Portfolio optimization - Revision history

Kevinpan156: /* Conclusion */

2021-12-17T23:55:18Z

Conclusion

← Older revision		Revision as of 19:55, 17 December 2021
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	== Conclusion ==		== Conclusion ==
	Linear programming is an applicable tool in the optimization of stock portfolios. With the sharpe ratio, the tools yield returns with less risks associated as the method selects stocks and allocates investments accordingly. The optimization method will assist with new stock portfolios. It ~~maximize~~ returns with the incorporation of the users risk tolerance.<ref>Overton, M. L. (1997). ''Linear Programming''</ref>		Linear programming is an applicable tool in the optimization of stock portfolios. With the sharpe ratio, the tools yield returns with less risks associated as the method selects stocks and allocates investments accordingly. The optimization method will assist with new stock portfolios. It maximizes returns with the incorporation of the users risk tolerance.<ref>Overton, M. L. (1997). ''Linear Programming''</ref>

	== Reference ==		== Reference ==
	<references />		<references />

Kevinpan156: /* Conclusion */

2021-12-17T23:53:59Z

Conclusion

← Older revision		Revision as of 19:53, 17 December 2021
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	== Conclusion ==		== Conclusion ==
	Linear programming is an applicable tool in the optimization of stock portfolios. With the sharpe ratio, the tools yield returns with less risks associated as the method selects stocks and allocates investments accordingly. It will ~~continue to be utilized in the future for~~ new stock portfolios to maximize returns with the incorporation of the users risk tolerance.<ref>Overton, M. L. (1997). ''Linear Programming''</ref>		Linear programming is an applicable tool in the optimization of stock portfolios. With the sharpe ratio, the tools yield returns with less risks associated as the method selects stocks and allocates investments accordingly. The optimization method will assist with new stock portfolios. It maximize returns with the incorporation of the users risk tolerance.<ref>Overton, M. L. (1997). ''Linear Programming''</ref>

	== Reference ==		== Reference ==
	<references />		<references />

Kevinpan156: /* Methodology */

2021-12-17T23:50:15Z

Methodology

← Older revision		Revision as of 19:50, 17 December 2021
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	== Methodology ==		== Methodology ==
	Portfolio optimization could be solved and analyzed by Linear Programming and statistics. Cut the relevant information and conditions in the portfolio optimization, as well as the final requirements into the relevant variables, constraints and linear functions of the linear programming problem.		Portfolio optimization could be solved and analyzed by Linear Programming and statistics. When figuring out the maximum return under the fixed risks investment projects it can be seen as follows:

	When figuring out the maximum return under the fixed risks investment projects:
	{\| class="wikitable"		{\| class="wikitable"
	!Function		!Function
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	Table 3: Constraints		Table 3: Constraints

	~~Table In one portfolio optimization analysis, several~~ types of constraints will be selected to formulate the final return. The difference of constraints depends on the differences in investment projects, and related policy requirements, etc.<ref>Ahmed, Shabbir. ''Isye 6669: Deterministic Optimization ... - Isye Home \| Isye''. 2002, <nowiki>https://www2.isye.gatech.edu/~sahmed/isye6669/notes/portfolio</nowiki>.</ref>		Several types of constraints will be selected to formulate the final return. The difference of constraints depends on the differences in investment projects, and related policy requirements, etc.<ref>Ahmed, Shabbir. ''Isye 6669: Deterministic Optimization ... - Isye Home \| Isye''. 2002, <nowiki>https://www2.isye.gatech.edu/~sahmed/isye6669/notes/portfolio</nowiki>.</ref>

	== Numerical Example ==		== Numerical Example ==
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	== Conclusion ==		== Conclusion ==
	Linear programming is an applicable tool in the optimization of stock portfolios. ~~In parallel with~~ the sharpe ratio the tools yield returns with less risks associated ~~with choosing stocks~~ as ~~it optimizes~~ the ~~percent allocation~~. It will continue to be utilized in the future for new stock portfolios to maximize returns with ~~tolerable risks~~.<ref>Overton, M. L. (1997). ''Linear Programming''</ref>		Linear programming is an applicable tool in the optimization of stock portfolios. With the sharpe ratio, the tools yield returns with less risks associated as the method selects stocks and allocates investments accordingly. It will continue to be utilized in the future for new stock portfolios to maximize returns with the incorporation of the users risk tolerance.<ref>Overton, M. L. (1997). ''Linear Programming''</ref>

	== Reference ==		== Reference ==
	<references />		<references />

Kevinpan156: /* Methodology */

2021-12-17T23:39:33Z

Methodology

← Older revision		Revision as of 19:39, 17 December 2021
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	~~Define~~ the ~~dolling~~ decision variables:
			Defining the following decision variables:

