# Questions tagged [markowitz]

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36 views

### Geometry of Efficient Frontier of Portfolios

I have been reading about Portfolio Theory, and though the algebra of it seems quite intuitive, I am having a hard time understanding it's geometry. For the sake of simplicity, I will only talk about ...
32 views

### Portfolio Optimization constrained to maximum N% of short selling portfolio weights

For mean-variance portfolio optimization with short-selling allowed, but restricted to a certain percentage of the portfolio weights (lets assume N), we can constrain it in the follwoing way: (from j=...
100 views

### Herfindahl-Hirsch-Index for FX Portfolios

Assume we have an FX portfolio with $n$ currency pairs, such that $w_i$ is the weight of currency pair $i \in \{1, \dots, n\}$ in the portfolio, and $\sum_{i=1}^n w_i = 1$. All pairs have USD as one ...
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### Construct portfolio with assets having expected negative returns

I have been asked to select a n stocks among N stocks, to construct a portfolio. Some of them have have negative weekly returns on average. If I want to select these n stocks by constructing an '...
101 views

### Mixed-integer programming approach for index tracking

Suppose you currently own a portfolio of eight stocks. Using the Markowitz model, you computed the optimal mean/variance portfolio. The weights of these two portfolios are shown in the following table:...
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### Statistical methodology for proving the stability in time of asset allocation weights

I am comparing the set of weights obtained by the classical Markowitz allocation process with those of another asset allocation technique I have devised. Markowitz's weights are unstable, as the ...
34 views

### How to deal with securities that has short historical data when performing mean-variance portfolio analysis?

I am trying calculate expected return and risk (stdev) based on historical data using Mean Variance Analysis framework. Let's say the portfolio has 10 stocks, 9 of them have more than 10 years history ...
57 views

### Optimal Portfolio Formulation

I'm currently studying Luenberg's Article "Projection Pricing" (Jrl of Optimization Theory and Applications, Vol. 109, No. 1, pp. 1–25, April 2001) and there is a claim that I can't prove. ...
41 views

### Is this efficient frontier graph reasonable?

Hi. The above image is taken from https://www.newfrontieradvisors.com/media/1166/optimization-with-non-normal-resampling.pdf. Is this a reasonable chart? The 4 different methods give 4 non-overlapping ...
88 views

### State-of-the-art MVO methods?

I learned Markowit'z mean-variance optimzation in school. Now I've been googling a bit, and to my surprise, Markowit'z is STILL being used by most people, AFAIK. Are there really not some state-of-the-...
80 views

### Optimal Portfolios with Skewed and Heavy-Tailed Distributions

I am learning about portfolio theory and been using Markowitz. I wondered, however, if I can use distributional and asymmetric information of the returns to solve the problem. For instance, I have a ...
60 views

### Portfolio variance $<=$ weighted average of individual variances [closed]

In portfolio theory, I often (with some justifications but the message is the same) come across the following statement: "The most important quality of portfolio variance is that its value is a ...
129 views

### Mean estimate in portfolio optimization (Markowitz) [duplicate]

The Markowitz mean-variance portfolio optimization problem is to find the optimal allocation, $w_{optimal}$ by solving: \begin{equation} w = \mathrm{argmax} \ \mu_{t}^Tw - \frac{\gamma}{2}w^{T}\...
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### Why isn't the asset with minimum variance given a 100% portfolio weight? [closed]

The maximum expected return portfolio is the one that assigns a 100% weight to the asset with the highest expected return amongst all assets under consideration. Shouldn't then the asset with the ...
128 views

### Standard deviation formula with Short selling- Markowitz model

I have 2 fast quastions. Before I begin I want to show you that I found minus before SD of bills in the book Principles of corporate finance(1.screen). I know SD of bills is zero and minus in this ...
108 views

### Why is there a $\frac{1}{2}$ in front of the portfolio variance formula? [closed]

Can someone explain me where $\frac12$ came from in the expression $\frac{1}{2} \omega'\Sigma \omega$? That is the expression to be minimized io order to get the minimal variance portfolio (with also ...
266 views

### Closed-form analytical solution for Markowitz efficient portfolio without short-selling

In a portfolio without risk-free assets I know that the efficient portfolio si given by: $\omega=\frac{1}{BC-A^2}[\mu(C\Sigma^{-1}R-A\Sigma^{-1}\mathbb{1})+B\Sigma^{-1}\mathbb{1}-A\Sigma^{-1}R]$, ...
178 views

### How to add the effect of skewness in the portfolio optimisation objective function?

I have the following risk adjusted portfolio which I optimise, where gamma is the risk return trade off, $r$ are the returns and $C$ is the covariance matrix which considers scenarios, so it is not ...
43 views

### Optimizing Portfolio Return by Targeting Variance

I understand Markowitz and targeting returns to minimize our variance. I know this optimization problem well and its constraints. However when the reverse scenario is to be considered I get very ...
1k views

### Portfolio Optimization sum of weights constraint with short selling

For mean-variance portfolio optimization with short-selling allowed I have seen 2 ways to specify the portfolio constraint. In most resources I've seen, such as https://www.coursera.org/learn/...
250 views

### Do normal returns make the mean-variance portfolio model perform properly?

The Markowitz mean-variance model is known to suffer from estimation error due to financial returns not meeting the assumptions of a normal distribution, providing portfolio weights that underperform ...
251 views

