The formula for portfolio variance is used to compare two assets. The Adaptive Asset Allocation (AAA) portfolio combines two different tactical approaches (momentum and minimum variance) into one algorithm. If the coefficient correlation is zero, then it means that the return on securities is independent of one another. Hi, I am going through the excel sheet (R8-P1-T1-Elton-CAPM-v3) in section 9, and I saw the formula to find out Wa for the Minimum variance portfolio. Equally-weighted risk contributions portfolios. Bride and Groom sing EPIC Thank You Song to Wedding Guests - Best Wedding speech - Duration: 14:54. Period Invested for ₹1000 SIP Started on Investments Latest Value Absolute Returns Annualised Returns; 1 Year: 07-Jun-19: 12000: 11948. Finding the Global Minimum Variance Portfolio. • min 𝒘 1 2 𝒘′ 𝒘 - :𝒘′𝝁= ℎ, 𝔼 = ℎ - :𝒘′𝟏=1, =1 • Result: Portfolio weights are linear in expected portfolio return. Calculating the Minimum Variance Portfolio in R, Pandas and IAP It reads from top to bottom like the original, with the main formula at the top and the various terms defined below. The global minimum variance portfolio is the only efficient stock portfolio whose weights do not depend on the expected returns. The authors show that regressing the negative excess returns of $\mathrm{n_A} - 1$ assets on the returns of the remaining asset results in coefficients that equal the $\mathrm{n_A} - 1$ respective portfolio weights; the remaining asset's weight is determined by the budget constraint. frontiers in addition to three optimal portfolios (Minimum risk portfolio, Max-imum portfolio return for a given level of risk, and Maximum Sharpe ratio portfolio). Equally weighted portfolio Assume n assets, equal weights: where wEW are the EW portfolio weights. P The section of the opportunity set above the minimum variance portfolio is the efficient frontier. In this study, we analyze three research items found in the literature on the comparative performance of the global minimum-variance to the tangent portfolio. The portfolio weights sum to one and the target return is λ. 50 per share, and its required rate of return is 10. While most brokerages will tell you the standard deviation for a mutual fund or ETF for the most recent three-year (36 months) period, you still might wish to calculate your overall portfolio standard deviation by factoring the standard deviation of your holdings. Find all Ways2Wealth Asset Management posts tagged with minimum variance portfolio formula. 3, then the portfolio return is shown in figure 15. First Order Condition of the problem $= wT w + wT1 1 = XN i=1 XN j=1 w iw j˙ ij + XN i=1 w i 1! FOC w. Finding the Global Minimum Variance Portfolio. how an asset covaries with the market. You own 20 shares, 40 shares and 50 shares, respectively. such as the Global Minimum Variance Portfolio or the Equally Risk Contribution Portfolio , . 3 The Markowitz Problem 4 2. In case of a two-asset portfolio, we can work out portfolio variance as follows: σ 2 = w 1 2 σ 1 2 + w 2 2 σ 2 2 + 2w 1 w 2 Covariance(1,2) Where w 1 is weight of first asset, w 2 is weight of second asset, σ 1 2 is variance of first asset and σ 2 2 is variance. The same formula applies for each weight, thus deriving the total optimized returns for each stock, as follows: Andrew calculates the portfolio variance by adding the individual values of each stocks: Portfolio variance = 0. Directional Variance Adjustment: Bias Reduction in Covariance Matrices Based on Factor Analysis with an Application to Portfolio Optimization Daniel Bartz , 1, * Kerr Hatrick , 2 Christian W. The minimum variance hedge ratio, also known as the optimal hedge ratio, is a formula to evaluate the correlation between the variance in the value of an asset or liability and that of the hedging instrument that is meant to protect it. Volatility is a statistical measure of a particular security's price movement (ups and downs). ASSETS To find portfolio variance, multiply each element in the covariance matrix by the pair of portfolio weights in its row and column borders. This simple example helps clarify how to solve for weights when constructing a minimum variance portfolio with 2 assets. When the button "Plot Optimal Portfolio Curve" buttons as described in section 1. Robust minimum variance portfolio optimization modelling under scenario uncertainty☆ Panos Xidonasa,⁎, Christis Hassapisb, John Soulisc, Aristeidis Samitasd a ESSCA Grande École, France b University of Cyprus, Cyprus c Imperial College London, United Kingdom d University of the Aegean, Greece ARTICLE INFO Keywords: Portfolio optimization. It is a formalization and extension of diversification in investing, the idea that owning different kinds of financial assets is less risky than owning only one type. price) - sum(s. Deviation just means how far from the normal. the variance of portfolio returns) at the rate of s ip, i. Add up the resultant terms, and you have the formula for portfolio variance given in equation 8. The aim is to assess portfolio performance using cumulative returns, the Sharpe ratio and the daily vola-tilities of each portfolio. As we can see that standard deviation is equal to 9. The problem can also be implemented as a regression, see Kempf and Memmel []. 3 Portfolio A is the minimum variance portfolio, see figure 1. FIN501 Asset Pricing Lecture 06 Mean-Variance & CAPM (14) Efficient frontier with n risky assets •A frontier portfolio is one which displays minimum variance among all feasible portfolios with the same expected portfolio return. So diversification does not reduce the portfolio variance in this case. Adding the risk-free asset to a risky asset portfolio that lies on the efficient frontier will have several effects. In Microsoft Excel, you can use the MMULT function to perform matrix multiplication. We talk a lot about the importance of diversification, asset allocation, and portfolio construction at Listen Money Matters. Minimum-Variance Portfolio with Constraints The OPTMODEL procedure in SAS is a powerful tool which can be used to solve many different types of optimization problems. Portfolio variance is a measure of dispersion of returns of a portfolio. The measure. 12) with variance portfolio, as follows: ( ) This figure show the global minimum variance portfolio, so the optimal portfolio can be chosen from efficient frontier with. The Variance is defined as:. Note that the minimum variance portfolio not only has a greater expected return, but also a lower risk than a portfolio consisting only of Asset B. Modern portfolio theory stipulates that you can achieve a portfolio on the efficient frontier with only two mutual funds. The minimum correlation algorithm is a relatively new asset allocation strategy. The portfolio variance is a weighted sum of the elements in the covariance matrix, with the products of the portfolio proportions as weights. Explanation. Most investors would like to strike a balance between the maximum returns and minimum variance portfolio. This is the set of risky assets which all MV investors would want to combine with a risk free to build a MV optimal portfolio. In Microsoft Excel, you can use the MMULT function to perform matrix multiplication. This formula can be used to calculate things like variance between this year and last year, variance between a budgeted