# Questions tagged [mean-variance]

The tag has no usage guidance.

136 questions
Filter by
Sorted by
Tagged with
51 views

### Maximum skewness portfolio solution derived from its Lagrangean formulation

$$\arg \min_w \quad w^\top \Sigma w$$ \begin{align}\text{s.t.} \quad \mathbf{1}^\top w = 1 \end{align} is the optimization problem for the minimum-variance portfolio weights, whose analytical solution,...
71 views

### Why isn't the asset with minimum variance given a 100% portfolio weight?

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 ...
66 views

### Conic programming for portfolio optimization instead of quadratic programming: Typo in source?

The benefit of using a cone constraint like $$\sqrt{\left(w[1]^{2}+w[2]^{2}\right)^{2}}<r$$ compared to a non-cone constraint like $$\sqrt{\left(w[1]^{2}-w[2]^{2}\right)^{2}}<r$$ is that the ...
51 views

112 views

### Do the minimum VaR and minimum ES portfolios lie on the mean-variance efficient frontier?

The mean-variance efficient frontier holds the minimum variance portfolio, but in the graph above it shows that the minimum VaR (Value-at-Risk) and minimum ES (CVaR) portfolios (expected shortfall/...
41 views

### Maximum return portfolio using linear programming with quadratic constraints

In the maximum return portfolio problem formulation above, is $A=\mu^\top \Sigma^{-1} \mu$? What is $b$ equal to, and is the second constraint required? An inequality constraint for target portfolio ...
22 views

### Symbol for the feasible set of portfolios in mean-variance analysis?

When we optimize some mean-variance efficient portfolio, it lies on the efficient frontier (blue line) which is considered superior to the feasible set of portfolios. The feasible set (red dots), on ...
109 views

### Is quadratic programming used to maximize portfolio skewness and kurtosis?

Quadratic programming, a type of convex optimization, is used to solve the minimum variance portfolio weights $$w = \arg \min_w \sigma_P^2 = w^\top \Sigma w$$ because the objective function coincides ...
163 views

### Do portfolio mean and portfolio variance have probability distributions?

If $X$ is a $T\times N$ matrix of multivariate asset returns, and $w$ is some optimal portfolio weight vector, then the portfolio return series is $r_p = X w \in\mathbb{R}^{T}$. This return series ...
87 views

### Should portfolios have zero or negative correlation between assets? [closed]

Is it more optimal to have a portfolio whose assets are negatively correlated? (I am not requiring all assets to be negatively correlated in this case, nor (-1) perfectly negative correlation either. ...
85 views

### Why does the likelihood of corner solutions in portfolios increase as the number of assets grows?

A three- asset portfolio doesn't seem prone to generating corner solutions, which are very high allocations to one of the assets and $0$ to the others. Instead, when the number of assets is low, these ...
65 views

### Contribution of an asset's variance, skewness and kurtosis to its portfolio weight?

The mean-variance model is known to assign higher weights to assets with high expected returns and low volatility, meaning that there is a direct link between the asset's weight within the portfolio ...
58 views

### Minimum variance portfolio's analytical solution, but assuming $t$-distribution

$$\boldsymbol{w} = \frac{\boldsymbol{\Sigma}^{-1} \boldsymbol{1} }{\boldsymbol{1}' \boldsymbol{\Sigma}^{-1} \boldsymbol{1}}$$ is the well known closed-form analytical solution to the minimum ...
41 views

### Mean-EVaR efficient frontier

Entropic Value-at-Risk (EVaR) is an alternative and more efficient risk measure than conditional Value-at-Risk (CVaR). EVaR serves as an upper bound to both VaR and CVaR. Below is a graph of the mean-...
122 views

### Cover's universal portfolio vs. Markowitz's mean-variance model

Cover's universal portfolio maximizes the wealth growth rate Markowitz's mean-variance model minimizes portfolio variance Both allocate assets based on historical returns. How do these two models ...
187 views

### Why is portfolio optimization a convex problem if variance is concave?

Variance is concave, so portfolio risk must be too. The mean-variance model employs quadratic programming to optimize (minimize) portfolio risk. My understanding is that quadratic programming requires ...
45 views

### Isn't portfolio optimization basically just feature selection?

Statistical learning has a large assortment of tools for conducting feature selection such as PCA analysis, ridge regression, LASSO, SVM and almost every other machine learning algorithm. In portfolio ...
205 views

### Why does portfolio optimization require a positive-definite covariance matrix?

Why does the portfolio optimization mean-variance model require the covariance matrix to be positive-definite? Does this requirement have to do with the need to be able to invert the matrix during ...
109 views

### Is non-linear correlation an issue in portfolio optimization?

Portfolio weights are linear combinations of assets. How can it be true then for there to be, and how can someone prove that there is any, non-linear correlation issues in portfolio optimization? Is ...
134 views

### Are mean-variance efficient portfolio weights random variables with probability distributions?

The mean-variance model outputs a portfolio weight vector whose elements are individual asset weights that sum to 1. Regardless of which portfolio along the efficient frontier is being solved, the ...
141 views

### James-Stein estimator for superior estimates of returns in m.v. portfolio optimization

I am currently learning about statistical techniques to enhance the estimation of input parameters in a m.v. optimization. Specifically I have some doubts about the James-Stein estimator applied as an ...
40 views

### Selecting the best characteristic portfolio per rebalance date

An investor typically decides a portfolio objective and sticks with that objective for every rebalance date in the portfolio's life. Common characteristic portfolios that the investor chooses are: ...
133 views

### Ledoit/Wolf covariance shrinkage in risk-parity optimisation

This is more of a theoretical question. I have been working on some mean-variance / Black-Litterman models and played around with Ledoit/Wolf's covariance shrinkage method (sklearn function in Python)....
152 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 ...
484 views

### Any portfolio theories not based on asset returns?

For data, the mean-variance model for portfolio optimization uses asset returns to minimize portfolio risk (covariance matrix), which is asset returns volatility, and sometimes simultaneously ...
92 views

### How do i find the covariance between two portfolios?

I know that the formula for covariance is But this is for two securities. How do I find the covariance between two portfolios? more specifically between the global minimum variance (GMV) and the mean-...
19 views

### Risk aversion coefficients for multi objective quadratic programming

I am solving different quadratic programming optimizations in a setting that involves the usual mean variance problem plus other objectives. My concern is how to choose the many so-called risk ...
302 views

### Any portfolio models not based on asset return moments?

The mean-variance model for portfolio optimization minimizes portfolio risk (covariance matrix), which is the second statistical moment of multivariate asset returns, and sometimes simultaneously ...
125 views

### Is this methodology for finding the minimum variance portfolio with no short-selling sound?

I have below here an excerpt from a book on (among other things) mean-variance analysis showing how to find the minimum variance portfolio (Risk and Portfolio Analysis: Principles and Methods, by Hult,...
46 views

### Mean variance portfolio - alternative formulations

From this lecture on YouTube the lecturer states that there are three ways to form the mean variance portfolio (minimize variance for a given return, maximize return for a given variance, maximize a "...
26 views

78 views

### Mean Semivariance Optimization VS PMPT

Mean Semivariance optimization defines semivariance, variance only below the benchmark/required rate of return, as: $(1/T).\sum_{t=1}^{T} [Min(R_{it}-B,0)]^2$ where $B$ is the benchmark rate, $R_{i}$...
60 views

### Surface plots of the mean-variance efficient frontier

3d surface plots contain an X, Y and Z axis. For the mean-variance efficient frontier: X axis is portfolio volatility ($\sigma_p$) Y axis is portfolio expected return ($\mu_p$) any ideas for what ...