	<math>x1 </math>: dollars invested in AAPL		<math>x1 </math>: dollars invested in AAPL

Kevinpan156: /* Introduction */

2021-12-16T04:25:28Z

Introduction

← Older revision		Revision as of 00:25, 16 December 2021
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	== Introduction ==		== Introduction ==
	[[File:Figure 1- Efficient Frontier Graph.jpg\|thumb\|~~333x333px~~\|Figure 1: Efficient Frontier Graph \|alt=]]		[[File:Figure 1- Efficient Frontier Graph.jpg\|thumb\|382x382px\|Figure 1: Efficient Frontier Graph \|alt=]]
	Portfolio optimization is a way to maximize net gains in a portfolio while minimizing risk. A portfolio is a set of selected stocks chosen by the investor. Risk is defined by the potential associated lost of some or all the original investment. Returns are the associated gains when the price of the stocks increases beyond the original investment. By applying probability statistics, linear algebra, optimization and other methods to redistribute the investment portfolio under the established target returns and risk limits to achieve the goal of reducing risks or obtaining higher benefits under the same risk conditions.<ref>Vanderbei, R. J. (2000). ''Linear programming: Foundations and extensions''. Boston: Kluwer. </ref>		Portfolio optimization is a way to maximize net gains in a portfolio while minimizing risk. A portfolio is a set of selected stocks chosen by the investor. Risk is defined by the potential associated lost of some or all the original investment. Returns are the associated gains when the price of the stocks increases beyond the original investment. By applying probability statistics, linear algebra, optimization and other methods to redistribute the investment portfolio under the established target returns and risk limits to achieve the goal of reducing risks or obtaining higher benefits under the same risk conditions.<ref>Vanderbei, R. J. (2000). ''Linear programming: Foundations and extensions''. Boston: Kluwer. </ref>

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	The x- axis represents the standard deviation of the data, which is also known as the risk tolerance. On the y- axis is the expected percent return. The points near the solid line are optimal, which is also known as the frontier line. Any points that fall below the curve are considered to be nonoptimal. This is due to the fact that with the same risk tolerance there are better expected returns. This method has its limitations as it relies on past data.<ref>Best, M. J. (2010). ''Champman & Hall/CRC Finance: Portfolio Optimization.'' Ontario: Canada.</ref>		The x- axis represents the standard deviation of the data, which is also known as the risk tolerance. On the y- axis is the expected percent return. The points near the solid line are optimal, which is also known as the frontier line. Any points that fall below the curve are considered to be nonoptimal. This is due to the fact that with the same risk tolerance there are better expected returns. This method has its limitations as it relies on past data.<ref>Best, M. J. (2010). ''Champman & Hall/CRC Finance: Portfolio Optimization.'' Ontario: Canada.</ref>
	== Theory ==		== Theory ==
	[[File:Nomenclature.jpg\|thumb\|Table 1:Nomenclature]]		[[File:Nomenclature.jpg\|thumb\|Table 1:Nomenclature\|387x387px]]
	Modern Portfolio Theory (MPT) or mean variance analysis is a mathematical framework introduced by the economist Harry Markowitz. MPT is used to combine a portfolio of assets in order to maximize the expected return under a given level of risk. Its main point is that investors can choose the best combination of asset risk and return based on the assessment of individual risk tolerance, so as to obtain the most ideal results.<ref>“Harry Markowitz's Modern Portfolio Theory [the Efficient Frontier].” ''Guided Choice'', 14 Feb. 2020, <nowiki>https://www.guidedchoice.com/video/dr-harry-markowitz-father-of-modern-portfolio-theory/</nowiki>. “Constraints: Portfolio Probe: Generate </ref>		Modern Portfolio Theory (MPT) or mean variance analysis is a mathematical framework introduced by the economist Harry Markowitz. MPT is used to combine a portfolio of assets in order to maximize the expected return under a given level of risk. Its main point is that investors can choose the best combination of asset risk and return based on the assessment of individual risk tolerance, so as to obtain the most ideal results.<ref>“Harry Markowitz's Modern Portfolio Theory [the Efficient Frontier].” ''Guided Choice'', 14 Feb. 2020, <nowiki>https://www.guidedchoice.com/video/dr-harry-markowitz-father-of-modern-portfolio-theory/</nowiki>. “Constraints: Portfolio Probe: Generate </ref>

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	With the following constraints:		With the following constraints:

	<math>E(r_p) = w* Er </math>		<math>E(r_p) = w* Er </math>[[File:Frontier.jpg\|thumb\|434x434px\|Figure 2: Efficient frontier graph of the 7 stocks from January 2018 to January 2019]]<math>\sigma_p = \surd(w\Sigmaw^T)</math>