and actual values, and so on. When the correlation is zero, an investor can expect deduction of risk by diversifying between two assets. m: ) w = 1 (P. Minimax mean-variance models for fuzzy portfolio selection Minimax mean-variance models for fuzzy portfolio selection Huang, Xiaoxia 2010-09-19 00:00:00 Soft Comput (2011) 15:251–260 DOI 10. Finding portfolio standard deviation under the Modern Portfolio theory using matrix algebra requires three matrices 1. Remember the asset allocation for the All Weather Portfolio: 40% long-term bonds, 30% stocks, 15% intermediate-term bonds, 7. Standard Deviation Formula. Calculate Sharpe Ratio of Portfolio in Excel Especially relevant is that we can calculate Sharpe ratio of portfolio in Excel using MarketXLS functions. Given a = 7 %, this formulation would select only. The rate of return on the asset is given by r. Again, plugging into the formula: V_p=. Standard Deviation. The estimation risk is huge. the upper tail of the efficient set. When run with the Solver add-in on MS Excel for Windows, this spreadsheet is capable of calculating the mean-variance efficient weights for a three asset portfolio. Portfolio Variance formula = w 1 * ơ 1 2 + w 2 * ơ 2 2 + 2. the standard deviation of the assets' returns. Beta given by the formula:. After finding the weights of both assets, we use them to estimate expected. A Bowen, 1984). These values are used as aid in the optimization of the three assets portfolio. Step 3: Next, determine the correlation among the assets and it basically captures the movement of each asset relative to another asset. Modern portfolio theory (MPT), or mean-variance analysis, is a mathematical framework for assembling a portfolio of assets such that the expected return is maximized for a given level of risk. the tangency portfolio. the variance of portfolio returns) at the rate of s ip, i. We assume here that both assets are risky, by which we mean that σ2 1 > 0 and σ22 > 0. Hence, this portfolio would be 1. 1981 X 13) = 8. Variance = [0. Lecture 4 - Portfolio Diversification and Supporting Financial Institutions Overview. (You can also use Solver to find the minimum variance portfolio) Label this portfolio on the scatterplot in (2). Portfolio can consist of many types of assets such as plant, property, real and financial assets (P. ) /: Expected return of the portfolio. The following are step by step instructions on how to compute the Minimum Variance Portfolio based on historical asset returns. Let P be the optimal portfolio for target expected return 0. • min 𝒘 1 2 𝒘′ 𝒘 - :𝒘′𝝁= ℎ, 𝔼 = ℎ - :𝒘′𝟏=1, =1 • Result: Portfolio weights are linear in expected portfolio return. A new promising approach introduced by Brandt, Santa-Clara and Valkanov (2009) avoid the di¢ culties in the estimation of asset returns moments by modelling directly the portfolio weight in each asset as a function. 05 x 35% x 2. In general the higher the expected return, the lower is the standard deviation or variance and lower is the correlation the better will be the security for investor choice. • Asset (portfolio) A mean-variance dominates asset (portfolio) B if μ A ≤μ B and σ A < σΒ or if μ A >μ B while σ A ≤σ B. A measure of the riskiness of an asset held in isolation is _____. The portfolio of risky assets with lowest variance. R i = Return on Respective Assets, for Assets 1 to n, in Portfolio W i = Weighting of the Assets, for Assets 1 to n, in Portfolio Enter your values: Return on Respective Assets: R 1: % R 2: % R 3: % Weighting of the Assets: W 1: % W. R Tools for Portfolio Optimization 3 stock price 80 85 90 95 100 Jan Mar IBM: 12/02/2008 - 04/15/2009 Maximum Mean-Variance Portfolio Optimization Function portfolio. † The equally-weighted (ew or 1/n) portfolio Another simple way is to attribute the same weight to all the assets of the portfolio (Bernartzi and Thaler, 2001). Portfolio Return Rates An investment instrument that can be bought and sold is often called an asset. The most risk-averse investor would construct the minimum variance portfolio which has an expected return of 4. 002341, and the variance for GE is 0. If correlation equals 0, standard deviation would have been 8. σ 2 = (64 + 1 + 16 + 36 + 16 + 36 + 4 + 81) / 8; σ 2 = 31. Variance of a Two-asset Portfolio. 23: Denotes the transpose of a matrix 2. When the correlation is zero, an investor can expect deduction of risk by diversifying between two assets. Above formula. The strategy is robust in its design, applies solid diversification principles and performs well in relation to other asset allocation strategies. minimum variance portfolio. The rest of the paper is proceed as follows. 0050, the correlation coefficient between. In order to locate the portfolio with minimum second moment, we draw a circle with its center at the origin, and the location of the minimum second moment portfolio [p. The minimum variance portfolio on the investment opportunity set is important because: A. pricing and hedging derivative securities based on a fundamental state model, the well-received. The Role of Real Estate in the Portfolio Allocation Process. Markowitz model and Sharpe ratio are briefly introduced in section 2. To help project managers understand the significance of schedule variance (SV), several authors have proposed a new element called time-based earned schedule (ES) for expressing SV in time units (i. the tangency portfolio. The Adaptive Asset Allocation (AAA) portfolio combines two different tactical approaches (momentum and minimum variance) into one algorithm. Jamie and Megan Recommended for you. Questions Assume that there are three assets available. Asset 2 makes up 72% of a portfolio has an expected return (mean) of 13% and volatility (standard deviation) of 6%. This simple example helps clarify how to solve for weights when constructing a minimum variance portfolio with 2 assets. The investment vectors of absolute minimum variance portfolios are scalar multiples of solutions of (3). it shows where all investors will invest because of its minimum risk. Every finance student learns the source of. Again, plugging into the formula: V_p=. The rest of the paper is proceed as follows. The Minimum Variance Portfolio is a portfolio where the weights of the different assets results in a portfolio with the minimum standard deviation. It makes maximum use of diversification to achieve the resultant risk level that is lower than the individual risk level of each of the stock it contains. leverage this freedom to create a new asset allocation pro-cedure we call the “mean-variance inverse optimization” (MV-IO) approach, which uses BL-type estimators in a set-ting motivated by the arbitrage pricing theory. Treasury bonds neither due or callable in less than 10 years (risk-free rate of return proxy). By optimizing the. † The minimum variance (mv) portfolio It is obtained for ` = 0 in the mean-variance problem and does not depend on the expected returns. same as the asset being hedged, a hedge ratio of 1. Auckland University of Technology. portfolio (w=g+hµ, see Lecture 5) into the objective function (w′∑w, see Lecture 5), then we get the minimum variance as a function of the target level of return. Calculating the Minimum Variance Portfolio in R, Pandas and IAP It reads from top to bottom like the original, with the main formula at the top