n Security Rashid Usman Ansari AMU Centre Jangipur Murshidabad Source Investment Analysis and Portfolio Management ID: 1002534
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1. InvestmentsUnit 2Efficient Frontier –n Security Rashid Usman Ansari AMU Centre Jangipur Murshidabad Source:Investment Analysis and Portfolio ManagementPrasanna ChandraFifth Edition
2. Efficient Frontier for n-Security Case:Figure 1
3. In a multi-security case, the collection of all possible portfolios is represented by a broken-egg shaped region. This region is called the feasible region as shown in the Figure 1. (AFXMNO) The number of possible portfolios in this region are endless. However, what really matters to the investor is the northwestern boundary of the feasible region.Continued
4. This boundary is referred to as the efficient frontier. It contains all the efficient portfolios. Consequently, in Figure 1, while all the feasible portfolios are contained in the region AFXMNO, efficient portfolios lie along the boundary AFX. AFX represents the efficient frontier. All other portfolios ( that is, not on AFX) are inefficient.A portfolio like Z is inefficient because portfolios like B and D dominate it. Efficient frontier is same for all investors as the portfolio theory is based on the assumption that investors have homogeneous expectations. Continued
5. A portfolio is efficient if ,and only if, there is no alternative with: the same expected return and a lower standard deviation the same standard deviation and higher expected return a higher expected return and lower standard deviation.What is an Efficient Portfolio?
6. The set of efficient portfolios may be determined with the help of: Graphical Analysis ( up to 3 securities only )Calculus (can’t handle constraints such as inequalities )Quadratic Programming ( can handle n number of securities as well as inequalities. Considered most suitable )Obtaining Efficient Portfolios
7. As can be observed in Figure 1, feasible region is shaped like a broken-egg or an umbrella. To analyse why it is so, we have to refer to the formula of calculating portfolio risk. We know that the portfolio expected return is not affected by ρ while standard deviation is affected by ρ. Let us consider two cases: Case 1: The returns of securities A and B are perfectly positively correlated. Case 2: The returns of securities A and B are less than perfectly positively correlated.Why is Feasible Region Broken-egg Shaped?
8. Continued
9. Continued Graphically, this means that the portfolio risk-return profile plots as the broken curved line. Since ρ is typically less than 1, most of the portfolio risk-return profiles plots are like the curved line. The implication is that feasible region would have an umbrella like shape.
10. Example