	~~<math>\sigma_p = \surd(w\Sigmaw^T)</math>[[File:Frontier.jpg\|thumb\|381x381px\|Figure 2: Efficient frontier graph of the 7 stocks from January 2018 to January 2019]]~~<math>x1 + x2 + x3 + x4 + x5 + x6 +x7 = 1</math> (percentage needs to add up to 100)		<math>x1 + x2 + x3 + x4 + x5 + x6 +x7 = 1</math> (percentage needs to add up to 100)

	<math>x1,x2,x3,x4,x5,x6 <= 0.05</math> (at least 5% weight on each stock)		<math>x1,x2,x3,x4,x5,x6 <= 0.05</math> (at least 5% weight on each stock)

Kevinpan156: /* Theory */

2021-12-16T04:22:47Z

Theory

← Older revision		Revision as of 00:22, 16 December 2021
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	The x- axis represents the standard deviation of the data, which is also known as the risk tolerance. On the y- axis is the expected percent return. The points near the solid line are optimal, which is also known as the frontier line. Any points that fall below the curve are considered to be nonoptimal. This is due to the fact that with the same risk tolerance there are better expected returns. This method has its limitations as it relies on past data.<ref>Best, M. J. (2010). ''Champman & Hall/CRC Finance: Portfolio Optimization.'' Ontario: Canada.</ref>		The x- axis represents the standard deviation of the data, which is also known as the risk tolerance. On the y- axis is the expected percent return. The points near the solid line are optimal, which is also known as the frontier line. Any points that fall below the curve are considered to be nonoptimal. This is due to the fact that with the same risk tolerance there are better expected returns. This method has its limitations as it relies on past data.<ref>Best, M. J. (2010). ''Champman & Hall/CRC Finance: Portfolio Optimization.'' Ontario: Canada.</ref>
	== Theory ==		== Theory ==
	Modern Portfolio Theory (MPT) or mean variance analysis is a mathematical framework introduced by the economist Harry Markowitz. MPT is used to combine a portfolio of assets in order to maximize the expected return under a given level of risk. Its main point is that investors can choose the best combination of asset risk and return based on the assessment of individual risk tolerance, so as to obtain the most ideal results.<ref>“Harry Markowitz's Modern Portfolio Theory [the Efficient Frontier].” ''Guided Choice'', 14 Feb. 2020, <nowiki>https://www.guidedchoice.com/video/dr-harry-markowitz-father-of-modern-portfolio-theory/</nowiki>. “Constraints: Portfolio Probe: Generate </ref>
	[[File:Nomenclature.jpg\|thumb\|Table 1:Nomenclature]]		[[File:Nomenclature.jpg\|thumb\|Table 1:Nomenclature]]
			Modern Portfolio Theory (MPT) or mean variance analysis is a mathematical framework introduced by the economist Harry Markowitz. MPT is used to combine a portfolio of assets in order to maximize the expected return under a given level of risk. Its main point is that investors can choose the best combination of asset risk and return based on the assessment of individual risk tolerance, so as to obtain the most ideal results.<ref>“Harry Markowitz's Modern Portfolio Theory [the Efficient Frontier].” ''Guided Choice'', 14 Feb. 2020, <nowiki>https://www.guidedchoice.com/video/dr-harry-markowitz-father-of-modern-portfolio-theory/</nowiki>. “Constraints: Portfolio Probe: Generate </ref>

	The mitigation of risk can be characterized as follows. Let '''<math>		The mitigation of risk can be characterized as follows. Let '''<math>
	A</math>''' be a matrix '''<math>		A</math>''' be a matrix '''<math>
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	When figuring out the maximum return under the fixed risks investment projects:		When figuring out the maximum return under the fixed risks investment projects:
	{\| class="wikitable"		{\| class="wikitable"
	\|+
	!Function		!Function
	!Max <math>\Sigma</math>ri*Xi		!Max <math>\Sigma</math>ri*Xi
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	\|The limit requirements		\|The limit requirements
	\|}		\|}
	~~Constraints~~:		Table 2: Fixed Risk Table

	The constraints of portfolio optimization are various, include<ref name=":0" />:		The constraints of portfolio optimization are various, include<ref name=":0" />:
	{\| class="wikitable"		{\| class="wikitable"
	\|+
	!Constraints		!Constraints
	!Explanation		!Explanation
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	\|Trade with assets of at lease a certain size		\|Trade with assets of at lease a certain size
	\|}		\|}
	In one portfolio optimization analysis, several types of constraints will be selected to formulate the final return. The difference of constraints depends on the differences in investment projects, and related policy requirements, etc.<ref>Ahmed, Shabbir. ''Isye 6669: Deterministic Optimization ... - Isye Home \| Isye''. 2002, <nowiki>https://www2.isye.gatech.edu/~sahmed/isye6669/notes/portfolio</nowiki>.</ref>		Table 3: Constraints

