Questions tagged [mean-variance]

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Is this equation correct for portfolio optimization for CARA normal with N risky and one riskless asset?

Suppose the consumer Solves $\max -e^{-\gamma W}$ where $W=X^T D -X^Tp R_f$ where $X$ is the vector invested in a risky asset and $D\sim N(E[D],\Sigma^2_D)$ and $R=\sim N(E[R],\Sigma^2_R)$. Then ${ X=(...
3 votes
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?...
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Why is the dynamic mean-variance problem time-inconsistent?

A lot of the literature in dynamic mean-variance problem states that the dynamic mean-variance problem is time-inconsistent. Now I was not able to find an example of why the problem is time ...
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1 vote
1 answer
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Mean-variance framework with endogenous correlations

In most mean-variance frameworks I have seen, once we clear markets in the model, it determines asset prices (and returns). However, all of these frameworks assume that the correlation matrix of the ...
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TDF(Target Date Fund) asset allocation strategy beyond glide path

I'm currently studying how target date fund(TDF) works in practice. I read a paper named 'How to Design Target-Date Funds' by Benjamin Bruder et al (https://papers.ssrn.com/sol3/papers.cfm?abstract_id=...
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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 ...
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1 vote
1 answer
80 views

Hedging with peer companies and optimize the weights

I am trying to long a security that is expected to outperform its peers after certain corporate actions, but want to hedge using the same group of peers (so short ~5 names). So the goal here is to ...
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Tangency portfolio negative maximum Sharpe ratio

Suppose I have three assets: the market, factor A and factor B. The market is in excess returns of the risk free rate. The other two factors are long-short portfolios. I have net returns for these ...
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Index Tracking Problem

I have set up a mean variance optimization problem, $$min:{W}^{\prime}{\Sigma_{\varepsilon}{W}}$$ $$s.t:{W}^{\prime}{\alpha}=R_B\;,\;\;W^{\prime}l={1},\;\;W'\beta=0,\;\;W'Z=\beta_p$$ where, $W$ is an (...
2 votes
1 answer
184 views

Utility Theory and Mean Variance Analysis

I was wondering if it's pertinent to use this interpretation of the expected utility function given by the Taylor series expansion, $${E(U(W)}\approx{U[E(W)}]+\frac{U''[E(W)]\sigma^2_W}{2}\tag{1}$$ to ...
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Closed form solution for Mean-Variance optimization under constraint

Is there a closed form solution for the vector weight $w$ for the following mean-variance optimization problem? $\max_w w'\mu - \frac{\gamma}{2}w'\Sigma w $ s.t. $w'z\geq \bar{z}$ where $w, z$ are N ...
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Questions about Merton's derivation of the security market line

In Merton's "An Analytic Derivation of the Efficient Frontier" (PDF), he derives the security market line for the CAPM using the definition of the tangency portfolio. He writes: Here, $m$ ...
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1 vote
1 answer
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Is this quadratic form the Sharpe ratio?

I'm reading Merton's An Analytic Derivation of the Efficient Portfolio Frontier. In section IV, he derives the efficient frontier with a riskless asset. Let $\mathbf{w}$ be a vector of portfolio ...
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3 votes
3 answers
529 views

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?
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2 answers
328 views

Why the market portfolio is the tangency portfolio in the Mean-Variance Optimization model?

I read in an explanation that the tangency portfolio has all securities with weights proportional to their market value because supply equal’s demand. But I can't understand why supply equals demand ...
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1 vote
1 answer
253 views

Monte Carlo vs. Block Bootstrapping vs. Bootstrapping

Because I can fit e.g. ~25 distributions via empirical cumulative distribution fitting to correlated data (including stable dist.), and then simulate the original data based on correlation (covariance)...
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1 vote
1 answer
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Mean-Variance Portfolio Axis Description

I'm currently looking into the mean-variance approach to portfolio theory and I wonder, why the standard deviation $\sigma$ is graphed on the x-axis and not the variance $\sigma^2$ as a measure of ...
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Proof that mean-variance opportunity set is closed

In the book Financial Economics (2010) by Hens and Rieger, on page 101 we find the following Lemma 3.1: If we have finitely many assets, the minimum-variance opportunity set is closed and connected. ...
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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=...
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Black-Litterman for quant portfolio

I have seen a lot of research around the Black-Litterman approach and I think theoretically, it is a nice framework. However, it appears that its main strength is from a practitioner's point of view, ...
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CVXOPT quadratic programming mean variance example

Trying to learn how to use CVXOPT to do quant finance optimization. For the example given on page https://cvxopt.org/userguide/coneprog.html#quadratic-programming . I feel confused how this "S&...
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1 answer
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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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3 votes
2 answers
399 views

Deriving the risk-aversion coefficient

By considering the parametrised formulation of the mean-variance criterion by Markowitz, the risk aversion coefficient $\lambda$ can be derived as follow. As suggested by Arrow and Pratt, given the ...
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2 votes
1 answer
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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. ...
-1 votes
1 answer
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Covariance Matrix for asset returns [closed]

Hey guys I'm pretty new here, not sure how to code my question so I'll include a picture reference instead. I'm a bit confused on how the standard deviation of F (commodity price) would affect the ...
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1 answer
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Corner portfolios

This is more a theoretical problem rather than a technical one. I am looking for a clear and rigorous definition of corner portfolios and I like to understand more precisely their relation with the ...
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1 answer
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Should a stock with high return autocorrelation be weighted more heavily in a portfolio?

Some say the presence of autocorrelation (aka serial correlation) in a stock's financial return time series helps with forecasting its next-day movements, unlike a stock that has low serial ...
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1 answer
190 views

Calculate weight of an asset

Suppose there are three assets, and the first asset has volatility 18%, the second asset has volatility 16%, and the third asset has volatility 16%. Suppose also that the first two assets' returns ...
-1 votes
1 answer
122 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 ...
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Prove norm $\frac{1}{p}\sum_{i=1}^n |w_i|^p$ of min-variance portfolio $\leq$ max-Sharpe portfolio

The minimum-variance portfolio weight vector is $$\boldsymbol{w}_{MV} = \frac{\boldsymbol{\Sigma}^{-1} \boldsymbol{1} }{\boldsymbol{1}' \boldsymbol{\Sigma}^{-1} \boldsymbol{1}}$$ whereas the maximum ...
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2 votes
1 answer
139 views

Downside deviation (semivariance) in m.v. portfolio optimization

Currently I am considering the downside deviation or semivariance in a m.v. optimization framework. For this specific measure of risk I have found in papers different formulae. The majority of them ...
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2 votes
1 answer
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Mean Absolute Deviation in m.v. portfolio optimization

I just read some articles about $MAD$ as a measure of risk in finance. Is the following formulation a correct way to implement a $MAD$ portfolio optimization model which minimizes risk without ...
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3 votes
1 answer
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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,...
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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 ...
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1 answer
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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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-2 votes
1 answer
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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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3 votes
2 answers
603 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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1 vote
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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. ...
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1 answer
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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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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{...
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2 votes
3 answers
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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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1 answer
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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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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 ...
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2 votes
1 answer
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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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1 vote
3 answers
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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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-2 votes
1 answer
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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. ...
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3 votes
1 answer
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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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1 answer
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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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1 vote
2 answers
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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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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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