and the various terms defined below. If the coefficient correlation is zero, then it means that the return on securities is independent of one another. how an asset covaries with the market. So now you ask, "What is the Variance?" Variance. the upper tail of the efficient set. The minimum correlation algorithm is a relatively new asset allocation strategy. It is interesting to note that the weights of the Tangency portfolio are a respectably close approximation of the actual sector weights in the S&P 500, when the data for this workbook were estimated. Definition: A minimum variance portfolio indicates a well-diversified portfolio that consists of individually risky assets, which are hedged when traded together, resulting in the lowest possible risk for the rate of expected return. The idea behind risk parity is more. cit) the mathematical derivation of a two-asset portfolio with minimum risk is quite straightforward. What are the investment proportions in the minimum-variance portfolio of the two risky funds, and what is the expected value and standard deviation of its rate of return?. Step 3: Next, determine the correlation among the assets and it basically captures the movement of each asset relative to another asset. the minimum variance portfolio. (You can also use Solver to find the minimum variance portfolio) Label this portfolio on the scatterplot in (2). Estimates of CER model for Microsoft, Nordstrom and Starbucks stock from monthly returns over the period January 1995 to January 2000. Beta measures: A. R Tools for Portfolio Optimization 3 stock price 80 85 90 95 100 Jan Mar IBM: 12/02/2008 - 04/15/2009 Maximum Mean-Variance Portfolio Optimization Function portfolio. 17%²) + (2 x 65% x 2. Economists often say that portfolios located below the GMV portfolio are dominated by others that have the same variances but higher expected returns. the weights of the minimum variance portfolio: w MVP(ρ)=f + a12 a11 g, (29) and the corresponding risk-return values: s MVP(ρ)= p 1/a11, (30) ρ MVP =a12/a11. Trending Questions. (A normal approximation will give a much tighter bound. but with the added (inequality) constraint that the weights must all be non-negative, that is, This is an example of an inequality constraint; in this case, short sales are prohibited. Source(s Ask Question + 100. Mean-Variance Optimization and the CAPM 2 Figure 1: Sample Portfolios and the E cient Frontier (without a Riskfree Security). global minimum-variance portfolio has zero weights in the test assets. Descriptive statistics can also be produced by using the Descriptive Statistics item from the Data Analysis dialog as shown in figure 3. The global minimum variance portfolio is the only efficient stock portfolio whose weights do not depend on the expected returns. the variance of portfolio returns) at the rate of s ip, i. Question 5. Investors use models of the movement of asset prices to predict where the price of an investment will be at any given time. 1 Portfolios with Three Risky Assets Portfolio characteristics using matrix notation Excel formula: MMULT(TRANSPOSE(xvec),MMULT(sigma,yvec)) 0for the global. A series of sample stocks are included, but the spreadsheet can be adapted to other stocks selections. , the portfolio with the minimum possible variance. Pair B (assets 1 and 3): Applying the above GMV weight formula to Pair B yields the following weights: 1 =3 X GMV (300%) and 2 X 3 =− (−200%) This means that the GMV portfolio of assets 1 and 3 involves short selling asset 3. Portfolio Variance formula = w 1 * ơ 1 2 + w 2 * ơ 2 2 + 2. Join Yahoo Answers and get 100. Show that the minimum variance portfolio has variance 14) Suppose a market consists of n risky assets and 1 risk free asset. CVaR Algorithm In this section we describe the algorithm used to calculate the CVaR and to find the optimal weights by minimizing that value. The portfolio, under the set of constraints that the manager chooses, that maximizes the value D(P) is the most diversified portfolio. The efficient frontier runs from Portfolio B, the minimum variance portfolio, to Portfolio E, the maximum return portfolio. A minimum variance portfolio is a well diversified portfolio of two or more assets with a low correlation to each other. Otherwise, as the formula for portfolio standard deviation shows, the portfolio standard deviation is less than the weighted average of the component-asset standard deviations. The total number of asset units is 20 plus 40 plus 50, which equals 110 shares. This means that, instead of using both risk and return information as in the Markowitz portfolio selection, the portfolio is constructed using only measures of risk. Beta measures: A. Minimum-Variance Portfolio with Constraints The OPTMODEL procedure in SAS is a powerful tool which can be used to solve many different types of optimization problems. When the button "Plot Optimal Portfolio Curve" buttons as described in section 1. Question 5. However, the following simple example shows that the minimum variance portfolio might be inefficient. It demonstrates the risk-and-return trade-off of a portfolio and helps us visualize the efficient portfolios, the global minimum variance portfolio and inefficient portfolios. So, I need to figure out the co variance of the two assets, given the portfolio weight, variances, and standard deviation. Stock A comprises 20% of the portfolio while stock B comprises 80% of the portfolio. Opportunity Set: n Risky Assets XII. If we want to find the exact minimum variance portfolio allocation for these two assets, we can use the following equation: x = (σ b ²-ρ ab σ a σ b) / (σ a ² + σ b ² - 2ρ ab σ a σ b) Plugging in the values from the first article in this series, we can see that x = 74. Finding portfolio standard deviation under the Modern Portfolio theory using matrix algebra requires three matrices 1. Cov The problem: To use this formula to -nd E(r i) r f, we need to know what the tangency portfolio is. Return On Portfolio (Three Securities) Calculator: Enter value and click on calculate. If r12 exceeds s1/s2, the minimum variance portfolio will require a short position in asset 1. The company provides the following information for the year 2015: 2 Jezebel Company incurred fixed costs of$300000. and the variance of the portfolio return is 2 =var( ) (1. Mathematics of Operations Research, Vol. The set of minimum variance portfolios is represented by a parabolic curve in the σ2 P− µPplane. The measure. Mean-variance efficient portfolios are. With the above inputs, we observe the decrease of portfolio return risk due to diversification when the number of assets increases. Given a constrained portfolio w~, it is possible to ﬁnd a covariance matrix §~ such that w~ is the solution of the global minimum variance portfolio. Given a = 7 %, this formulation would select only. Use the following two portfolios to answer parts one to four. Multiplying the three matrices gives us the variance of the portfolio. The expected return for a portfolio that includes a risk-free asset and a risky asset portfolio is the weighted average of the two returns. Minimum Variance Portfolio. They are called e–cient funds. Stock C has a weight value of 15% of your total portfolio. Efficient frontier comprises investment portfolios that offer the highest expected return for a specific level of risk. 