12. Estabilidad 111
12.2. Stern
The Markowitz Portfolio Theory also inspects the arch called the efficient frontier. The thought behind this arch is a graphic appearance of a set of portfolios that offer the highest rate of return for any given stage of risk. However, the efficient frontier identifies portfolios that recommend the least amount of risk for any given stage of return.
43 When a portfolio is capable of gathering its expected returns with the smallest possible risk, it is called an optimal portfolio. By altering the portion of investments in a portfolio, we can construct a number of diverse optimal portfolios with varying returns. All of these portfolios offer the lowest stage of risk possible for each stage of return. They also offer the highest returns for each risk stage.
This whole set of possible optimal portfolios is known as the efficient frontier. The Efficient Frontier is the compilation of all efficient portfolios and is represented on the attached graph as a solid line. Given that our objective is to raise expected return and to decrease risk, we will only be concerned with those portfolios that lie near the solid line. Because diversification is an influential way of accomplishing risk reduction, the investment verdict is not merely which securities to own, but how to segregate the assets among securities.
For each risk level, the efficient frontier locates the portfolio with the highest returns. It also reveals the portfolio with the smallest amount of risk for each level of return. An efficient frontier is charted as a curve as below. The Efficient Frontier graph below shows the relationship between risk and return. From this graph, the line area is the efficient frontier. The investor then uses the efficient frontier to choose a portfolio that matches his or her risk tolerance.
44
Figure 0.1 The Efficient frontier
Source: Elton Edwin J., Gruber Martin J., Brown Stephan J., and Goetzmann William N., Modern Portfolio Theory and Investment Analysis, 6th ed, Wiley, 2005.
The portfolio optimization process starts after the initial selection of asset classes. Minimum and maximum holding ranges are established for each asset class to ensure adequate diversification before running the optimization program using Microsoft Excel. We use historical and or forecasted returns, standard deviations, correlations, and covariances in calculating an optimal mix of assets for a portfolio at any level of desired volatility (risk). The optimization process requires adding and deleting asset classes and or changing holding constraints until an optimal mix of assets is achieved that meets the
45 investor's risk tolerance and rate of return. The optimal portfolio is then further compared against other portfolios and/or independent variables to calculate beta, alpha coefficient, Sharpe ratios values etc. These values fairly indicate how the portfolio will react in different "what if" scenarios.
The core centre is given to portfolio composition rather than individual security analysis. Risk and reward restrictions are quantified for portfolios then contrasted and optimized with past data. Dissimilar combinations of asset class construct an efficient frontier curve that gives the highest possible rate of return for every level of risk that the investor is willing to take. Any further portfolio not on the efficient frontier curve, which displays the same standard deviation (risk), will produce lower returns and, consequently, will be considered inefficient.
In Markowitz’s investigation of investor activities, Markowitz identified diverse categories of investment knowledge as positive or negative investment operations – distinctiveness of return and volatility – at dissimilar times. From this, he also concluded that when merging two investments that achieve in a different way together, the portfolio revealed a lesser amount of volatility than expected. Taking this inspection significantly advance, Markowitz developed a resourceful computational model to recognize all potential portfolios that recommends to investors both highest expected return for diverging stages of risk and lowest risk for diverging stages of expected return. Once illustrated graphically, this set of efficient portfolios form a locus line referred to as the efficient frontier – this is where the most excellent portfolios are.
46 The Markowitz efficient investor will try to find his or her optimum portfolio someplace along the efficient frontier arch, depending on their personal view of the return-risk link. Every portfolio on the arch will either have a greater rate of return for the same or lower risk, or lower risk for an equal or better rate of return when matched to portfolios or securities that are not on the efficient frontier.
As portfolios have the benefit of diversification due to the improperly correlated assets contained within them, the efficient frontier is in actuality composed of portfolios rather than individual securities or assets. The two potential exceptions would be the efficient frontier curve’s end points, at the start of which could be the asset with the lowest risk and at the end of which could be the asset with the highest return.
Assume there are no short selling and no risk less lending and borrowing, this yields the following formula: Minimize j ij n i n i j j i i n i i x x x
1 1 2 1 2 ( ) ( ) (7) Subject to: 1 1
n i i x p i n i iR R x
) ( 1 , 0 i x i =1,…,n47 Where
p
R = total return to the portfolio
i
x = fraction of portfolio represented by asset i
i
R return to asset i, i=1,…,n
2
i
= variance of asset i and
ij
= covariance of asset i and j, i=1,…,n, j=1,…,n, i≠j
In summary, Modern Portfolio Theory quantifies the benefits of diversification. Through an arithmetical method called mean-variance optimization, Markowitz illustrated precisely how an investor could lessen the volatility (risk as measured by standard deviation) of portfolio returns by deciding assets that do not move closely simultaneously. When he charted volatility (risk) against expected return, Markowitz developed a technique to analyse the efficiency of a portfolio. A portfolio is believed to be optimally efficient if there is no portfolio comprising the identical volatility (risk) with a larger expected return and there is no portfolio having the identical return with a smaller volatility.
Going beyond Markowitz, Tobin (1958), argues that investors would diversify saving between a risk free asset and a single portfolio of risky assets. By joining a risk free asset with risky assets, there is the potential to build portfolios whose risk return profiles are superior to those of the portfolios on the efficient frontier. By doing this, what is called the capital market line has been constructed as a tangent line to the efficient frontier that passes through the risk free rate.
48 An immense amount of investigation presenting confirmation of the benefits of international diversification has been published. Grubel (1968), and Levy and Sarnat (1970) were among the earliest to demonstrate that intensifying the investment universe from only US stocks to take account of foreign stock enhanced portfolio diversification. Grubel (1968) published the first theoretical paper on international portfolio theory employing the Markowitz model. Levy and Sarnat (1970) gave further details on Grubel’s work. Both of these papers employed the price indices of the common stocks of different countries in testing the benefits of Markowitz diversification on an international level. Both papers concluded that when an American investor diversified his portfolio to incorporate securities from other countries he was able to obtain a superior rate of return or a lower standard deviation.