			Table In one portfolio optimization analysis, several types of constraints will be selected to formulate the final return. The difference of constraints depends on the differences in investment projects, and related policy requirements, etc.<ref>Ahmed, Shabbir. ''Isye 6669: Deterministic Optimization ... - Isye Home \| Isye''. 2002, <nowiki>https://www2.isye.gatech.edu/~sahmed/isye6669/notes/portfolio</nowiki>.</ref>

	== Numerical Example ==		== Numerical Example ==
	~~'''Example 1:'''~~		[[File:Table1 .jpg\|thumb\|394x394px\|Table 4: Annual Returns of Sample Portfolio]]'''Example 1:'''
	[[File:Table1 .jpg\|thumb\|394x394px\|Table 2: Annual Returns of Sample Portfolio]]

	Now the investor is looking for investment opportunities for $200000, and the financial analyst identified seven investment opportunities and projected their annual rates of return. The investor is required to figure out a portfolio that could receive maximum returns.		Now the investor is looking for investment opportunities for $200000, and the financial analyst identified seven investment opportunities and projected their annual rates of return. The investor is required to figure out a portfolio that could receive maximum returns.
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	Using the same 7 stocks as the first example, matrix A represents the daily closing value of each stock collected. It is a matrix the size of 7x251 where 251 is the number of available trading days in the year 2018.		Using the same 7 stocks as the first example, matrix A represents the daily closing value of each stock collected. It is a matrix the size of 7x251 where 251 is the number of available trading days in the year 2018.
	[[File:Varcovar.jpg\|thumb\|Table 3:Variance - Covariance Table\|489x489px]]		[[File:Varcovar.jpg\|thumb\|Table 5:Variance - Covariance Table\|489x489px]]


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	<math>E(r_p) = w* Er </math>		<math>E(r_p) = w* Er </math>

	<math>\sigma_p = \surd(w\Sigmaw^T)</math>		<math>\sigma_p = \surd(w\Sigmaw^T)</math>[[File:Frontier.jpg\|thumb\|381x381px\|Figure 2: Efficient frontier graph of the 7 stocks from January 2018 to January 2019]]<math>x1 + x2 + x3 + x4 + x5 + x6 +x7 = 1</math> (percentage needs to add up to 100)

	<math>x1 + x2 + x3 + x4 + x5 + x6 +x7 = 1</math> (percentage needs to add up to 100)[[File:Frontier.jpg\|thumb\|381x381px\|Figure 2: Efficient frontier graph of the 7 stocks from January 2018 to January 2019]]<math>x1,x2,x3,x4,x5,x6 <= 0.05</math> (at least 5% weight on each stock)		<math>x1,x2,x3,x4,x5,x6 <= 0.05</math> (at least 5% weight on each stock)

	Solution:		Solution:

Kevinpan156: /* Reference */

2021-12-16T04:16:19Z

Reference

← Older revision		Revision as of 00:16, 16 December 2021
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	== Introduction ==		== Introduction ==
	[[File:Figure 1- Efficient Frontier Graph.jpg\|thumb\|333x333px\|Figure 1: Efficient Frontier Graph\|alt=]]		[[File:Figure 1- Efficient Frontier Graph.jpg\|thumb\|333x333px\|Figure 1: Efficient Frontier Graph \|alt=]]
	Portfolio optimization is a way to maximize net gains in a portfolio while minimizing risk. A portfolio is a set of selected stocks chosen by the investor. Risk is defined by the potential associated lost of some or all the original investment. Returns are the associated gains when the price of the stocks increases beyond the original investment. By applying probability statistics, linear algebra, optimization and other methods to redistribute the investment portfolio under the established target returns and risk limits to achieve the goal of reducing risks or obtaining higher benefits under the same risk conditions. ~~[1]~~		Portfolio optimization is a way to maximize net gains in a portfolio while minimizing risk. A portfolio is a set of selected stocks chosen by the investor. Risk is defined by the potential associated lost of some or all the original investment. Returns are the associated gains when the price of the stocks increases beyond the original investment. By applying probability statistics, linear algebra, optimization and other methods to redistribute the investment portfolio under the established target returns and risk limits to achieve the goal of reducing risks or obtaining higher benefits under the same risk conditions.<ref>Vanderbei, R. J. (2000). ''Linear programming: Foundations and extensions''. Boston: Kluwer. </ref>

	Risk is a major factor in choosing stocks in a portfolio. In order to mitigate these risks, investors typically use an efficient frontier graph as shown in figure 1.		Risk is a major factor in choosing stocks in a portfolio. In order to mitigate these risks, investors typically use an efficient frontier graph as shown in figure 1.