### how to relate risk aversion and sharpe ratio in optimisation

I am trying to optimise the following: U(w)=w′μ−λ/2w′Σw which is the typical risk aversion problem. I would like to set lambda in order to have the max sharpe but I cannot find in literature what is ...
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### Markowitz portfolio in reality

I am in academia and begin to work on topics including portfolio optimization. I just read lots of paper discussing different extensions to the Markowitz approach, given different (possibly ...
346 views

### What does the concept “standard Markowitz approach” include?

Does "standard Markowitz approach" include only mean-variance approach or does it also include other approach such as minimum-variance approach?
51 views

### Spot Rates on Treasuries

I am trying to find the spot rates for 1mo, 3mo, and 6mo tbills. This would just be their yields as listed on the treasury website, correct or am I missing something?
348 views

### Is a more robust Covariance estimation possible?

I'm working on a mean-variance optimization problem, but instead of financial securities I'm choosing a 'portfolio' of N athletes. It is a 1-period optimization problem over one generic statistic ...
44 views

### financial markets

Let's suppose the following model of financial markets : Market-Maker : the sell financial derivatives, the hedge all the risk after calculating their sensibilities to market risk factors. Thus ...
123 views

### Tangency portfolio with two additional constraints so that portfolio weights are unconstrained

I know that the formula for determining the weights of the Tangency portfolio is given as $w_{tan}$ = $\frac{\Sigma \mu}{\iota^{\prime}\Sigma\mu }$, but I was wondering how to derive the weights in ...
10k views

### Where to get MSCI World Index constituents (+ weights)

Where can I download The MSCI World index constituents and their weights (daily update) The current prices of the constituents plus three years of EOD (or weekly) history Corporate actions of the ...
628 views

### Max allowable return in Markowitz model

The Markowitz model solves the following problem: The portfolio with the smallest variance among attainable portfolios with expected return µV. Here we have to choose µV to get the optimal portfolio ...
167 views

### Estimated betas and optimal portfolio

I ran a regression on 20 assets to estimate their beta with different methods. I would like to see the differences of these estimation differences in terms of mean-variance optimal portfolio. How can ...
130 views

### Risk/Return Paradox in Markowitz Optimization?

It is possible that from the efficient frontier obtained varying the "lambda" parameter of the risk-appetite coefficient, in the Mean Variance Parametric Quadratic programming problem, it results that ...
34 views

### fPortfolio specify our constraints for efficientPortfolio [closed]

I am working on the library fPortfolio in R and I have a question. How can we fix for a portfolio the sum of weights equal to 1 ? When I study the code, I see that we cannot choose which constraints ...
95 views

(hope this is not too basic, I'm new to this forum) Im struggling to understand the optimization problem (global minimum variance portfolio) formula in Markowitz Theory: $$\arg\ \min\ Var(Return\ x) =... 1answer 222 views ### Mean-variance maximization I denote by W_0 and W_1 the wealth of an investor at t=0 and t=1, respectively. Let r_f be the risk free rate, r the vector of returns of the risky assets in excess of the risk free rate, ... 2answers 2k views ### Relationship between CML and SML I am referring to the book Sharpe et al. (1998), Investments, 6th Edition. I am trying to wrap my head around some lines from the book, pertaining to Security Market Line. It reads: Earlier it was ... 1answer 783 views ### Optimal Portfolio from Efficient Frontier I found this code on plotly site, using CVXOPT to find the efficient frontier, and then, the optimal Portfolio. The optimal function is ... 0answers 78 views ### Markowitz models with uncertain returns I am analyzing the Markowitz models with uncertain returns as follows: after calculating the expected returns and the covariances of 30 monthly historical series of 30 stocks, I resolve the Markowitz ... 1answer 128 views ### Markowitz portfolio optimization and CAL [closed] Just had some questions regarding the efficient frontier and the CAL. As i understand it the point where the CAL is tangent to the efficient frontier is the optimal mix of risky assets. However I also ... 1answer 207 views ### Market Portfolio Optimization Consider the minimization problem$$\min\left\{\frac{1}{2}x^T\Sigma x - \lambda(\mu-r_f)^Tx\right\}$$and assume the CAPM model, i.e.$$r_i-r_f = \beta_i(r_m-r_f) + \varepsilon_i$$Assuming \... 1answer 148 views ### Markowitz portfolio risk with PV01 instead of variance As the PV01 (= dpdy \times notional) of a bond is a measure of its risk, as well as its price return variance, could we measure the risk of a bonds portfolio with the Markovitz portfolio variance ... 0answers 84 views ### Mean-cVaR model: How can one include transaction cost$$ \min \delta CVaR - (1-\delta) \sum_i^{n} \mu_i x_i \\ \sum x_i = \sum x^{old}_i \\ Losses(s) = \sum x_i - \sum_i^{n} (R(s,i))x_i \\ VaRDev(s) = Losses(s) - VaR \\ CVaR = VaR + \frac{\sum_s^{} ...
Referencing Wei Jiao (2003) p. 8, formula (1.12), for $Ax = b$ set of linear constraints in a portfolio, the solution for the optimum weights to maximize the utility is: w^* = \Sigma^{-1}A^T \left( ...