attainable Combination

38
Iso-Mean Curve

39
Results

• Max Return vs. Min ADR Max Return

40
Results

• Min ADR vs. Max Return Min ADR