	The x- axis represents the standard deviation of the data, which is also known as the risk tolerance. On the y- axis is the expected percent return. The points near the solid line are optimal, which is also known as the frontier line. Any points that fall below the curve are considered to be nonoptimal. This is due to the fact that with the same risk tolerance there are better expected returns. This method has its limitations as it relies on past data. ~~[2]~~		The x- axis represents the standard deviation of the data, which is also known as the risk tolerance. On the y- axis is the expected percent return. The points near the solid line are optimal, which is also known as the frontier line. Any points that fall below the curve are considered to be nonoptimal. This is due to the fact that with the same risk tolerance there are better expected returns. This method has its limitations as it relies on past data.<ref>Best, M. J. (2010). ''Champman & Hall/CRC Finance: Portfolio Optimization.'' Ontario: Canada.</ref>
	== Theory ==		== Theory ==
	Modern Portfolio Theory (MPT) or mean variance analysis is a mathematical framework introduced by the economist Harry Markowitz. MPT is used to combine a portfolio of assets in order to maximize the expected return under a given level of risk. Its main point is that investors can choose the best combination of asset risk and return based on the assessment of individual risk tolerance, so as to obtain the most ideal results.		Modern Portfolio Theory (MPT) or mean variance analysis is a mathematical framework introduced by the economist Harry Markowitz. MPT is used to combine a portfolio of assets in order to maximize the expected return under a given level of risk. Its main point is that investors can choose the best combination of asset risk and return based on the assessment of individual risk tolerance, so as to obtain the most ideal results.<ref>“Harry Markowitz's Modern Portfolio Theory [the Efficient Frontier].” ''Guided Choice'', 14 Feb. 2020, <nowiki>https://www.guidedchoice.com/video/dr-harry-markowitz-father-of-modern-portfolio-theory/</nowiki>. “Constraints: Portfolio Probe: Generate </ref>
	[[File:Nomenclature.jpg\|thumb\|Table 1:Nomenclature]]		[[File:Nomenclature.jpg\|thumb\|Table 1:Nomenclature]]
	The mitigation of risk can be characterized as follows. Let '''<math>		The mitigation of risk can be characterized as follows. Let '''<math>
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	m		m
	</math>'''. <math>\sigma_p </math> is the standard deviation of the individual the data set of an individual stock, where '''<math>\sigma_p^2		</math>'''. <math>\sigma_p </math> is the standard deviation of the individual the data set of an individual stock, where '''<math>\sigma_p^2
	</math>''' is the variance. ~~[3]~~		</math>''' is the variance.<ref>Andre F. “Large-Scale Portfolio Optimization.” ''Management Science'', vol. 30, no. 10, 1984, pp. 1143–1160., <nowiki>https://doi.org/10.1287/mnsc.30.10.1143</nowiki>.</ref>

	<math>X = A - Er</math>		<math>X = A - Er</math>
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	w		w

	</math>''' is the percent of the initial capital being allocated into each stock. ~~[4]~~		</math>''' is the percent of the initial capital being allocated into each stock.<ref name=":0">Random Portfolios. Fund Management Software by Burns Statistics.” ''Portfolio Probe \| Investment Technology for the 21st Century'', 19 Apr. 2012, <nowiki>https://www.portfolioprobe.com/features/constraints/</nowiki>.</ref>



	<math>\sigma_p = \surd(w\Sigmaw^T)</math>		<math>\sigma_p = \surd(w\Sigmaw^T)</math>

	<math>\sigma_p </math> represents the standard deviation of the portfolio. ~~[5]~~		<math>\sigma_p </math> represents the standard deviation of the portfolio.<ref>Disatnik, David, and Saggi Katz. ''“Portfolio Optimization Using a Block Structure for the Covariance Matrix.” Journal of Business Finance & Accounting, 2012,'' <nowiki>https://doi.org/10.1111/j.1468-5957.2012.02279.x</nowiki>.</ref>



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	Constraints:		Constraints:

	The constraints of portfolio optimization are various, include~~[4]~~:		The constraints of portfolio optimization are various, include<ref name=":0" />:
	{\| class="wikitable"		{\| class="wikitable"
	\|+		\|+
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	\|Trade with assets of at lease a certain size		\|Trade with assets of at lease a certain size
	\|}		\|}
	In one portfolio optimization analysis, several types of constraints will be selected to formulate the final return. The difference of constraints depends on the differences in investment projects, and related policy requirements, etc.		In one portfolio optimization analysis, several types of constraints will be selected to formulate the final return. The difference of constraints depends on the differences in investment projects, and related policy requirements, etc.<ref>Ahmed, Shabbir. ''Isye 6669: Deterministic Optimization ... - Isye Home \| Isye''. 2002, <nowiki>https://www2.isye.gatech.edu/~sahmed/isye6669/notes/portfolio</nowiki>.</ref>

