Questions tagged [mean-variance]

Mean-variance is the starting point of most portfolio optimisation techniques.

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What methods do you use to improve expected return estimates when constructing a portfolio in a mean-variance framework?

One of the main problems when trying to apply mean-variance portfolio optimization in practice is its high input sensitivity. As can be seen in (Chopra, 1993) using historical values to estimate ...
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Does mean-variance portfolio optimization provide a real edge to those who use it?

Mean-variance optimization (MVO) is a 50+ year concept, and perhaps the first seminal idea of quantitative finance. Still, as far as I know, less than 25% of AUM in the US is quantitatively managed. ...
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Derivation of the tangency (maximum Sharpe Ratio) portfolio in Markowitz Portfolio Theory?

I have seen the following formula for the tangency portfolio in Markowitz portfolio theory but couldn't find a reference for derivation, and failed to derive myself. If expected excess returns of $N$ ...
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Portfolios from Sorts

Some time ago Almgren and Chriss proposed a method for portfolio optimization based on sorting criteria such as $r_1 > r_2 >... > r_N$ instead of explicit expected returns: see portfolios ...
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Typical risk aversion parameter value for mean-variance optimization?

What are typical values for risk aversion parameters $\lambda$ used in mean-variance optimization? Please provide references. Just to be clear, I'm talking about the $\lambda$ in $U(w) = w'\mu - \...
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Minimum Variance and Minimum Tracking Error portfolio as second order cone program

The quadratic optimization (min variance) $$ w^{T} \Sigma w \rightarrow \text{min}, $$ where $w$ is the vector of portfolio weights and $\Sigma$ is the covariance matrix of asset returns, is a well ...
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Formula for the efficient portfolios in mean-variance optimisation?

Consider the setting of mean-variance portfolio optimisation: $n$ assets with expected returns $\overline{r}_1,...,\overline{r}_n$ and standard deviations $\sigma_1,...\sigma_n$. For a certain fixed $\...
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Markowitz mean-variance optimization as "error maximization"

I hear it said a lot that standard MV optimization "maximizes errors". But I can't find a good explanation for what exactly they mean by this "maximization" of estimation error. I understand that if ...
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9 votes
2 answers
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Comparing MVO with Resampled Efficient Frontier

My question: How can I compare the Resampled Frontier (REF) to the standard MVO frontier when I have been provided with $\mu$, $\Omega$, and don't have access to true future data to test real out of ...
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Why does the Markowitz mean-variance model require the assumption of normality?

Given $N$ assets, the Markowitz mean-variance model requires expected returns, expected variances and a $N \times N$ covariance matrix. The joint distribution is fully defined by these measures. ...
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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 ...
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Mean Variance Portfolio theory and real-world problem?

There are many assumptions on mean-variance portfolio theory and they seem to be very unrealistic, for example 1) investors have the same information at the same time: calculating expected returns ...
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Generalized Mean Variance Portfolio

Utility based portfolio optimization deals with the problem of finding the optimal portfolio $x_T$ by maximizing the utility/objective function $O(x_T,x_0)$ where $x_0$ is the current portfolio. In ...
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Michaud's Resampled Efficient Frontier - Out of Sample Simulation Testing

I will be putting ALL my account points on bounty to whoever answers this question [if your answer is crap but it's the only answer, you're getting the 165 points]. You will have to wait 2 days or so ...
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6 votes
5 answers
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Fastest solver possible for portfolio optimization

I am using quadprog in MATLAB for very simple mean-variance optimization, with less than 100 assets. It is quite fast but if I run a strategy with daily ...
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6 votes
2 answers
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Matlab Portfolio Optimization with bid ask spread

I'm trying to find the optimal portfolio of options and stock which minimizes the standard deviation of the portfolio returns but also taking into consideration the bid and ask prices of the assets. ...
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How to calculate expected return based on historical data for Mean Variance Analysis

I've recently started reading some books on asset allocation and portfolio theory but I don't work in the field and don't have much knowledge yet. So I've been reading up on mean-variance analysis ...
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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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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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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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Robust-Bayesian optimization in Markowitz framework

Suppose we are in the mean-variance optimization setting with a vector of returns $\alpha$ and a vector of portfolio weights $\omega$. In a robust setting, the returns are assumed to lie in some ...
Geraldine Bailey's user avatar
5 votes
1 answer
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Quasi Random Monte Carlo in m.v. portfolio optimization

Not specifying a correlation matrix for the Monte Carlo Simulation's random returns is equivalent to assuming no correlation or a correlation coefficient of zero, which will seriously and adversely ...
Nipper's user avatar
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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 ...
Giovanni Nicolazzo's user avatar
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2 answers
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mean variance optimization vs max sharpe ratio