4 Mean-Variance E cient Portfolio The maximum returns portfolio and minimum variance portfolio give the two extremes of returns and risk. As we can see, the more assets that are combined in a portfolio, the more cumbersome the formula of its standard deviation. with risky-investment weights w. How to Determine Weights in an Investment Portfolio by Hunkar Ozyasar & Reviewed by Ashley Donohoe, MBA - Updated May 21, 2019 The weight of an asset in an investment portfolio is a representation of what percentage of the portfolio's total value is tied up in that specific asset. Hint: consider a 2-assets portfolio made of an arbitrary portfolio p and the MVP, with weights a and 1-a. The variance for a portfolio consisting of two assets is calculated using the following formula: Where: w i – the weight of the ith asset; σ i 2 – the variance of the ith asset; Cov 1,2 – the covariance between assets 1 and 2 Note that covariance and correlation are mathematically related. User simply get the historical price information with matlab and transfer it to Excel. Determine the total number of asset units in your portfolio. The risk parity approach asserts that when asset allocations are adjusted (leveraged or deleveraged) to the same risk level, the risk parity portfolio can achieve a higher Sharpe ratio and can be more. Portfolio variance is a measure of dispersion of returns of a portfolio. If a test question asks for the standard deviation then you will need to take the square root of the variance calculation. Definition: A minimum variance portfolio indicates a well-diversified portfolio that consists of individually risky assets, which are hedged when traded together, resulting in the lowest possible risk for the rate of expected return. By drawing on ideas from robust optimization, we generalize the traditional Markowitz portfolio problem to. This saves the investor a lot of time when choosing securities to add to his or her portfolio. Trending Questions. Furthermore assume that the risk free rate is 5%. A minimum variance portfolio is a collection of securities that combine to minimize the price volatility of the overall portfolio. When run with the Solver add-in on MS Excel for Windows, this spreadsheet is capable of calculating the mean-variance efficient weights for a three asset portfolio. Stock C has a weight value of 15% of your total portfolio. the minimum variance portfolio. - Minimum variance frontier is a ggraph of the expected return/variance combinations for all minimum variance portfolios. updating formula takes the weighted average of the unconditional variance, the squared residual for the first observation and the starting variance and estimates the variance of the second observation. thevoltreport. the upper tail of the efficient set. Portfolio utility depends on both portfolio characteristics and the risk tolerance of the Investor in question. It optimizes asset allocation by finding the stock distribution that minimizes the standard deviation of the portfolio while maintaining the desired return. HANSEN ©2000, 20201 University of Wisconsin Department of Economics This Revision: June 2020 Comments Welcome 1This manuscript may be printed and reproduced for individual or instructional use, but may not be printed for. Source(s Ask Question + 100. Beta given by the formula:. From this table we conclude that the minimum variance portfolio is given by setting w 1 equal to (roughly) 0. $\ \ \$1) there is not riskless asset (cash account or treasury bond), $\ \ \$2) you are fine with any expected return. ALSO as n gets large =(1/n * σ²i_) approaches 0. Inputs on three stocks: mean, standard deviation, and correlation matrix Standard Expected Stock Deviation A B C Correlation Matrix B. This portfolio is depicted at the outmost left point on the efficient part of the mean variance frontier (MSCI Index Research, 2010). ( ) 1 11 2 1 * * * 1 1 1' 11' 1 1' 1 1' 1 V V V with w V V µ µ σ − −− − − − = = = 32. The process to construct the CAL for a collection of portfolios is described below. The company provides the following information for the year 2015: 2 Jezebel Company incurred fixed costs of $300000. Determine the total number of asset units in your portfolio. Formula for Portfolio Variance. Optimal Risky Portfolio with a Risk-Free Asset. Two-asset portfolio Consider two assets with known means R1 and R2, variances ¾12 and ¾2 2, of the expected rates of returns R1 and R2, together with the correlation coe-cient ‰. Jamie and Megan Recommended for you. Prior to the Black-Litterman model, investors used to input expected returns of the assets. Estimation Step - estimate the expected return + variance for each individual asset and the correlation of each pair of assets. and shows e (˙^ 2) 5:2: 4(log2) 3. Despite the lower risk, the average return on the minimum variance portfolio over time actually exceeds the average return on the. Variance of any portfolio: The formula shows that the overall variance in a portfolio is the sum of each individual variance along with the cross-asset correlations. ) From the two-asset portfolio variance. 03) is combined with a risk-free investment. Another way to reduce the overall risk of a portfolio is to diversify the risks of its assets and to look for the assets weights that maximise a diversiﬁcation indicator such as the variety (or. Fur-thermore, using a speci c reparameterization we obtain non-informative and informative priors for the portfolio weights directly. *] can be obtained from the point where the circle is tangent to the minimum-variance frontier of the risk-free and risky assets. 3 Denote Σ = V22 − V21V −1 11 V12 and Q = [0N×K, IN] where IN is an N by N identity matrix. =((n-1)/n) approaches 1. Let E = n §~ > 0 : ~w = w? ‡ §~ ·o denotesthecorrespondingset. the minimum variance portfolio. 3 is clicked on, the minimum Variance Portfolio will be determined using Microsoft Excel Solver as one of the steps. A B Portfolio weight: 0. P invests in the same risky assets as the Market Portfolio and in the same proportions! The only di erence is the total weight, w. The variance (or risk) of a portfolio is not simply a weighted average of the variances of the individual securities in the portfolio. According to standard portfolio theory, the tangency portfolio is the only efficient stock portfolio. Returns are dependent on the investment combinations that make up the portfolio. Portfolio 3: Minimum Variance Portfolios with No Short Sales. The correct answer is B. Deviation just means how far from the normal. It is calculated the portfolio structure and it is noticed that the investor prefers to invest the less in the first two assets which shows the lowest return and risk and prefers to invest an appropriate proportion in the latter two assets of the highest risk. The standard deviation of a portfolio is a function of: The weighted average of the individual variances, plus; The weighted covariances between all the assets in the portfolio. One of the investment strategies gaining popularity is the Minimum Variance (MV) portfolio. Descriptive statistics can also be produced by using the Descriptive Statistics item from the Data Analysis dialog as shown in figure 3. 99% ctp = [wWD + wE^E + lwDwECov(rD, r£)]1/2. If a test question asks for the standard deviation then you will need to take the square root of the variance calculation. R Tools for Portfolio Optimization 19. 