	== Numerical Example ==		== Numerical Example ==
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	== Applications ==		== Applications ==
	Portfolio optimization is often used for investment screening and investment amount allocation. The operation of this model gives investors the opportunity to avoid risks as much as possible while obtaining the maximum relevant benefits. At the same time, after entering the constraints, the model will give reasonable investment allocation suggestions and related optimal returns.		Portfolio optimization is often used for investment screening and investment amount allocation. The operation of this model gives investors the opportunity to avoid risks as much as possible while obtaining the maximum relevant benefits. At the same time, after entering the constraints, the model will give reasonable investment allocation suggestions and related optimal returns. <ref>DeMiguel, Victor, et al. “A Generalized Approach to Portfolio Optimization: Improving Performance by Constraining Portfolio Norms.” ''Management Science'', vol. 55, no. 5, 2009, pp. 798–812., <nowiki>https://doi.org/10.1287/mnsc.1080.0986</nowiki>.</ref>

			Retirement fund investment portfolio is one of the specific applications of investment portfolio optimization in modern times. Retirement planning needs to determine the source of income, estimate expenses, implement savings plans, and manage assets and risks. In the field of asset management and risk, modern financial theory advocates focusing on the total return of retirement-oriented investment portfolios rather than income. When the investment portfolio needs to be allocated, investors can choose between different asset classes and adjust their investment shares. <ref>Armstrong, Frank. “How to Create a Retirement Portfolio Strategy.” ''Investopedia'', Investopedia, 22 Aug. 2021, <nowiki>https://www.investopedia.com/articles/retirement/11/implement-effective-retirement-income-strategy.asp#toc-a-balanced-portfolio</nowiki>.</ref>

	== Conclusion ==		== Conclusion ==
	Linear programming is an applicable tool in the optimization of stock portfolios. In parallel with the sharpe ratio the tools yield returns with less risks associated with choosing stocks as it optimizes the percent allocation. It will continue to be utilized in the future for new stock portfolios to maximize returns with tolerable risks. ~~[8]~~		Linear programming is an applicable tool in the optimization of stock portfolios. In parallel with the sharpe ratio the tools yield returns with less risks associated with choosing stocks as it optimizes the percent allocation. It will continue to be utilized in the future for new stock portfolios to maximize returns with tolerable risks.<ref>Overton, M. L. (1997). ''Linear Programming''</ref>

	== Reference ==		== Reference ==
	~~[1] Vanderbei, R. J. (2000). ''Linear programming: Foundations and extensions''. Boston: Kluwer.~~		<references />

	~~[2] Best, M. J. (2010). ''Champman & Hall/CRC Finance: Portfolio Optimization.'' Ontario: Canada.~~

	~~[3] Perold, Andre F. “Large-Scale Portfolio Optimization.” ''Management Science'', vol. 30, no. 10, 1984, pp. 1143–1160., <nowiki>https://doi.org/10.1287/mnsc.30.10.1143</nowiki>.~~

	[4] “Constraints: Portfolio Probe: Generate Random Portfolios. Fund Management Software by Burns Statistics.” ''Portfolio Probe \| Investment Technology for the 21st Century'', 19 Apr. 2012, <nowiki>https://www.portfolioprobe.com/features/constraints/</nowiki>.

	[5] Disatnik, David, and Saggi Katz. ''“Portfolio Optimization Using a Block Structure for the Covariance Matrix.” Journal of Business Finance & Accounting, 2012,'' <nowiki>https://doi.org/10.1111/j.1468-5957.2012.02279.x</nowiki>.

	~~[6] Ahmed, Shabbir. ''Isye 6669: Deterministic Optimization ... - Isye Home \| Isye''. 2002, <nowiki>https://www2.isye.gatech.edu/~sahmed/isye6669/notes/portfolio</nowiki>.~~

	[7] DeMiguel, Victor, et al. “A Generalized Approach to Portfolio Optimization: Improving Performance by Constraining Portfolio Norms.” ''Management Science'', vol. 55, no. 5, 2009, pp. 798–812., <nowiki>https://doi.org/10.1287/mnsc.1080.0986</~~nowiki~~>.