I keep reading/hearing that the results from mean-var optimization is max Sharpe ratio. It seems making sense if you fix either target return or target risk, but in general, it doesn't seems right, ...
starx's user avatar
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Multi-period portfolio allocation: Time-inconsistent approach

Consider a multi-period mean-variance portfolio optimization so that at time $t$ I find the strategy that maximizes my expected terminal wealth $X_T$, subject to a constraint on risk, \begin{align*} \...
arni's user avatar
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Derivation of portfolio skewness and portfolio kurtosis

Where can I find derivation of formula for portfolio skewness and kurtosis? I can find formulas everywhere, but not their derivations? For example, the portfolio variance formula, $\sigma_P = w^\top \...
mary's user avatar
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Analyzing the angle between vector of weights and vector of returns in mean-variance optimization

I am using the paper "A Sharper Angle on Optimization" by Golts and Jones (2009) as a basis for my (minor) masters thesis in mathematical finance. The paper focuses on the mean-variance analysis of ...
Geraldine Bailey's user avatar
4 votes
2 answers
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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{...
develarist's user avatar
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4 votes
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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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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 ...
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mean-variance optimization === max sharpe ratio portfolio?

Noobie here. I just wanna ask a simple question: in the context of portfolio optimization, is Mean-Variance optimization the same as the max sharpe ratio portfolio?
Nygen Patricia's user avatar
4 votes
6 answers
525 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 ...
George's user avatar
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1 answer
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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)....
Riskay's user avatar
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Dollar-Neutral in addition to Market-Neutral?

What is the point/benefit of using a dollar-neutral strategy in addition to a Beta-neutral strategy? What exactly does a dollar-neutral strategy buy the investor? What's useful about balancing long ...
Josh's user avatar
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3 answers
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Monte Carlo based mean variance optimization

I was asked this question in an interview some years ago. It struck me as a poorly formed question. I thought I would put it out there to the community to see if I just simply missed something. ...
Eric Bruce's user avatar
4 votes
1 answer
524 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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Monte Carlo (resampling) in m.v. portfolio optimization

The instability and high sensitivity of optimisation results can be augmented by adding another layer of quantitative methodology in the form of Monte Carlo Simulation. The name Monte Carlo alludes to ...
Nipper's user avatar
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1 answer
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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 ...
John's user avatar
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Methods for superior estimates of returns in m.v. portfolio optimization

Leaving aside the aspects related to the estimation of the variance component (all the latest techniques to compute a stable covariance matrix of a given set of assets such as simple shrinkage, Ledoit-...
Nipper's user avatar
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3 answers
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Is this realized "efficient" frontier reasonable?

I have performed some out-of-sample analysis of mean-variance optimization with monthly rebalancing. Studying the "realized efficient frontier", I am worried that something is wrong. Since the ...
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Markowitz Mean-Variance Implied Returns

What is the closed form solution for the following inverse Markowitz problem? Given a mean-variance optimized fully invested portfolio $X$, a risk aversion parameter $\lambda$ and a var-covar ...
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Mean-variance portfolio & quadratic programming

I am somewhat confused when it comes to modern portfolio theory, mean-variance portfolio optimization and its quadratic programming formulation. Issue 1: Formulation of mean-variance portfolio ...
Mel's user avatar
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Evaluating estimate of covariance matrix

I am testing out different methods / shrinkages to estimate a covariance matrix and I am wondering what is the best method of comparing the estimated covariance matrix to the true covariance matrix (...
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Why does the mean term have a higher effect than the covariance term in MV optimization? [closed]

I am trying to use the mean-variance (MV) optimization framework. When I change the mean term using future-ground-truth return (I am not supposed to do so), it has a higher effect on the MV ...
randy's user avatar
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5 answers
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Efficient frontier doesn't look good

Hi I'm trying to draw an efficient frontier. Below is what I used. returns parameter consists of 9 column returns of portfolio. I selected 10,000 portfolios and this is how my efficient frontier ...
Hiru's user avatar
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1 answer
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Covariance Between Two Frontier Portfolios

Based on the definitions of A, B, C, and D in "An Analytic Derivation Of The Efficient Portfolio Frontier" by Robert Merton (1972), how can I prove the following in a line-by-line derivation?...
David 's user avatar
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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,...
develarist's user avatar
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3 votes
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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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3 votes
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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 ...
secretsanta's user avatar
3 votes
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Mean Variance portfolio optimisation (Long Only) CVXPY including cardinality constraint

I am working on a portfolio optimisation that requires me to constrain on the number of assets used, e.g from S&P500 build a 20 asset portfolio that is feasible. After doing some research I came ...
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