2 Unsystematic Risk 11 3. Money › Investment Fundamentals Portfolios Returns and Risks. where ωs are proportions of the respective assets in the portfolio. thevoltreport. 99 +$33,122 / $35,210 × 1. CVaR Algorithm In this section we describe the algorithm used to calculate the CVaR and to find the optimal weights by minimizing that value. The other two assets, B and S are risky asset with the following attributes. How about the portfolio variance and then ultimately the portfolio standard deviation? Well, when we're looking at the variance, our portfolio here is w in large cap stocks, 1 - w in the risk-free asset. Mean-Variance Optimization and the CAPM 2 Figure 1: Sample Portfolios and the E cient Frontier (without a Riskfree Security). 9 February 2011 Portfolios Under Construction Minimum Variance: Exposing the "magic" Minimum Variance Portfolios The minimum variance portfolio has been touted as the "magic" formula that generates superior risk-adjusted returns. Question 4. The minimum correlation algorithm is a relatively new asset allocation strategy. Portfolio with the Riskless Asset VIII. Portfolio Beta =$2,088 / $35,210 × 0. Pg 1-3 Portfolio Optimization Version 1. Its expected return and standard deviation are. 2) Derive the minimum variance frontier for various levels of expected returns, without short-selling of risky assets 3) Derive the minimum variance frontier for various levels of expected returns with restricted short-selling (apply whatever restriction you consider appropriate) of risky assets allowed. Portfolio Problem. Inputs on three stocks: mean, standard deviation, and correlation matrix Standard Expected Stock Deviation A B C Correlation Matrix B. From this follows the formula for the efficient frontier. It uses the variance of asset prices as a proxy for risk. Handout 7: Optimal portfolios when there is a riskfree asset CorporateFinance,Sections001and002 How does the set of possible portfolios change when you have access to a riskfree. The third matrix is the transpose of the first matrix. The source portfolio is then levered according to a leverage rule. As the name suggests, minimum variance portfolio is a portfolio with diversified securities that consists of risky assets on an individual basis, which are hedged in case they are traded together which in return results in the lowest possible risk for the expected rate of return. This study explores the role of direct real estate investment in a portfolio context incorporating the real estate imperfections of indivisible assets and no short sales. The Variance is defined as:. As a result, Bulgarian investors can select their own optimal portfo-lio that maximizes portfolio rate of return with respect to their risk preference. Jamie and Megan Recommended for you. The complete portfolio is composed of Treasury bills that pay 5% and a risky portfolio, P, constructed with two risky securities, X and Y. One may think that all possible values have to fall inside the convex hull. From a ﬁnancial point. A is a risk-free asset with that yields a rate of 8%. Figure 3: Implied alphas for minimum variance portfolios divided by assets in the portfolio and those out of the portfolio. I need to create a global minimum variance portfolio and am completely confused after spending days doing other things in Excel which have worn my brain out. Uploaded By ProfessorGullMaster84. What are the investment proportions in the minimum-variance portfolio of the two risky funds, and what is the expected value and standard deviation of its rate of return?. The total number of asset units is 20 plus 40 plus 50, which equals 110 shares. The company provides the following information for the year 2015: 2 Jezebel Company incurred fixed costs of$300000. Stock A comprises 20% of the portfolio while stock B comprises 80% of the portfolio. The correct answer is B. Remark 6 — The optimal portfolio in risky assets for the investor with mean-variance model can be divided into three parts. 3 Characteristics of Minimum Variance Portfolio MVP W XOM 9569 W GE 431. 0): ''' Minimum Variance Portfolio weights. Interpret the standard deviation. the actual return on an asset. The expected return on a portfolio of assets is the market-weighted average of the expected returns on the individual assets in the portfolio. The relationship between each security in the portfolio with every other security as measured by the covariance of return has also to be considered. The formula for portfolio variance is used to compare two assets. As part of producing a demo for FP Complete's new IAP product, I wound up implementing the Minimum Variance Portfolio calculation for a stock portfolio in R, then in Haskell for the IAP, and finally in Python using the NumPy and SciPy extension libraries. The variance of a portfolio is not just the weighted average of the variance of individual assets but also depends on the covariance and correlation of the two assets. The first button computes Mean Return and Standard Deviation of each assets. 3 characteristics of minimum variance portfolio mvp w School Saint Louis University; Course Title FIN 363; Type. Again, the variance can be further extended to a portfolio of more no. The n risky assets are independent with mean rate of return 휇 푖 and standard deviation of 휎 푖 > 0, 푖 = 1. Hint: consider a 2-assets portfolio made of an arbitrary portfolio p and the MVP, with weights a and 1-a. • Asset (portfolio) A mean-variance dominates asset (portfolio) B if μ A ≤μ B and σ A < σΒ or if μ A >μ B while σ A ≤σ B. The excel page consists of 2 buttons. If there is a risk-free asset with return of 5%, there is the possibility of an arbitrage opportunity since we can synthesize risk-free securities that oﬀer a higher expected return. The section of the opportunity set above the minimum-variance portfolio is the efficient frontier. Find all Ways2Wealth Asset Management posts tagged with minimum variance portfolio formula. The variance of the returns are calculated using the variance formula, The weight of this portfolio is made of three (3) assets (IVM, IVP and LR. This portfolio variance statistic is calculated using the. The curve passing point V and point A shows all possible efficient portfolios and is identified as the efficient frontier. You should actually be able to come up with this relationship on your own, as it is not a complicated 'derivation' to understand. As a result, Bulgarian investors can select their own optimal portfo-lio that maximizes portfolio rate of return with respect to their risk preference. We have reduced risk. The process to construct the CAL for a collection of portfolios is described below. Hence,somerisk-neutralvolatilityisobserved. The portfolio weights sum to one and the target return is λ. Above formula. 