Questions tagged [mean-variance]

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63 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 ...
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1answer
51 views

Correlation between mean-variance efficient portfolios

If the covariance solution between the returns series of the minimum-variance portfolio ($A$) and any other portfolio along the efficient frontier ($B$) is $$Cov_{A, B} = \frac{1}{\mathbf{1}^T\mathbf{\...
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39 views

How to prove that the portfolio mean $\mu_P$ is biased in the presence of skewness in the portfolio return distribution?

Portfolio returns are a vector that have a statistical distribution, which in turn has moments. $$s_P = w^\top M_3 (w\otimes w)$$ is the formula for portfolio skewness, where $M_3$ is the coskewness ...
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1answer
64 views

Efficient frontier portfolio's analytical solution for a given expected return $r$

$$\begin{equation} \boldsymbol{w}(r) = \frac{r\mathbf\Sigma^{-1} \boldsymbol{\mu}}{\boldsymbol{\mu}^{\top} \mathbf{\Sigma}^{-1}\boldsymbol{\mu}} \end{equation} $$ is the closed-form analytical ...
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17 views

Is the feasible set of portfolios an epigraph?

In mathematics, the epigraph of a function is the set of points lying on or above its graph, in this case a convex function: The efficient frontier from mean-variance portfolio analysis encloses an ...
2
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2answers
145 views

Contribution of an asset's variance to portfolio variance

How can an asset's variance, $\sigma_i^2$, be shown to contribute to portfolio variance, $\sigma_p^2$? I was thinking of taking the derivative (first order conditions $\frac{\partial L_{\sigma_p^2}(w,\...
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0answers
34 views

Mathematical proof of out-of-sample disappointment in portfolio performance being a function of a portfolio's variance

The minimum-variance portfolio is considered more optimal than the maximum Sharpe ratio (tangency) portfolio on the grounds that its in-sample performance is less likely to disappoint out-of-sample. ...
2
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1answer
50 views

Derivation of mean-variance portfolio weights as closed-form analytical solution from Lagrangean equations

I am trying to find a closed form solution for the constrained MVO problem below. $\max_w w'\mu - \frac{\lambda}{2}w'\Sigma w $ s.t. $w'$1 = 1 The Lagrange for the objective is $L(w, \gamma) = w'\mu ...
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2answers
288 views

Closed-form analytical solution for the variance of the minimum-variance portfolio?

The portfolio weights vector of the minimum-variance portfolio has a closed-form analytical solution, $$\boldsymbol{w} = \frac{\boldsymbol{\Sigma}^{-1} \boldsymbol{1} }{\boldsymbol{1}^\top \boldsymbol{...
2
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2answers
110 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/...
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1answer
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 ...
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21 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 ...
2
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1answer
108 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 ...
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3answers
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 ...
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1answer
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. ...
3
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1answer
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 ...
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1answer
63 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 ...
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0answers
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 ...
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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-...
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1answer
121 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 ...
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2answers
180 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 ...
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0answers
44 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 ...
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2answers
202 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 ...
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1answer
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 ...
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3answers
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 ...
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0answers
139 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 ...
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0answers
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: ...
3
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1answer
131 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)....
4
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1answer
151 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 ...
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3answers
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 ...
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1answer
90 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-...
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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 ...
4
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3answers
297 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 ...
1
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1answer
118 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,...
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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 "...
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26 views

Calculating R* in a two-asset world

In chapter 5 of John Cochrane's Asset pricing, we derive a state-space interpretation of the mean variance frontier by defining $R^*$ and $R^{e*}$. A little forward, we have this formulation: $$R^* = \...
2
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1answer
183 views

Mean-Variance optimization with no short selling

I am wondering how I can find the vector of Lagrange multipliers $\mu$ for the non-negativity constraint of the following problem: $$ L(w,\lambda, \mu) = w^{T}\Sigma w - \lambda(w -1) + \mu w $$ So ...
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512 views

How good is the inverse-volatility portfolio?

Heuristic portfolio construction techniques include the equally-weighted portfolio (1/N) and the inverse volatility portfolio (IVP), which is based on the low-volatility effect. They can be assembled ...
4
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1answer
683 views

Can a capital market line have a negative slope?

I am struggling to interpret my mean-variance / efficient frontier / capital market line results. I have no issues calculating the efficient frontier. However, I do increase the risk-free rate from ...
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1answer
101 views

How to stress test a correlation matrix

As part of a mean variance portfolio task, I am calculating portfolio risk and optimal allocations between assets given required level of return. Input: expected returns, volatility and correlation ...
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0answers
137 views

Finding the Efficient Frontier in Tensorflow

I have two assets A and B. Asset A has an expected return of 0.92% with a standard deviation of 2.10. Asset B has an expected return of 1.39% and a standard deviation of 2.20. I want to find the set ...
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48 views

How to obtain tangency portfolio of the resampled efficient frontier in MATLAB?

I have generated the resampled frontier according to Michaud's approach. In order to compare it with the classical mean variance approach I want to invest in the respective tangency portfolios. While ...
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1answer
87 views

What if all the weights are negative in mean-variance optimization during a crisis?

Usually the constraint is that all weights sum up to 1. But in a crisis when all assets are falling in prices, intuitively, all the weights should be negative in the optimization. But it contradicts ...
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1answer
145 views

Fixes of quadratic utility when probability of decreasing utility is large

In finance and specifically portfolio theory, a popular utility function is quadratic utility $$ u(x)=x-\frac{\lambda}{2}(x-\mu_X)^2 $$ where $x$ is wealth and $\lambda$ is the parameter of risk ...
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0answers
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, $R_{i}$...
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1answer
58 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 ...
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2answers
228 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?
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6answers
273 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 ...
2
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1answer
50 views

ARMA moments proof

Consider a standard ARMA(1,1) process such as $$x_t - \beta x_{t-1} = \theta u_{t-1} + u_t$$ where $u_t$ is i.i.d. $u_t \sim N(0,\sigma^2)$. I know how to derive mean and variance with stationary ...
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1answer
93 views

Prove that the portfolio that maximizes utility lies on the efficient frontier

When maximizing mean-variance utility in a portfolio optimization framework $max \{R - \lambda \sigma ^2\}$ where R is portfolio return, $\lambda$ is a risk aversion parameter, and $\sigma^2$ is ...