	~~[8] Overton, M. L. (1997). ''Linear Programming''~~

Kevinpan156: /* Introduction */

2021-12-16T03:07:46Z

Introduction

← Older revision		Revision as of 23:07, 15 December 2021
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	The simplex table can be derived as the following:		The simplex table can be derived as the following:

	[[~~index.php%3Ftitle=~~File:P1.jpg\|~~alt=\|821x821px~~]]		[[File:Step 1.jpg\|820x820px]]

	The least coefficient is found to be row 2, where the constant is divided by <math>x1</math>. Once the pivot is found then the row reduction can begin.		The least coefficient is found to be row 2, where the constant is divided by <math>x1</math>. Once the pivot is found then the row reduction can begin.

	[[File:P2.jpg]]		[[File:Step2.jpg\|820x820px]]

	There are 3 negative terms in the last row, row reduction will occur for column <math>x1</math>.		There are 3 negative terms in the last row, row reduction will occur for column <math>x1</math>.

	[[~~index.php%3Ftitle=~~File:P3.jpg~~\|alt=~~]]		[[File:Step 3.jpg]]

	There are 2 negative terms in the last row, row reduction will occur for column <math>x3</math>.		There are 2 negative terms in the last row, row reduction will occur for column <math>x3</math>.

	[[~~index.php%3Ftitle=~~File:P4.jpg~~\|alt=~~]]		[[File:Ste p4.jpg]]

	There is now 1 negative terms in the last row, row reduction will occur for column <math>x2</math>		There is now 1 negative terms in the last row, row reduction will occur for column <math>x2</math>

	[[~~index.php%3Ftitle=~~File:P5.jpg~~\|alt=~~]]		[[File:Ste p5.jpg]]

	There is no need to conduct further calculations as all values in the last row are now positive. Therefore, the optimal solution is achieved:		There is no need to conduct further calculations as all values in the last row are now positive. Therefore, the optimal solution is achieved:
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	<math>\sigma_p = \surd(w\Sigmaw^T)</math>		<math>\sigma_p = \surd(w\Sigmaw^T)</math>

	<math>x1 + x2 + x3 + x4 + x5 + x6 +x7 = 1</math> (percentage needs to add up to 100)		<math>x1 + x2 + x3 + x4 + x5 + x6 +x7 = 1</math> (percentage needs to add up to 100)[[File:Frontier.jpg\|thumb\|381x381px\|Figure 2: Efficient frontier graph of the 7 stocks from January 2018 to January 2019]]<math>x1,x2,x3,x4,x5,x6 <= 0.05</math> (at least 5% weight on each stock)

	~~<math>x1,x2,x3,x4,x5,x6 <= 0.05</math> (at least 5% weight on each stock)[[File:Frontier.jpg\|thumb\|381x381px\|Figure 2: Efficient frontier graph of the 7 stocks from January 2018 to January 2019]]~~Solution:		Solution:

	<math>x1,x2,x3 = 5%</math>		<math>x1,x2,x3 = 5%</math>

Kevinpan156: /* Methodology */

2021-12-16T02:57:10Z

Methodology

← Older revision		Revision as of 22:57, 15 December 2021
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	{\| class="wikitable"		{\| class="wikitable"
	\|+		\|+
	! ~~colspan="2" \|Types of~~ Constraints		!Constraints
			!Explanation
	\|-		\|-
	\|~~Long- only constraint~~		\|Monetary Value Constraints
	\|~~Maximum weight constraints~~		\|Control the amount of money in the portfolio
	\|-		\|-
	\|~~Asset trade constraints~~		\|Maximum Weight Constraints
	\|~~Risk fraction constraints~~		\|Limit the proportion of assets in the portfolio.
	\|-		\|-
	\|~~Number of Assets Held Constraint~~		\|Risk Fraction Constraints
	\|~~Number~~ of ~~Assets trade Constraint~~		\|Control the number of variances that each asset may have.
	\|-		\|-
	\|~~Trade Universe Constraint~~		\|Number of Assets Held/ Traded Constraints
	~~\|Threshold Constraints~~		\|Limit the number of assets held/traded in the portfolio.
	\|-
	~~\|Forced Trade~~ Constraints
	\|~~Linear Constraints~~
	\|-		\|-
	\|Count Constraints		\|Count Constraints
	\|~~Tracking Error Constraints~~		\|Limit the number of assets that show in each category of a category variable
	\|-		\|-
	\|~~Volatility constraints~~		\|Cost Constraints
	\|~~Expected return constraints~~		\|Limit the transaction cost
	\|-		\|-
	\|~~Distance constraints~~		\|Expected Return Constraints
	\|~~Sum~~ of ~~largest weights constraints~~		\|Control the predicted expected return of the portfolio
	\|-		\|-
	\|~~Cost constraint~~		\|Volatility Constraints
	\|~~Number~~ of ~~closing positions constraint~~		\|Constrain the predicted volatility of the portfolio(Calculate by predicted variance matrix)
	\|-		\|-
	\|~~Quadratic constraints~~		\|Forced Trade Constraints
	\|~~Round Lot constraints~~		\|Trade with assets of at lease a certain size
	\|-
	~~\|Monetary value constraints~~
	~~\|Turnover constraints~~
	\|}		\|}
	In one portfolio optimization analysis, several types of constraints will be selected to formulate the final return. The difference of constraints depends on the differences in investment projects, and related policy requirements, etc.		In one portfolio optimization analysis, several types of constraints will be selected to formulate the final return. The difference of constraints depends on the differences in investment projects, and related policy requirements, etc.