88 × 100 + $331. By optimizing the. It optimizes asset allocation by finding the stock distribution that minimizes the standard deviation of the portfolio while maintaining the desired return. a)Find the expected return and variance of the minimum-variance portfolio. The section of the opportunity set above the minimum-variance portfolio is the efficient frontier. Lecture 4 –Mean Variance Portfolio Choice in Excel –Manuela Pedio Problem one: the stock-bond asset allocation 11 Now, we can compute also the Global Minimum Variance Portfolio, i. Given the following data, calculate the market portfolio™s standard deviation. These values are used as aid in the optimization of the three assets portfolio. Table 3 shows the Global Minimum result obtained from the returns. Portfolio theories propose how rational and prudent investors should use their due diligence to diversify their investments to optimize their portfolios, and how a risky asset should be priced as compared to less risky asset. Standard Deviation. Question 5. † The equally-weighted (ew or 1/n) portfolio Another simple way is to attribute the same weight to all the assets of the portfolio (Bernartzi and Thaler, 2001). The balances in some of the asset accounts will be combined and presented as a single amount when the balance sheet is prepared. Modern portfolio theory (MPT), or mean-variance analysis, is a mathematical framework for assembling a portfolio of assets such that the expected return is maximized for a given level of risk. A is a risk-free asset with that yields a rate of 8%. The output of your calculation of the efficient frontier was a series of two vectors, vpm (vector of portfolio means), and vpsd (vector of standard deviations, or volatilities), and a matrix of. Section 4 concludes. I would try to ration it out rather than worrying about its origin, it is a basic formula. 5 x V x 0) =. Hence,somerisk-neutralvolatilityisobserved. How this formula works. The authors show that regressing the negative excess returns of$\mathrm{n_A} - 1$assets on the returns of the remaining asset results in coefficients that equal the$\mathrm{n_A} - 1$respective portfolio weights; the remaining asset's weight is determined by the budget constraint. 3 Solution 3 The market beta must satisfy 1 = X A + X B B + X C C + X D D = X A ˙ AM ˙ 2 M + X B ˙ BM ˙ M + X C. Result will be displayed. It demonstrates the risk-and-return trade-off of a portfolio and helps us visualize the efficient portfolios, the global minimum variance portfolio and inefficient portfolios. The minimum correlation algorithm is a relatively new asset allocation strategy. 2) Long-term prepayment speeds of 20% for ARMS and 10% for fixed rate. The problem can also be implemented as a regression, see Kempf and Memmel []. Show that the minimum variance portfolio has variance 14) Suppose a market consists of n risky assets and 1 risk free asset. Here we add box constraints for the asset weights so that the minimum weight of any asset must. such as the Global Minimum Variance Portfolio or the Equally Risk Contribution Portfolio , . Covariance Matrix C. the actual return on an asset. Therefore, this paper analyzes the optimization of the investment portfolio of Mean-Variance liability assets, in which both asset return and liability returns mean and volatility. Effect of Diversification with n Risky Assets XI. A portfolio w⁄ is said to be mean. Bride and Groom sing EPIC Thank You Song to Wedding Guests - Best Wedding speech - Duration: 14:54. return 100% stocks. The other two assets, B and S are risky asset with the following attributes. Two Asset Portfolio Calculator Payback Period Calculator Simple Interest Calculator Price of Zero-Coupon Bond Calculator Breakeven Point Calculator Mean-variance criterion Calculator Balloon Loan Calculator Fisher Effect Calculator - Nominal Interest Rate Fisher Effect Calculator - Real Interest Rate. the actual return on an asset. As 4 moves between 2 and. I need to create a global minimum variance portfolio and am completely confused after spending days doing other things in Excel which have worn my brain out. Over the past few weeks, all of us have been impacted by the ever-changing events and challenges of COVID-19. 5% commodities. Given 4 assets' risk and return as following, what could be the risk-return for any portfolio built with the assets. The variance (or risk) of a portfolio is not simply a weighted average of the variances of the individual securities in the portfolio. Modern portfolio theory stipulates that you can achieve a portfolio on the efficient frontier with only two mutual funds. -The "s" for volatility in the formula is the standard deviation of each asset. Here is how to optimize a portfolio using Excel: Import asset return data. The rate of return on the asset is given by r. A Bowen, 1984). (It does not consider cash as an investing asset. The data collected for the research is secondary data of monthly prices of ETFs listed on BSE and is for the period Jan 2012 to June 2017. PyPortfolioOpt is a library that implements portfolio optimisation methods, including classical mean-variance optimisation techniques and Black-Litterman allocation, as well as more recent developments in the field like shrinkage and Hierarchical Risk Parity, along with some novel experimental features like exponentially-weighted covariance matrices. 3 is clicked on, the minimum Variance Portfolio will be determined using Microsoft Excel Solver as one of the steps. Finding portfolio standard deviation under the Modern Portfolio theory using matrix algebra requires three matrices 1. 0 The hedger chooses a value for the hedge ratio that minimises the variance of the value of the hedged position. The second part, due to unsystematic risk is smaller. Minimum Variance Portfolios in the U. This portfolio variance statistic is calculated using the. An alternative way to find the minimum variance portfolio is to use the result established above. Inputs on three stocks: mean, standard deviation, and correlation matrix Standard Expected Stock Deviation A B C Correlation Matrix B. Suppose we purchase an asset for x 0 dollars on one date and then later sell it for x 1 dollars. One may think that all possible values have to fall inside the convex hull. Table 3 shows the Global Minimum result obtained from the returns. Let’s take the case of a three asset portfolio. A weighted average is one that takes into account the importance, or weight, of each value. updating formula takes the weighted average of the unconditional variance, the squared residual for the first observation and the starting variance and estimates the variance of the second observation. Determine the total number of asset units in your portfolio. R Tools for Portfolio Optimization 19. With the above inputs, we observe the decrease of portfolio return risk due to diversification when the number of assets increases. portfolio comprising of four ETFs (Exchange Traded Funds) listed on BSE. Chapter 4: Mean-Variance Analysis Modern portfolio theory identifies two aspects of the investment problem. Let P be the optimal portfolio for target expected return 0. The stock sells for$21. To calculate the beta of a portfolio, you must first use a formula to calculate the beta of each asset it contains. 