A qualification to this analysis is in order. A strategy involving short positions may require that additional capital be pledged to cover possible shortfalls between ending asset and liability values.
Absent this, a higher rate will typically be charged for the short position so the income to the lender in states of the world in which the borrower is solvent will be sufficiently high to compensate for the shortfalls associated with the states of the world in which the borrower is insolvent.
Here the locus of the ep,sp combinations plots as two lines, the first associated with the lower lending rate, the second with the higher borrowing rate. The efficient frontier is shown by the solid lines. While points on it are infeasible, those plotting on its extension to the right of the point representing the risky asset are both feasible and efficient.
In practice the rate charged for borrowing may increase with the amount borrowed. Under these conditions there will eventually be decreasing returns to risk-taking. When a portfolio includes two risky assets, the Analyst needs to take into account expected returns, variances and the covariance or correlation between the assets' returns.
The differences from the earlier case in which one asset is riskless occur in the formula for portfolio variance.
In terms of risks and correlations it is:. To begin, consider the case in which two returns are perfectly positively correlated. Under these conditions:. As long as x1 and x2 are both non-negative the expression itself will be non-negative since neither s1 nor s2 can ever be negative. However, if one of the two x values is sufficiently negative, the absolute value must be utilized explicitly. For any such combination:. This relationship can be extended by allowing x2 to be either greater than one or less than zero.
Of particular interest is the combination that gives the smallest possible risk: the minimum-variance portfolio. In this case it is possible to achieve a variance of zero! We seek a value x2 for which:. This can be accomplished by taking a short position in asset 2 equal to one-half the Investor's funds and investing the proceeds as well as the original amount of money in asset 1.
In practice this may require the pledging of some other collateral to provide a sufficient guarantee to the lender of asset 2 that the short position can be covered when needed. The ability to form a riskless portfolio by taking offsetting positions in two perfectly positively correlated assets leads directly to a figure similar to that derived earlier when a riskless asset was combined with a risky one.
Let the expected return on the zero-variance portfolio be:. The set of portfolio risks and returns can then be derived by considering combinations of this riskless asset portfolio and either asset 1 or asset 2. Either view will provide the familiar graph associated with risky and a riskless asset. In this case:. While all the combinations shown are feasible, only those on the upper line are efficient -- a point emphasized by the use of a dashed line for the dominated portion of the relationship.
While perfectly positively correlated risky assets do exist, they are the exception rather than the rule. In most cases correlation coefficients are less than 1. The implications of this fact for risk are central to an understanding of the effects of diversification.
The standard deviation of a security is synonymous with risk. Lower covariance between portfolio securities results in lower portfolio standard deviation.
Successful optimization of the return versus risk paradigm should place a portfolio along the efficient frontier line. Optimal portfolios that comprise the efficient frontier tend to have a higher degree of diversification. Why is the efficient frontier important?
What is the optimal portfolio? How is the efficient frontier constructed? How can an investor use the efficient frontier A risk-seeking investor would select investments that lie on the right end of the efficient frontier which is populated with securities that are expected to have a high degree of risk coupled with high potential returns. Article Sources. Investopedia requires writers to use primary sources to support their work.
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This compensation may impact how and where listings appear. Investopedia does not include all offers available in the marketplace. Related Terms Inefficient Portfolio An inefficient portfolio is one that delivers an expected return that is too low for the amount of risk taken on. Modern Portfolio Theory MPT The modern portfolio theory MPT looks at how risk-averse investors can build portfolios to maximize expected return based on a given level of risk.
Excess Returns Excess returns are returns achieved above and beyond the return of a proxy. Excess returns will depend on a designated investment return comparison for analysis. Markowitz Efficient Set The Markowitz efficient set is a portfolio with returns that are maximized for a given level of risk based on mean-variance portfolio construction.
What Is the Treynor-Black Model? The Treynor-Black model is a portfolio optimization model that consists of an active portfolio and a passively managed market portfolio. Finally, we use excel to implement Mean-Variance optimization and construct a portfolio with the highest Sharpe ratio. In the assignment, you will be required to apply Mean-Variance Analysis to do portfolio selection, Sharpe ratio computation, and risky asset pricing, etc. Optimal Portfolios and Efficient Frontier. Optimization Methods in Asset Management.
Enroll for Free. This Course Video Transcript. Model Setup Since in the efficient frontier plot, all the assets on the frontier and below frontier are all well diversified, the risk plotted on the X-axis is in a way can be called as systematic risk only. Although we plot the total risk std dev but since we have already diversified, the total risk in a way represents the systematic risk only. I am stucked with this particular thought.
Could you anyone please help me out. Thanks, Kaushik. Hi Kaushik did you ask on our YouTube?
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