Kevinpan156: /* Numerical Example */

2021-12-16T02:52:30Z

Numerical Example

← Older revision		Revision as of 22:52, 15 December 2021
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	Authors: ~~Ainsely~~ Li, Kevin Pan, Qizeng Sun, Hanshen Li, Eric Luo, SYSEN 5800 Fall 2021		Authors: Fanghan Li, Kevin Pan, Qizeng Sun, Hanshen Li, Eric Luo, SYSEN 5800 Fall 2021

	== Introduction ==		== Introduction ==
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	The simplex table can be derived as the following:		The simplex table can be derived as the following:

	[[File:P1.jpg\|~~left~~\|821x821px]]		[[index.php%3Ftitle=File:P1.jpg\|alt=\|821x821px]]

	The least coefficient is found to be row 2, where the constant is divided by <math>x1</math>. Once the pivot is found then the row reduction can begin.		The least coefficient is found to be row 2, where the constant is divided by <math>x1</math>. Once the pivot is found then the row reduction can begin.
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	There are 3 negative terms in the last row, row reduction will occur for column <math>x1</math>.		There are 3 negative terms in the last row, row reduction will occur for column <math>x1</math>.
	~~[[File:P3.jpg\|left\|Part 3]]~~






			[[index.php%3Ftitle=File:P3.jpg\|alt=]]

	There are 2 negative terms in the last row, row reduction will occur for column <math>x3</math>.		There are 2 negative terms in the last row, row reduction will occur for column <math>x3</math>.
	~~[[File:P4.jpg\|left\|part 4]]~~






			[[index.php%3Ftitle=File:P4.jpg\|alt=]]

	There is now 1 negative terms in the last row, row reduction will occur for column <math>x2</math>		There is now 1 negative terms in the last row, row reduction will occur for column <math>x2</math>
	~~[[File:P5.jpg\|left\|part 5]]~~






			[[index.php%3Ftitle=File:P5.jpg\|alt=]]

	There is no need to conduct further calculations as all values in the last row are now positive. Therefore, the optimal solution is achieved:		There is no need to conduct further calculations as all values in the last row are now positive. Therefore, the optimal solution is achieved:
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	<math>x1 + x2 + x3 + x4 + x5 + x6 +x7 = 1</math> (percentage needs to add up to 100)		<math>x1 + x2 + x3 + x4 + x5 + x6 +x7 = 1</math> (percentage needs to add up to 100)

	<math>x1,x2,x3,x4,x5,x6 <= 0.05</math> (at least 5% weight on each stock)		<math>x1,x2,x3,x4,x5,x6 <= 0.05</math> (at least 5% weight on each stock)[[File:Frontier.jpg\|thumb\|381x381px\|Figure 2: Efficient frontier graph of the 7 stocks from January 2018 to January 2019]]Solution:

	~~Solution:~~		<math>x1,x2,x3 = 5%</math>

	~~<math>x1,x2,x3 = 5%</math>~~
	~~[[File:Frontier.jpg\|thumb\|381x381px\|Figure 2: Efficient frontier graph of the 7 stocks from January 2018 to January 2019]]~~
	<math>x4,x5 = 40%</math>		<math>x4,x5 = 40%</math>

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	== Applications ==		== Applications ==
	Portfolio optimization ~~can be~~ used ~~to screen~~ investment ~~projects that meet investors, rationally allocate~~ investment ~~amounts, etc~~. The operation of this model gives investors the opportunity to avoid risks as much as possible while obtaining the maximum relevant benefits. At the same time, after entering the ~~relevant conditions~~, the model will give reasonable investment allocation suggestions and related optimal returns.		Portfolio optimization is often used for investment screening and investment amount allocation. The operation of this model gives investors the opportunity to avoid risks as much as possible while obtaining the maximum relevant benefits. At the same time, after entering the constraints, the model will give reasonable investment allocation suggestions and related optimal returns.

	== Conclusion ==		== Conclusion ==