2) Derive the minimum variance frontier for various levels of expected returns, without short-selling of risky assets 3) Derive the minimum variance frontier for various levels of expected returns with restricted short-selling (apply whatever restriction you consider appropriate) of risky assets allowed. Money › Investment Fundamentals Portfolios Returns and Risks. of assets, for instance, a 3-asset portfolio can be represented as, Portfolio variance formula = w 1 2 * ơ 1 2 + w 2 2 * ơ 2 2 + w 3 2 * ơ 3 2 + 2 * ρ 1,2 * w 1 * w 2 * ơ 1 * ơ 2 + 2 * ρ 2,3 * w 2 * w 3 * ơ 2 * ơ 3 + 2 * ρ 3,1 * w 3 * w 1 * ơ 3 * ơ 1. Figure 3 shows implied alphas (using the same variance as in the optimization) for assets that are in the portfolio (positive weight) and out of the portfolio (zero weight). price)*sum(m. Weights for each asset class are either positive or zero, but there are definitely no negative weights (so life isn't really worth living). The Standard Deviation is a measure of how spread out numbers are. Ask Question + 100. Beta measures: A. In section 3, empiricial analysis is explained. The problem can also be implemented as a regression, see Kempf and Memmel []. Otherwise, as the formula for portfolio standard deviation shows, the portfolio standard deviation is less than the weighted average of the component-asset standard deviations. R-by-N matrix of R portfolio weights for N securities. 15 (portfolio variance:. All of these. Variance as a measure of concentration in sampling theory <4. Calculate the weights associated with the Minimum Variance Portfolio using formula in footnote i on p. equal to -1. In a world with a zero risk-free rate, the Tangency portfolio should be the market portfolio, assuming all assets are represented. The risk parity approach asserts that when asset allocations are adjusted (leveraged or deleveraged) to the same risk level, the risk parity portfolio can achieve a higher Sharpe ratio and can be more. The mean-variance frontier. It is built up a mean-variance portfolio with fixed return and minimum risk conditions. 3 characteristics of minimum variance portfolio mvp w School Saint Louis University; Course Title FIN 363; Type. Portfolio Rules with a Risk-free Asset In this section, we discuss the portfolio choice problem for a mean-variance investor when both risky assets and a risk-free asset are available. 31 Global Minimum Variance Portfolio In a similar fashion, we can solve for the global minimum variance portfolio: The global minimum variance portfolio is the efficient frontier portfolio that displays the absolute minimum variance. [Solution] IPA Week 3 - Chapter 7. 88% 1 points Question 18 1. it shows where the minimum return is. The problem of minimizing the covariance risk for a given target return with optional box and group constraints is a quadratic programming problem with linear constraints. The portfolio variance is a weighted sum of the elements in the covariance matrix, with the products of the portfolio proportions as weights. A minimum variance portfolio is a portfolio model made up of investments that are volatile individually but are seen by some as low risk when put together. Estimation step = E(R) and σ² for each asset and Corr for each pair of assets. The most left point on the minimum variance set is called the minimum variance point. The global minimum variance portfolio has a standard deviation that is always. Find all Ways2Wealth Asset Management posts tagged with minimum variance portfolio formula. Variance portfolio . Earnings –Asset Yield • Formula –Annualized Loan & Investment Income ÷Average Total Assets • Industry Standard –Determined by market conditions. Where a proportion of funds x is invested in Project A and (1 -x) in Project B, the portfolio variance can be defined by the familiar equation:. With a covariance of 6%, calculate the expected return, variance, and standard deviation of the portfolio. The global minimum variance portfolio is the only efficient stock portfolio whose weights do not depend on the expected returns. minimum variance portfolio. pricing and hedging derivative securities based on a fundamental state model, the well-received. Portfolio variance is a measurement of how the aggregate actual returns of a set of securities making up a portfolio fluctuate over time. • Asset (portfolio) A mean-variance dominates asset (portfolio) B if μ A ≤μ B and σ A < σΒ or if μ A >μ B while σ A ≤σ B. So given our assumptions for large and small stocks, let's calculate the minimum variance portfolio, the efficient portfolio, and the portfolio with the highest Sharpe Ratio. σ²p = (1/n * σ²i_) + ((n-1)/n * Cov_) σ²i_ = average variance of all asset in portfolio Cov_ = average covariance of all pairings in the portfolio. Mean-variance model and Sharpe ratio are testing on. This video details how to calculate a minimum variance two-asset portfolio by hand. The minimum correlation algorithm is a relatively new asset allocation strategy. Investment and Portfolio Analysis (FINA602) Uploaded by. leverage this freedom to create a new asset allocation pro-cedure we call the “mean-variance inverse optimization” (MV-IO) approach, which uses BL-type estimators in a set-ting motivated by the arbitrage pricing theory. If we want to find the exact minimum variance portfolio allocation for these two assets, we can use the following equation: x = (σ b ²-ρ ab σ a σ b) / (σ a ² + σ b ² – 2ρ ab σ a σ b) Plugging in the values from the first article in this series, we can see that x = 74. 3 −50% Mean = [0. Given a constrained portfolio w~, it is possible to ﬁnd a covariance matrix §~ such that w~ is the solution of the global minimum variance portfolio. An investor is supposed to be risk-averse, hence he/she wants a small variance of the return (i. ) "The total stock market fund covers large-cap, mid-cap and small-cap stocks. - Minimum variance frontier is a ggraph of the expected return/variance combinations for all minimum variance portfolios. Consider now the fully invested, minimum variance portfolio given by the solution to. Modern portfolio theory (MPT), or mean-variance analysis, is a mathematical framework for assembling a portfolio of assets such that the expected return is maximized for a given level of risk. Davis Pham. Given 4 assets’ risk and return as following, what could be the risk-return for any portfolio built with the assets. For two assets, it is easy to solve for the minimum variance portfolio. I need to create a global minimum variance portfolio and am completely confused after spending days doing other things in Excel which have worn my brain out. Join Yahoo Answers and get 100. This portfolio is called the zero-beta portfolio. the upper tail of the efficient set. The correct answer is B. To do so, we have created a template, where you can enter your stock portfolio and it will automatically calculate the Sharpe Ratio of the portfolio. The expected return on a portfolio of assets is the market-weighted average of the expected returns on the individual assets in the portfolio. Details are left to the reader who will carry out most of the analysis in a Homework Set 3. Pg 1-3 Portfolio Optimization Version 1. sids as sorted([list of sids]), then they will have the same ordering. This portfolio model might not be right for individual investors though. On the other hand, it is assumed that every investor should hold both domestic and international stocks. Finding portfolio standard deviation under the Modern Portfolio theory using matrix algebra requires three matrices 1. If we want to find the exact minimum variance portfolio allocation for these two assets, we can use the following equation: x = (σ b ²-ρ ab σ a σ b) / (σ a ² + σ b ² - 2ρ ab σ a σ b) Plugging in the values from the first article in this series, we can see that x = 74. 