Investment Essay Example: Calculating the Efficient Frontier

2021-07-28 08:13:39
5 pages
1356 words
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Wesleyan University
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Term paper
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The list below presents the stock exchange markets in Thailand which will be used in the analysis of the Efficient frontiers.

The Steel Public Company Limited

Tipco Foods Public Company Limited

Ticon Industrial Connection Public Company Limited

TPARK Logistics Property Fund

TMB Bank Public Company Limited

TS Flour Mill Public Company Limited

Thai Oil Public Company Limited

TPBI Public Company Limited

Thai Rayon Public Company Limited

TRC Construction Public Company Limited

Triton Holding Public Company Limited

Thai Unique Coil Center Public Public Company Limited

Thoresen Thai Agencies Public Company Limited

TWZ Corporation Public Company Limited

Thai Wah Public Company Limited

Thai Vegetable Oil Public Company

Talaad Thai Leasehold Property Fund

Tata Steel Public Company Limited

Thiensurat Public Company Limited

Three Sixty Five Public Company Limited

Creating Efficient Frontier Using the monthly returns covering a period of five years.

Calculating the efficient frontier.

Basically, the efficient frontier reveals the standard deviation or the minimum risk which can be obtained given a particular level of risk.Mostly, the concept makes use of software like Mat lab and Octave using a matrix approach of mean-variance or co variance optimization. In calculating the efficient frontier, the expected returns estimates and the co variance matrix between securities should be available. Basically, these parameters are important because the optimal optimal portfolio is dependent on them. Another plausible and more accurate method of calculating the efficient frontier points is the two-fund separation method, a theorem which also takes into account the mean variance and expected returns.

To compute efficient frontier, two inputs are required; the expected returns of the securities and the matrix illustrating the variance -co-variance for the assets in question. The expected returns vector is represented as and the matrix form represented as . In addition, a unity vector with the same magnitude as vector is required. With these factors, a mathematical program can be used to compute the efficient frontier basing on the following equations.

A = 1 S1>0

B= 1S1

C= S1

= AC - B^2 > 0

Using these values to compute the variance () of expected returns (b) the following formula is used.

= (Ab^2 - 2Bb + C)

Basing from the equation above, the efficient frontier is parabola in co-variance space.Taking for instance two firms illustrating two portfolios of different assets, say Thai Wah public limited (TWPL)and the Three Sixty Five Public Company Limited (TSFPCL). The following data basing from their monthly returns can be used to construct efficient frontier.

security of (TWPL) security of (TSFPCL)

Expected returns 13% 19%

standard deviation 20% 40%

correlation -0.3 These two securities can be combined to form a portfolio with an infinite number of distinct risk returns profiles. Using mathematical programs risk profiles can be calculated at various levels. The following table would illustrates risk returns profiles of two securities combining them using distinct weights.

Portfolio TWPL proportion TSFPCL proportion Expected return Standard deviation

1 1 0 13.00% 20%

2 0.9 0.1 13.30% 17.64%

3 0.5 0.5 16.40% 20.49%

4 0 1 20% 40%

Basically, the risk and returns profiles of these two securities can be plotted into a efficient frontier curves by use of mathematical programs with returns on the Y-axis and risk on the Y-axis. The graph below is an illustration of efficient frontier of the two securities.

However, the efficient frontier can be computed and plotted using various number of assets or securities. There is no limiting factor on how many security variables can be used. The efficient frontier designates all the feasible points at which an investor would like to take into consideration. In addition, there is MVP which means minimum variance portfolio where there is minimum risk. Literally, an investor wold not like to invest at this point because of the underlying notion that low risk portfolios potentially projects low returns. Below the MVP is the most unfeasible region in which virtually all the investors tends to avoid because it represents zero risk tolerance a situation which is uncommon with investors. The curve bends backwards to annotate the benefits of a diversified portfolio as a result of negative correlation. This relationship between risk and diversification has notably led to development of capital asset pricing models.

Although the efficient frontier models shows feasible investments points, the model is usually lacking in its assumptions which do not properly represent the reality. For instance, the underlying assumption that returns assumes a normal distribution is misguided because sometimes returns are more than three standard deviations. Conversely, it is assumed that securities follow a leptorkurtic distribution which is a biased assumption.

From Both data provided, the modern portfolio theory can be used in maximizing a portfolios expected return for a given risk level. The theory holds that through diversification of assets portfolio, a higher return per risk unit can be obtained through holding of a particular asset. In addition, through the adjusting of each portfolios asset, an optimal portfolio can be made for the risk aversion level of each investor. Taking the assumption that there is efficiency of markets and that the portfolios asset are not correlated perfectly, a portfolios total variance can be reduced at any single return through a combination of assets in different weights.

For instance, a graph can be formed with X axis having risks, that are measured at standard deviation of the historical returns of the asset as well as return that is dividend-adjusted on the Y axis, which is measured as the historical returns average. Demonstrated below, therefore, is the manner in which the capital market, by use of Stock Exchange of Thailand can be modeled in an Excel spreadsheet so as to produce portfolios that are efficient.For instance, one can choose to come up with a portfolio that is risky with all 20 stocks in the Stock Exchange of Thailand. Given in the first figure below is a screenshot of each months closing stock price for the last 5 years. Data is divided to be on monthly and weekly basis. From the provided data, repetitive compounded monthly returns can be calculated through taking the log that is natural of each months closing price divided by the prior months closing price. In an excel spreadsheet, such is entered as =LN(Prices!B4/Prices!B5). A table is created for monthly returns that are continuously compounded for the last five years, then computation of standard deviation and mean is done for each particular stock over the given time horizon. Covariance between stocksreturns is a key factor in optimization of the portfolio as the diversification value emanates from assets that do not have a perfect correlation as the larger the covariance, diversification can be done in a more effective manner so as to reduce variability of portfolio. Covariance that exists between returns on both assets A and B is defined as

This leads to the need of finding every stocksmonthly price change less its average for the monthly returns in five years, multiplied by each stockschange per month in price minus the specific mean for the monthly returns in the five years. The number is then divided by the amount of months given in the table. Computation of the covariance can be done between every specific stock in such a model by use of matrix algebra. The covariance between each individual stock in the model by use of matrix algebra is done through subtracting the returnsmatrix through the means vector and multiplying it by the version that is transposed of itself subtracted by the means vector. In the excel sheet, such is entered as =MMULT(TRANSPOSE(B3:AF62-B64:AF64),B3:AF62-B64:AF64)/COUNT(A3:A62) where the command of count is a representation of the quantity of returns per month in the table given below. The covariance table therefore looks as shows in the diagram below. At such a point, a portfolio of arbitrary weights that area assigned to specific stock can be contructed.Cells can also be made for storing the mean of the portfolio, that is the mean returns vector, multiplied by weights assigned vector to every stock so as to obtain an average mean that is weighted, Portfolio variance, which is the weights vector multiplied by the matrix covariance, multiplied by the weights vector, as well as standard deviation of the portfolio, which is the variance square root. According to the way the spreadsheet is structured below, there is need of transposing vectors so as to multiply the specific matrices.

 

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