3 The Markowitz Problem 4 2. our formula with his or her forecast of the ve determinants in order to generate a forecast for return to the levered strategy. Multiplying the three matrices gives us the variance of the portfolio. 1 Introduction. the minimum variance portfolio. Weights of the assets in the portfolio, in row format = W 2. In this paper we present a adaptive Mean-Variance-CVaR (MVC) model of multi-portfolio optimization. The estimation risk is huge. Therefore, one can avoid extreme estimation risk by investing into this portfolio. equal to -1. ) From the two-asset portfolio variance. The expected return on a portfolio of assets is the market-weighted average of the expected returns on the individual assets in the portfolio. equal to zero. 22--asset portfolio opportunities asset portfolio opportunities with with nono risk free asset Will show that there is a Will show that there is a singlesingle optimal risky portfolio Will derive the Will derive the Minimum Variance Minimum Variance Frontier (MVF) The MVF is the set of portfolios with the. Matlab fetch -> Excel. Beta measures: A. Additional Readings Buzz Words: Minimum Variance Portfolio, Mean Variance. Given 4 assets' risk and return as following, what could be the risk-return for any portfolio built with the assets. 5 Portfolio Optimization (2 Assets) In the "PortfolioOptimization2Assets" worksheet, we will use Markowitz theory to optimize the proportions of the 2 normal risky assets and the riskless asset in the portfolio. It is calculated the portfolio structure and it is noticed that the investor prefers to invest the less in the first two assets which shows the lowest return and risk and prefers to invest an appropriate proportion in the latter two assets of the highest risk. The methods used to make these predictions are part of a field in statistics known as regression analysis. The measure. (31) An important investment preference on the "efﬁcient fron-tier" is the portfolio with the maximum Sharpe ratio [1-3]: ξ =ρ/s. An example of this approach might be a two-fund portfolio that divides assets equally between stocks and bonds:. From the day Money Education started, taking care of our customers (and team) has been our top priority. 05%²) + (35%² x 2. Pair B (assets 1 and 3): Applying the above GMV weight formula to Pair B yields the following weights: 1 =3 X GMV (300%) and 2 X 3 =− (−200%) This means that the GMV portfolio of assets 1 and 3 involves short selling asset 3. The portfolio variance is a weighted sum of the elements in the covariance matrix, with the products of the portfolio proportions as weights. where ωs are proportions of the respective assets in the portfolio. Academic year. Figure 3 shows implied alphas (using the same variance as in the optimization) for assets that are in the portfolio (positive weight) and out of the portfolio (zero weight). standard deviation) that can be achieved at each level of expected return for a given set of risky securities. equal to -1. This is the set of risky assets which all MV investors would want to combine with a risk free to build a MV optimal portfolio. a)Find the expected return and variance of the minimum-variance portfolio. stock index fund, a total international stock fund and a total bond index fund, where "total" means it includes a wide range of equities, in the case of the stock funds, or bonds, getting you the most diversity for your buck. More speciﬁcally, in part 1. Pages 5 Ratings 100% (3) 3 out of 3 people found. Quantifying risk allows investors to build a portfolio that suits their tolerance for loss, while trying to. Use the following two portfolios to answer parts one to four. Asset 1 makes up 28% of a portfolio and has an expected return (mean) of 25% and volatility (standard deviation) of 6%. To emphasize this one could write: pu(p,k) = e(p) - v(p)/t(k) where e(p) is the expected value (or return) of portfolio p, v(p) is its variance, t(k) is Investor k's risk tolerance, and pu(p,k) is the utility of portfolio p for. Minimum Variance Portfolio [Complete Proof] Ask Question Asked 5 years, The problem that I was considering was how to get the weights over the assets directly during the optimization process. Portfolio variance is calculated by multiplying the squared weight of each asset by its equivalent variance and adding two times. a combination of the risk-free asset and the Market Portfolio. A weighted average is one that takes into account the importance, or weight, of each value. leverage this freedom to create a new asset allocation pro-cedure we call the “mean-variance inverse optimization” (MV-IO) approach, which uses BL-type estimators in a set-ting motivated by the arbitrage pricing theory. It refers to the total returns of the portfolio over a particular period of time. The Variance is defined as:. As we can see, the more assets that are combined in a portfolio, the more cumbersome the formula of its standard deviation. Modern portfolio theory (MPT), or mean-variance analysis, is a mathematical framework for assembling a portfolio of assets such that the expected return is maximized for a given level of risk. This article will show you how to use Excel’s SUMPRODUCT and SUM functions individually and how to combine the two to calculate a weighted average. The first two here-- the minimum variance portfolio, efficient portfolio for a given level of risk, let's use the risk of large stocks-- these first two we've already done. Figure 3 shows implied alphas (using the same variance as in the optimization) for assets that are in the portfolio (positive weight) and out of the portfolio (zero weight). how an asset covaries with the market. Join Yahoo Answers and get 100. I will use Minimum Variance portfolio as an example for this post. , no leverage, no shorting, max concentration, etc. Suppose we purchase an asset for x 0 dollars on one date and then later sell it for x 1 dollars. Stock Y has a higher β, so owning more of stock Y gives you higher beta, but the variance is minimum when the portfolio is more evenly split between stock X and stock Y. Most portfolios are diversified to protect against the risk of single securities or class of securities. 71% of the portfolio is invested in asset A, 29% in B) Variances: 0. Finally, the minimum variance portfolio is obtained through the following matrix formula: σ 2 = w'∑w where w is the weights vector with sum equal to 100% and w( i ) ≥ 0 and ∑ equals the. The first two here-- the minimum variance portfolio, efficient portfolio for a given level of risk, let's use the risk of large stocks-- these first two we've already done. The measure. a combination of the risk-free asset and the Market Portfolio. Most Popular Terms: Add a Symbol My Portfolio Edit / Your Watchlist is empty. Speciﬁcally, if there are N risky assets, Minimum Variance Condition for M E R i E R. From what I gather, I need all of the closing price data for each of the 5 stocks within a single Workbook, alongside some other criteria which I'm not fully aware how to input to get this. Example: Consider a 2-asset portfolio consisting of GlaxoSmithKline and General Electric. The section of the opportunity set above the minimum-variance portfolio is the efficient frontier. The following sequence of examples highlights features of the Portfolio object in the Financial Toolbox™. Portfolio Choice: n Risky Assets and a Riskless Asset XIII.